Experiments and theory in cold and ultracold collisionsyast/Articles/MyArt/RM.pdf · Experiments and theory in cold and ultracold collisions John Weiner Laboratory for Atomic, Molecular,

Experiments and theory in cold and ultracold collisions

John Weiner

Laboratory for Atomic, Molecular, and Optical Science and Engineering,University of Maryland, College Park, Maryland 20742

Vanderlei S. Bagnato and Sergio Zilio

Instituto de Fısica de Sao Carlos Universidade de Sao Paulo, Sao Carlos,Sao Paulo 13560, Brazil

Paul S. Julienne

Atomic Physics Division, National Institute of Standards and Technology,Gaithersburg, Maryland 20899

The authors review progress in understanding the nature of atomic collisions occurring attemperatures ranging from the millidegrees Kelvin to the nanodegrees Kelvin regime. The reviewincludes advances in experiments with atom beams, light traps, and purely magnetic traps.Semiclassical and fully quantal theories are described and their appropriate applicability assessed. Thereview divides the subject into two principal categories: collisions in the presence of one or more lightfields and ground-state collisions in the dark. [S0034-6861(99)00101-4]

CONTENTS

I. Foreword 2II. General Introduction 2

III. Introduction to Cold Collision Theory 5A. Basic concepts of scattering theory 5B. Quantum properties as E→0 8C. Collisions in a light field 10

IV. Experimental Methods for the Study of ColdCollisions 12A. Atom traps 12

1. Light forces 122. The magneto-optical trap (MOT) 14

a. Basic notions 14b. Densities in a MOT 15

3. Dark SPOT MOT 154. The far-off resonance trap (FORT) 155. Magnetic traps 16

B. Atom beams 17V. Inelastic Exoergic Collisions in MOTS 18

A. Excited-state trap-loss collisions 181. Early quasistatic models 18

a. Gallagher-Pritchard model 18b. Julienne-Vigue model 20

2. Theoretical approaches to excited-state traploss 21

a. Quasistatic theories 21b. Method of complex potentials 21c. Two-photon distorted wave theory 22d. Optical Bloch equations 23e. Quantum Monte Carlo methods 23

3. Assessment of theoretical approaches 24a. Quasistatic vs dynamical 24b. Small vs large detuning 24

4. Excited-state trap-loss measurements 26a. Sodium trap loss 28b. Cesium trap loss 30c. Rubidium trap loss 30d. Lithium trap loss 34e. Potassium trap loss 36

Reviews of Modern Physics, Vol. 71, No. 1, January 1999 0034-6861/99

f. Sodium-potassium mixed-species trap loss 37g. Rare-gas metastable loss in MOTs and

optical lattices 38B. Ground-state trap-loss collisions 39

1. Hyperfine-changing collisions 39a. Cesium and rubidium 39b. Sodium 40

2. Summary comparison of trap-loss rateconstants 40

VI. Photoassociation Spectroscopy 41A. Introduction 41B. Photoassociation at ambient and cold

temperatures 42C. Associative and photoassociative ionization 43

1. Photoassociative ionization at small detuning 462. PAI and molecular hyperfine structure 473. Two-color PAI 47

D. Photoassociation spectroscopy in MOTs andFORTs 501. Sodium 502. Rubidium 523. Lithium 534. Potassium 54

E. Atomic lifetimes from photoassociationspectroscopy 55

F. Precision measurement of scattering length byphotoassociative spectroscopy 571. Lithium 582. Sodium 593. Rubidium 59

VII. Optical Shielding and Suppression 59A. Introduction 59B. Optical suppression of trap loss 61C. Optical shielding and suppression in

photoassociative ionization 62D. Optical shielding in xenon and krypton

collisional ionization 64E. Optical shielding in Rb collisions 65F. Theories of optical shielding 66

VIII. Ground—State Collisions 69A. Early work 69

1/71(1)/1(85)/$32.00 ©1999 The American Physical Society

2 John Weiner et al.: Experiments and theory in cold and ultracold collisions

B. Bose-Einstein condensation 70IX. Future Directions 75

Acknowledgments 76References 76

I. FOREWORD

Cold and ultracold collisions occupy a strategic posi-tion at the intersection of several powerful themes ofcurrent research in chemical physics, in atomic, molecu-lar, and optical physics, and even in condensed matter.The nature of these collisions bears critically on opticalmanipulation of inelastic and reactive processes, preci-sion measurement of molecular and atomic properties,matter-wave coherences and quantum-statistical con-densates of dilute, weakly interacting atoms. This crucialposition explains the wide interest and explosive growthof the field since its inception in 1987, but presents theauthors of a review with both a blessing and curse. Theblessing is self-evident. Exciting physics speaks for itselfand will capture the reader’s attention in spite of errorsand flaws in the presentation. The curse is unfortunatelyjust as self-evident. Any review attempting to encom-pass the breadth of scientific developments engenderedby such wide appeal runs the risk of discouraging all butthe most determined from reading it serially from startto finish. We have, therefore, tried to minimize this riskby organizing the review so subsets of it might conformto readers’ more focused interests. For example, readersinterested in broadening their general culture of physicsin this area could read Secs. II, III, IV, VI.A,B, VII.A,and VIII.B. Readers interested primarily in experimen-tal developments could concentrate on Secs. II, IV,V.A.4, V.B, VI, VII.A–VII.E, VIII, and IX. Theoreticalwork is concentrated in Secs. III, V.A.1–V.A.3, VII.F,and VIII. Ground-state collisions, of primary impor-tance to Bose-Einstein condensates, are discussed inSec. VIII. Control of collisional processes by opticalfields is concentrated in Secs. VI and VII. We have alsoreluctantly decided to omit many very important ad-vances in the determination of molecular potentials, andtheir calculation and experimental determination, be-cause it would render an already large review com-pletely unmanageable. For the same reason we haveomitted review of the exciting advances in Bose-Einsteincondensation except where they touch explicitly on col-lisions. Within this more modest scope we have at-tempted to be comprehensive; but of course we takeresponsibility for all errors and omissions and would liketo hear from readers who detect them.

II. GENERAL INTRODUCTION

Just over a decade ago the first successful experimentsand theory1 demonstrating that light could be used to

1A good introduction to the early physics of laser cooling andtrapping can be found in two special issues of the Journal ofthe Optical Society of America B: ‘‘The Mechanical Effects ofLight,’’ J. Opt. Soc. Am. B 2, No. 11, November 1985 and

Rev. Mod. Phys., Vol. 71, No. 1, January 1999

cool and confine atoms to submillikelvin temperatures(Phillips et al., 1985; Dalibard and Cohen-Tannoudji,1985) opened several exciting new chapters in atomic,molecular, and optical physics. Atom interferometery(Adams, Carnal, and Mlynek, 1994; Adams, Sigel, andMlynek, 1994), matter-wave holography (Morinagaet al., 1996), optical lattices (Jessen and Deutsch, 1996),and Bose-Einstein condensation in dilute gases2 all ex-emplify startling new physics where collisions betweenatoms cooled with light play a pivotal role. These colli-sions have become the subject of intensive study notonly because of their importance to these new areas ofatomic, molecular, and optical physics but also becausetheir investigation has lead to new insights into how coldcollision spectroscopy can lead to precision measure-ments of atomic and molecular parameters and how ra-diation fields can manipulate the outcome of a collisionitself. As a general orientation Fig. 1 shows how a typi-cal atomic de Broglie wavelength varies with tempera-ture and where various physical phenomena situatealong the scale. With de Broglie wavelengths on the or-der of a few thousandths of a nanometer, conventionalgas-phase chemistry can usually be interpreted as theinteraction of classical nuclear point particles movingalong potential surfaces defined by their associated elec-tronic charge distribution. At one time liquid helium wasthought to define a regime of cryogenic physics, but it isclear from Fig. 1 that optical and evaporative coolinghave created cryogenic environments many orders ofmagnitude below the temperature of liquid helium. Atthe level of Doppler cooling and optical molasses1 the deBroglie wavelength becomes comparable to or longerthan the chemical bond, approaching the length of thecooling optical light wave. Here we can expect wave andrelativistic effects such as resonances, interferences, andinteraction retardation to become important. FollowingSuominen (1996), we will term the Doppler cooling andoptical molasses temperature range, roughly between 1mK and 1 mK, the regime of cold collisions. Most colli-sion phenomena at this level are studied in the presenceof one or more light fields used to confine the atoms andto probe their interactions. Excited quasimolecularstates often play an important role. Below about 1 mK,where evaporative cooling and Bose-Einstein condensa-tion (see footnote 2) become the focus of attention, thede Broglie wavelength grows to a scale comparable tothe mean distance separating atoms at the critical con-densation density; quantum degenerate states of the

‘‘Laser Cooling and Trapping of Atoms,’’ J. Opt. Soc. Am. B 6,No. 11, November 1989. Two more recent reviews (Metcalfand van der Straten, 1994; Adams and Riis, 1997) update adecade of developments since the early work recounted in theJ. Opt. Soc. Am. B special issues.

2For an introduction to current research in alkali-atom Bose-Einstein condensation see the special issue on Bose-Einsteincondensation in the Journal of Research of the National Insti-tute of Standards and Technology 101, No. 4, July-August1996.

3John Weiner et al.: Experiments and theory in cold and ultracold collisions

atomic ensemble begin to appear. In this regime groundstate collisions only take place through radial (not angu-lar) motion and are characterized by a phase shift, orscattering length, of the ground-state wave function.Since the atomic translational energy now lies below thekinetic energy transferred to an atom by recoil from ascattered photon, light can play no further role in atomcooling, and collisions in a temperature range from 1 mK→0 must occur in the dark. These collisions are termedultracold. We and others have used the terms ‘‘cold’’and ‘‘ultracold’’ in various ways in the past but now pre-fer to use the terminology of Suominen (1996) in distin-guishing between the two. This review will recountprogress in understanding collision processes in both thecold and ultracold domains.

We start with cold collisions because many experi-ments carried out in optical traps and optically slowedatomic beams take place in this temperature range. Soonafter the first successful experiments reported opticalcooling and trapping in alkali gases, Vigue (1986) andPritchard (1986) discussed the possible consequences ofbinary collisions in a cold or ultracold gaseous medium.Figure 2 shows schematically the general features of acold binary collision. Two S ground-state atoms interactat long range through electrostatic dispersion forces, andapproach along an attractive C6 /R6 potential. The first

FIG. 1. Illustrative plot of various physical phenomena along ascale of temperature (energy divided by Boltzmann’s constantkB) plotted against the de Broglie wavelength for atomic so-dium. The term a0 represents the Bohr radius of the H atom.


excited states, correlating to an S1P asymptote andseparated from the ground state by the atomic excitationenergy \v0 , interact by resonance dipole-dipole forcesand approach either along an attractive or repulsiveC3 /R3 curve. If the colliding atoms on the ground statepotential encounter a red-detuned optical field \v1 , theprobability of a free-bound transition to some vibration-rotation level v of the attractive excited state will maxi-mize around the Condon point RC where \v1 matchesthe potential difference. The quasimolecule finds itselfphotoassociated and vibrating within the attractive well.A second field \v2 can then further excite or even ion-ize the photoassociated quasimolecule, or it can relaxback to some distribution of the continuum and boundlevels of the ground state with a spontaneous emissionrate g. Photoassociation and subsequent inelastic pro-cesses comprise the discussion of Secs. V and VI. If theatoms colliding on the ground state potential interactwith a blue-detuned optical field \v, the probability oftransition to some continuum level of the repulsive ex-cited state will maximize around the Condon point RC ,which will also be quite close to the turning point of thenuclear motion on the repulsive excited state. The atomsapproach no further and begin to separate along the re-pulsive curve leading to the S1P asymptote. The blue-detuned field ‘‘shields’’ the atoms from further interac-tion at more intimate internuclear separation and‘‘suppresses’’ the rate of various inelastic and reactiveprocesses. Optical shielding and suppression is the topicof Sec. VII. Cold collisions in the presence of opticalfields or in the ground state reveal new physics in do-mains where atomic scales of length, time, and spectralline width are reversed from their conventional rela-tions. In addition to the atomic de Broglie wave increas-ing to a length several hundred times that of the chemi-cal bond, the collisional interaction time grows to

FIG. 2. Schematic of a cold collision. Light field \v1 , reddetuned with respect to the S1P asymptote, excites the qua-simolecule in a free-bound transition around the Condon pointRC and can lead to excitation or ionization with the absorptionof a second photon \v2 . Light field \v1 , blue detuned withrespect to the S1P asymptote, prevents atoms from approach-ing significantly beyond the Condon point. Blue-detuned exci-tation leads to optical shielding and suppression of inelasticand reactive collision rates.


several times the spontaneous emission lifetime, and theinhomogeneous Doppler line width at cold and ultracoldtemperatures narrows to less than the natural width ofan atomic dipole transition. The narrow, near-thresholdcontinuum state distribution means that at most a fewpartial waves will contribute to a scattering event. Aver-aging therefore does not obscure matter wave effectssuch as resonances, nodes, and antinodes in scatteringwave function amplitudes, and channel-opening thresh-old laws. In the cold collision regime, Doppler broaden-ing is narrow compared to the radiative natural width,therefore permitting ultrahigh precision, free-bound mo-lecular spectroscopy, and efficient participation of theentire atom ensemble in the excitation process. Longcollision duration ensures interacting partners sufficienttime to exchange multiple photons with modes of anexternally applied radiation field. The frequency, inten-sity, and polarization of the optical field can in turnmodify effective interaction potentials and control theprobability of inelastic, reactive, and elastic final productchannels.

Three questions have motivated significant develop-ments in cold collisions: (1) How do collisions lead toloss of atom confinement in traps? (2) How can photo-association spectroscopy yield precision measurementsof atomic properties and insight into the quantum na-ture of the scattering process itself? (3) How can opticalfields be used to control the outcome of a collisionalencounter?

This third question has had an enduring appeal tochemical physicists, and we review here some of theearly history to put in perspective current developmentsand to emphasize the importance of the ultracold re-gime. In the decade of the seventies, after the develop-ment of the CO2 laser, researchers in atomic and chemi-cal physics immediately thought to use it to influence orcontrol inelastic processes and chemical reactions. How-ever, early attempts to induce reactivity by exciting well-defined, localized molecular sites such as double bondsor functional groups failed because the initial optical ex-citation diffused rapidly into the rotations, vibrations,and torsional bending motions of the molecular nuclei.The unfortunate result was that the infrared light of theCO2 laser essentially heated the molecules much as thefamiliar and venerable Bunsen burner. Enthusiasm forlaser-controlled chemistry cooled when researchers real-ized that this rapid energy diffusion throughout the mo-lecular skeleton blocked significant advance along theroad to optical control of reactivity. In the early eightiesthe development of the pulsed dye laser, tunable in thevisible region of the spectrum, together with new pro-posals for ‘‘radiative collisions,’’ in which the electronsof the molecule interact with the light rather than thenuclei, revived interest. A second round of experimentsachieved some success in optically transforming reac-tants to products; in general, however, the high peakpowers necessary to enhance reactivity significantly in-terfered with the desired effects by inducing nonlinear,multiphoton excitation and ionization in the moleculeitself. The necessity for high peak power in turn arose


from two crucial factors: (1) Doppler broadening at am-bient temperature permits only a few percent of all mo-lecular collisions to interact with an applied optical fieldmode so the product ‘‘yield’’ is low, and (2) the opticalfield had to influence the strong chemical binding inter-action in order to affect atomic behavior during the col-lisional encounter. This requirement implied the needfor power densities greater than a few megawatts persquare centimeter. Power densities of this order are wellabove the threshold for multiphoton absorption and ion-ization, and these processes quickly convert atoms ormolecules from a neutral gas into an ionized plasma. Itappeared depressingly difficult to control collisions withlight without first optically destroying them.

However, atomic deceleration and optical coolingbrightened this discouraging picture, by narrowing theinhomogeneous Doppler broadening to less than a natu-ral line width of an atomic transition and transferringthe optical-particle interaction from the ‘‘chemical’’zone of strong wave-function overlap to an outer regionwhere weak electrostatic terms characterize the colli-sion. In this weakly interacting outer zone only hundredsof milliwatts per square centimeter of optical powerdensity suffice to profoundly alter the inelastic and reac-tive collision rate constant. Furthermore, although aconventional atomic collision lasts only a few hundredfemtoseconds, very short compared to the tens of nano-seconds required before excited molecules or atomsspontaneously emit light, in the ultracold regime par-ticles move much more slowly, taking up to hundreds ofnanoseconds to complete a collisional encounter. Thelong collision duration leaves plenty of time for the twointeracting partners to absorb energy from an externalradiation field or emit energy by spontaneous or stimu-lated processes.

To the three earlier questions motivating studies ofcold collisions may now be added a fourth relevant tothe ultracold regime: What role do collisions play in theattainment and behavior of boson and fermion gases inthe quantum degenerate regime? Quantum statistical ef-fects have been observed and studied in the superfluidityof liquid helium and in the phenomena of metallic andhigh-temperature superconductivity. These dramaticand significant manifestations of quantum collective ef-fects are nevertheless difficult to study at the atomiclevel because the particles are condensed and stronglyinteracting. Observation and measurement of weakly in-teracting dilute gases, however, relate much more di-rectly to the simplest, microscopic models of this behav-ior. The differences between model ideal quantum gasesand real quantum gases begin with binary interactionsbetween the particles, and therefore the study of ultra-cold collisions is a natural point of departure for inves-tigation. Collisions determine two crucial aspects ofBose-Einstein condensation experiments: (1) the evapo-rative cooling rate necessary for the attainment of Bose-Einstein condensation depends on the elastic scatteringcross section, proportional to the square of the s-wavescattering length, and (2) the sign of the scatteringlength indicates the stability of the condensate: positive


scattering lengths lead to large, stable condensates whilenegative scattering lengths do not. The ability to pro-duce condensates by sympathetic cooling also dependscritically on the elastic and inelastic (loss and heating)rate constants among different states of the collidingpartners. Although a confined Bose atom condensatebears some analogy to an optical cavity (photons arebosons), the atoms interact through collisions; these col-lisions limit the coherence length of any atom lasercoupled out of the confining cavity or Bose-Einsteincondensation trap. Another important point, relatingback to optical control, is that the amplitude and sign ofthe scattering length depends sensitively on the fine de-tails of the ground potentials. The possibility of manipu-lating these potentials and consequently collision rateswith external means, using optical, magnetic, or radiofrequency fields, holds the promise of tailoring the prop-erties of the quantum gas to enhance stability and coher-ence. Finally it now appears that the limiting loss pro-cess for dilute gaseous Bose-Einstein condensations arethree-body collisions that also measure the third-ordercoherence, thus providing a critical signature of truequantum statistical behavior. Understanding the quan-tum statistical collective behavior of ultracold dilutegases will drive research in ultracold collisions for yearsto come.

III. INTRODUCTION TO COLD COLLISION THEORY

A. Basic concepts of scattering theory

Let us first consider some of the basic concepts thatare needed to describe the collision of two ground stateatoms. We initially consider the collision of two distin-guishable, structureless particles a and b with interac-tion potential Vg(R) moving with relative momentum k,where R is the vector connecting a and b . We will gen-eralize below to the cases of identical particles and par-ticles with internal structure. The collision energy is E5\2k2/2m , where m is the reduced mass of the two par-ticles. If there is no interaction between the particles,Vg50, the wave function describing the relative motionof the two particles in internal states u0a& and u0b& is

Cg1~R!5eik–Ru0a0b&. (1)

If the interaction potential is nonzero, the collision be-tween the particles results in a scattered wave, and atlarge R beyond the range of the potential the wave func-tion is represented as

Cg1~R!;H eik–R1

eikR

Rf~E ,k,ks!J u0a0b&, (2)

where k is a unit vector indicating the direction of k andks is a unit vector indicating the direction of the scat-tered wave with amplitude f(E ,k,ks). The overall effectof the collision is described by a cross section s(E). In agas cell, for which all directions k are possible, s(E) isdetermined by integrating over all scattered directionsand averaging over all values of initial k:


s~E !5E4p

dk4p E

4pdksuf~E ,k,ks!u2. (3)

Since f has units of length, the cross section has units oflength25area. Equation (3) simplifies for a sphericallysymmetric potential, for which f depends only on theangle u between k and ks:

s~E !52pE0

p

du sin~u!uf~E ,u!u2 (4)

The object of scattering theory is to calculate the scat-tering amplitude and cross section, given the interactionpotentials between the two atoms. The first step in re-ducing the problem to practical computation is to intro-duce the partial wave expansion of the plane wave:

eik–R54p(l50

`

(m52l

l

i lYlm* ~ k!Ylm~ ks!j l~kR !, (5)

where Ylm is a spherical harmonic and the function j lhas the following form as R→` :

j l~kR !;sin@kR2 ~p/2!l#

kR. (6)

The complete wave function at all R is also expanded ina partial wave series:

Cg1~R!54p(

l50

`

(m52l

l

i lYlm* ~ k!Ylm~ ks!

3Fl

1~E ,R !

Ru0a0b&, (7)

where Fl1(E ,R) is determined from the Schrodinger

equation,

d2Fl1~E ,R !

dR2 12m

\2 S E2Vg~R !1\2l~ l11 !

2mR2 DFl1~E ,R !

50. (8)

By imposing the following boundary condition onFl

1(E ,R) as R→` , the asymptotic wave function hasthe desired form, Eq. (2), representing an incident planewave plus scattered wave:

Fl1~E ,R !

R;sinS kR2

p

2l1h lD eih l

kR(9)

;j l~kR !1i

2ei@kR2~p/2!l#

kRTl~E !. (10)

Here h l is the phase shift induced by the interactionpotential Vg(R), and Tl(E)512e2ih l is the T-matrixelement from which the amplitude of the scattered waveis determined,

f~E ,k,ks!52pi

k (l50

`

(m52l

l

i lYlm* ~ k!Ylm~ ks!Tl~E !

(11)

Using the simpler form of Eq. (4), the cross section be-comes


s~E !5p

k2 (l50

`

~2l11 !uTl~E !u2 (12)

54p

k2 (l50

`

~2l11 !sin2h l . (13)

The cross section has a familiar semiclassical interpre-tation. If the interaction potential Vg(R) vanishes, theparticle trajectory is a straight line with relative angularmomentum R3p5bp , where p is linear momentum andb is the distance of closest approach. If we take the an-gular momentum to be bp5\Al(l11)'\(l1 1

2 ), wherethe ‘‘classical’’ l here is nonquantized, then b is the clas-sical turning point of the repulsive centrifugal potential\2l(l11)/2mR2 in Eq. (8). The semiclassical expressionfor the cross section, analogous to Eq. (12), has the formof an area,

s~E ,semiclassical!52pE0

`

bP~b ,E !db (14)

weighted by P(b ,E)5uTl(E)u2. An important featureof cold collisions is that only a very few values of l cancontribute to the cross section, because the classicalturning points of the repulsive centrifugal potential areat large values of R . Only collisions with the very lowestl values allow the atoms to get close enough to one an-other to experience the interatomic interaction poten-tial. We will discuss below the specific quantum proper-ties associated with discrete values of l. Semiclassicaltheory is useful for certain types of trap loss collisions inrelatively warm traps where light is absorbed by atompairs at very large R , but a quantum treatment alwaysbecomes necessary at sufficiently low collision energy.

In the simple introduction above, the cross section inEq. (13) represents elastic scattering, for which the in-ternal states of the particles do not change, and theirrelative kinetic energy E is the same before and afterthe collision. In general, the atoms will have nonzerointernal angular momentum due to hyperfine structurein the case of alkali atoms or due to electronic structurein the case of rare gas metastable atoms. In a field-freeregion, these internal states are characterized by totalangular momentum F and projection M on a space-fixedquantization axis. Often an external field, either a mag-netic or an optical or radio frequency electromagneticfield, is present in cold collision experiments, and theseFM states are modified by the external field. Species likehydrogen and alkali atoms, which have 2S ground statesand nonvanishing nuclear spin I , have two hyperfinecomponents with F5I1 1

2 and F5I2 12 . Figure 3 shows

the Zeeman splitting of the two F51 and F52 hyper-fine components of ground state 87Rb atoms, and Fig. 4shows the ground-state interaction potentials for two in-teracting Rb atoms. If the atomic field-modified states,commonly called field-dressed states, are represented byua i& for atom i5a ,b , then the general collision processrepresents a transition from the state uaa&uab& to thestate uaa8&uab8&, represented by the general transition am-plitude f(E ,k,ks,aaab→aa8ab8) and the T-matrix ele-


ment T(E ,aaab→aa8ab8). These are the most completetransition amplitudes that describe an observable transi-tion, specified in terms of quantities that can be selectedexperimentally. The most complete T-matrix elementsthat can be calculated from the multicomponent versionof Eq. (8) are also specified by the initial and final par-tial waves: T(E ,aaablm→aa8ab8 l8m8). There are threedistinct physical axes which define these amplitudes: aspace-fixed axis which defines the space projection quan-tum numbers Ma and Mb , the asymptotic direction ofapproach k, and the asymptotic direction of separationks. In beam experiments all three axes can be different.Most all the work on cold atom collisions is carried outin a homogeneous gas, where neither k nor ks are se-lected or measured, and a cell average cross section de-fined as in Eq. (12) for the general s(E ,aaab→aa8ab8) isappropriate. Some experiments, especially in the contextof Bose-Einstein condensation, have a well-defined localspace-fixed axis, and select particular values of Ma andMb , but most of the work we will describe is for unpo-larized gases involving an average over Ma and Mb .

Instead of cross sections, it is usually preferable togive a rate coefficient for a collision process. The ratecoefficient is directly related to the number of collisionevents occurring in a unit time in a unit volume. Con-sider the reaction

aa1ab→aa81ab8 , (15)

where the quantum numbers are all assumed to be dif-ferent. If the density of species a i in a cell is na i

, the rateof change of the density of the various species due tocollision events is

naa

dt1

nab

dt52

naa8

dt2

nab8

dt

5K~T ,aaab→aa8ab8 !naanab

. (16)

The rate coefficient K(T ,aaab→aa8ab8) is related to thecross section through

K~T ,aaab→aa8ab8 !5^s~E ,aaab→aa8ab8 !v&, (17)

where the brackets imply an average over the distribu-tion of relative collision velocities v .

The general theory of collisions of degenerate speciesis well understood. The basic multichannel theory can befound, for example, in Mies (1973a, 1973b) or Bayliset al. (1987). There is a considerable body of work onthe theory of collisions of cold spin-polarized hydrogenatoms, as a consequence of the still unfulfilled quest toachieve Bose-Einstein condensation in such a system.This will be discussed in Sec. VIII. The Eindhovengroup led by B. Verhaar has been especially active indeveloping the theory. Stoof, Koelman, and Verhaar(1988) give an excellent introduction to the subject. Thetheory of cold collisions in external magnetic fields andoptical or radiofrequency electromagnetic fields has alsobeen developed, as we will describe in more detail in thelater sections of this review.


It is important to distinguish between two differentkinds of collisions: elastic and inelastic. As mentionedabove, an elastic collision is one in which the quantumstates aa ,ab of each atom remain unchanged by the col-lision. These collisions exchange momentum, therebyaiding the thermalizing of the atomic sample. These are‘‘good’’ collisions that do not destroy the trapped states,and they are necessary for the process of evaporativecooling we will describe later. An inelastic collision isone in which one (or more) of these two quantum num-bers changes in the collision. Most cold collision studieshave dealt with inelastic events instead of elastic ones,that is, the collision results in hot atoms or untrappedspecies or even ionic species. As we will see in the nextsection, the quantum threshold properties of elastic andinelastic collisions are very different.

The basic difference between collisions of differentatomic species and identical atomic species is the needto symmetrize the wave function with respect to ex-change of identical particles in the latter case. Otherthan this symmetrization requirement, the theory is thesame for the two cases. Symmetrization has two effects:the introduction of factors of two at various points in thetheory, and the exclusion of certain states since they vio-late the exchange symmetry requirement. Such symme-try restrictions are well known in the context of diatomicmolecular spectroscopy, leading to, for example,orthospecies and paraspecies of molecular hydrogen andto every other line being missing in the absorption spec-trum of molecular oxygen, due to the zero nuclear spinof the oxygen atom (Herzberg, 1950). In the case ofatomic collisions of identical species, if the two quantumnumbers are identical, aa5ab , only even partial waves lare possible if the particles are composite bosons, andonly odd partial waves are possible if the particles arecomposite fermions. If the two quantum numbers arenot identical, aaÞab , both even and odd partial wavescan contribute to collision rates. The effect of this sym-metry is manifestly present in photoassociation spectra,where for example, half the number of lines appear in adoubly spin polarized gas (where all atoms are in thesame quantum state) as contrasted to an unpolarized gas(where there is a distribution of quantum states). Thesespectra will be described in Section VI below.

Stoof, Koelman, and Verhaar (1988) give a good dis-cussion of how to modify the theory to account for ex-change symmetry of identical particles. Essentially, theyset up the states describing the separated atoms, the so-called channel states of scattering theory, as fully sym-metrized states with respect to particle exchange.T-matrix elements and event rate coefficients, defined asin Eq. (17), are calculated conventionally for transitionsbetween such symmetrized states. The event rate coeffi-cients are given by

K~$gd%→$ab%!

5K p\

mk (l8m8

(lm

uT~E ,$gd%l8m8,$ab%lm !u2L(18)


where the braces $¯% signify symmetrized states, and theT matrix as defined in this review is related to the uni-tary S matrix by T512S. Then collision rates are un-ambiguously given by

FIG. 3. Ground hyperfine levels of the 87Rb atom versus mag-netic field strength. The Zeeman splitting of each manifold isevident. The energy has been divided by the Boltzmann con-stant in order to express it in temperature units.

FIG. 4. The ground-state potential energy curves of the Rb2molecule. The potentials have been divided by the Boltzmannconstant in order to express them in units of temperature. Thefull figure shows the short range potentials on the scale ofchemical bonding. The inset shows an enlargement at longrange, showing the separated atom hyperfine levels Fa1Fb

5111, 112, and 212. The upper two potentials in the insetcorrelate adiabatically with the 3Su .


dna

dt5(

b($gd%

~11dab!

3@K~gd→ab!ngnd2K~ab→dg!nanb# . (19)

This also works for the case of elastic scattering of iden-tical particles in identical quantum states: K(aa→aa)must be multiplied by a factor of 2 to get the rate ofmomentum transfer k scatters to (ksÞk) since two at-oms scatter per collision event. Gao (1997) has also de-scribed the formal theory for collisions of cold atomstaking into account identical particle symmetry.

B. Quantum properties as E→0

Cold and ultracold collisions have special propertiesthat make them quite different from conventional roomtemperature collisions. This is because of the differentscales of time and distance involved. The effect of thelong collision time will be discussed in Sec. V. Here wewill examine the consequence of the long de Brogliewavelength of the colliding atoms. The basic modifica-tion to collision cross sections when the de Brogliewavelength becomes longer than the range of the poten-tial was described by Bethe (1935) in the context of coldneutron scattering, and has been widely discussed in thenuclear physics literature (Wigner, 1948; Delves, 1958).Such quantum threshold effects only manifest them-selves in neutral atom collisions at very low tempera-ture, typically !1 K (Julienne and Mies, 1989; Julienne,Smith, and Burnett, 1993), but they play a very impor-tant role in the regime of laser cooling and evaporativecooling that has been achieved.

Figure 1, in the Introduction, gave a qualitative indi-cation of the range of l as temperature is changed. Evenat the high temperature end of the range of laser cool-ing, l on the order of 100a0 is possible, and at the lowerend of evaporative cooling, l.1 mm (20 000a0). Thesedistances are much larger than the typical lengths asso-ciated with chemical bonds, and the delocalization of thecollision wave function leads to characteristic behaviorwhere collision properties scale as some power of thecollision momentum k5A2mE52p/l as k→0, depend-ing on the inverse power n of the long range potential,which varies as R2n.

In the case of elastic scattering, the phase shift h l inEqs. (9) and (13) has the following property as k→0(Mott and Massey, 1965): if 2l,n23,

limk→0

k2l11cot h l521

Al, (20)

where Al is a constant, whereas, if 2l.n23,

limk→0

kn22cot h l5const. (21)

For neutral ground state atoms, this ensures that thephase shift vanishes at least as fast as k3 for all l>1.Thus all contributions to the cross section vanish when kbecomes sufficiently small, except the contribution fromthe s wave, l50. Since the s-wave phase shift varies as


2kA0 as k→0, we see from Eq. (13) that the elasticscattering cross section for identical particles approaches

s~E !→8pA02 , (22)

where the factor of 8 instead of 4 occurs due to identicalparticle symmetry (two particles scatter per collision).Thus, the cross section for elastic scattering becomesconstant in the low energy limit. The quantity A0 is thes-wave scattering length, an important parameter in thecontext of Bose-Einstein condensation. Note that therate coefficient for elastic scattering vanishes as T1/2 inthe limit of low temperature, since K5^sv&.

Figure 5 illustrates the properties of a very low colli-sion energy wave function, taken for a model potentialwith a van der Waals coefficient C6 and mass character-istic of Na atom collisions. The upper panel illustratesthe wave function at long range, on the scale of 1 mm.The lower panel shows an enlargement of the short-range wave function for the case of three different po-tentials, with three different scattering lengths, negative,zero, and positive. The figure shows the physical inter-pretation of the scattering length, as the effective pointof origin of the long wavelength aymptotic sine wave atR5A0 , since the long range wave function is propor-tional to sin@k(R2A0)#.

Inelastic collisions have very different threshold prop-erties than elastic ones. As long as the internal energy ofthe separated atoms in the exit channel is lower than forthe entrance channel, so that energy is released in thecollision, the transition matrix element varies as (Bethe,1935; Wigner, 1948; Julienne and Mies, 1989)

T~E !}kl11/2, (23)

where l is the entrance channel partial wave index. Us-ing this form in Eq. (12) shows that the cross sectionvanishes at least as fast as k for all l.0, but it varies as1/k for the s wave. This variation (sometimes called the1/v law) was given by Bethe (1935) and is well known innuclear physics. Although the cross section for an inelas-tic, energy-releasing, collision becomes arbitrarily largeas k→0, the rate coefficient K remains finite, and ap-proaches a nonvanishing constant. Although Cote et al.(1996) speak of a quantum suppression of cold inelasticcollisions, the ‘‘suppression’’ is only relative to a semi-classical process, and actual rate constants can be sur-prisingly large near zero temperature, as will be dis-cussed in Sec. VIII B.

The range of k where these limiting threshold lawsbecome valid depends strongly on the particular species,and even on the specific quantum numbers of the sepa-rated atoms. Knowledge of the long range potentialalone does not provide a sufficient condition to deter-mine the range of k in which they apply. This range, aswell as the scattering length itself, depends on the actualphase shift induced by the whole potential, and is verysensitive to uncertainties in the short range part of thepotential in the chemical bonding region (Gribakin andFlambaum, 1993). On the other hand, a necessary con-dition for threshold law behavior can be given basedsolely on the long-range potential (Julienne and Mies,


1989; Julienne, Smith, and Burnett, 1993). This condi-tion is based on determining where a semiclassical,WKB connection breaks down between the long-rangeasymptotic s wave and the short-range wave functionthat experiences the acceleration of the potential. Con-sider the ground-state potential Vg(R) as a function ofR. The long-range potential is

Vg~R !52Cn

Rn , (24)

where n is assumed to be >3. Let us first define

EQ5\2

m F S 2n11

3 D 2nS n226n D nS 2n12

n22•

\2

mCnD 2G ~1/n22 !

(25)

RQ5S n222n12

•

Cn

EQD 1/n

. (26)

The properties of the wave function C(E ,R) depend onthe values of E and R relative to EQ and RQ . WhenE@EQ , the energy is high enough that it is always pos-sible to make a semiclassical Wentzel-Kramers-Brillouin(WKB) connection between the wave function in thelong-range zone, R@RQ , and the wave function in theshort-range zone, R!RQ . In this case, the WKB repre-sentation of the s-wave wave function is a good approxi-mation at all R , and there are no special threshold ef-fects. On the other hand, E!EQ satisfies the conditionthat there is some range of distance near R'RQ wherethe WKB approximation fails, so that no semiclassicalconnection is possible between long- and short-rangewave functions. This is the regime where a quantumconnection, with its characteristic threshold laws, is nec-essary. When E!EQ the wave function for R@RQ isbasically the asymptotic wave with the asymptotic phaseshift (Julienne, 1996), whereas the wave function for R!RQ is attenuated in amplitude, relative to the normalWKB amplitude, by a factor proportional to kl11/2 (Juli-enne and Mies, 1989).

Julienne, Smith, and Burnett (1993) calculated thevalues of EQ and RQ for a number of alkali and rare gasmetastable species that can be laser cooled. These areshown in Table I. The alkali values were based onknown values of C6 for the n56 van der Waals poten-tials, and estimates were made for the metastable raregas C6 coefficients, for which the weak C5 /R5

quadrupole-quadrupole contribution to the potential formetastable rare gases was ignored in comparison to theC6 contribution. The value of RQ is small enough thatrelativistic retardation corrections to the ground statepotential are insignificant. The RQ ,EQ parameters foreach rare gas are similar to those of the neighboringalkali species in the periodic table. Figure 6 shows a plotof the ground-state potential expressed in units of RQand EQ . The position of the maximum in the barrierdue to the centrifugal potential for lÞ0, as well as themaximum binding energy and minimum outer classicalturning point of the last bound state in the van derWaals potential, scale with C6 and mass as RQ and EQdo, scaling as (mC6)1/4 and m23/2C6

21/2 , respectively.


These other scalable parameters are also indicated onFig. 6, which gives a universal plot for any of the speciesin Table I. Figure 6 also shows lQ /2p50.857RQ for E5EQ . The centrifugal barriers for the lowest few partialwaves are lower than EQ , but lie outside RQ . The po-sitions RC(l) and heights EC(l) of the centrifugal barri-ers are RC(l)52.67RQ /@ l(l11)#1/4 and EC(l)50.0193EQ@ l(l11)#3/2 respectively. The p- and d-wavebarriers have heights of 0.055EQ and 0.28EQ , respec-tively. EQ is small enough that several partial wavestypically contribute to ground-state collisions of theheavier species at typical magneto-optical trap tempera-tures of 100 mK to 1 mK.

The distance RQ , closely related to the mean scatter-ing length a50.967RQ defined by Gribakin and Flan-baum (1993), is solely a property of the long-range po-tential and atomic mass. On the other hand, theapproach to the quantum threshold laws is in a range 0,E!EQ , where the actual range depends on the effectof the whole potential, as measured by the thresholdphase shift proportional to the actual s-wave scatteringlength A0 . Let Ab be defined by Eb5\2/2mAb

2 , whereEb is the position of the last bound state of the potentialjust below threshold (Ab.0) or the position of a virtualbound state just above threshold (Ab,0). When Eb,EQ , the range over which the threshold laws applycan be estimated from 0,k!uAbu21, or 0,E!Eb .

FIG. 5. The upper panel illustrates the long de Broglie wave atlong range, on the scale of 1 mm. The lower panel shows anenlargement of the short range wave function for the case ofthree different potentials, with three different scatteringlengths, negative, zero, and positive.


Thus, the threshold laws apply only over a range com-parable to the lesser of EQ and Eb . In the limit wherethe last bound state is very near threshold, Gribakin andFlambaum (1993) find that A05 a1Ab ; this reduces tothe result of Mies et al. (1996), A0'Ab , when uA0u@ a .The special threshold properties of the last bound statein the potential has recently been described by Boisseau,Audouard, and Vigue (1998) and Trost et al. (1998). Theusual quantization rules in a long-range potential (Le-Roy and Bernstein, 1970) do not adequately character-ize the threshold limit of the last bound state.

Since the scattering length can be arbitrarily large,corresponding to the case where the last bound state isarbitrarily close to threshold, Eb can be arbitrarily small.Therefore, the actual range over which the threshold lawexpressions in Eqs. (22) and (23) apply can in principlebe less than the actual temperature range in cold or evenultracold experiments. In fact, the Cs atom in its doublyspin polarized hyperfine level seems not yet to be in thethreshold limit even at a few mK (Arndt et al., 1997; Leoet al., 1998).

To show how the interatomic potential affects thescattering length, we can imagine artificially modifyingthe ground-state potential by adding some small correc-tion term lVcorr(R) to it which has a short range com-pared to RQ . Figure 7 shows that we can make A0 varybetween 2` and 1` by making relatively smallchanges in the potential. The two singularities corre-spond to the case where the potential just supports an-other bound state. Thus, when lVcorr changes across asingularity, the number of bound states in the potentialchanges by exactly one. The scattering length is negativewhen a new bound state is just about to appear, andpositive when a new bound state has just appeared. Scat-tering lengths are sufficiently sensitive to potentials that,with rare exceptions, it is impossible to predict them ac-curately from first principles, and therefore they must bemeasured. How this can be done accurately will be de-scribed in the section on photoassociation spectroscopy.

C. Collisions in a light field

Much of the work on collisions of cooled and trappedatoms has involved collisions in a light field. This is thesubject of Secs. V, VI, and VII of this review and hasbeen the subject of two earlier reviews. (Walker andFeng, 1994; Weiner, 1995). Therefore, it is useful to pro-vide an overview of some of the concepts that are usedin understanding cold collisions in a light field. Figure 2gives a schematic description of the key features of thesecollisions, showing ground and excited state potentials,Vg(R) and Ve(R), and indicating the optical couplingbetween ground and excited states induced by a lightfield at frequency v1 , which is detuned by D5v12v0from the frequency of atomic resonance v0 . Detuningcan be to the red or blue of resonance, corresponding tonegative or positive D respectively. The figure illustratesthat the excited state potential, having a resonant-dipoleform that varies as C3 /R3, which can be attractive orrepulsive, is of much longer range than the ground-state


van der Waals potential. Figure 2 also indicates the Con-don point RC for the optical transition. This is the pointat which the quasimolecule comprised of the two atomsis in optical resonance with the light field, i.e.,

Ve~RC!2Vg~RC!5\v1 . (27)

The Condon point is significant because in a semiclassi-cal picture of the collision, it is the point at which thetransition from the ground state to the excited state isconsidered to occur. The laser frequency v1 can bereadily varied in the laboratory, over a wide range of redand blue detuning. Thus, the experimentalist has at hisdisposal a way of selecting the Condon point and theupper state which is excited by the light. By varying thedetuning from very close to atomic resonance to very farfrom resonance, the Condon point can be selected torange from very large R to very small R .

Although the semiclassical picture implied by usingthe concept of a Condon point is very useful, a propertheory of cold collisions should be quantum mechanical,accounting for the delocalization of the wave function. Ifthe light is not too intense, the probability of a transi-tion, Pge(E ,v1), is proportional to a Franck-Condonoverlap matrix element between ground- and excited-state wave functions:

TABLE I. Characteristic threshold parameters RQ and EQ .

SpeciesRQ

(Bohr)EQ

(mK) SpeciesRQ

(Bohr)EQ

(mK)

Li 32 120 He* 34 180Na 44 19 Ne* 40 26K 64 5.3 Ar* 60 5.7Rb 82 1.5 Kr* 79 1.6Cs 101 0.6 Xe* 96 0.6

FIG. 6. The position of the maximum in the barrier due to thecentrifugal potential for lÞ0, as well as the maximum bindingenergy and minimum outer classical turning point of the lastbound state in the van der Waals potential, all have identicalscalings with C6 and mass as RQ and EQ , with scaling as(mC6)1/4 and m3/2C6

1/2 , respectively.


Pge~E ,v1!}u^Ce~R !uVeg~R !uCg~E ,R !&u2 (28)

'uVeg~RC!u2u^Ce~R !uCg~E ,R !&u2, (29)

where the optical Rabi matrix element Veg(R) mea-sures the strength of the optical coupling. Julienne(1996) showed that a remarkably simple reflection ap-proximation to the Franck-Condon factor, based on ex-panding the integrand of the Franck-Condon factorabout the Condon point, applies over a wide range ofultracold collision energies and red or blue detunings.This approximation, closely related to the usual station-ary phase approximation of line broadening theory,shows that the Franck-Condon factor (in the caseswhere there is only one Condon point for a given state)is proportional to the ground-state wave function at theCondon point:

Fge~E ,D!5u^Ce~R !uCg~E ,R !&u2 (30)

51

DCuCg~E ,RC!u2 blue detuning (31)

5hnv

1DC

uCg~E ,RC!u2 red detuning. (32)

Here DC5ud(Ve2Vg)/dRuRCis the slope of the differ-

ence potential, and the wave functions have been as-sumed to be energy normalized, that is, the wave func-tions in Eqs. (7) to (9) are multiplied by A2mk/p\2 sothat

u^Cg~E ,R !uCg~E8,R !&u25d~E2E8!. (33)

The formulas for red and blue detuning differ becausethey represent free-bound and free-free transitions re-spectively; here nv is the vibrational frequency of thebound vibrational level excited in the case of red detun-ing. Figure 8 shows the validity of this approximationover a range of Condon points and collision energies fora typical case. We thus see that it is legitimate to use thesemiclassical concept of excitation at a Condon point indiscussing cold collisions in a light field.

The excited-state potential can be written at longrange as (Meath, 1968)

Ve~R !5f\GAS lA

2pR D 3

, (34)

where lA and GA are the respective transition wave-length and spontaneous decay width of the atomic reso-nance transition, and where f is a dimensionless factoron the order unity. Thus, in a magneto-optical trap,where uDu is on the order of GA , the Condon point willhave a magnitude on the order of lA/2p , that is, on theorder of 1000a0 for typical laser cooling transitions. Thisis the regime of the trap loss experiments described inSec. V. Optical shielding experiments are typically donewith detuning up to several hundred GA , correspondingto much smaller Condon points, on the order of a fewhundred a0 . Photoassociation spectroscopy experimentsuse quite large detunings, and typically sample Condonpoints over a range from short range chemical bondingout to a few hundred a0 .


Because of the very different ranges of D and RC inthe various kinds of cold collisions in a light field, a num-ber of different kinds of theories and ways of thinkingabout the phenomena have been developed to treat thevarious cold collision processes. Since RC for magneto-optical trap-loss collisions is typically much larger thanthe centrifugal barriers illustrated in Fig. 6, many partialwaves contribute to the collision, the quantum thresholdproperties are not evident at all, and semiclassical de-scriptions of the collision have been very fruitful in de-scribing trap loss. The main difficulty in describing suchcollisions is in treating the strong excited-state spontane-ous decay on the long time scale of the collision. We willsee that a good, simple quantum mechanical descriptionof such collisions is given by extending the above Con-don point excitation picture to include excited state de-cay during the collision. Quantitative theory is still madedifficult by the presence of complex molecular hyperfinestructure, except in certain special cases. Since the Con-don points are at much smaller R in the case of opticalshielding, excited state decay during the collision in nota major factor in the dynamics, and the primary chal-lenge to theory in the case of optical shielding is how totreat the effect of a strong radiation field, which doesstrongly modify the collision dynamics. Simple semiclas-sical pictures seem not to work very well in accountingfor the details of shielding, and a quantum description isneeded. The theory of photoassociation is the mostquantitative and well developed of all. Since the Condonpoint is typically inside the centrifugal barriers and RQ ,only the lowest few partial waves contribute to the spec-trum. Although the line shapes of the free-bound tran-sitions exhibit the strong influence of the quantumthreshold laws, they can be quantitatively described byFranck-Condon theory, which can include ground andexcited state hyperfine structure. We will show how pho-toassociation spectroscopy has become a valuable toolfor precision spectroscopy and determination of groundstate scattering lengths.

FIG. 7. The scattering length A0 varies between 2` and 1`by making relatively small changes (lVcorr) in the ground-state potential. The two singularities correspond to the casewhere the potential just supports another bound state.


IV. EXPERIMENTAL METHODS FOR THE STUDYOF COLD COLLISIONS

A. Atom traps

1. Light forces

It is well known that a light beam carries momentumand that the scattering of light by an object produces aforce. This property of light was first demonstrated byFrish (1933) through the observation of a very smalltransverse deflection (331025 rad) in a sodium atomicbeam exposed to light from a resonance lamp. With theinvention of the laser, it became easier to observe effectsof this kind because the strength of the force is greatlyenhanced by the use of intense and highly directionallight fields, as demonstrated by Ashkin (1970) with themanipulation of transparent dielectric spheres sus-pended in water. The results obtained by Frish and Ash-kin raised the interest of using light forces to control themotion of neutral atoms. Although the understanding oflight forces acting on atoms was already established bythe end of the 1970s, unambiguous demonstration ofatom cooling and trapping was not accomplished beforethe mid-80s. In this section we discuss some fundamentalaspects of light forces and schemes employed to cooland trap neutral atoms.

The light force exerted on an atom can be of twotypes: a dissipative, spontaneous force and a conserva-tive, dipole force. The spontaneous force arises from theimpulse experienced by an atom when it absorbs oremits a quantum of photon momentum. When an atomscatters light, the resonant scattering cross section canbe written as s05l0

2/2p where l0 is the on-resonantwavelength. In the optical region of the electromagneticspectrum the wavelengths of light are on the order ofseveral hundreds of nanometers, so resonant scatteringcross sections become quite large, ;1029 cm2. Eachphoton absorbed transfers a quantum of momentum \kWto the atom in the direction of propagation (\ is thePlanck constant divided by 2p, and k52p/l is the mag-nitude of the wave vector associated with the opticalfield). The spontaneous emission following the absorp-tion occurs in random directions; over many absorption-emission cycles, it averages to zero. As a result, the netspontaneous force acts on the atom in the direction ofthe light propagation, as shown schematically in Fig. 9.The saturated rate of photon scattering by spontaneousemission (one half the reciprocal of the excited-statelifetime) fixes the upper limit to the force magnitude.This force is sometimes called radiation pressure.

The dipole force can be readily understood by consid-ering the light as a classical wave. It is simply the time-averaged force arising from the interaction of the tran-sition dipole, induced by the oscillating electric field ofthe light, with the gradient of the electric field ampli-tude. Focusing the light beam controls the magnitude ofthis gradient, and detuning the optical frequency belowor above the atomic transition controls the sign of theforce acting on the atom. Tuning the light below reso-nance attracts the atom to the center of the light beam


while tuning above resonance repels it. The dipole forceis a stimulated process in which no net exchange of en-ergy between the field and the atom takes place, butphotons are absorbed from one mode and reappear bystimulated emission in another. Momentum conserva-tion requires that the change of photon propagation di-rection from initial to final mode imparts a net recoil tothe atom. Unlike the spontaneous force, there is in prin-ciple no upper limit to the magnitude of the dipole forcesince it is a function only of the field gradient and de-tuning.

We can bring these qualitative remarks into focus byconsidering the amplitude, phase, and frequency of aclassical field interacting with an atomic transition di-pole. A detailed development of the following results isout of the scope of the present work, but can be foundelsewhere (Stenholm, 1986; Cohen-Tannoudji et al.,1992). The usual approach is semiclassical and consistsin treating the atom as a two-level quantum system andthe radiation as a classical electromagnetic field (Cook,1979). A full quantum approach can also be employed(Cook, 1980; Gordon and Ashkin, 1980), but it will notbe discussed here.

The basic expression for the interaction energy is

U52mW •EW , (35)

where mW is the transition dipole and EW is the electricfield of the light. The force is then the negative of thespatial gradient of the potential,

FW 52¹W R•U5m¹W RE , (36)

where we have set ¹W R•m equal to zero because there isno spatial variation of the dipole over the length scale ofthe optical field. The optical-cycle average of the force isexpressed as

^FW &5^m¹W RE&5m@~¹W RE0!u1„E0¹W R~kLR !…v#, (37)

where u and v arise from the steady-state solutions ofthe optical Bloch equations,

FIG. 8. The plots show the validity of the reflection approxi-mation over a range of Condon points and collision energiesfor a typical case.


u5V

2DvL

~DvL!21~G/2!21V2/2(38)

and

v5V

2G/2

~DvL!21~G/2!21V2/2. (39)

In Eqs. (38) and (39) DvL5v2v0 is the detuning of theoptical field from the atomic transition frequency v0 , Gis the natural width of the atomic transition, and V istermed the Rabi frequency and reflects the strength ofthe coupling between field and atom,

V52mW •EW 0

\. (40)

In writing Eq. (37) we have made use of the fact that thetime-average dipole has in-phase and in-quadraturecomponents,

^mW &52mW ~u cos vLt2v sin vLt !, (41)

and the electric field of the light is given by the classicalexpression,

E5E0@cos~vt2kLR !# . (42)

The time-averaged force, Eq. (37), consists of two terms:the first term is proportional to the gradient of the elec-tric field amplitude; the second term is proportional tothe gradient of the phase. Incorporating Eqs. (38) and(39) into Eq. (37), we have for the two terms,

^FW &5m~¹W RE0!V

2 F DvL

~DvL!21~G/2!21V2/2G1m„E0¹W R~2kLR !…

V

2 F G/2

~DvL!21~G/2!21V2/2G .

(43)

The first term is the dipole force, sometimes called thetrapping force, FT , because it is a conservative force andcan be integrated to define a trapping potential for theatom:

FW T5m~¹W RE0!V

2 F DvL

~DvL!21~G/2!21V2/2G (44)

and

UT52E FTdR5\DvL

2lnF11

V2/2

~DvL!21~G/2!2G .

(45)

The second term is the spontaneous force, sometimescalled the cooling force, FC , because it is a dissipativeforce and can be used to cool atoms,

FW C52mE0kW L

V

2 F G/2

~DvL!21~G/2!21V2/2G . (46)

Note that in Eq. (44) the line-shape function in squarebrackets is dispersive and changes sign as the detuningDvL changes sign from negative (red detuning) to posi-tive (blue detuning). In Eq. (46) the line-shape functionis absorptive, peaks at zero detuning and exhibits a


Lorentzian profile. These two equations can be recast tobring out more of their physical content. The dipoleforce can be expressed as

FW T521

2V2 ¹W V2\DvLF s

11s G , (47)

where s , the saturation parameter, is defined to be

s5V2/2

~DvL!21~G/2!2 . (48)

In Eq. (48) the saturation parameter essentially defines acriterion to compare the time required for stimulatedand spontaneous processes. If s!1 then spontaneouscoupling of the atom to the vacuum modes of the field isfast compared to the stimulated Rabi oscillation, and thefield is considered weak. If s@1 then the Rabi oscilla-tion is fast compared to spontaneous emission and thefield is said to be strong. With the help of the definitionof the Rabi frequency, Eq. (40), and the light beam in-tensity,

I512

e0cE02 , (49)

Eq. (47a) can be written in terms of the gradient of thelight intensity, the saturation parameter, and the detun-ing,

FW T5214I

~¹W I !\DvLF s

11s G . (50)

Note that negative DvL (red detuning) produces a forceattracting the atom to the intensity maximum while posi-tive DvL (blue-detuning) repels the atom away from theintensity maximum.

The spontaneous force or cooling force can also bewritten in terms of the saturation parameter and thespontaneous emission rate,

FW C5\kW LG

2 F s

11s G , (51)

which shows that this force is simply the rate of absorp-tion and reemission of momentum quanta \kL carriedby a photon in the light beam. Note that as s increases

FIG. 9. Spontaneous emission following absorption occurs inrandom directions, but absorption from a light beam occursalong only one direction.


beyond unity, FW C approaches \kW LG/2 , the photon scat-tering rate for a saturated transition.

2. The magneto-optical trap (MOT)

a. Basic notions

Pritchard et al. (1986) originally suggested that thespontaneous light force could be used to trap neutralatoms. The basic concept exploited the internal degreesof freedom of the atom as a way of circumventing theoptical-Earnshaw theorem proved by Ashkin and Gor-don (1983). This theorem states that if a force is propor-tional to the light intensity, its divergence must be nullbecause the divergence of the Poynting vector, whichexpresses the directional flow of intensity, must be nullthrough a volume without sources or sinks of radiation.This null divergence rules out the possibility of an in-ward restoring force everywhere on a closed surface.However, when the internal degrees of freedom of theatom are considered, they can change the proportional-ity between the force and the Poynting vector in aposition-dependent way such that the optical-Earnshawtheorem does not apply. Spatial confinement is then pos-sible with spontaneous light forces produced by counter-propagating optical beams. Using these ideas to circum-vent the optical-Earnshaw theorem, Raab et al. (1987)demonstrated a trap configuration that is presently themost commonly employed. It uses a magnetic field gra-dient produced by a quadrupole field and three pairs ofcircularly polarized, counterpropagating optical beams,detuned to the red of the atomic transition and inter-cepting at right angles in the position where the mag-netic field is zero. The magneto-optical trap exploits theposition-dependent Zeeman shifts of the electronic lev-els when the atom moves in the radially increasing mag-netic field. The use of circularly polarized light, red-detuned ;G results in a spatially dependent transitionprobability whose net effect is to produce a restoringforce that pushes the atom toward the origin.

To make clear how this trapping scheme works, con-sider a two-level atom with a J50→J51 transitionmoving along the z direction (the same arguments willapply to the x and y directions). We apply a magneticfield B(z) increasing linearly with distance from the ori-gin. The Zeeman shifts of the electronic levels areposition-dependent, as shown in Fig. 10(a). We also ap-ply counterpropagating optical fields along the 6z di-rections carrying opposite circular polarizations and de-tuned to the red of the atomic transition. It is clear fromFig. 10 that an atom moving along 1z will scatter s2

photons at a faster rate than s1 photons because theZeeman effect will shift the DMJ521 transition closerto the light frequency.

F1z52\k

2G

3V2/2

@D1kvz1~mB /\!~dB/dz !z#21~G/2!21V2/2.

(52)


In a similar way, if the atom moves along 2z it willscatter s1 photons at a faster rate from the DMJ511transition.

F2z51\k

2

3GV2/2

S D2kvz2mB

\

dB

dzz D 2

1~G/2!21V2/2.

(53)

The atom will therefore experience a net restoring forcepushing it back to the origin. If the light beams are red-detuned ;G , then the Doppler shift of the atomic mo-tion will introduce a velocity-dependent term to the re-storing force such that, for small displacements andvelocities, the total restoring force can be expressed asthe sum of a term linear in velocity and a term linear indisplacement,

FMOT5F1z1F2z52a z2Kz . (54)

Equation (54) expresses the equation of motion of adamped harmonic oscillator with mass m ,

z12a

mz1

K

mz50. (55)

FIG. 10. MOT schema: (a) Energy level diagram showing shiftof Zeeman levels as atom moves away from the z50 axis.Atom encounters a restoring force in either direction fromcounterpropagating light beams. (b) Typical optical arrange-ment for implementation of a magneto-optic trap.


The damping constant a and the spring constant K canbe written compactly in terms of the atomic and fieldparameters as

a5\kG16uD8u~V8!2~k/G!

@112~V8!2#2$114~D8!2/@112~V8!2#%2 (56)

and

K5\kG16uD8u~V8!2~dv0 /dz !

@112~V8!2#2$114~D8!2/@112~V8!2#%2 ,

(57)

where V8, D8, and dv0 /dz5@(mB /\)(dB/dz)#/G are Gnormalized analogues of the quantities defined earlier.Typical MOT operating conditions fix V851/2, D851,so a and K reduce (in mks units) to

a~Nm21s !.~0.132!\k2 (58)

and

K~Nm21!.~1.1631010!\kdB

dz. (59)

The extension of these results to three dimensions isstraightforward if one takes into account that the quad-rupole field gradient in the z direction is twice the gra-dient in the x ,y directions, so that Kz52Kx52Ky . Thevelocity dependent damping term implies that kineticenergy E dissipates from the atom (or collection of at-oms) as E/E05e2(2a/m)t where m is the atomic massand E0 the kinetic energy at the beginning of the coolingprocess. Therefore the dissipative force term cools thecollection of atoms as well as combining with the dis-placement term to confine them. The damping time con-stant t5m/2a is typically tens of microseconds. It is im-portant to bear in mind that a MOT is anisotropic sincethe restoring force along the z axis of the quadrupolefield is twice the restoring force in the xy plane. Further-more a MOT provides a dissipative rather than a con-servative trap, and it is therefore more accurate to char-acterize the maximum capture velocity rather than thetrap ‘‘depth.’’

Early experiments with mgneto-optical trapped atomswere carried out by slowing an atomic beam to load thetrap (Raab et al., 1987; Walker et al., 1990). Later, a con-tinuous uncooled source was used for that purpose, sug-gesting that the trap could be loaded with the slow at-oms of a room-temperature vapor (Cable et al., 1990).The next advance in the development of magneto-optical trapping was the introduction of the vapor-cell magneto-optical trap (VCMOT). This variationcaptures cold atoms directly from the low-velocity edgeof the Maxwell-Boltzmann distribution always presentin a cell background vapor (Monroe et al., 1990).Without the need to load the MOT from an atomicbeam, experimental apparatuses became simpler;and now many groups around the world use theVCMOT for applications ranging from precision spec-troscopy to optical control of reactive collisions.

b. Densities in a MOT

The VCMOT typically captures about a million atomsin a volume with less than one millimeter diameter, re-


sulting in densities ;1010 cm23. Two processes limit thedensity attainable in a MOT are (1) collisional trap lossand (2) repulsive forces between atoms caused by reab-sorption of scattered photons from the interior of thetrap (Walker et al., 1990; Sesko et al., 1991). Collisionalloss in turn arises from two sources: hot background at-oms that knock cold atoms out of the MOT by elasticimpact, and binary encounters between the cold atomsthemselves. Trap loss due to cold collisions is the topicof Sec. V. The ‘‘photon-induced repulsion’’ or photontrapping arises when an atom near the MOT centerspontaneously emits a photon which is reabsorbed byanother atom before the photon can exit the MOT vol-ume. This absorption results in an increase of 2\k in therelative momentum of the atomic pair and produces arepulsive force proportional to the product of the crosssections for absorption of the incident light beam andfor the scattered fluorescence. When this outward repul-sive force balances the confining force, a further increasein the number of trapped atoms leads to larger atomicclouds, but not to higher densities.

3. Dark SPOT MOT

In order to overcome the ‘‘photon-induced repulsion’’effect, Ketterle et al. (1993) proposed a method that al-lows the atoms to be optically pumped to a ‘‘dark’’ hy-perfine level of the atom ground state that does not in-teract with the trapping light. In a conventional MOTone usually employs an auxiliary ‘‘repumper’’ lightbeam, copropagating with the trapping beams but tunedto a neighboring transition between hyperfine levels ofground and excited states. The repumper recovers popu-lation that leaks out of the cycling transition betweenthe two levels used to produce the MOT. As an ex-ample, Fig. 11 shows the trapping and repumping tran-sitions usually employed in a conventional Na MOT.The scheme proposed by Ketterle et al. (1993), known asa dark spontaneous-force optical trap (dark SPOT),passes the repumper through a glass plate with a smallblack dot shadowing the beam such that the atoms at thetrap center are not coupled back to the cycling transitionbut spend most of their time (;99%) in the ‘‘dark’’hyperfine level. Cooling and confinement continue tofunction on the periphery of the MOT but the centercore experiences no outward light pressure. The darkSPOT increases density by almost two-orders of magni-tude.

4. The far-off resonance trap (FORT)

Although a MOT functions as a versatile and robust‘‘reaction cell’’ for studying cold collisions, light fre-quencies must tune close to atomic transitions, and anappreciable steady-state fraction of the atoms remainexcited. Excited-state trap-loss collisions and photon-induced repulsion limit achievable densities.

A far-off resonance trap (FORT), in contrast, uses thedipole force rather than the spontaneous force to con-fine atoms and can therefore operate far from resonancewith negligible population of excited states. The firstatom confinement (Chu et al., 1986) was reported using


a dipole-force trap. A hybrid arrangement in which thedipole force confined atoms radially while the spontane-ous force cooled them axially was used by a NIST-Maryland collaboration (Gould et al., 1988) to studycold collisions, and a FORT was demonstrated by Milleret al. (1993a) for 85Rb atoms. The FORT consists of asingle, linearly polarized, tightly focused Gaussian-modebeam tuned far to the red of resonance. The obviousadvantage of large detunings is the suppression of pho-ton absorption. Note from Eq. (46) that the spontaneousforce, involving absorption and reemission, falls off asthe square of the detuning while Eq. (45) shows that thepotential derived from dipole force falls off only as thedetuning itself. At large detunings and high field gradi-ents (tight focus) Eq. (45) becomes

U.\V2

4DvL, (60)

which shows that the potential becomes directly propor-tional to light intensity and inversely proportional to de-tuning. Therefore at far detuning but high intensity thedepth of the FORT can be maintained but most of theatoms will not absorb photons. The important advan-tages of FORTS compared to MOTS are (1) high den-sity (;1012 cm23) and (2) a well-defined polarizationaxis along which atoms can be aligned or oriented (spinpolarized). The main disadvantage is the small numberof trapped atoms due to small FORT volume. The bestnumber achieved is about 104 atoms (Cline et al., 1994a,1994b).

5. Magnetic traps

Pure magnetic traps have also been used to study coldcollisions, and they are critical for the study of dilutegas-phase Bose-Einstein condensates in which collisionsfigure importantly. We anticipate therefore that mag-

FIG. 11. Usual cooling (carrier) and repumping (sideband)transitions when optically cooling Na atoms. The repumperfrequency is normally derived from the cooling transition fre-quency with electrooptic modulation. Dashed lines show thatlasers are tuned about one natural linewidth to the red of thetransition frequencies.


netic traps will play an increasingly important role infuture collision studies in and near Bose-Einstein con-densation conditions.

The most important distinguishing feature of all mag-netic traps is that they do not require light to provideatom containment. Light-free traps reduce the rate ofatom heating by photon absorption to zero, an appar-ently necessary condition for the attainment of Bose-Einstein condensation. Magnetic traps rely on the inter-action of atomic spin with variously shaped magneticfields and gradients to contain atoms. Assuming the spinadiabatically follows the magnetic field, the two govern-ing equations for the potential U and force FW B are

U52mW S•BW 52gsmB

\SW •BW 52

gsmB

\MSB (61)

and

FW B52gsmB

\MS¹W B . (62)

where mS and mB are the electron spin dipole momentand Bohr magneton respectively, MS the z-componentof SW , and gs the Lande g-factor for the electron spin. Ifthe atom has nonzero nuclear spin IW then FW 5SW 1IW sub-stitutes for SW in Eq. (61), the g factor generalizes to

gF>gS

F~F11 !1S~S11 !2I~I11 !

2F~F11 !, (63)

and

FW B52gFmB

\MF¹W B . (64)

Depending on the sign of FW B with respect to BW , atoms instates whose energy increases or decreases with mag-netic field are called ‘‘weak-field seekers’’ or ‘‘strong-field seekers,’’ respectively. For example, see Fig. 3. Onecould, in principle, trap atoms in any of these states,needing only to produce a minimum or a maximum inthe magnetic field. Unfortunately only weak-field seek-ers can be trapped in a static magnetic field because sucha field in free space can only have a minimum. Dynamictraps have been proposed to trap both weak- and strong-field seekers (Lovelace et al., 1985). Even when weak-field seeking states are not in the lowest hyperfine levelsthey can still be used for trapping because the transitionrate for spontaneous magnetic dipole emission is;10210 sec21. However, spin-changing collisions canlimit the maximum attainable density.

The first static magnetic field trap for neutral atomswas demonstrated by Migdall et al. (1985). An anti-Helmholtz configuration (see Fig. 10), similar to a MOT,was used to produce an axially symmetric quadrupolemagnetic field. Since this field design always has a cen-tral point of vanishing magnetic field, nonadiabatic Ma-jorana transitions can take place as the atom passesthrough the zero point, transferring the population froma weak-field to a strong-field seeker and effectivelyejecting the atom from the trap. This problem can be


overcome by using a magnetic bottle with no point ofzero field (Gott et al., 1962; Pritchard, 1983; Bagnatoet al., 1987; Hess et al., 1987). The magnetic bottle, alsocalled the Ioffe-Pritchard trap, was recently used toachieve a Bose-Einstein condensation in a sample of Naatoms pre-cooled in a MOT (Mewes et al., 1996). Otherapproaches to eliminating the zero field point are thetime-averaged orbiting potential trap (Anderson et al.,1995) and an optical ‘‘plug’’ (Davis et al., 1995) that con-sists of a blue-detuned intense optical beam alignedalong the magnetic trap symmetry axis and producing arepulsive potential to prevent atoms from entering thenull field region. Trap technology continues to developand the recent achievement of Bose-Einstein condensa-tion will stimulate more robust traps containing greaternumbers of atoms. At present ;107 atoms can betrapped in a Bose-Einstein condensate loaded from aMOT containing ;109 atoms.

B. Atom beams

Although MOTs and FORTs have proved productiveand robust experimental tools for the investigation ofultracold collisions, they suffer from two shortcomingsof any ‘‘reaction cell’’ technique: (1) collision kinetic en-ergy or temperature cannot be varied over a wide rangeand (2) the isotropic distribution of molecular collisionaxes in the laboratory frame masks effects of light-fieldpolarization on inelastic and reactive processes. Further-more, probe beams in a MOT cannot explore close tothe cooling transition without severely perturbing or de-stroying the MOT.

An alternative approach to atom traps is the study ofcollisions between and within atomic beams. In fact theapplication of thermal atomic and molecular beams toinvestigate inelastic and reactive collisions dates fromthe early sixties (Herschbach, 1966). It was not until theearly eighties, however, that polarization and velocity-group selection led to detailed measurements of colli-sion rate constants as a function of velocity, alignment,and orientation of the colliding partners (Wang andWeiner, 1990). Beginning the decade of the nineties,Thorsheim, et al. (1990) reported the first measurementsof subthermal collisions in an atomic beam. In this ex-periment a single-mode laser beam excited a narrow ve-locity group from a thermal sodium atomic beam, andthe rate of associative ionization within the selected ve-locity group was measured as a function of light polar-ization perpendicular and parallel to the beam axis. Theeffective collision temperature of 65 mK was just on thethreshold of the subthermal regime, defined as TS , thetemperature at which the atomic radiative lifetime be-comes comparable to the collision duration. For sodiumexcited to the 3p2P3/2 , TS is 64 mK (Julienne and Mies,1989). Building on this experiment, Tsao et al. (1995)used a much narrower optical transition (5p2P3/2) to se-lect a velocity group from a collimated atomic beam andreported a rate constant for photoassociative ionizationat 5.3 mK, well within the subthermal regime. Figure 12shows a diagram of the experimental setup. Atoms


emerging from a mechanically collimated thermalsource interact with light in three stages: first, the atombeam crosses an optical ‘‘pump’’ beam tuned to transferpopulation from F52 to F51 of the ground state. Sec-ond, a velocity-group-selecting optical beam, tuned tothe 3s2S1/2 (F51)→5p2P3/2 transition crosses theatomic beam at ;45° and pumps a very narrow group tothe previously emptied F52 level of the ground state.Finally, a third light beam, detuned ;G to the red of theD2 transition, excites collision pairs from within the F52 velocity group to form the detected photoassociativeionization product ions. The production rate of the Na2

1

product ions is then measured as a function of intensityand polarization. Although the optical-velocity-selectiontechnique achieves subthermal collision temperatures, itsuffers from the disadvantage that only a small fractionof the atoms in the atomic beam participate, and signallevels are low.

Subjecting the axial or transverse velocity componentsof the beam to dissipative cooling dramatically com-presses the phase space of the atom flux, resulting indense, well-collimated atomic beams suitable for thestudy of atom optics, atom holography, or ultracold col-lision dynamics. Prodan et al. (1982) first demonstratedthe importance of this phase-space compression. In factatomic beams can now achieve ‘‘brightness’’ (atombeam flux density per unit solid angle) many timesgreater than the phase-space conservation limit imposedby the Liouville theorem (Pierce, 1954; Sheehy et al.,1989; Kuyatt, 1967). Several groups have used MOTphysics to make ‘‘atom funnels’’ or ‘‘2D MOTS’’ to cooland compress the transverse components of atom beams(Nellessen et al., 1990; Riis et al., 1990; Faultsich et al.,1992; Hoogerland, 1993; Yu et al., 1994; Swanson et al.,1996; Tsao et al., 1996) or have used ordinary magneto-optical traps as sources of slow, well-collimated beams(Myatt et al., 1996; Lu et al., 1996). Tsao (1996) has usedaxial phase-space compression to increase atom densityin a velocity group prior to optical selection. Figure 13shows how a cooling light beam counterpropagating to a

FIG. 12. Schematic of atomic beam experiment in which anarrow velocity group is selected (laser beam 2) and intrav-elocity group collisions are probed (laser beam 3). From Thor-sheim et al. (1990).


thermal atomic beam can compress the Maxwell-Boltzmann distribution into a relatively narrow range ofvelocities. Tsao et al. (1998) used this technique to inves-tigate the polarization properties of optical suppression,an optically controlled collisional process described inSec. VII. To date atom beams have not contributed asimportantly to cold and ultracold collision dynamics asMOTs and FORTs, but their full potential has yet to beexploited.

V. INELASTIC EXOERGIC COLLISIONS IN MOTS

An exoergic collision converts internal atomic energyto kinetic energy of the colliding species. When there isonly one species in the trap (the usual case) this kineticenergy is equally divided between the two partners. Ifthe net gain in kinetic energy exceeds the trapping po-tential or the ability of the trap to recapture, the atomsescape, and the exoergic collision leads to trap loss.

Of the several trapping possibilities described in thelast section, by far the most popular choice for collisionstudies has been the magneto-optical trap (MOT). AMOT uses spatially dependent resonant scattering tocool and confine atoms. If these atoms also absorb thetrapping light at the initial stage of a binary collision andapproach each other on an excited molecular potential,then during the time of approach the colliding partnerscan undergo a fine-structure-changing collision or relaxto the ground state by spontaneously emitting a photon.In either case electronic energy of the quasimoleculeconverts to nuclear kinetic energy. If both atoms are intheir electronic ground states from the beginning to theend of the collision, only elastic and hyperfine-changingcollisions can take place. Elastic collisions (identicalscattering entrance and exit states) are not exoergic butfigure importantly in the production of Bose-Einsteincondensates. At the very lowest energies only s-wavescontribute to the elastic scattering, and in this regimethe collisional interaction is characterized by the scatter-ing length. The sign and magnitude of the scatteringlength determine the properties of a weakly-interactingBose gas and controls the rate of evaporative cooling,needed to achieve BEC. The hyperfine-changing colli-sions arise from ground state splitting of the alkali atomsinto hyperfine levels due to various orientations of thenonzero nuclear spin. A transition from a higher to alower molecular hyperfine level during the collisionalencounter releases kinetic energy. In the absence of ex-ternal light fields hyperfine-changing collisions oftendominate trap heating and loss.

If the collision starts on the excited level, the long-range dipole-dipole interaction produces an interatomicpotential varying as 6C3 /R3. The sign of the potential,attractive or repulsive, depends on the relative phase ofthe interacting dipoles. For the trap-loss processes thatconcern us in this chapter we concentrate on the attrac-tive long-range potential, 2C3 /R3. Due to the ex-tremely low energy of the collision, this long-range po-tential acts on the atomic motion even when the pair areas far apart as l/2p (the inverse of the light field wave


vector k). Since the collision time is comparable to orgreater than the excited state life time, spontaneousemission can take place during the atomic encounter. Ifspontaneous emission occurs, the quasimolecule emits aphoton red-shifted from atomic resonance and relaxes tothe ground electronic state with some continuum distri-bution of the nuclear kinetic energy. This conversion ofinternal electronic energy to external nuclear kinetic en-ergy can result in a considerable increase in the nuclearmotion. If the velocity is not too high, the dissipativeenvironment of the MOT is enough to cool this radiativeheating, allowing the atom to remain trapped. However,if the transferred kinetic energy is greater than the re-capture ability of the MOT, the atoms escape the trap.This process constitutes an important trap loss mecha-nism termed radiative escape, and was first pointed outby Vigue (1986). For alkalis there is also another exoer-gic process involving excited-ground collisions. Due tothe existence of a fine structure in the excited state (P3/2and P1/2), the atomic encounter can result in fine-structure-changing collision, releasing DFS of kinetic en-ergy, shared equally between both atoms. For examplein sodium 1

2 DEFS /kB.12 K, which can easily cause theescape of both atoms from the MOT, typically 1 K deep.

These three effects, hyperfine-changing collisions, ra-diative escape, and fine-structure-changing collisions,are the main exoergic collisional processes that takeplace in a MOT. They are the dominant loss mecha-nisms which usually limit the maximum attainable den-sity and number in MOTs. They are not, however, theonly type of collision in the trap, as we shall see later.

A. Excited-state trap-loss collisions

1. Early quasistatic models

a. Gallagher-Pritchard model

Soon after the first observations of ultracold collisionsin traps, Gallagher and Pritchard (1989) proposed asimple semiclassical model to describe the fine-structure-changing and radiative escape trap-loss pro-cesses. Figure 14 shows the two mechanisms involved.When atoms are relatively far apart they absorb a pho-ton, promoting the system from the ground state to along-range attractive molecular state. We denote the rel-evant molecular levels by their asymptotic energies.Thus, S1S represents two ground-state atoms, and S1P1/2 or S1P3/2 represents one ground and one excitedstate atom which can be in either one of the fine-structure levels.

For radiative escape, the process is described by

A1A1\v→A2*→A1A1\v8 (65)

with energy \(v2v8)/2 transferred to each atom. Thefine-structure-changing collision is represented by

A1A1\v→A* ~P3/2!1A→A* ~P1/2!1A1DEFS(66)

with DEFS/2 transferred to each atom.The Gallagher-Pritchard model begins by considering

a pair of atoms at the internuclear separation R0 . If this


pair is illuminated by a laser of frequency vL and inten-sity I , the rate of molecular excitation is given by

R~R0 ,vL ,I !5F ~GM/2!2

@DM#21~GM/2!2G I

\vL

l2

2p

5e~vL ,R0!I

\vL

l2

2p, (67)

where DM5@vL2v(R0)# , and v(R0)5vA2C3 /\R03 is

the resonant frequency at R0 , and vA is the atomic reso-nant frequency. The constant C3 characterizes the ex-cited molecular potential, l2/2p is the photoabsorptioncross section to all attractive molecular states (one halfthe atomic value), and I and vL are the laser intensityand frequency. Finally GM is the molecular spontaneousdecay rate, here taken as twice the atomic rate, GA .

After excitation at R0 , the atom pair begins to accel-erate together on the 2C3 /R3 potential, reaching theshort-range zone where the fine-structure-changing col-lision and radiative escape processes leading to trap lossoccur. The time to reach the short-range zone is essen-tially the same as to reach R50. This time is easily cal-culated by integrating the equation of motion,

t~R0!5FM

4 G1/2E0

R0dRFC3

R3 2C3

R03G21/2

50.747FMR05

4C3G 1/2

, (68)

where M is the atomic mass. Defining D5vA2vL as thedetuning, Rt the interaction separation reached by theatoms (in one life time t5GM

21), and Dt the detuning atRt . With these definitions, the time for the atoms toreach the short-range zone can be written as

t~R0!5S Dt

D D 5/6

GM21 , (69)

and the probability of survival against spontan-eous emission is given by g5exp@2GMt(R0)# or g5exp@(2Dt /D)5/6# .

If the A* -A pair reaches the short-range region(where fine-structure-changing collisions take place) be-fore radiating, it rapidly traverses this zone twice, yield-

FIG. 13. The trace with the sharp peak is the velocity profile ofaxially cooled atomic beam. The broad low-amplitude traceshows the residual thermal velocity distribution from which thesharp peak was compressed. From Tsao et al. (1998).


ing a probability for fine-structure-changing collisions(FCC) equal to hJ52P(12P), where P is the Landau-Zener single-transit curve-crossing probability. Re-peated oscillations of the atoms through the crossing re-sults in a probability PFCC which is the sum of theprobabilities at each traversal,

PFCC5hJg1hJ~12hJ!g31¯5hJg

@12g21hJg2#

.

(70)

If n is the total atomic density, the number of pairs withseparation R0→R01dR0 is n24pR0

2dR0/2, and the totalrate of fine-structure-changing collisions per unit of vol-ume is the integration over all internuclear separations,

RFCC5n2

2 E0

`

dR04pR02R~R0 ,vL ,I !PFCC~R0!. (71)

The rate constant between two colliding ground stateatoms is then

kFCC512 E

0

`

dR04pR02R~R0 ,vL ,I !PFCC~R0!. (72)

Alternatively the rate constant can be expressed interms of a binary collision between one excited and oneunexcited atom,

RFCC5nn* kFCC8 , (73)

where n* is the excited atom density. At steady state,the rate of atom photo-excitation expressed as the prod-uct of photon flux, atom density, and absorption crosssection (including line shape function) equals the rate offluorescence,

~I/\vL!n~l2/p!F GA2 /4

~DM!21GA2 /4G5n* GA , (74)

and combining Eqs. (67), (71), (73), and (74),

kFCC8 514 F ~DM!21GA

2 /4

GA/4

3E0

`

dR04pR02e~vL ,R0!PFCC~R0!G . (75)

The fine-structure change is not the only loss mecha-nism in the trap. Radiative escape is also important. Therate of radiative escape (RE), per unit of volume, can beobtained from Eqs. (72) or (75), when PFCC(R0) is re-placed by PRE(R0), the probability of spontaneousemission during transit in an inner region, R,RE . HereRE is defined as the internuclear separation at which thekinetic energy acquired by the colliding pair is enough toescape from the magneto-optical trap. The probabilityPRE(R0) can be obtained as a series sum similar to Eq.(70). Defining the time spent at R,RE as 2tE(R0) ob-tained from an integration similar to Eq. (69), one writesPRE(R0) as

PRE~R0!52tE~R0!GMg

@12g21hJg2#

. (76)


The Gallagher-Pritchard model predicts that fine-structure-changing collisions dominate over radiative es-cape. Gallagher and Pritchard (1989) estimate hJ;0.2for sodium collisions, resulting in a prediction ofPFCC /PRE;20 for Na–magneto-optical trap loss.

b. Julienne-Vigue model

Julienne and Vigue (1991) proposed an elaboration,hereafter termed the JV model, which introduced therole of angular momentum and a thermal averaging pro-cedure, employed realistic molecular states, and tookinto account the effects of retardation on molecularspontaneous emission decay rates. Like Gallagher andPritchard, they neglected molecular hyperfine structure.The rate of fine-structure-changing collisions or radia-tive escape transition per unit of volume is written as

ratevolume

5Kn251

2dg2

pvk2 (

l ,e~2l11 !PTL~e ,l !

3PES~R ,e ,l ,D ,I !n2. (77)

where n is the ground state density, v the asymptoticvelocity for the atoms when they are far apart, k thewave vector associated with the momentum of the re-duced particle, I the light intensity, dg the ground statedegeneracy, and the factor of 1/2 accounts for homo-nuclear symmetry. The summation is performed over allcontributing attractive excited states indexed by e. Theterm PTL(e ,l) represents the probability that trap loss(either fine-structure-changing collisions or radiative es-cape) occurs at small internuclear separation RTL afterthe atoms have drawn together. This PTL(e ,l) factor wasevaluated using different methods and was shown to benearly independent of the asymptotic energy of the at-oms over a wide range of temperature. The JV expres-sion Eq. (77) corresponds to the Gallagher-Pritchard ex-pression Eq. (71). Note however that excitation, survivalagainst spontaneous decay, and the collisional interac-tion are factored differently. In Gallagher and Pritchard,R(R0 ,vL ,I), represents excitation and PFCC or PRE

FIG. 14. Schematic diagram showing mechanisms of radiativeescape and fine-structure-changing collisions. An \v photon,red detuned ;GA , transfers the population to long-range at-tractive curve. Atoms accelerate radially along a molecular po-tential until either the molecule emits a photon \v8 (RE) orundergoes a crossing to a potential curve asymptotically disso-ciating to 2P1/21

2S1/2 (FCC).


represents survival and collisional interaction together;in JV, PES represents excitation and survival and PTLthe probability of the collisional process itself. The exci-tation and survival probability, PES(R ,e ,D ,I), repre-sents the probability that an excited state « produced ata rate G(R8) at some R8.R by light detuned D andintensity I will survive without spontaneous radiativedecay during the motion from R8 to R . In Julienne-Vigue theory, PES can be written as

PES~R ,e ,l ,L ,I !5ER

`

G~R8,e ,D ,I !SEil ~R ,R8,e!

dR8

vEil ~R8!

(78)where SEi

l (R ,R8,e) is the survival factor for the atompair moving from R8 to R on the excited state e, withangular momentum l and the initial kinetic energy Ei ,starting from the ground state. The excitation rate isG(R8,e ,D ,I) and the excitation probability is given byG(R8,e ,D ,I)dR8/vEi

l (R8).Trap-loss collision rates in alkali magneto-optical

traps have been measured extensively, and the specificmolecular mechanisms for fine-structure-changing colli-sions and radiative escape trap loss for this class of at-oms have been discussed theoretically by Julienne andVigue (1991). The molecular structure of the alkali met-als are qualitatively similar, and the long range poten-tials are well known. The long-range potentials comingfrom the P3/21S1/2 separated Na atoms are shown in Fig.15 and give rise to five attractive states (1u , 0g

2 , 2u , 1g ,and 0u

1). Those states have different decay rates whichmakes the survival effect very sensitive to the initial ex-citation. As to the collisional interaction itself, Julienneand Vigue (1991) verified, using quantum scattering cal-culations, that the predictions of Dashsevskaya et al.(1969) about fine-structure-changing collisions werequalitatively correct. The calculations by Dashevskayaet al. (1969) show that fine-structure-changing collisionsarise from two principal mechanisms: spin-orbit (radial)coupling at short range and Coriolis (angular) couplingat long range and that gerade states contribute very littleto the fine-structure-changing collisions process. There-fore Julienne and Vigue (1991) conclude that, of all the2P3/21

2S1/2 entrance channel states, only 0u1 and 2u con-

tribute significantly to the FCC cross section. Da-shevskaya (1979) points out three basic pathways forFCC: (1) spin-orbit mixing of the 0u

1 components of theA1Su and b3Pu where they cross at short range, (2)Coriolis mixing of the V50,1,2 components of b3Pu ,and Coriolis coupling of 0u

1 and 1u states at long range.The dominant mechanism for the light alkalis with rela-tively small spin-orbit terms (Li, Na, K) are the Corioliscouplings. For the heavy alkalis (Rb, Cs) the short-rangespin-orbit mechanism prevails. Julienne and Vigue(1991) calculated fine-structure-changing collisions andradiative escape for all alkalis and obtained rate coeffi-cients for trap loss combining fine-structure-changingcollisions and radiative escape as shown in Fig. 16. Be-cause the spin-orbit interaction in Li is small, fine-structure-changing collisions do not lead to appreciable


trap loss. It becomes important only if the trap depth isless than 250 mK. The radiative escape contribution forLi is also small due to the fast decay rate of the excitedstates, preventing close approach of the two partnersalong the accelerating 2C3 /R3 excited-state potentials.As a result, Li is predicted to have the lowest trap lossrate of all the alkalis.

2. Theoretical approaches to excited-state trap loss

a. Quasistatic theories

Both the Gallagher-Pritchard and Julienne-Viguetheories develop rate expressions that derive from the‘‘quasistatic’’ picture of light-matter interaction. Theidea is that the atoms approach each other sufficientlyslowly such that at each internuclear separation Rc thereis enough time to establish a steady state between ratesof quasimolecule stimulated absorption and spontane-ous emission. This steady state sets up the familiarLorentzian line-shape function, (GM/2)2/„DM

2

1(GM/2)2… , for the probability of excited state popula-

tion. This line-shape function permits off-resonant exci-tation around the Condon point Rc where DM50. For aflat ground state Vg50 and excited state Ve52C3 /R3

the detuning is D5C3 /\Rc3 and udD/dRcu53C3 /\Rc

4.Thus at relatively large detuning the line-shape functionis still rather narrow and the excitation is fairly well con-fined around Rc . At small detunings however, off-resonant excitation contributes significantly to the qua-sistatic rate constant, and further theoretical studiesusing quantum scattering with complex potentials(Boesten et al., 1993; Boesten and Verhaar, 1994; Juli-enne et al., 1995), optical Bloch equations (Band andJulienne, 1992; Band et al., 1994) or quantum MonteCarlo methods (Lai et al., 1993; Holland et al., 1994)have shown that this off-resonant excitation picture is ofdubious validity. These issues have been reviewed in de-tail by Suominen (1996). It is also worth noting that boththe Gallagher-Pritchard and Julienne-Vigue theoriesfactor the rate constant into two terms: the first termdescribing excitation and survival at long range, and thesecond term expressing the probability of the short-range trap-loss process itself. This decoupling of opticalexcitation and survival from the short-range collisionalinteraction provides a useful factorization even for the-oretical approaches which do not invoke the quasistaticpicture.

b. Method of complex potentials

At weak field, where optical cycling of population be-tween ground and excited states can be safely ignored, atractable quantum mechanical treatment of collisionalloss processes can be conveniently introduced by inser-tion of an imaginary term in the time-independentHamiltonian. Boesten et al. (1993) and Julienne et al.(1995) have each developed complex potential modelsof ultracold collisions in order to test the semiclassicaltheories. Both found serious problems with existingsemiclassical theories at small detuning. Again the ap-proach is to factor the probability for trap loss into two


parts: (1) long-range excitation and survival,J(E ,l ,D ,I ;G), and (2) the short-range loss process itself,PX(E ,l), where the subscript X denotes fine-structureor radiative escape and E ,l are the total collision energyand angular momentum, respectively. The terms D ,Irepresent the detuning and intensity of the optical exci-tation with G the spontaneous decay rate. The overallprobability for the process is therefore,

P~E ,l ,D ,I ;G!5PX~E ,l !J~E ,l ,l ,D ,I ;G!. (79)

In principle the inclusion of a dissipative term like spon-taneous decay means that the problem should be set upusing the optical Bloch equations or quantum MonteCarlo methods which take dissipation into account natu-rally over a wide range of field strengths. However in theweak-field limit exact quantum scattering models with acomplex potential suffice for testing the semiclassicalGallagher-Pritchard and Julienne-Vigue theories anddeveloping alternative semiclassical models. Julienneet al. (1994), for example, studied a three-state modelcomprising the ground state entrance channel, an opti-cally excited molecular state, and a fictitious probe statewhich simulates the collisional loss channel of the fine-structure or radiative escape processes. The trap-lossprobability is then expressed in terms of the S matrixconnecting the ground state (g) to the probe state (p),

P~E ,l ,D ,I ;G!5uSgp~E ,l ,D ,I ;G!u2

5PXQ~E ,l !•JQ~E ,l ,D ,I ;G!. (80)

The subscript Q denotes quantum close coupling, andJulienne et al. (1994) compare JQ to four semiclassicalapproximations for J : (1) JJV from the semiclassicalJulienne-Vigue model, (2) JBJ from an optical-Bloch-equation calculation of Band and Julienne (1992), (3)JBV from the Landau-Zener model of Boesten and Ver-haar (1994), and (4) JLZ from the Landau-Zener model

FIG. 15. Long-range potential curves of the first excited statesof the Na dimer labeled by Hund’s case (c) notation. The num-ber gives the projection of electron spin plus orbital angularmomentum on the molecular axis. The g/u and 1/2 labelscorrespond to conventional point-group symmetry notation.


of Julienne et al. (1995). Both JBV and JLZ are variantsof the celebrated Landau-Zener formula,

JLZ.Sae2A, (81)

where Sa is the survival factor and A is the familiarLandau-Zener argument,

A52puVag~I !u2

\UdVa

dR2

dVg

dR URc

va

. (82)

In Eq. (82) Vag is the optical coupling, proportional tothe laser intensity, and the denominator is the productof the difference in slopes evaluated at the Condonpoint RC and the local velocity va . The inner-regioncollisional interaction PXQ(E ,l) is insensitive to the in-tensity and detuning of the outer-region optical excita-tion D ,I , the decay rate G, or the collision energy E .Therefore the overall trap-loss probability, Eq. (80), iscontrolled by the behavior of JQ(E ,l ,D ,I ;G). Figure 17shows Cs trap-loss probabilities for the various J’s as afunction of collision temperature and reveals two impor-tant points. First, the quantal close coupling JQ confirmsthe conclusion of Boesten et al. (1993) which firstpointed out a ‘‘quantum suppression’’ effect, i.e., that Jfor a Cs model falls by about four orders of magnitudebelow the semiclassical prediction, JBJ , as the tempera-ture decreases from 1 mK to 10 mK. Second, JLZ tracksthe close-coupling results very closely even down intothis ‘‘quantum suppression’’ region. At first glance theability of the semiclassical Landau-Zener formula to fol-low the quantum scattering result even where the semi-classical optical-Bloch-equation and Julienne-Viguetheories fail may seem surprising. The very good agree-ment between JQ and JLZ simply means that the dra-matic reduction in trap-loss probability below 1 mK canbe interpreted essentially as poor survival against spon-taneous emission when the approaching partners are lo-cally excited at the Condon point at long range. Thecalculations in Fig. 17 were carried out with a detuning Dof one atomic line width GA . To check this interpreta-

FIG. 16. Total trap-loss rate constants b as a function of tem-perature for the different alkali species. These rates are calcu-lated by Julienne and Vigue (1991), and the solid points aremeasurements from Sesko et al. (1989). The arrow indicatesthe experimental uncertainty in temperature.


tion, further calculations were carried out at greater reddetunings and with lighter alkalis. In both cases the‘‘quantum suppression’’ effect was greatly diminished asan increased survival interpretation would lead one toexpect. An experimental measurement of the reductionin trap-loss rate with temperature has been reported byWallace et al. (1995) in 85Rb collisions.

c. Two-photon distorted wave theory

Solts et al. (1995) have developed a distorted wavetheory for evaluating weak-field, two-photon spectra,collision-induced energy redistribution, and radiative es-cape rates in cold atom traps. This theory complementsthe method of complex potentials discussed above andillustrates the connection between trap-loss spectra andphotoassociation spectra. The two-photon distorted-wave collision theory is a rigorous quantum mechanicaldevelopment, valid in the weak-field limit, and capableof treating multichannel collisional effects such as hyper-fine structure. The theory concentrates on radiative es-cape trap-loss collisions,

A1A1\vL→@A1A* #→A1A1\vS (83)

treating the radiative coupling as weak perturbations,while fully taking into account the collision dynamics.The A term stands for an alkali atom or metastable rare-gas atom, @A1A* # is a bound excited quasimolecule,and vL ,vS are the exciting laser photon and reemittedfluorescent photon, respectively. When vL is close tov0 , the A↔A* atomic resonance frequency, so that thered-detuning DL5vL2v0 , is within a few natural linewidths of the atomic transition, the spectrum of \vS willappear continuous, and its asymmetrically peaked formis controlled by the pair distribution function depen-dence on DL and on radiative damping. For an exampleof this line shape see the corresponding trap-loss spec-trum in Fig. 37. As uDLu increases, and the density ofexcited bound states decreases, the spectrum resolvesinto the discrete vibration-rotation progressions charac-teristic of photoassociation spectra. The state-to-statecross section for energy-specific (Ei→Ef) transitions isgiven by

ds~Ei ,Ef!

dEf5

2p2

ki2 \guVLu2Y~Ei ,Ef!, (84)

where ki252mEi /\2 and VL is the exciting laser cou-

pling strength. The ‘‘characteristic strength’’ functionY(Ei ,Ef) can be expressed as a partial wave expansion,

Y~Ei ,Ef!5(l50

lmax

~2l11 !uQ fi~ l !u2, (85)

where Q fi(l) is the two-photon Tfi matrix element

(Ef ,luTfiuEi ,l) divided by the product of DL and DS5vS2v0 . The summation over partial waves is effec-tively cut off by a centrifugal barrier. The trap-loss rateconstant bRE(T) is calculated by averaging over thethermal distribution of atoms at the temperature of thetrap and integrating over the final energies exceedingthe trap escape threshold DRE ,


bRE~T !5S 2p\2

mkBT D 3/2

guVLu2E0

`

dEiI~DRE!e2Ei /kBT,

(86)

where kB is the Boltzmann constant and

I~DRE!5EDRE

`

dEfY~Ei ,Ef!. (87)

At weak field the loss rate constant is directly propor-tional to the field intensity through the uVLu2 term.Equation (84) shows that the state-to-state redistribu-tion cross section is proportional to the ‘‘characteristicstrength function.’’ Figure 18 shows the ‘‘spectrum’’ ofY(Ei ,Ef) as a function of detuning DL , taking 0u

1 stateof Na2 as an example of an excited-state potential. Thefigure shows how the density of resonances decreaseswith increasing detuning. These lines, typical of the pho-toassociation spectra to be discussed in Sec. VI, areclearly resolved even to very small detuning becauseonly one spinless excited state is involved. If other ex-cited states and hyperfine structure were included, theoverlapping resonances would blend together into an ef-fective continuum at small detuning, but would resolveinto typical photoassociation spectra at large detuning(see, for example, Fig. 57). In principle the two-photondistorted-wave theory could be applied to a real photo-association spectrum (at weak field) including hyperfinestructure and excited states. This theory presents a uni-fied view of trap-loss and photoassociation spectroscopythat shows them to be two aspects of the same cold-collision interaction with the laser field, differing only inthe detuning regime.

d. Optical bloch equations

Both the Gallagher-Pritchard and Julienne-Viguesemiclassical models as well as the complex potentialand two-photon distorted wave quantum scattering cal-culations assume an initial optical excitation and subse-quent spontaneous decay, after which the decayed popu-lation no longer participates in the collisional process.This picture is valid as a weak-field limit, but at higherfields the decayed population can be reexcited, resultingin important modification of the overall probability ofexcitation at various points along the collision trajectoryas well as changes in the ground-state kinetic energy.The need to treat dissipation at high field requires adensity matrix approach; by introducing semiclassicaltrajectories, the time-dependent equations of motiongoverning the density matrix transform to the optical-Bloch equations. Although the optical Bloch equationstreat population recycling properly in principle, prob-lems arise when the time dependence of these equationsmaps to a spatial dependence. This mapping convertsthe time coordinate to a semiclassical reference trajec-tory; however, because the potentials of the ground andexcited states are quite different, the state vectors evolv-ing on these two potentials, in a diabatic representation,actually follow very different trajectories. Band and Juli-enne (1992) formulated two versions of an optical-


Bloch-equation trap-loss theory in a diabatic representa-tion and calculated the excitation-survival factor,J(E ,l ,D ,I ;g) of Eq. (79). One version applies correctionfactors to the initial reference trajectory so as to accu-rately reflect the actual semiclassical path followed byeach state vector component. The other version makesno such corrections. Figure 17 shows that the ‘‘not cor-rected’’ version JBJ

nc actually works much better than the‘‘corrected’’ version JBJ , although neither tracks thequantal results as well as JLZ . The problem appears tobe that the correction factors can successfully handle thediagonal terms of the density matrix, but fail when ap-plied to the field-induced off-diagonal terms. Theseterms are important in a diabatic representation. Bandet al. (1994) have recast the optical-Bloch-equation cal-culations in an adiabatic basis in which the field interac-tion is diagonalized and the interaction coherences aretherefore eliminated. This approach yields much betterresults at low temperature, as can be seen in Fig. 19. Acomplete derivation of the adiabatic optical-Bloch-equation method is given by Suominen, Band, et al.(1998) including interpretative Landau-Zener modelsadapted to the strong field case.

Although numerical results using the optical-Blochequations show reasonable agreement with experimentsperformed in Cs by Sesko et al. (1989) and Na by Mar-cassa et al. (1993), the quantum calculations of Boestenet al. (1993), Boesten and Verhaar (1994), Julienne et al.(1994), and Suonimen et al. (1994) indicate that theseresults are fortuitous. A proper quantum treatment us-ing two-state models would give rate coefficients morethan one order of magnitude smaller than measured.

e. Quantum Monte Carlo methods

Suominen et al. (1994) have carried out a quantumMonte Carlo two-state model study of survival in thevery low temperature regime in which JQ , JBV , JLZ ofFig. 17 all show a dramatic reduction in the trap-loss

FIG. 17. Cs trap-loss probabilities for various J’s as a functionof collision temperature. Notice the steep drop for JBV , JLZ ,and JBJ

nc . Notice also that semiclassical theory JJV completelymisses this drop below about 1 mK. From Julienne et al.(1994).


probability. The approach is to apply the Monte Carlostate vector method of Castin et al. (1993) to the timeevolution of wave packets undergoing random quantumjumps (Lai et al., 1993) during the course of the coldcollision. By repeating the simulation many times, a sta-tistical sample that approaches the density matrix isgradually built up. This approach treats spontaneousemission rigorously at all field strengths, taking into ac-count not only population recycling but also ground-state kinetic energy redistribution. The motivation forthe study was to provide definitive quantum mechanicalbenchmark calculations against which all approximate(but numerically more manageable) theories could becompared. Further motivation was to demonstrate, froman entirely different wave-packet, time-dependent ap-proach, the surprising resilience and robustness of theLandau-Zener formula in the very low energy regimewhere a semiclassical expression would be normally sus-pect. Figure 20 shows the results of a diabatic optical-Bloch equation, a Monte Carlo wave packet, and aLandau-Zener calculation of the excited-state flux sur-vival for Cs collisions at an energy of 100 mK. The di-abatic optical-Bloch equation clearly fails, and theLandau-Zener result appears as the semiclassical aver-age of the wave packet result. The conclusions of thisbenchmark study are, therefore, (1) semiclassical, diaba-tic optical-Bloch-equation calculations cannot be trustedbelow 1 mK, (2) the quantal wave-packet calculationsconfirm previous findings that trap-loss collision prob-abilities drop off dramatically with temperature, and (3)the Landau-Zener results are in excellent accord withthe oscillation-averaged fully quantal results.

Holland et al. (1994) have applied the wave-packetquantum Monte Carlo technique to consider radiativeheating of an ensemble of two-level Cs atoms; i.e., thereduced mass of the two-body collision was chosen to bethat of Cs and the excited state was chosen to have theC3 parameter of the attractive 0u

1 state. Figure 21 showshow the Monte Carlo method reveals momentum distri-bution spreading due to reexcitation and population re-cycling. Note that the spreading is especially marked forlower values of the initial momentum and results in en-semble heating even if trap loss does not occur. Signifi-cant heating without trap loss may have an importantbearing on the limiting temperatures and densities at-tainable in optical traps.

3. Assessment of theoretical approaches

a. Quasistatic vs dynamical

The quasistatic theories of GP and JV assume that,for any pair separation R0 , the colliding partners aremoving slowly enough that the field-quasimolecule cou-pling sets up a steady state population distribution at R0governed by the Lorentzian of Eqs. (67) or (74). Thecharacteristic time to establish this distribution is on theorder of the spontaneous emission lifetime. Under theinfluence of an attractive C3 /R3 potential the collidingnuclei rapidly pick up kinetic energy; therefore, at largedetunings the time requirement to achieve the steady-


state distribution is never fulfilled. At small detunings,where the nuclei accelerate more slowly, there may besufficient time if the excitation is not too weak; but thegoverning Lorentzian distribution then implies verystrong delocalization of excitation about the Condonpoint, RC . This delocalization of excitation has beentested by quantum Monte Carlo wave packet studies(Suominen et al., 1994). The results demonstrate that ex-citation is a ‘‘dynamical’’ process, sharply localizedaround RC for all detunings, and established on a timescale of the order of the inverse of the excitation Rabifrequency. The quantum scattering complex potentialstudies and adiabatic OBE calculations confirm the dy-namical picture, at least in the weak field limit. Further-more the excitation-survival probability distribution it-self can be factored into a survival term and anexcitation term expressed by the Landau-Zener formula,JLZ.Sae2A with A given by Eq. (82). Figure 17 showsthat all dynamical theories correctly demonstrate the‘‘quantum suppression’’ effect on trap-loss probabilitiesas temperature descends from the mK to the mK regime,while the quasistatic steady-state theories completelymiss it. At higher excitation fields, where theory mustcorrectly take into account spontaneous emission, popu-lation recycling between ground and excited states, andkinetic energy redistribution on the ground state, so faronly the quantum Monte Carlo wave packet studies ofSuominen et al. (1994) provide a reliable guide to thephysics of collision processes. The calculational burdenof these statistical samplings of the full density matrixhas restricted these studies to two-level model systems.Studies of realistic potentials including hyperfine inter-action are still beyond the scope of present calculationalresources.

b. Small vs large detuning

The various theoretical approaches to trap loss atsmall detuning, i.e., D<10 GA from atomic resonance,

FIG. 18. Spectrum of the characteristic strength function, Y ,as a function of detuning DL . The units of detuning in thefigure are given in mK where 1 mK'20 MHz. Note that reso-nance peaks are resolvable even at small detuning due to thesimplicity of the model calculation. See Solts et al. (1995).


can at best provide a qualitative guide for understandingthe process. No theory yet exists that is fully quantitativebecause small detuning implies an RC typically on theorder of l/2p , where l is the wavelength of the coolingtransition. The large value of RC leads to two majorcomplications: (1) the long time scale for the atoms to beaccelerated to small R where fine-structure-changingcollisions or radiative escape energy exchange occursgives a prominent role to spontaneous decay, and (2) themolecular hyperfine structure introduces great complex-ity into the long range potential curves. All theoreticaltreatments to date ignore (2), because of the computa-tional intractability of the full problem, which is cur-rently beyond the scope of any conceivable numericalcomputational solution. A full and proper treatment ofthe small detuning case requires the solution to the vonNeumann equation of motion for the quantum densitymatrix r ij(R ,R8) (Julienne, Smith, Burnett, 1993; Suom-inen et al., 1994). This is now feasible with Monte Carlosimulations for two-state models, as demonstrated bySuominen et al. (1994), whose calculations provide thebenchmarks against which any other two-state theoriesmust be tested.

Unfortunately, all current theories of trap loss forsmall detuning suffer from the limitation of being re-stricted to two (or at least a very few) scattering chan-nels. There are some general conclusions that can beextracted from these calculations, however. The first isthat of the great usefulness of the semiclassical pictureof localized excitation at the Condon point. As discussedin Sec. V.A.3.a above, the local equilibrium models ofGallagher and Pritchard and Julienne and Vigue givesimple pictures for interpreting trap loss. As detuningchanges from small (order of a few natural atomicwidths) to large (many natural widths), these theories allshow that the trap-loss rate first increases with the mag-nitude of the detuning (as survival improves due to ex-citation at smaller R) then decreases (survival ap-proaches unity, but the available phase space of pairs

FIG. 19. Trap-loss probability as a function of temperature forcomplex potential, Landau-Zener, and adiabatic optical-Blochequation approaches. Notice that the adiabatic optical-Blochequation is much better than the diabatic version, from Bandet al. (1994).


proportional to R2 decreases). Unfortunately, all of thequantum calculations show that in the small detuningregime the quasistatic picture with delocalized, Lorentz-ian excitation is fundamentally incorrect. The correctpicture shows localized excitation at the Condon pointfollowed by spontaneous decay as the atoms are accel-erated together on the excited state. The Monte Carlocalculations even demonstrate that such a picture caneven be extended to the strong field saturation regime oftwo state models: a semiclassical delayed decay model(Suominen et al., 1997) works where decay is only calcu-lated from the inner edge of a region of saturation aboutthe Condon point.

In contrast, at large detuning the Gallagher-Pritchardand Julienne-Vigue theories do result in excitation local-ized near the Condon point because of the steep slope ofthe upper potential curve, and in fact Band and Julienne(1992) showed that the Gallagher-Pritchard theory be-comes equivalent to the Landau-Zener1survival modelwhich has now been demonstrated by the quantum cal-culations. In practice, this will be true for detunings onthe order of 10 atomic line widths or more. TheGallagher-Pritchard theory is especially useful in this re-gime. The validity of the simple Gallagher-Pritchard for-mula has been verified in the trap-loss studies to the redof the S1/21P1/2 limit by Peters et al. (1994). Unfortu-nately, the semiclassical theories leave out the role ofbound states of the excited potential, which are now wellknown from the detailed spectra measured by photoas-sociation spectroscopy at sufficiently large detuning,typically more than 100 linewidths from atomic reso-nance. There is a need for the development of a theorythat bridges the gap between the large detuning photo-association region and the intermediate detuning regimewhere the density of bound vibrational states is so highthat the quasicontinuum approach of Gallagher andPritchard applies. This bridging could be done by apply-ing the complex potential method, which uses well-understood scattering algorithms, or the perturbative

FIG. 20. Probability as a function of temperature for the di-abatic optical-Bloch equation (dotted line), a wave packet(solid line), and Landau-Zener calculation (dashed line) ofexcited-state flux survival at collision energy of 100 mK.


Franck-Condon approach of Solts, Ben-Reuven, andJulienne (1995).

Another main complication at small detuning is theextreme complexity of the molecular hyperfine structure(Walker and Pritchard, 1994; Julienne et al., 1994; Lett,Julienne, and Phillips, 1996). A number of the experi-mental studies described below indicate a prominentrole for hyperfine structure in determining the trap-lossrate. For example Fig. 22 shows the molecular hyperfinepotentials originating from 2S12P3/2 Na atoms. The re-cent work of Fioretti et al. (1997) suggests that hyperfineoptical pumping in a molecular environment during acollision may also be significant (see also Julienne andMies, 1989). The fundamental difficulty with the two-state treatments in the older Gallagher-Pritchard,Julienne-Vigue, and optical-Bloch-equation studies isthat the apparent agreement with experiment is onlyfortuitous. According to the quantum calculations, withtheir dynamical picture of the excitation, the predictedtrap-loss rates will be much lower (an order of magni-tude or more) for these same two-state models thanrates given by the older semiclassical theories. Thereforethere is a fundamental discrepancy that remains to beunderstood. There is considerable likelihood that theresolution of this problem will require understanding therole of the complex molecular hyperfine structure, withmolecular optical pumping and spontaneous decay. Al-though the semiclassical optical Bloch equation methodsshowed clear problems in the diabatic formulation, thereis some promise in an adiabatic formulation of the semi-classical optical-Bloch-equation method that appearsmuch more accurate than the diabatic formulation(Suominen, Band, et al., 1998). Since it is unlikely that afully quantum density matrix treatment including hyper-fine structure will be available in the foreseeable future,a possible approach to understanding small-detuningtrap loss may lie with adiabatic optical-Bloch-equationor Landau-Zener semiclassical models (Boesten andVerhaar, 1994; Julienne et al., 1994; Suominen, Band,et al., 1998).

The situation is much better at somewhat larger de-tuning, i.e., large compared to the atomic spontaneousdecay line width. When the detuning becomes suffi-

FIG. 21. Momentum distribution spreading due to reexcitationand population recycling. From Holland et al. (1994).


ciently large, the trap-loss process gives rise to a discretespectrum of trap loss due to the bound states in theexcited attractive molecular potentials. The Condonpoint is sufficiently small that many vibrational cyclesare necessary before spontaneous decay can occur, andthe spacing between vibrational levels becomes largecompared to their natural linewidth. We will discuss thiscase in detail in Sec. VI on photoassociation spectros-copy. In this limit of large detuning, the theory is inexcellent shape, because spontaneous emission can betreated as a perturbation on the normal conservativemultichannel scattering theory, and fully quantitativecalculations including all molecular hyperfine structurehave been carried out in great detail, and with superbagreement with experiment.

It is within the realm of possible calculations to studya realistic system without hyperfine structure, such asGroup II species like Mg, Ca, or Sr. All of these have1S→1P transitions. We will show in Sec. VII on opticalshielding how a full three-dimensional scattering treat-ment taking into account the degeneracy of the excitedP level is possible. Monte Carlo simulations, as well assimpler models, of such a system appear to be feasiblewith advanced computation methods. Experimentalstudies of such a simple system would be very helpful inunraveling the complexities of small detuning trap loss.There is a realistic hope of testing proper theory againstexperimental measurements in such systems.

4. Excited-state-trap loss measurements

In general, collisions in traps are investigated by ei-ther detecting the product resulting from the binary en-counter or through loss of the colliding species from thetrap. When an ultracold collision releases kinetic energy,the reaction rate can be measured by observing the timedependence of the number of atoms N either as the traploads or as it decays. The equation governing the netloading rate of the trap is given by

dN

dt5L2gN2bE

vn2~r ,t !d3r , (88)

where L is the capture rate, g the rate constant for col-lisions of trapped atoms with thermal background gas,and b the loss-rate constant due to collisions betweentrapped atoms. In general, spatial variation of theatomic density n(r ,t) requires integration over thewhole volume occupied by the atoms; however, the spa-tial density distribution of the trapped atoms can be inone of two possible limits: at low-density radiation trap-ping is negligible, and the trap spatial distribution isclose to Gaussian; in the high-density limit Walker et al.(1990) showed that radiation trapping dominates andproduces a constant spatial atom density. Thus in thelow density limit the distribution n(r ,t) can be ex-pressed as n(r ,t)5n0(t)e2r2/w2

, and Eq. (88) becomes

dN

dt5L2gN2

bN2

~2p!3/2w3 . (89)


This equation has a complicated solution as shown byPrentiss et al. (1988), but N(t) can be measured directlyand b determined by curve fitting.

In the high-density limit, trap loading proceeds at con-stant density. As the number of atoms in the trap in-creases, the volume expands so that the density in thetrap remains unchanged, and Eq. (88) takes the form

dN

dt5L2~g1bnc!N , (90)

where nc is the constant atomic density obtained in thetrap. The value of nc is characteristic of the trappingconditions. The factor g1bnc is determined by begin-ning with an empty trap, turning it on, and measuringthe number of trapped atoms as a function of time.Equation (90) is then fit to the measurement. Examplesof these N(t) transients obtained in a sodium MOT byMarcassa et al. (1993) are presented in Fig. 23, at severaldifferent MOT laser intensities. A semilog plot of theabsolute value of 12N/N0 versus t determines the slopeequal to 2(g1bnc) as shown in Fig. 24. The linearity ofthose plots confirms that trap loading takes place withinthe regime of constant density, nc . These loading-curvefits ignore the very early times where the constant den-sity regime has not yet been reached. The density nc isobtained by first measuring the trap diameter with, forexample, a charge-coupled-device (CCD) camera to de-termine the volume. Then fluorescence detection or ab-sorption with an independent probe laser determines thenumber of trapped atoms. To extract b, the two-bodytrap-loss rate constant, it is necessary to devise an inde-pendent measurement of g. This rate can be determinedeither by observing variations of (g1bnc) for differentbackground pressures of the trapped gas or by operatingthe MOT with a very low number of trapped atoms suchthat the term bnc is negligible compared with g. Figure24 shows the light-field-intensity dependence of 2(g

FIG. 22. Hyperfine molecular states near the 2S12P3/2 asymp-tote. The dense number of states gives rise to a large numberof avoided crossings, state mixing, and congested spectra.From Lett, Julienne, and Phillips (1996).


1bnc) that is associated with the intensity-dependenceof b since g and nc are insensitive to this parameter.

If the MOT is loaded from a slow atomic beam, thebackground-loss rate g is normally very small and a con-venient way to investigate b is to observe the decay ofthe trapped atoms as a function of time after the loadingprocess has been completed and the source of atomsinterrupted. In this case the capture rate L is no longerpresent in Eq. (88). Typical decay of a cesium MOTatomic fluorescence (Walker and Feng, 1994) as a func-tion of time is shown in Fig. 25. In the initial stage, Fig.25 shows a constant-density collision rate resulting inexponential decay. As the number of atoms decreases,the radiation trapping effect becomes weak enough thatthe density begins to drop while the trapped-atom dis-tribution retains a Gaussian shape. In this regime thedecay is non-exponential. Finally, when the number ofatoms become very small, the intra-trap collisions are nolonger important and the collisions with background gascause the decay to once again become exponential intime, with a rate smaller than the first stage.

The trap-loss parameter b contains the probabilitiesfor inelastic processes such as fine-structure-changingcollisions, radiative escape, and photoassociation. Tobetter understand the influence of applied optical fieldson these processes, several research groups have per-formed studies of the dependence of b on various pa-rameters, especially MOT laser frequency and intensity.Both the Gallagher-Pritchard and Julienne-Vigue mod-els predict a b increasing linearly with MOT intensity;initial experiments sought to verify this behavior.

Investigation of the variation of b with MOT lightintensity is straightforward to carry out. One can eitherload or unload the trap at different intensities and ex-tract b from the transient behavior. If the experiment isperformed by loading a MOT from a slow atomic beam,this technique works well over a wide range of intensi-ties (Sesko et al., 1989; Wallace et al., 1992; Kawanaka,et al., 1993; Ritchie, et al., 1995). However, for vapor-cellloading where the g term in Eq. (88) becomes signifi-cant, the steady-state number of atoms in the trap at lowMOT intensity becomes too small for reliable measure-ments. In this case, the study of b as a function of inten-sity can be implemented by a sudden change of intensi-ties as demonstrated by Santos et al. (1996). The basicidea is to switch between two different intensities in ashort time interval. A large number of atoms is loadedinto the trap at high intensity after which the intensity issuddenly attenuated by introducing a calibrated neutraldensity filter into the laser beam. At lower intensity thenew loading and loss rates determine a new (lower)steady state number of atoms. The transient variation ofthe number of trapped atoms between the two limitsdetermines b in the low intensity regime. Figure 26shows a typical fluorescence-time spectrum using a sud-den change of intensity, starting from a high steady-statenumber of atoms. The fast drop observed at the firstinstant of time corresponds to the decrease in the pho-ton scattering rate due to the change in light intensitywithout variation in the number of trapped atoms. The


subsequent slow decay must be analyzed using Eq. (89)or Eq. (90) already described.

The investigation of b with MOT detuning is moredifficult because the trap only operates over a narrowrange of detunings ;G to the red of the atomic reso-nance. Hoffmann et al. (1992) overcame this limitationby introducing an extra probe laser, which they called a‘‘catalysis’’ laser. This idea was first introduced by Seskoet al. (1989). Although the added optical field does notact as a catalyst in the conventional chemical sense ofincreasing reaction rates without net consumption of thecatalysis photon ‘‘reagent,’’ the term has been widelyadopted anyway. As long as the frequency of this laser isnot too close to the MOT or repumping transition, itdoes not affect the trapping and cooling processes; how-ever, it can produce changes in the trap-loss rate. Hoff-mann et al. (1992) assumed the effect of the catalysislaser on the trap-loss rate to be additive, b5b t1bc ,where b t represents the rate constant with only the trap-ping laser present and bc the contribution of the cataly-sis laser. It is important to note, however, that Sanchez-Villicana et al. (1996) have demonstrated a ‘‘fluxenhancement effect’’ (see discussion below in the sec-tion on rubidium trap loss and Fig. 38) that call the badditivity assumption into question. Detuning the ca-talysis laser (Dc) affects b and the number of atoms inthe trap, N ,

N5L

g1bncf, (91)

where L ,g have their usual meaning—see Eq. (88)—andf5*n2(r,t)d3r/ncN is a measure of the deviation of thedensity distribution from the uniform density nc . As theintensity Ic and detuning Dc of the catalysis laser vary,the trap-loss constant b varies as,

b5b t1bc~Dc!Ic~Dc!

Iref, (92)

where Iref is a reference laser intensity taken to be10 mW cm22. Equation (91) shows that varying Ic andDc in turn will alter N and consequently the trap density,n . Unfortunately, the most difficult quantity to deter-mine accurately in these experiments is the trap density,but Hoffmann et al. (1992) avoided this problem by ad-justing Ic at each Dc such that N and n remained con-stant over the entire range of detunings in the experi-ment. With the condition that the product bc(Dc)Ic(Dc)be held invariant to maintain constant N , it was onlynecessary to measure the loading curve, Eq. (88), withand without the catalysis laser at one fixed detuning andintensity in order to obtain the absolute value of thebcIc product. Once known, measures of Ic(Dc) deter-mined absolute values of bc(Dc) over the entire detun-ing range of the experiment. The reliability of the cataly-sis laser method depends strongly on the assumptionthat the extra laser light affects only the atomic colli-sions and not the performance and characteristics of theMOT. This assumption is best justified if the catalysislaser is not tuned too close to the trapping and repump-


ing transitions and the power is not so high that lightshifts begin to affect the MOT functioning.

Using these techniques to study trap loss, several spe-cific cases have been investigated.

a. Sodium trap loss

Prentiss et al. (1988) carried out the first measure-ments of collisional losses in a MOT loaded from a coldsodium beam. In this work sudden interruption of theatomic beam stopped the loading at a well-defined in-stant of time and the subsequent fluorescence decay wasrecorded. At early times a nonexponential fluorescencedecay was observed at a trap density ;73109 cm23. Fit-

FIG. 23. Examples of trap loading rates in MOT (type-I trapwith trapping laser tuned approximately one linewidth to thered of 2S (F52)→3P3/2(F853) for several different MOTintensities: (a) 96 mW cm22, (b) 70 mW cm22, (c)36 mW cm22, (d) 23 mW cm22, (e) 16 mW cm22. Curves arefrom Marcassa et al. (1993).

FIG. 24. Semilog plot of the integrated loading curves: abso-lute value of 12N/N0 versus t . The slope, 2(g1bnc), deter-mines the trap loss rate constant b. Results are for a type-Itrap with trapping laser tuned approximately one linewidth tothe red of 2S (F52)→3P3/2(F853). Curves are from Mar-cassa et al. (1993).


ting of the nonexponential decay with an equation likeEq. (89) permitted extraction of a value for b. Investi-gation of the variation of b with laser intensity is shownin Fig. 27. The data yielded a b invariant with intensity(within 640%) over a range of 5 to 50 mW cm22.Variations of the trap depth, realized through changes inthe MOT magnetic field, indicated that b was insensitiveto trap-depth changes as well. The absolute values of bwere measured to be about 4310211 cm3 s21 (within afactor of 5). Although Prentiss et al. (1988) could notreadily explain the unexpected insensitivity of b to lightintensity, they offered several possibilities for the obser-vation including the effect of mF hyperfine levels mixingand shifts of the atomic energy levels that occur during acollision. The surprising lack of intensity dependence intheir results underscored the importance of investigatingbinary collisions in MOTs.

Marcassa et al. (1993) investigated b over a widerrange of intensity, from 20 to 300 mW cm22, using a Navapor-cell-loaded MOT. In their report a definite lightintensity dependence was indeed observed, and the ap-parent independence previously reported by Prentisset al. (1988) was explained by the relatively narrowrange over which the earlier experiments had been car-ried out. In the experiment of Marcassa et al. (1993) theMOT laser was tuned about 10 MHz to the red of theatomic cooling transition, and b extracted from the fluo-rescence growth curve recorded during MOT loading,starting from an empty trap. Densities in the MOTranged from 53109 to 231010 cm23, which is in thehigh-density limit where Eq. (90) governs the loadingprocess. In order to distinguish g from bnc trap loss wasmeasured using two tuning conditions, the type I andtype II traps, described in Prentiss et al. (1988). The typeII trap was operated in a density regime low enough sothat trapped-atom collisions could be ignored and theslope of the fluorescence growth exponential deter-mined g. A calibrated telescope measured the MOT di-ameter from which the assumed spherical volume wascalculated, and the atom number in the MOT was deter-mined from fluorescence imaged onto a calibrated pho-tomultiplier detection train. These two measurementsdetermined nc , and b was then calculated from Eq. (90).

FIG. 25. Time evolution of MOT decay. From Walker andFeng (1994).


The measured intensity dependence of b is shown in Fig.28. Despite error bars of 50% or more, due essentially touncertainties in the volume measurement, the expectedincrease of b with MOT intensity is unmistakable.

Band and Julienne (1992) proposed a new theory fortrap loss, explicitly avoiding the local equilibrium as-sumption of the Gallagher-Pritchard and Julienne-Viguemodels, and based on the optical-Bloch equations. Theoptical-Block-equation density-matrix approach to traploss is appropriate to studies of optical field dependencebecause it naturally takes into account population recy-cling (optical excitation, decay, and reexcitation) at highfield. In contrast Gallagher-Pritchard and Julienne-Vigue are weak-field theories in which spontaneousemission is treated as a simple loss term, and the de-cayed population never recycles. Marcassa et al. (1993)applied this optical-Bloch-equation theory to theirintensity-dependence studies of Na trap loss by calculat-ing b as a function of MOT intensity and comparing it tothe measured values. A difficulty in making this com-parison is knowledge of the minimum velocity that mustbe imparted to the atoms in order to escape from theMOT. Assuming single-atom escape velocities of 20 and30 ms21, Marcassa et al. (1993) calculated a radiative es-cape contribution to the total loss rate of 38% and 24%respectively with the remaining fraction due to fine-structure-changing collisions. Figure 28 plots the resultsfrom calculation and measurement. Although the modeldid not include hyperfine structure of the excited states,the agreement between theory and experiment is rea-sonably satisfactory, with theory somewhat higher thanexperiment. However, the assessment in Sec. V.A.3 in-dicates that this agreement is likely to be fortuitous. It isworthwhile to note that Marcassa et al. (1993), in thecourse of carrying out these b calculations, also reevalu-ated the PXQ(E ,l) factor of Eq. (79) for the fine-structure-changing collision probability, using quantumclose-coupling calculations and accurate potential curvesof Magnier et al. (1993). The results showed a fine-structure-changing collisions probability smaller by a

FIG. 26. Fluorescence-time spectrum. The abrupt drop at theinitial moment is due to a change in light intensity withoutvariation in the number of trapped atoms. From Santos et al.(1996).


factor of 2 than that estimated in Julienne and Vigue(1991) and underscores the necessity of using accuratemolecular potential data in collisions calculations. Du-lieu et al., 1994 showed that fine-structure-changing col-lisions probabilities can be quite sensitive to the molecu-lar potentials.

In a followup experiment Marcassa et al. (1997) used acatalysis laser to study Na trap loss as a function of reddetuning. As discussed above, the number of atoms ob-served in the presence of the catalysis or probe laser isN5L/@g1b(D ,I)ncf# where b(D ,I)5b t1bc(D)I/Iref .As in Hoffmann et al. (1992) the intensity of the extralaser is adjusted at each detuning to keep N constant.From this condition Marcassa et al. (1997) determinedbc(D). Figure 29 shows the trap-loss spectrum for thecatalysis laser detuned to the red of both the 3s2S1/213p2P3/2 asymptote and the 3s2S1/213p2P1/2 asymp-tote. The reason for measuring the trap-loss spectrumfrom both asymptotes is that both fine-structure-changing collisions and radiative escape contribute fromthe 3s2S1/213p2P3/2 level, while only radiative escapecontributes from the lower 3s2S1/213p2P1/2 level. Forcomparison, the Gallagher-Pritchard theory is shown inthe same figure as the solid line. The overall behavior isin good qualitative agreement with the model. Compari-son between the absolute values shows that the ampli-tude of the trap-loss spectrum to the red of the 3s2S1/213p2P1/2 asymptote is only 20% lower than the trap-loss spectrum near the 3s2S1/213p2P1/2 . ThereforeMarcassa et al. (1997) concluded that fine-structure-changing collisions is responsible for only 20% of traploss, with radiative escape responsible for the remaining80%. This result is in sharp disagreement with the cal-culations reported in Marcassa et al. (1993) in which ra-diative escape contributes about 30%. These disagree-ments indicate that the small-detuning trap-loss processand its detailed mechanisms are still poorly understood.

b. Cesium trap loss

Sesko et al. (1989) carried out the first collisional traploss measurements on cesium using a MOT loaded froman atomic beam. In their experiment fluorescence decaywas observed after loading. To ensure a constant half-width Gaussian profile for the whole investigated inten-sity range, the number of trapped atoms was kept low

FIG. 27. Early experiment, Prentiss et al. (1988), measuringvariation of trap loss rate constant in a Na MOT as a functionof MOT intensity. The dashed line traces the calculated varia-tion of the excited-state population.


(;33104). Figure 30 shows the measured b as a func-tion of trap laser intensity obtained by Sesko et al.(1989). Above 4 mW cm22 b increases with intensity dueto the energy transfer collisions as radiative escape andfine-structure-changing collisions. Below 4 mW cm22 thedramatic increase in b is due to hyperfine-changing col-lisions between ground-state atoms. For Cs, a changefrom 6s2S1/2(F54) to 6s2S1/2(F53) in one of the atomsparticipating in the collision transfers about 5 ms21 ofvelocity to each atom. The Gallagher-Pritchard model isalso included in Fig. 30 (solid line) showing a value a fewtimes smaller than the measurement.

In the same experiment Sesko et al. (1989) also inves-tigated the trap loss spectrum using a ‘‘catalysis’’ laser.They did not adjust the catalysis laser intensity at differ-ent detunings, as did Hoffmann et al. (1992) and Mar-cassa (1995), to maintain constant trap density, but in-stead measured the trap density at each point on thetrap-loss curve, and measured the b-decay curve withand without the catalysis laser present. Their results arepresented in Fig. 31, together with the Gallagher-Pritchard model (solid line). The scatter in the data re-flects the difficulty of obtaining accurate measurementof the trap density.

In a recent work Fioretti et al. (1997) directly mea-sured fine-structure-changing collisional losses from acesium type I and type II MOT (Raab et al., 1987), withthe trapping light near the 6s2S1/2(F54)→6p2P3/2(F55) and the repumping light near 6s2S1/2(F53)→6p2P3/2(F54) for the type I trap and 6s2S1/2(F53)→6p2P3/2(F52) for the type II trap, by detecting D1fluorescence emitted on the 6p2P1/2→6s2S1/2 transition.Measuring the counting rate of D1 photons emitted, theefficiency of detection of both D2 and D1 light, anddetermining the trap density from the D2 count rate andthe spatial extent of the trap, Fioretti et al. (1997) derivean absolute value for fine-structure-changing collisionsin the cesium MOT. They have examined the sensitivityof fine-structure-changing collisions to different excited-state hyperfine components by modifying the ground-state hyperfine population that starts the process. Theyfit their data with a model that requires molecular hy-perfine optical pumping during the collision. Their mea-surement of bFCC (2310212 cm3 s21) and comparisonwith the previous measurement, Sesko et al. (1989), ofthe total b (8310212 cm3 s21) leads to the conclusionthat fine-structure-changing collisions contribute about25% of the total (FCC1RE) loss rate constant. TheJulienne-Vigue theory predicts about equal contribu-tions from fine-structure-changing collisions and radia-tive escape, but it must be born in mind that the Fiorettiet al. (1997) experiment was carried out under strong-field conditions (total trap intensity as high as150 mW cm22) and Julienne-Vigue is a weak-fieldtheory that neglects hyperfine structure. Therefore it isnot too surprising that theory and experiment are not inaccord.

c. Rubidium trap loss

Wallace et al. (1992) carried out the first measure-ments of trap loss in rubidium, using a MOT similar to


the one employed by Sesko et al. (1989) for the study ofcesium. The MOT was loaded from a rubidium atomicbeam slowed with a frequency-chirped diode lasersource, and the trap loss determined by monitoring theatomic fluorescence decay. The trap-loss rate constant bwas obtained by fitting the data to

dN

dt52gN2bE

vn2~r ,t !d3r , (93)

which is just Eq. (88) with the loading rate L set tozero. The trap operates at sufficiently low densities(,231010 cm3 s21) that radiation trapping effects canbe ignored (Walker et al., 1990). The rate constant forloss due to collisions with thermal background gas, g,was constant and fixed by the background pressure of;10210 Torr. The number of atoms in the trap and thedensity of the trap are determined from (1) absorptionmeasurements with a weak probe beam, (2) measure-ment of the excited-state population fraction obtainedfrom selective photoionization of the Rb (5p2P3/2) level,and (3) trap size determined by a digitized charge-coupled device (CCD) camera image. Figure 32 shows bplotted as a function of total trap intensity at a fixeddetuning of 4.9 MHz to the red of the trapping transitionfor both rubidium isotopes, 85Rb and 87Rb. The behav-ior is similar to that of Cs (Sesko et al., 1989), with bdecreasing approximately linearly from high trap inten-sities to a minimum at a few mW cm22, and then in-creasing sharply as the trap intensity is further reduced.On the high-intensity side of the curve b behavior indi-cates trap loss due to a combination of radiative escapeand fine-structure-changing collisions, although there isno way to separate the contributions of the two pro-cesses in this experiment. At low intensity the sharp in-crease of b is the signature of hyperfine-changing colli-sion. In addition to this general behavior, the most

FIG. 28. Measurement of the trap loss rate constant as a func-tion of MOT light intensity. The wider range of light intensityreveals an increase in b with MOT intensity. Dotted, full, anddot-dash curves are theory calculations using the optical-Blochequation method of Band and Julienne (1992) with differentassumptions about the maximum escape velocity of the MOT.The curve Vesc5V(I) results from a simple model in which theescape velocity is a function of the MOT intensity. Data arefrom Marcassa et al. (1993).


striking feature of the high-intensity branch of the trap-loss curve is a strong isotope effect: the value of b for85Rb is always a factor of 3.3 (60.3) higher than for87Rb. Wallace et al. (1992) point out that this isotopeeffect was not anticipated by either the Gallagher-Pritchard or Julienne-Vigue theories; furthermore, sincethe only significant difference between the two isotopesis their hyperfine structure, it probably results from theinfluence of long-range molecular hyperfine interactionson the fine-structure-changing collisions and radiative-escape loss mechanisms. The Connecticut group, led byP. L. Gould, also compared their determination of theabsolute value of b for both isotopes at a trap intensityof 10 mW cm22 to the Julienne-Vigue theory. The mea-sured values are b53.4310212 cm3 s21 and 1.0310212

cm3 s21 for 85Rb and 87Rb, respectively. The Julienne-Vigue theory calculates b51.7310211 cm3 s21, factorsof 5 and 17, respectively, in disagreement with eitherexperimental measurement. Since Wallace et al. (1992)quote an uncertainty of only 640% in their absolute bmeasurements, the disagreement between experimentand theory is large and real. In addition to ignoring hy-perfine structure, it is important to remember theJulienne-Vigue is a weak-field theory and that the calcu-lation of the fine-structure-changing collisions contribu-tion is quite sensitive to PXQ(E ,l), the accurate calcu-lation of which, in turn, relies on accurate molecularpotential energy curves. The strong sensitivity of thefine-structure-changing collisions probability to the mo-lecular potentials has been examined by Dulieu et al.

FIG. 29. Trap-loss spectrum in a Na MOT for catalysis laserdetuned to the red from both atomic fine-structure asymptotes.From Marcassa et al. (1997).


(1994). At this writing the accord between experimentand theory must be considered unsatisfactory.

On the low-intensity branch of Fig. 32, where b startsto increase due to hyperfine-changing collisions, the iso-topic ordering of b reverses, and the trap intensity atwhich b reaches a minimum differs between the two iso-topes by a factor of ;1.5. Since the ratio of the kineticenergy release due to hyperfine-changing collisions from87Rb and 85Rb is 6835 MHz/3036 MHz 52.25, the factorof 1.5 at first appears surprising. However Wallace et al.(1992) point out that a MOT contains the atoms byposition-dependent forces and velocity-dependentforces. If the velocity-dependent forces control, and ifthese forces vary linearly with trap intensity, then onewould expect the trap-loss ratio to be equal to the ratioof velocities or A6835 MHz/3036 MHz51.5 in accordwith observation. At the very lowest trap intensities b

FIG. 30. Trap-loss rate constant b as a function of MOT in-tensity for Cs collisions: comparison of experiment with GPtheory, from Sesko et al. (1989).

FIG. 31. Trap-loss spectrum in a Cs MOT using a ‘‘catalysis’’laser (points), and comparison with GP theory (solid line). Thedotted line shows the value of b without the ‘‘catalysis’’ laser.From Sesko et al. (1989).


levels off, and the Connecticut group concludes that un-der these conditions all the atoms undergoing thehyperfine-changing collisions process can escape. There-fore the b of this plateau is a measure of the hyperfine-changing collision rate. The measurements show a valueof ;2310211 cm3 s21.

At about the same time the Connecticut group wascarrying out these rubidium trap-loss isotope studies, an-other group at the University of Wisconsin, led by T.Walker, started investigating 85Rb. In contrast to Wal-lace et al. (1992), the results reported by Hoffmann et al.(1992) were carried out in a vapor-loaded trap, not anatomic-beam loaded trap, and b was obtained by fittingEq. (90) to the observed fluorescence loading curves.The Wisconsin group also used a catalysis laser to mea-sure the trap-loss spectrum, i.e., b as a function of reddetuning of the catalysis laser from the trapping transi-tion. Their results are displayed in Fig. 33. They com-pare their rubidium results to the earlier cesium study(Sesko et al., 1989), and find the following differences:(1) b at the peak of the 85Rb trap-loss spectrum is abouta factor of 3.5 smaller than b at the peak of the Cstrap-loss spectrum, (2) these peaks do not appear at thesame detuning for the two species. Cesium b peaks at adetuning .450 MHz while 85Rb peaks .160 MHz. (3)Extrapolating their data to 10 mW cm22 and small de-tunings, Hoffmann et al. (1992) determine b5(3.661.5)310212 cm3 s21, a factor of 5 below the Julienne-Vigue calculation but in quite good agreement with themeasurement of Wallace et al. (1992).

The striking and unexpected isotope effect reportedby Wallace et al. (1992) stimulated further investigationby the Wisconsin group. Feng et al. (1993) carried outtrap-loss studies on both rubidium isotopes similar tothose already described by Hoffmann et al. (1992) for85Rb. The trap-loss spectra show that for catalysis-laserdetunings outside the excited-state hyperfine structureregime, about 400 MHz to the red of the trapping tran-sition, b for both isotopes is the same. Within the hyper-fine region, Feng et al. (1993) found approximately thesame ratio of b85 /b87.3 as Wallace et al. (1992). Fur-thermore, they observed a double-peaked structure inthe 87Rb b at 400 MHz red detuning, and difficulty inmaking reliable measurements left gaps or holes in thespectrum at detunings near the atomic hyperfine transi-tions. Figure 34 shows these features of 87Rb trap lossspectrum. These results clearly demonstrate that the iso-tope effect as well as the spectral ‘‘holes’’ are associatedwith trap-loss processes strongly modulated by molecu-lar hyperfine structure. The Wisconsin group detailstheir experimental method and summarizes the resultsin Hoffmann et al. (1994).

The important role of hyperfine structure in trap-lossdynamics demonstrated by the experiments of Wallaceet al. (1992) and Feng et al. (1993) prompted the devel-opment of models proposed by Walker and Pritchard(1994) and Lett et al. (1995). In essence these studiesextend the Gallagher-Pritchard picture of a single attrac-tive excited state to multiple excited states with bothattractive and repulsive branches. Figures 35(a) and


35(b) show the simplest, illustrative case. Potentialcurves 0 and E1 represent two closely spaced hyperfinelevels of the excited state. The upper curve, 0, shows thefamiliar V(R)52C3 /R3 long-range behavior while thelower level splits into 6C3 /R3 corresponding to attrac-tive and repulsive dipole-dipole interaction. Each of theattractive curves gives rise to a Gallagher-Pritchard-liketrap-loss spectral profile resulting in an observed ‘‘dou-blet’’ as a catalysis (probe) laser detunes to the red, in-dicated in Fig. 35(b). The repulsive curve arising fromE1 mixes with the attractive curve from 0 in a localizedregion around the crossing point. As the catalysis laserscans through the crossing, population in the attractivelevel leaks onto the repulsive curve, effectively reroutingscattering flux away from radiative escape and fine-structure-changing trap-loss processes occurring atsmaller R . The result is hole burned in the trap-lossspectrum shown schematically by the dotted curve inFig. 35(b). Lett et al. (1995) generalized this model tomultiple levels with multiple crossings. In the case of87Rb, for example, three hyperfine levels (F853,2,1)can be excited from the F52 ground state. Very carefulnew trap-loss measurements were carried out in 85Rbright around the detuning region where structure in thetrap-loss spectrum was expected. Figure 36 shows theresults of these new measurements together with theformer data of Hoffmann et al. (1992) and the hyperfine-structure model using 2u potential parameters and theLandau-Zener avoided-crossing probability PLZ50.6.The agreement between the experimental points and themodel appears quite satisfactory. Attempts to fit thedata using long-range potentials other than the 2u in themodel cannot reproduce consistently the data of Hoff-mann et al. (1992) and these new results reported byLett et al. (1995). Further application of the model to the87Rb trap-loss data of Feng et al. (1993) also shows goodagreement, and the model is successful with Cs data (al-though a printing error in Fig. 8 of Lett et al., (1995),does not actually permit the comparison in that article;see, however, Corrigendum, 1995). The application ofthis hyperfine-level avoided crossing model leads to twomajor conclusions: (1) structure in the trap-loss spec-

FIG. 32. Trap-loss spectrum for two isotopes of rubidium.Right-hand branch shows trap loss due FCC and RE. Left-hand branch shows trap loss due to HCC. From Wallace et al.(1992).


trum is due to excitations of multiple hyperfine levelsand avoided crossings between them, and (2) the majortrap-loss mechanism is long-range radiative escape onthe 2u potential curve. However, we must caution thatusing the molecular Hund’s case (c) symmetry label 2ufor potential curves in a region of strong hyperfine-induced mixing is unwarranted. These results do suggestthat a state characterized by weak optical coupling isimplicated in the trap-loss mechanism.

Peters et al. (1994) greatly simplified the interpreta-tion of trap-loss data by measuring b from collision part-ners excited near the Rb(5p2P1/2)1Rb(5s2S1/2) asymp-tote. Since most hyperfine-level curve crossings areeliminated and the spacing between the levels is large,all of the attractive hyperfine levels converging on theRb(5s2S1/2 ;F52)1Rb(5p2P1/2 ;F851) will closely fol-low a C3 /R3 power law and their contributions to thetotal trap-loss rate constant should be simply additive.Furthermore, trap loss contributions to the red of the5p2P1/2 asymptotes can only be due to the radiative es-cape process since energy conservation closes the fine-structure-changing collisions channel. The results shownin Fig. 37 confirm that under these simplified conditionsthere is no discernible isotope effect between 85Rb and87Rb, that the Gallagher-Pritchard model works fairlywell, but that multiple orbiting of the collision partnersmust be included to obtain good agreement between themeasurements and the model. Another benchmark isthe position of the peak in the trap-loss spectrum. Forhyperfine levels converging on the Rb(5p2P3/2)1Rb(5s2S1/2) asymptote the Gallagher-Pritchard modelpredicts this peak to be near a red detuning of about 115MHz. Unfortunately this is also the region where curve-crossing and interactions among the hyperfine levelscomplicate the trap-loss spectrum. From the calculatedshapes of the long-range potentials converging on theRb(5s2S1/2 ;F52)1Rb(5p2P1/2 ;F851) asymptote,however, Peters et al. (1994) calculate that the trap-losspeak should occur at about 25 MHz red detuning withno complicating hyperfine interactions to cloud themodel prediction. Probing the trap with the ‘‘catalysis’’laser so close to the trapping transition unfortunatelyleads to such strong trap perturbations that trap-lossmeasurements become unreliable. Therefore the Wis-consin group could not apply their measurement tech-nique to test this particular prediction of the Gallagher-Pritchard model.

Very recently Sanchez-Villicana et al. (1996) havedemonstrated a flux enhancement effect in ultracold col-lisions by comparing trap-loss rate constants in anatomic-beam-loaded rubidium MOT with trap andprobe lasers present simultaneously or alternately andwith the probe laser tuned as far as 1 GHz to the red ofthe MOT trapping transition. Figure 38 outlines the ba-sic process. Scattering flux enters the collision on the5S15S ground state and is excited by the MOT laser atRt (red detuning ;G) to one of the attractive 2C3 /R3

potentials converging on the 5S15P asymptote. The ex-cited flux accelerates and spontaneously relaxes back tothe ground state with increased kinetic energy, enabling


higher partial-wave components to overcome their an-gular momentum barriers at closer internuclear separa-tion. At some near-zone Condon point RC (red detuning;50–150 G) a probe laser reexcites the flux to the2C3 /R3 potential where it may undergo an inelastictrap-loss interaction (e.g., a short-range curve crossingleading to a fine-structure change). Sanchez-Villicanaet al. (1997) measure an enhancement in the trap-lossconstant b due to the combined effect of the trap andprobe lasers by defining an enhancement factor h,

h5b t1p2b t

bp, (94)

which is the difference between the trap-loss constantwith both beams present simultaneously or alternately.Note that if there is no combined effect, then h wouldremain equal to one. This flux enhancement process is

FIG. 33. Trap-loss spectrum of 85Rb measured by the Wiscon-sin group and compared to trap-loss spectrum of Cs. FromHoffmann et al. (1992).

FIG. 34. Trap-loss spectrum of 87Rb showing ‘‘holes’’ due tohyperfine structure. From Feng et al. (1993).


significant because it might influence the results of anyexperiment where trap and probe beams are present si-multaneously. The Connecticut group has also very re-cently reported a comprehensive study of trap-loss col-lisions in the 85Rb and 87Rb covering a wide range oftrap parameters (Gensemer et al., 1997).

d. Lithium trap loss

Although collisional trap loss occurs for all alkali spe-cies, lithium shows two unique features. First, theexcited-state hyperfine structure is inverted with2p2P3/2(F853) lying lower than 2p2P3/2(F852,1,0).Therefore long-range attractive molecular states corre-lating to the 2s2S1/2(F52)12p2P3/2(F853) asymptoteare less likely to be perturbed by hyperfine interactionsand avoided crossings with higher-lying molecular states.Second, the fine-structure splitting, DEFS , between the2p2P3/2 and 2p2P1/2 levels is sufficiently small that, byvarying the trapping laser intensity, the MOT trap depthcan be made comparable to the fine-structure-changingcollisions energy release. In temperature units DEFS /kBis only 0.48 K. A fine-structure-changing collision resultsin each atom leaving the binary encounter with one-halfthe exoergicity. If the trap depth ET is shallower thanthis 1

2 DEFS , then each atom can escape the trap andfine-structure-changing collisions becomes an importantloss mechanism. However, if the trap deepens below12 DEFS , the atoms cannot escape and the fine-structure-changing collisions mechanism no longer functions as atrap-loss process. In this situation, the remaining lossfrom radiative escape may be studied without the influ-ence of fine-structure-changing collisions.

Kawanaka et al. (1993) reported the first measure-ment of b for 7Li. The experiment implemented an un-usual four-beam MOT in which three of the beams pointtoward the vertex of an equilateral pyramid and thefourth beam counterpropagates along the axis of thepyramid, itself aligned with a trap-loading Li beam. Theaxial counterpropagating laser beam serves the doublerole of atomic beam slower and MOT confining laser.Total intensity in the MOT laser was 30 mW cm22, and

FIG. 35. Model of the influence of hyperfine structure on trap-loss spectra. From Lett et al. (1995).


to change the trap depth, the MOT beams were on-offmodulated at 500 kHz with the ‘‘on’’ time varying from;30 to 100% duty ratio. Values of b as a function ofMOT duty ratio were extracted from curve fitting fluo-rescence decay. The trap was operated with a smallnumber of atoms, such that the spatial density profilewas always close to a Gaussian profile. As the duty ratio(or trap depth) was varied, b exhibited the behaviorshown in Fig. 39. The value of b appears to show a pla-teau ;3310212 m3 s21 at the minimum duty ratio of;30% (shallow trap). As the duty ratio increases(deeper trap) this value decreases rapidly about one or-der of magnitude, reaching a minimum value of;3.2310213 cm3 s21 at ;65% duty ratio. As the MOTlaser ‘‘on’’ duty ratio increases above this minimumpoint to 100%, b increases only slightly to ;5310213

cm3 s21. Kawanaka et al. (1993) interpret this b behavioras showing the contributions of both fine-structure-changing collisions and radiative escape below the mini-mum, but at higher duty ratios (deeper traps) the fine-structure-changing collisions process no longercontributes. Therefore above about 65% duty ratio theonly source of collisional trap loss is radiative escape. Ifthis interpretation is correct, then the duty ratio at whichthe minimum b occurs must be equivalent to ET5 1

2 DEFS . Kawanaka et al. (1993) carried out a directmeasurement of the trap depth at 100 percent duty ratioby applying the kick-and-recapture technique describedby Raab et al. (1987). This measurement showed thatthe MOT is anisotropic with a depth of 0.64 K along theradial direction and 1.3 K along the axial direction. Thetrap-modulation technique effectively varies the averageMOT intensity and therefore the average trap depth. AMOT operates with two kinds of restoring forces: pro-portional to position and proportional to velocity. If theposition-dependent conservative force dominates, thenthe trap depth should be linear with the average MOTintensity. If the velocity-dependent dissipative forcedominates then the trap capture ability should vary asthe square root of the average MOT intensity. Given thedirect measurement of the trap depth at 100% duty ratioand assuming a square-root dependence, 1

2 DEFS corre-sponds to about 60% duty ratio, near the minimum band roughly consistent with the interpretation of aradiative-escape-only trap loss at duty ratios above 60–65 %. It should be born in mind that the anisotropy of

FIG. 36. Structure in the trap-loss spectrum of 85Rb, showingthe influence of hyperfine structure. Solid curve is a modelcalculation. Full circles are the measurements of Lett et al.(1995), and the open squares are measurements from Hoff-mann et al. (1992).


the trap was not explicitly taken into account in thesemeasurements. It should be further noted that valueof b reported in the radiative-escape-only regime,;5310213 cm3 s21, is about one order of magnitudegreater than predicted either by the Gallagher-Pritchardor Julienne-Vigue models.

Ritchie et al. (1995) carried out another study of col-lisional trap loss in a 7Li MOT, measuring b over awider range of intensities and several detunings. Thisstudy carefully took account of the anisotropy of theMOT trap depth by modeling and simulating atom es-cape velocities as a function of angle, intensity, and de-

FIG. 37. Rubidium trap loss spectra from the Wisconsin groupshowing that no isotope effect is measurable for trap-loss col-lisions from the 5p2P1/215s2S1/2 asymptote and that multipleorbiting must be included in any model of the trap loss process.Solid line is a model calculation including multiple orbits;dashed line shows the same model with a single orbit. FromPeters et al. (1994).

FIG. 38. Schematic of flux enhancement. The first excitation atRt puts flux on the long-range C3 /R3 attractive potential. Thisexcitation effectively pulls added flux into the small R regionwhere it is probed by the second excitation at Rp . FromSanchez-Villicana et al. (1996).


tuning. In a separate study Ritchie et al. (1994) developand examine this model in detail. The MOT setup usedthe conventional six-way orthogonal laser beam ar-rangement and was loaded from an optically-slowedatomic beam. The trap-loss rate constant b was ex-tracted from fits to fluorescence decay after the loadingwas shut off. Figure 40 shows the measured b for fourvalues of laser detuning and a range of intensities. Foreach of the four detunings the measured loss rate waslargest at small intensities and decreased with MOT in-tensity until reaching a distinct minimum, which oc-curred at a different intensity for each detuning. Similarto the interpretation of Kawanaka et al. (1993), this bminimum is attributed to the suppression of fine-structure-changing collisions loss. At higher values of in-tensity the loss is due entirely to radiative escape. Also,shown in Fig. 40 are the results of an optical-Bloch-equations-theory calculation (Julienne et al., 1994) forradiative escape trap-loss rates in Li. The optical-Bloch-equations approach is appropriate because the MOT in-tensities in this study reach as high as 120 mW cm22

where weak-field theories such as Gallagher-Pritchardor Julienne-Vigue cannot be applied. In addition theoptical-Bloch-equations theory was found not to fail forLi as it does for the heavier alkalis (Julienne et al.,1994). The optical-Bloch-equations theory predicts astrong dependence on trap depth with bRE scaling asET

23.0 , and therefore a thorough understanding of trapdepth spatial and intensity dependence is crucial. Tothat end Ritchie et al. (1994) developed a model andcarried out trajectory simulations of Li atoms subject tovarious conditions of intensity and detuning in theMOT. These simulations revealed that the trap depthET was highly dependent on u,f the polar and azimuthalangles with respect to the MOT symmetry axis. Fromthe model spatial dependence Ritchie et al. (1995) wereable to calculate b averaged over the different trap di-rections,

b5b0

4p E S E0

ETDdV (95)

where b0 , E0 are the trap-loss rate constant and trapdepth in the shallowest direction. The result showed arather startling anisotropy with b/b050.2360.02, rela-tively independent of intensity and detuning. The graphsin Fig. 40 plot this averaged b as well as the calculatedintensity Ic required to recapture an atom released inthe shallowest direction with a kinetic energy equal to12 DEFS . Except for the largest detuning, the agreementbetween the calculated Ic and the breakpoint in the b vsI plots (where bFCC→0) inspires confidence in the basiccorrectness of the model of Ritchie et al. (1994). In con-trast, while the optical-Bloch-equation calculation getsabout the correct value for bRE near Ic , the predictedand observed trap depth dependences go in opposite di-rections. The culprit is probably molecular hyperfinestructure, ignored in the application of the optical-Bloch-equation theory. At this writing the disagreementremains unresolved. It is, however, interesting to notethat Ritchie et al. (1994) find that the increase in bRE


with MOT intensity above the breakpoint can be well fitto a function proportional to the product of ground andexcited state atomic fractions and that this fitting worksover the range of detunings from D522.3G to 24.1G .Such a fitting is more consistent with a Gallagher-Pritchard or Julienne-Vigue picture of radiative escapesince the quasistatic theories should not be very sensi-tive to trap depth or detuning. At the high trap intensi-ties of these experiments, however, the basic weak-fieldassumption of Gallagher-Pritchard and Julienne-Viguecannot possibly be appropriate and the ‘‘agreement’’ isprobably just fortuitous. Further work in application oftheory is clearly needed.

e. Potassium trap loss

Potassium is the last stable atom in Group IA of theperiodic table to be successfully confined in a MOT andon which cold-collisions have been investigated. The pe-culiarities of K atomic structure have precluded a con-ventional approach to trap-loss measurements. William-son and Walker (1995) reported the first successfulpotassium MOT and the first studies of exoergic traploss. The Wisconsin group trapped two isotopes, 39K and41K, with natural isotopic abundances of 93 and 7 %respectively; however, the very small excited-state hy-perfine splittings of these isotopes required a modifica-tion of conventional MOT technique. The unusual fact isthat, in the case of 39K, the splitting of the entireexcited-state hyperfine manifold is only 33 MHz, and theseparation between 4p2P3/2(F853) and 4p2P3/2(F852) is only 21 MHz. In the case of 41K the excited hy-perfine manifold covers only 17 MHz. With a naturallinewidth of 6.2 MHz, overlapping transitions from the4s2S1/2(F52) ground state to adjacent hyperfine levelswould lead to considerable optical pumping and heatingas spontaneous emission populated the lower F51

FIG. 39. Trap-loss rate constant b as function of ‘‘trap depth’’(MOT duty ratio) for lithium atoms. Rapidly rising left-handbranch thought to be due to fine-structure-changing collisionsand radiative escape, but as trap becomes deeper (increasingduty ratio) fine-structure-changing collisions no longer contrib-utes. From Kawanaka et al. (1993).


ground state. To overcome this problem Williamson andWalker (1995) tuned the trapping and repumper lasersto the red of the entire hyperfine manifold, essentiallytreating the 2P3/2 fine-structure level of 39K as an unre-solved line with ;30 MHz width. Although opticalpumping ‘‘population leakage’’ will certainly be higherwhen the maximum FuMFu state cannot be populateduniquely, this tactic has at least the advantage that thecapture range of K atom velocity extends to 30 ms21 anincrease of about a factor of five over the capture rangefor Na. The increase in velocity capture range increasesthe MOT loading rate and partially compensates for op-tical pumping losses. The Wisconsin potassium MOToperates at high light intensity (470 mW cm22 for 39K)and with a large number of trapped atoms such that theloading proceeds mostly at constant density. For the 41Kthe MOT intensity goes as high as 530 mW cm22 withlarge beam diameter of 0.6 cm. Fitting trap-loading fluo-rescence curves to Eq. (90) yields the total loss rate con-stant G(n)5g1bn where g is the rate constant for col-lisions with hot background gas and b is the familiar traploss rate constant for ultracold potassium collisions. Theb rate constant is then determined by measuring thedensity dependence of the total trap loss rate, G at fixeddetuning. The slope of this plot, dG/dn , yields b; andthis program is carried out over a range of detunings.Figure 41 shows the variation of b with the detuning Dfor the 39K MOT. Although some variation of b withdetuning is evident, the range is considerably narrowerthan the typical trap-loss spectra reported earlier by theConnecticut and Wisconsin groups for rubidium. Ob-taining the detuning dependence of b from the densityplots is clearly much more arduous than the using a ca-talysis laser so the difference in detuning range is notsurprising. Williamson and Walker (1995) report that41K shows even less detuning dependence than 39K. Thetrap-loss collision dynamics of K is much less well-studied than the other alkalis and investigations mustcontinue to clarify the role played by the differences inhyperfine structure.

Another study of the dependence of trap-loss rate of apotassium MOT as a function of trap laser intensity wasrecently performed by Santos et al. (1996) using a MOTloaded from vapor. The loading measurements were car-ried out under conditions of constant trap density. To beable to measure a large range of intensities without com-promising the trap performance, the sudden-change-of-intensity technique was used: the trap was loaded at highintensity, after which a neutral density filter was me-chanically introduced, producing a low intensity condi-tion. The transition between the two intensities regimecauses a transient decay in the number of atoms thatpermits determination of b. The values of b measuredfor several intensities are plotted in Fig. 42, where eachpoint is a result of fifteen independent measurements.From high intensities down to 10 mW cm22, b showsonly small variation with a possible tendency of increas-ing with intensity. In this regime, indicated by a dashedline on Fig. 42, radiative escape and fine-structure-changing collisions are dominant losses. At about


10 mW cm22 an abrupt increase in b is observed, whichis associated with hyperfine-changing collisions as ob-served in other systems. The HCC regime for trap loss isindicated in Fig. 42 as a dotted line.

f. Sodium-potassium mixed-species trap loss

All experiments and theories about exoergic collisionsin atomic traps have considered only homonuclear sys-tems. Santos et al. (1995) recently demonstrated that twodifferent alkalis (Na/K) could be cooled and confined inthe same MOT. This experiment opens the study of het-eronuclear ultracold exoergic collisions.

The mixed-alkali MOT of Santos et al. (1995) uses aglass vapor cell that contains a mixture of Na/K vapor atroom temperature. The trapping laser beams for bothalkalis (589 nm for Na and 766 nm for K) are combinedusing dichroic mirrors and enter the MOT cell copropa-gating. The optimum trapping conditions are similar toeach alkali individually considered. Fluorescence detec-tion measures the number of trapped atoms and a CCDcamera measures the spatial distribution of each species.The sodium atomic cloud is located within the largerpotassium cloud. To investigate the inter-species coldcollision effects, Santos et al. (1995) measured the load-ing of the sodium trap in the presence and absence ofthe cold potassium cloud. As the sodium trap loaded,the net loading rate is expressed as

FIG. 40. Trap-loss rate constant b for four different detuningsand a range of intensities in the Rice group’s Li MOT. Thevertical lines in each plot denote Ic the calculated critical in-tensity required to recapture an atom released in the shallow-est direction. From Ritchie et al. (1995).


dNNa

dt5L2gNNa2bnNaNNa2b8nKNNa . (96)

where gNNa is the collision rate of trapped Na with hotbackground gas (both Na and K). Observing the tran-sient behavior in a regime of constant density for bothspecies, one can measure b8 which is the cross-speciestrap-loss collision rate. The measured value for Na/Kwas b853.061.5310212 cm3 s21. The loss-rate constantfor Na/Na collisions was previously reported by Mar-cassa et al. (1993) to be b5361310211 cm3 s21, aboutone order of magnitude larger than b8 due to Na/K col-lisions. The Brazilian group, led by V. S. Bagnato, ratio-nalized the marked difference by noting that the long-range, excited-state molecular potential in NaK arisesfrom a nonresonant dipole-dipole interaction and variesonly as 2C6 /R6 instead of the familiar 2C3 /R3 as inthe homonuclear case. They argued that the Na/K atompairs excited at long range will experience much lessacceleration along their line of centers with consequentreduction in the survival probability factor [see, for ex-ample, Eq. (78) or (81)] of the rate constant. An alter-native interpretation could be that the 2C6 /R6 interac-tion implies a much reduced phase space available forthe initial excitation due to the much smaller Condonpoint.

g. Rare-gas metastable loss in MOTs and optical lattices

Rage gas metastable atoms can also be trapped inMOTs and give rise to trap loss by Penning or associa-tive ionization collisions. Bardou et al. (1992) have car-ried out an unusual variation on the trap-loss experi-ment by decelerating a beam of helium metastableatoms, He(23S1), and capturing the slowed atoms in aMOT. The cooling transition is between the 23S1‘‘ground state’’ and the 23P2 excited state (l51.083 mm). They carried out trap-decay measure-ments by shutting off the loading beam and observingthe non-exponential decay of the ions rather than theusual fluorescence. By fitting the usual trap-loss equa-tions to the ion decay curves, they determined a trap lossrate from Penning ionization,

He*1He*→He1He11e . (97)

The MOT contains a significant fraction of 23P2 as wellas 23S1 so the He* collisions could be between any com-bination of these two species. Bardou et al. determineb5725

12131028 cm3 sec21 which is a remarkably bignumber. In fact the formula—Eq. (48), given in Julienneand Mies (1989)—calculated from the S-matrix unitaritycondition, shows that an upper bound to the Penningionization rate constant for collisions between He(3S1)atoms approaching on the 1(g

1 molecular state will be231029 cm3 sec21 at TMOT51 mK. The fact that themeasured b is more than an order of magnitude higherthan the s-wave unitarity limit may mean that collisionsbetween He(3S1) and He(3P2) play an important role.Julienne et al. (1993) confirmed this conjecture by show-ing that the unitarity limit increases by a factor of ( lmax11)2 where lmax partial waves are captured by the ex-


cited state. At 1 mK lmax is about 5. Bardou et al. probedthe role of the excited state by modulating the trappinglight and observing the effect on the ion production rate.The decline was dramatic, and by varying the density inthe MOT they were able to determine bP51431028 cm3 sec21. Therefore their measured b5725

121

31028 cm3 sec21 is an average of bS and bP , weightedby the product of relative ground-state population forS-S collisions and the product of ground- and excited-state population for S-P collisions. Both kinds of colli-sions contribute a loss-rate constant several orders ofmagnitude greater than fine-structure-changing colli-sions or radiative escape in the alkalis. As a conse-quence Bardou et al. report a steady state density in theMOT of about 108 cm23, about two orders of magnitudelower than the density of a typical alkali MOT. Veryrecently Mastwijk et al. (1998) have reported a new mea-surement of the rate constant for Penning and associa-tive ionization in metastable He trapped in a MOT.Their reported result, (1.960.8)31029 cm3 s21, is abouttwo orders of magnitude smaller than that of Bardou. Atthis writing the discrepancy has not been resolved. Ka-tori and Shimizu (1994) and Wahlhout et al. (1995) havealso measured trap-loss rates of ionizing collisions inmetastable Kr and Xe traps.

An interesting variation on ‘‘trap loss’’ in ionizing col-lisions is the modification of collision rates in opticallattices which constrain atomic motion in a periodic po-tential. Kunugita et al. (1997) measured a reduced colli-sion rate in a Kr lattice and Lawall et al. (1998) mea-sured both suppression and enhancement in a Xe lattice.Boisseau and Vigue (1996) discuss how very long-rangemolecular interactions in a light field might be modifiedin optical lattices. Finally we note that Katori et al.(1995) have measured a quantum statistical effect onionizing collision rates of metastable Kr. The collisionrate for the spin-polarized fermionic isotope 83Kr de-creased relative to the collision rate in the unpolarizedgas. The decrease was attributed to the threshold lawsfor p-waves (which contribute to the spin-polarizedcase) that differed from those for s waves (which only-

FIG. 41. Trap-loss spectrum of 39K in a the potassium MOT ofthe Wisconsin group. From Williamson and Walker (1995).


contribute in the spin-unpolarized case). For a discus-sion of these threshold laws see Sec. III.B.

B. Ground-state trap-loss collisions

Trap-loss processes such as radiative escape and fine-structure-changing collisions always start with collisionpartners approaching on an excited long-range attractivemolecular potential. Collisions on the molecular groundstate divide into two categories: elastic collisions andhyperfine-changing collisions. Exoergic hyperfine-changing collisions can also lead to loss if the trap ismade sufficiently shallow such that the velocity gainedby each atom from the hyperfine-changing collisions ki-netic energy release exceeds the maximum capture ve-locity of the trap.

1. Hyperfine-changing collisions

If atoms have nuclear spin IW then the coupling of thisnuclear spin to the total electronic angular momentum JW

results in hyperfine states FW 5JW1IW with projection MFalong some axis of quantization. In binary collisions thisaxis is always taken to be the line joining the two nucleiwith the hyperfine states of the two partners labeledF1MF1 and F2MF2 . The manifold of molecular hyper-fine states FW m is constructed from vector pairing of theatomic states in the usual way, FW m5FW 11FW 2 . In theasymptotic limit of internuclear separation the molecu-lar hyperfine states are products of the atomic hyperfinestates, uF1MF1&uF2MF2&, and their energy levels are thesum of their constituent atomic hyperfine energies. Fig-ure 4 illustrates these molecular potentials near theirasymptotic limit for the case of 87Rb. It is customary tolabel the asymptotic energy levels by F1F2 . For examplethe ground state of Na splits into two hyperfine levels,F51,2 by the vector coupling of J51/2 to the nuclearspin I53/2. The energy difference between the two hy-perfine levels is 1772 MHz. The molecular hyperfine as-ymptotes are labeled F1F25111, 112, 212 with eachasymptote separated by 1772 MHz. If during a colli-sional encounter the exit channel lies on a lower hyper-

FIG. 42. Intensity dependence of the trap-loss rate constant bin the Sao Carlos potassium MOT. From Santos et al. (1996).


fine level than the entrance channel, the energy differ-ence appears as the kinetic energy of the recedingpartners. In homonuclear collisions each partner re-ceives half the kinetic-energy release and an incrementin velocity vHCC5ADEHCC /m . If this velocity incrementexceeds the capture velocity of the MOT, the hyperfine-changing collision leads to trap loss.

Two physical processes govern the change of hyper-fine level: spin exchange and spin dipole-dipole interac-tion. Spin exchange occurs when the charge clouds ofthe two colliding partners begin to overlap. The electronspins SW 1 ,SW 2 decouple from the nuclear spins and re-couple to form molecular electron spin states SW 5SW 1

1SW 2 . The atomic nuclear spins also recouple to formmolecular nuclear spin states IW5IW11IW2 such that FW m

5SW 1IW is conserved and MF5MF11MF2 . In the ex-change region the asymptotic hyperfine statesuF1MF1&uF2MF2& transform to linear combinations thatlead to a finite probability that the postcollisional popu-lation of asymptotic states differs from the entrancechannel. In contrast to the exchange interaction thelong-range spin-dipole interaction is usually weaker byseveral orders of magnitude, because it derives fromrelativistic spin-dependent forces proportional to thesquare of the fine-structure constant. Later we will en-large the discussion of ground-state collisions and thesephysical processes in the context of Bose-Einstein con-densates (Stoof et al., 1988). For the present discussionof trap-loss processes it suffices to note that in a MOTendoergic inelastic channels are generally closed by en-ergy conservation, but the exoergic hyperfine-changingcollisions channels can lead to trap loss.

a. Cesium and rubidium

Earlier we noted that Sesko et al. (1989) observed arapid increase in b as the intensity of their Cs MOTdropped below 4 mW cm22. They interpreted this loss asdue to hyperfine-changing collisions, but could not inferdirectly bHCC , the rate constant for the hyperfine-changing collisions process itself, because even at thelowest MOT laser intensity the trap loss b does not be-come independent of the trap depth. Sesko et al. (1989)state a bHCC estimation between 10210 and10211 cm3 sec21. Wallace et al. (1992) also observed arapidly rising branch of b as a function of decreasingMOT laser intensity below about 2.5 mW cm22 in a 85RbMOT and below about 4 mW cm22 in a 87Rb MOT.They also reported a strong isotope effect in b andshowed that the ratio of b85 /b87 at constant trap inten-sity supported the view that MOT velocity-dependentforces rather than position-dependent forces dominateatom capture and trapping in the MOT environment.Furthermore, at the lowest MOT intensities b showssome sign of becoming independent of MOT capturevelocity which would imply a direct measure of bHCC .Wallace et al. (1992) estimated this bHCC to be about thesame for both isotopes, ;2310211 cm3 sec21. If MOTsfunctioned as spatially isotropic, conservative potentials,


then one would expect a very sharp threshold for loss asthe trapping laser intensity decreases below the mini-mum required to contain the hyperfine-changing colli-sions kinetic energy release. In fact, to first approxima-tion, the MOT restoring forces are the sum of a velocity-dependent dissipative term and a spatially-dependentterm, so it is more accurate to characterize a MOT by itscapture velocity rather than the trap ‘‘depth.’’ This cap-ture velocity is a complicated function of trapping laserdetuning, intensity, spot size, magnetic field gradient,and, as Ritchie et al. (1994) have shown in detail for a LiMOT, exhibits spatial anisotropy. Therefore, as theMOT laser intensity decreases, the trap will start to leakin the shallowest directions (along the MOT symmetryaxis for a six-beam MOT) first. As MOT laser intensitycontinues to decrease, the leak rate will increase until breaches a constant value where atoms are no longer con-tained in any direction. Ritchie et al. (1994, 1995) haveshown how a detailed trap simulation can be used toextract bFCC from the intensity dependence of MOTperformance for fine-structure-changing collisions in Li,but a similar analysis to extract bHCC has not yet beenattempted.

b. Sodium

Marcassa et al. (1993) could not maintain their NaMOT at intensities sufficiently low to measure b on therapidly rising branch where HCC is thought to becomethe dominant trap-loss process. Although similarhyperfine-changing collisions losses have been observedin Cs (Sesko et al., 1989), Rb (Wallace et al., 1992), andK (Santos et al., 1996), the low-intensity branch is moredifficult to observe in Na because the threshold for itsoccurrence starts at MOT intensities already weak com-pared to the saturation intensity of the atomic coolingtransition. At such low intensities the vapor-loadedMOT barely functions. However, starting from a slowatomic beam Shang et al. (1994) succeeded in efficientlyloading a Na MOT to investigate trap losses over a widerange of MOT intensity, including the low-intensity bbranch. Another novel aspect of their work is the inves-tigation of trap loss with all ground-state trapped atomsprepared in either the 3s2S1/2(F51) hyperfine level, orthe 3s2S1/2(F52) level. The motivation was to show di-rectly that, since only collisions between atoms preparedin the F52 level can lead to hyperfine-changing colli-sions, only they exhibit the low-intensity b branch. Theirtrap-loss measurements presented in Fig. 43 confirm thatF51 collisions do not exhibit the rapidly-rising b signa-ture of hyperfine-changing collisions. In contrast, for at-oms trapped in the F52 ground state, Shang et al.(1994) did observe the conventional decrease of b asMOT intensity decreases, until a minimum is reached atabout 4 mW cm22. Below 4 mW cm22 hyperfine-changing collisions dominate, and the loss rate increasesrapidly. For atoms in the F51 hyperfine level thehyperfine-changing collisions exoergic channel is closed,and b exhibits no rapidly rising branch below the4 mW cm22 minimum. To obtain a trap with atoms pre-dominantly in the F51 ground hyperfine level, Shang


et al. (1994) used the type II trap described by Prentisset al. (1988) and later used by Marcassa et al. (1993). Thevalues measured by Shang et al. (1994) are in reasonableagreement with theory presented in Marcassa et al.(1993) in which b from fine-structure-charging and ra-diative escape were calculated using the optical-Bloch-equation approach of Band and Julienne (1992) and ac-curate Na2 potential curves from Magnier et al. (1993).We have previously discussed the limitations of theoptical-Bloch-equation approach in Sec. V.A.3. Thesolid line in Fig. 43 shows the theory calculation. Com-parison between bF1 and bF2 on the high intensitybranch shows that trap-loss rate from F51 collisions isgreater than F52 collisions by about a factor of threeover the intensity range from 20 to 90 mW cm22. Fromabout 4 to 20 mW cm22 bF2 drops by about two ordersof magnitude from ;2310210 cm3 sec21 to ;2310212 cm3 sec21. Shang et al. (1994) speculate thatsub-Doppler cooling in this intensity range may slow thecollision rate. Differences in molecular hyperfine struc-ture between F52 collisions and F51 collisions mayalso play a role. At present the dramatic drop in bF2 isnot clearly explained.

The most recent contribution to ground-state trap lossin sodium corresponds to a study of collisional loss rateof Na in a MOT operating on the D1 line (Marcassaet al., 1996). Although the idea of trapping alkalis usingthe D1 resonance is straightforward, only recently has itbeen demonstrated by Flemming et al. (1996). The per-formance of a D1 trap alone in a vapor cell is very poorand requires loading from an external source of coldatoms. Therefore, Marcassa et al. (1996) used a conven-tional MOT operating on the D2 line for loading the D1MOT. Atoms are loaded by capturing and cooling themin a conventional D2 MOT with the D1 optical beamssuperposed in the MOT configuration. After loading,the D2-line laser is turned off, but the cold atoms remaintrapped in the D1 MOT. The number of trapped atomsis determined during the D2 MOT. As previously, bD1 isextracted from the fluorescence decay during the D1-line cycle. The measured bD1 from several intensities oflaser at a detuning of about 210 MHz from 3S1/2(F52)→3P1/2(F52), and the repumper close to 3S1/2(F51)→3P1/2(F52), is presented in Fig. 44. In the rangefrom 200 mW cm22 to about 65 mW cm22 no significantvariation in b is observed. At about 65 mW cm22 thecollisional loss rate increases considerably. At this pointof rapid increase, the trap has become shallow enoughto make hyperfine-changing collisions an important lossmechanism. For a trap operating at the D2 line the in-tensity at which this rapid increase occurs is about5 mW cm22 (Shang et al., 1994), about 10 times smallerthan for the D1-line trap. As a consequence the trapdepth of the D1-line trap is about 1/10 of the D2-linetrap at comparable intensities. This considerable de-crease in trap depth is attributed to the several possible‘‘dark states’’ that can occur in the D1-line trap.

2. Summary comparison of trap-loss rate constants

Many research groups have carried out hyperfine-changing collisions trap loss experiments in MOTs with


varied instrumental design parameters, different MOTlaser intensities, spot sizes, and detunings, and a host ofatoms all with their own resonance lines and lifetimes. Itwould be desirable to organize these measurements on acommon basis so as to identify the critical factors result-ing in a particular determination of b. We present here asimple model of MOT atom confinement from which wederive a parameter Q that correlates well with the mea-surements discussed in the preceding subsections.

We assume that an atom pair subject to a hyperfine-changing collisions collision receives an extra quantity ofenergy DEHCC corresponding to the exoergicity of thetransition and that each atom of mass m carries awayhalf the velocity increase,

v5FDEHCC

m G1/2

. (98)

We further assume that the atom travels outward fromthe center of the trap to the radius of the MOT w sub-ject only to the dissipative velocity-dependent decelerat-ing force. Wallace et al. (1992) cite evidence from theirisotopic studies in a Rb MOT that the velocity-dependent force dominates over the position-dependentrestoring force. Thus

F52av (99)

and

v~z !5FDEHCC

m G1/2

2a

mz . (100)

The maximum capture velocity is defined by

v~w !50, (101)

so

a

mw5FDEHCC

m G1/2

5vmax . (102)

Now, from Eq. (56),

a516\k2V2DG

~4D21G212V2!2 , (103)

so at relatively large detunings and low intensities thesecond two terms in the denominator of Eq. (103) canbe ignored and writing the MOT intensity I in terms ofthe Rabi frequency V,

V25I

IsatG2, (104)

we have

vmax5\k2IwG3

D3Isatm. (105)

Removing all the constant terms we define a trap cap-ture parameter Q as

Q51l2

G3

D3

I

Isat

w

m. (106)

Figure 45 shows b plotted against Q for all the alkalisdiscussed above. Note that for heteronuclear collisions


Eq. (106) must be modified by substituting (m1/2)(11m1 /m2) for m where m1 is the mass of the species ofthe observed trap loss.

VI. PHOTOASSOCIATION SPECTROSCOPY

A. Introduction

If, while approaching on an unbound ground state po-tential, two atoms absorb a photon and couple to anexcited bound molecular state, they are said to undergophotoassociation. Figure 46 diagrams the process. Atlong range electrostatic dispersion forces give rise to theground-state molecular potential varying as C6 /R6. Ifthe two atoms are homonuclear, then a resonant dipole-dipole interaction sets up 6C3 /R3 excited-state repul-sive and attractive potentials. Figure 47 shows the actuallong-range excited potential curves for the sodiumdimer, originating from the 2S1/21

2P3/2 and 2S1/212P1/2 separated atom states. For cold and ultracoldphotoassociation processes the long-range attractive po-tentials play the key role; the repulsive potentials figureimportantly in optical shielding and suppression, thesubject of Sec. VII. In the presence of a photon withfrequency vp the colliding pair with kinetic energy kBTcouples from the ground state to the attractive molecu-lar state in a free-bound transition near the Condonpoint RC , the point at which the difference potentialjust matches \vp .

Scanning the probe laser vp excites population ofvibration-rotation states in the excited bound potentialand generates a free-bound spectrum. This general classof measurements is called photoassociative spectroscopyand can be observed in several different ways. The ob-

FIG. 43. Trap-loss due to hyperfine changing collisions is pos-sible for F52 collisions but not possible for F51 collisions.The absence of a rapidly rising branch at low intensity for F51 collisions confirms the trap-loss process at low trap depth.From Shang et al. (1994).


servation may consist of bound state decay by spontane-ous emission, most probably as the nuclei move slowlyaround the outer turning point, to some distribution ofcontinuum states on the ground potential controlled bybound-free nuclear Franck-Condon overlap factors. Thefree atoms then recede on the ground state with somedistribution of kinetic energy. In a MOT or a FORT,this radiative escape process can be an important sourceof loss if each atom receives enough kinetic energy toovercome trap restoring forces. As vp passes throughthe free-bound resonances, the loss channel opens, andthe number of atoms in the trap sharply diminishes.Thus scanning vp generates a high-resolutionfluorescence-loss spectrum. The excited bound quasi-molecule may also decay around the inner turning pointto bound levels of the ground molecular state. Since thestable dimers are usually not confined by MOT fields,this cold-molecule production is another trap-loss chan-nel, although the branching ratio to ground-state boundmolecules is usually small, only a few percent of the totalloss rate.

Instead of observing the spectrum by spontaneous de-cay and trap loss, a second probe photon may excite thelong-range bound quasimolecule to a higher excitedstate (or manifold of states) from which the moleculemay undergo direct photoionization or autoionizationwithin some range of internuclear separation. If this sec-ond photon comes from the same light source as thefirst, the process is called ‘‘one-color’’ photoassociativespectroscopy; if from an independently tunable sourcethen it is called ‘‘two-color’’ photoassociative spectros-copy. When the ultimate product of the photoassocia-tion is a molecular ion, the process is called photoasso-ciative ionization. Heinzen (1995) has provided a goodintroductory review of photoassociation and opticalshielding (the subject of the next section).

B. Photoassociation at ambient and cold temperatures

The first measurement of a free-bound photoassocia-tive absorption appeared long before the development

FIG. 44. Trap-loss from a Na MOT formed on the D1 reso-nance line as a function of MOT laser intensity (trap depth).From Marcassa et al. (1996).


of optical cooling and trapping, about two decades ago,when Scheingraber and Vidal (1977) reported the obser-vation of photoassociation in collisions between magne-sium atoms. In this experiment fixed uv lines from anargon ion laser excited free-bound transitions from thethermal continuum population of the ground X1Sg

1

state to bound levels of the A1Su1 state of Mg2. Schein-

graber and Vidal analyzed the subsequent fluorescenceto bound and continuum states from which they inferredthe photoassociative process. The first unambiguousphotoassociation excitation spectrum, however, was mea-sured by Inou et al. (1982) in collisions between Xe andCl at 300 K. In both these early experiments the excita-tion was not very selective due to the broad thermaldistribution of populated continuum ground states.Jones et al. (1993) with a technically much improved ex-periment reported beautiful free-bound vibration pro-gressions in KrF and XeI X→B transitions; further-more, from the intensity envelope modulation they wereable to extract the functional dependence of the transi-tion moment on the internuclear separation. Althoughindividual vibrational levels of the B state were clearlyresolved, the underlying rotational manifolds were not.Jones et al. (1993) simulated the photoassociation struc-ture and line shapes by assuming a thermal distributionof rotational levels at 300 K. Photoassociation and dis-sociation processes prior to the cold and ultracold epochhave been reviewed by Tellinghuisen (1985).

A decade after Scheingraber and Vidal reported thefirst observation of photoassociation, Thorsheim et al.(1987) proposed that high-resolution free-bound mo-lecular spectroscopy should be possible using opticallycooled and confined atoms. Figure 48 shows a portion oftheir calculated X→A absorption spectrum at 10 mKfor sodium atoms. This figure illustrates how cold tem-peratures compress the Maxwell-Boltzmann distributionto the point where individual rotational transitions inthe free-bound absorption are clearly resolvable. Themarked differences in peak intensities indicate scatter-ing resonances, and the asymmetry in the line shapes,tailing off to the red, reflect the thermal distribution ofground-state collision energies at 10 mK. Figure 49 plotsthe photon-flux-normalized absorption rate coefficientfor singlet X1Sg

1→A1Su1 and triplet a3Su

1→13Sg1 mo-

lecular transitions over a broad range of photon excita-tion, red-detuned from the Na (2S→2P) atomic reso-nance line. The strongly modulated intensity envelopesare called Condon modulations, and they reflect theoverlap between the ground state continuum wave func-tions and the bound excited vibrational wave functions.These can be understood from the reflection approxima-tion described in Eq. (32). We shall see later that theseCondon modulations reveal detailed information aboutthe ground state scattering wave function and potentialfrom which accurate s-wave scattering lengths can bedetermined. Thorsheim et al. (1987), therefore, pre-dicted all the notable features of ultracold photoassocia-tion spectroscopy later to be developed in many experi-ments: (1) precision measurement of vibration-rotationprogressions from which accurate excited-state potential


parameters can be determined, (2) line profile measure-ments and analysis to determine collision temperatureand threshold behavior, and (3) spectral intensity modu-lation from which the ground state potential, the scatter-ing wave function, and the s-wave scattering length canbe characterized with great accuracy.

An important difference distinguishes ambient tem-perature photoassociation in rare-gas halide systems andsub-millidegree-Kelvin temperature photoassociation incooled and confined alkali systems. At temperaturesfound in MOTs and FORTs (and within selected veloc-

FIG. 45. Summary plot of the trap-loss constant b as a func-tion of the trap-loss parameter Q for all the alkali systemsstudied to date.

FIG. 46. Diagram of the photoassociation process. First exci-tation (\vp) is followed by (a) spontaneous or stimulatedemission back to bound or continuum levels of the groundstate or (b) second excitation to the molecular ion (photoasso-ciative ionization).


ity groups in atomic beams) the collision dynamics arecontrolled by long-range electrostatic interactions, andCondon points RC are typically at tens to hundreds ofa0 . In the case of the rare-gas halides the Condon pointsare in the short-range region of chemical binding, andtherefore free-bound transitions take place at muchsmaller internuclear distances, typically less than ten a0 .For the colliding A ,B quasimolecule the pair density nas a function of R is given by

n5nAnB4pR2e2V~R !/kT (107)

so the density of pairs varies as the square of the inter-nuclear separation. Although the pair-density R depen-dence favors long-range photoassociation, the atomic re-actant densities are quite different with the nAnBproduct on the order of 1035 cm26 for rare-gas halidephotoassociation and only about 1022 cm26 for opticallytrapped atoms. Therefore the effective pair densityavailable for rare-gas halide photoassociation greatly ex-ceeds that for cold alkali photoassociation, permittingfluorescence detection and dispersion by high resolution(but inefficient) monochromators.

C. Associative and photoassociative ionization

Conventional associative ionization occurring at ambi-ent temperature proceeds in two steps: excitation of iso-lated atoms followed by molecular autoionization as thetwo atoms approach on excited molecular potentials. Insodium for example (Weiner et al., 1989),

Na1\v→Na* (108)

Na*1Na*→Na211e (109)

The collision event lasts a few picoseconds, fast com-pared to radiative relaxation of the excited atomic states(;tens of nanoseconds). Therefore the incoming atomic

FIG. 47. Long-range potential curves of the first excited statesof the Na dimer labeled by Hund’s case (c) notation. The num-ber gives the projection of electron spin plus orbital angularmomentum on the molecular axis. The g/u and 1/2 labelscorrespond to conventional point-group symmetry notation.


excited states can be treated as stationary states of thesystem Hamiltonian, and spontaneous radiative lossdoes not play a significant role. In contrast, cold andultracold photoassociative ionization must always starton ground states because the atoms move so slowly thatradiative lifetimes become short compared to the colli-sion duration. The partners must be close enough at theCondon point, where the initial photon absorption takesplace, so that a significant fraction of the excited scatter-ing flux survives radiative relaxation and goes on topopulate the final inelastic channel. Thus photoassocia-tive ionization is also a two-step process: (1) photoexci-tation of the incoming scattering flux from the molecularground-state continuum to specific vibration-rotationlevels of a bound molecular state; and (2) subsequentphoton excitation either to a doubly-excited molecularautoionizing state or directly to the molecular photoion-ization continuum. For example, in the case of sodiumcollisions, the principal route is through doubly-excitedautoionization (Julienne and Heather, 1991),

Na1Na1\v1→Na2*1\v2→Na2**→Na211e , (110)

whereas for rubidium atoms the only available route isdirect photoionization in the second step (Leonhardtand Weiner, 1995),

Rb1Rb1\v1→Rb2*1\v2→Rb211e . (111)

Collisional ionization can play an important role inplasmas, flames, and atmospheric and interstellar phys-ics and chemistry. Models of these phenomena dependcritically on the accurate determination of absolute crosssections and rate coefficients. The rate coefficient is thequantity closest to what an experiment actually mea-sures and can be regarded as the cross section averagedover the collision velocity distribution,

K5E0

`

vs~v !f~v !dv . (112)

The velocity distribution f(v) depends on the conditionsof the experiment. In cell and trap experiments it is usu-ally a Maxwell-Boltzmann distribution at some well-defined temperature, but f(v) in atomic beam experi-ments, arising from optical excitation velocity selection,deviates radically from the normal thermal distribution(Tsao et al., 1995). The actual signal count rated(X2

1)/dt relates to the rate coefficient through

1Va

d~X21!

dt5K@X#2, (113)

where V is the interaction volume, a the ion detectionefficiency, and @X# the atom density. If rate constant orcross section measurements are carried out in crossed orsingle atomic beams (Weiner et al., 1989; Thorsheimet al., 1990; Tsao et al., 1995) special care is necessary todetermine the interaction volume and atomic density.

Photoassociative ionization was the first measuredcollisional process observed between cooled andtrapped atoms (Gould et al., 1988). The experiment wasperformed with atomic sodium confined in a hybrid laser


trap, utilizing both the spontaneous radiation pressureand the dipole force. The trap had two counterpropagat-ing, circularly polarized Gaussian laser beams broughtto separate foci such that longitudinal confinement alongthe beam axis was achieved by the spontaneous forceand transversal confinement by the dipole force. Thetrap was embedded in a large (;1 cm diameter) conven-tional optical molasses loaded from a slowed atomicbeam. The two focused laser beams comprising the di-pole trap were alternately chopped with a 3 msec ‘‘trapcycle,’’ to avoid standing-wave heating. This trap cyclefor each beam was interspersed with a 3 msec ‘‘molassescycle’’ to keep the atoms cold. The trap beams weredetuned about 700 MHz to the red of the 3s2S1/2(F52)→3p2P3/2(F53) transition while the molasses wasdetuned only about one natural linewidth (;10 MHz).The atoms captured from the molasses (;107 cm23)were compressed to a much higher excited atom density(;53109 cm23) in the trap. The temperature was mea-sured to be about 750 mK. Ions formed in the trap wereaccelerated and focused toward a charged-particle de-tector. To assure the identity of the counted ions, Gouldet al. carried out a time-of-flight measurement; the re-

FIG. 48. Calculated free-bound photoassociation spectrum at10 mK. From Thorsheim et al. (1987).

FIG. 49. Calculated absorption spectrum of photoassociationin Na at 10 mK, showing Condon fluctuations. The curve la-beled ‘‘310’’ refers to 3Sg→3Su while the other curve refersto 1Sg→1Su transitions. From Thorsheim et al. (1987).


sults of which, shown in Fig. 50, clearly establish the Na21

ion product. The linearity of the ion rate with the squareof the atomic density in the trap, supported the view thatthe detected Na2

1 ions were produced in a binary colli-sion. After careful measurement of ion rate, trap volumeand excited atom density, the value for the rate coeffi-cient was determined to be K5(1.120.5

11.3)310211 cm3 sec21. Gould et al. (1988), following con-ventional wisdom, interpreted the ion production ascoming from collisions between two excited atoms,

dNI

dt5KE ne

2~rW !d3rW5KneNe , (114)

where dNI /dt is the ion production rate, ne(rW) theexcited-state density, Ne the number of excited atoms inthe trap @5*ne(rW)d3rW# , and ne the ‘‘effective’’ excited-state trap density. The value for K was then determinedfrom these measured parameters. Assuming an averagecollision velocity of 130 cm s21, equivalent to a trap tem-perature of 750 mK, the corresponding cross section wasdetermined to be s5(8.623.8

110.0)310214 cm2. In contrast,the cross section at ;575 K had been previously deter-mined to be ;1.5310216 cm2 (Bonnano et al., 1983,1985; Wang et al., 1986; Wang et al., 1987). Gould et al.(1988) rationalized the difference in cross section size byinvoking the difference in de Broglie wavelengths, thenumber of participating partial waves, and the tempera-ture dependence of the ionization channel probability.The quantal expression for the cross section in terms ofpartial wave contributions l and inelastic scatteringprobability S12 is

s12~e!5S p

k2D(l50

`

~2l11 !uS12~e ,l !u2

>pS ldB

2p D 2

~ lmax11 !2P12 , (115)

where ldB is the entrance channel de Broglie wave-length and P12 is the probability of the ionizing collisionchannel averaged over all contributing partial waves ofwhich lmax is the greatest. The ratio of the (lmax11)2

between 575 K and 750 mK is about 400 and the deBroglie wavelength ratio factor varies inversely withtemperature. Therefore in order that the cross sectionratio be consistent with low- and high-temperatureexperiments, s12(575 K)/s12(750 mK);1.731023,Gould et al. (1988) concluded that P12 must be aboutthree times greater at 575 K than at 750 mK.

However, it soon became clear that the conventionalpicture of associative ionization, starting from the ex-cited atomic states, could not be appropriate in the coldregime. Julienne (1988) pointed out the essential prob-lem with this picture. In the molasses cycle the opticalfield is only red-detuned by one linewidth, and the at-oms must therefore be excited at very long range, near1800 a0 . The collision travel time to the close internu-clear separation where associative ionization takes placeis long compared to the radiative lifetime, and most ofthe population decays to the ground state before reach-


ing the autoionization zone. During the trap cycle, how-ever, the excitation takes place at much closer internu-clear distances due to a 70-linewidth red detuning andhigh-intensity field dressing. Therefore one might expectexcitation survival to be better during the trap cycle thanduring the molasses cycle, and the NIST group, led byW. D. Phillips, set up an experiment to test the predictedcycle dependence of the ion rate.

Lett et al. (1991) performed a new experiment usingthe same hybrid trap. This time however, the experi-ment measured ion rates and fluorescence separately asthe hybrid trap oscillated between ‘‘trap’’ and ‘‘molas-ses’’ cycles. The results from this experiment are shownin Fig. 51. While keeping the total number and densityof atoms (excited atoms plus ground state atoms) essen-tially the same over the two cycles and while the excitedstate fraction changed only by about a factor of two, theion rate increased in the trapping cycle by factors rang-ing from 20 to 200 with most observations falling be-tween 40 and 100. This verified the predicted effectqualitatively even if the magnitude was smaller than theestimated 104 factor of Julienne (1988). This modulationratio is orders of magnitude more than would be ex-pected if excited atoms were the origin of the associativeionization signal. Furthermore, by detuning the trappinglasers over 4 GHz to the red, Lett et al. (1991) continuedto measure ion production at rates comparable to thosemeasured near the atomic resonance. At such large de-tunings reduction in atomic excited state populationwould have led to reductions in ion rate by over fourorders of magnitude had the excited atoms been the ori-gin of the collisional ionization. Not only did far off-resonance trap cycle detuning maintain the ion produc-tion rate, Lett et al. (1991) observed evidence of peakstructure in the ion signal as the dipole trap cycle de-tuned to the red.

To interpret the experiment of Lett et al. (1991), Juli-enne and Heather (1991) proposed a mechanism whichhas become the standard picture for cold and ultracoldphotoassociative ionization. Figure 52 details the model.Two colliding atoms approach on the molecular ground-state potential. During the molasses cycle with the opti-cal fields detuned only about one linewidth to the red ofatomic resonance, the initial excitation occurs at verylong range, around a Condon point at 1800 a0 . A secondCondon point at 1000 a0 takes the population to a 1udoubly excited potential that, at shorter internuclear dis-tance, joins adiabatically to a 3(u

1 potential, thought tobe the principal short-range entrance channel to associa-tive ionization (Dulieu et al., 1991; Henriet et al., 1991).More recent calculations suggest other entrance chan-nels are important as well (Dulieu et al. 1994). The long-range optical coupling to excited potentials in regionswith little curvature implies that spontaneous radiativerelaxation will depopulate these channels before the ap-proaching partners reach the region of small internu-clear separation where associative ionization takesplace. The overall probability for collisional ionizationduring the molasses cycle remains therefore quite low.In contrast, during the trap cycle the optical fields are


detuned 60 linewidths to the red of resonance, the firstCondon point occurs at 450 a0 ; and, if the trap cyclefield couples to the 0g

2 long-range molecular state(Stwalley et al., 1978), the second Condon point occursat 60a0 . Survival against radiative relaxation improvesgreatly because the optical coupling occurs at muchshorter range where excited-state potential curvature ac-celerates the two atoms together. Julienne and Heather(1991) calculate about a three-order-of-magnitude en-hancement in the rate constant for collisional ionizationduring the trap cycle. The dashed and solid arrows inFig. 52 indicate the molasses cycle and trap cycle path-ways, respectively. The strong collisional ionization rateconstant enhancement in the trap cycle calculated byJulienne and Heather (1991) is roughly consistent withthe measurements of Lett et al. (1991), although the cal-culated modulation ratio is somewhat greater than whatwas actually observed. Furthermore, Julienne andHeather calculate structure in the trap detuning spec-trum. As the optical fields in the dipole trap tune to thered, a rather congested series of ion peaks appear whichJulienne and Heather ascribed to free-bound associationresonances corresponding to vibration-rotation boundlevels in the 1g or the 0g

2 molecular excited states. Thedensity of peaks corresponded roughly to what Lettet al. (1991) had observed; these two tentative findingstogether were the first evidences of a new photoassocia-tion spectroscopy. In a subsequent full paper expandingon their earlier report, Heather and Julienne (1993) in-troduced the term, ‘‘photoassociative ionization’’ to dis-tinguish the two-step optical excitation of the quasimol-ecule from the conventional associative ionizationcollision between excited atomic states. In a very recentpaper, Pillet et al. (1997) have developed a perturbativequantum approach to the theory of photoassociationwhich can be applied to the whole family of alkali homo-nuclear molecules. This study presents a useful table ofphotoassociation rates which reveals an important trendtoward lower rates of molecule formation as the alkali

FIG. 50. Time-of-flight spectrum clearly showing that the ionsdetected are Na2

1 and not the atomic ion. From Gould et al.(1988).


mass increases and provides a helpful guide to experi-ments designed to detect ultracold molecule production.

1. Photoassociative ionization at small detuning

When near-resonant light couples ground and excitedmolecular states of colliding atoms, spontaneous emis-sion occurs during the collision with high probability.Collision problems involving strong coupling to one ormore optical field modes and with radiative dissipationcan only be properly treated by solving the Liouvilleequation of motion for the quantum density matrix ofparticles plus field,

d

dtr~ t !5

1i\

@H ,r~ t !# , (116)

where r(t) is the density matrix describing the states ofthe quasimolecule and the radiation field and H5Hmolec1Hfield1H int is the Hamiltonian for the systemof quasimolecule, radiation field, and the coupling be-tween the two. Photoassociative ionization during themolasses cycle of the NIST hybrid trap or within theconfines of a magneto-optical trap correspond to thissituation. Bagnato, Marcassa, Wang, et al. (1993), there-fore, undertook to investigate the photoassociative ion-ization (PAI) rate constant in a sodium MOT as a func-tion of MOT optical field intensity. The essential ideawas to measure the dependence of the PAI rate constanton MOT intensity and compare it to rate constants cal-culated from an optical-Bloch-equation theory, based onEq. (116), proposed earlier by Band and Julienne(1992). Bagnato, Marcassa, Wang et al. (1993) also com-pared the experimental results to the Gallagher-Pritchard model or ‘‘local equilibrium,’’ model discussedat length in Sec. V. The results are shown in Fig. 53. Thesemiclassical optical-Bloch-equation theory agrees wellwith the experiment at the low end of the intensity scale,but the measured rate constants appear to increase lessrapidly with increasing intensity than the calculationsand to flatten around 100 mW cm22. The local equilib-

FIG. 51. Trap modulation experiment showing much greaterdepth of ion intensity modulation (by more than one order ofmagnitude) than fluorescence or atom number modulation,demonstrating that excited atoms are not the origin of the as-sociative ionizing collisions. From Lett et al. (1991).


rium model builds on the quasistatic theory of collisionbroadening, treats the molecule-field interaction pertur-batively, and does not attempt to solve a density matrixequation of motion. The term ‘‘local equilibrium’’ comesfrom the characteristic of this theory that, at any Con-don point, the detuning dependence of the optical exci-tation takes the form of a Lorentzian line shape. Thekey assumption, therefore, of the local equilibriummodel is that at any internuclear separation the quasi-molecule has time to reach ‘‘equilibrium’’ with themodes of the radiation field. In both the optical-Bloch-equation and local equilibrium models, calculation ofthe rate constant requires determining two probabilities:first the probability that a two-step optical excitation-survival process (first to the 1g or 0g

2 , then to the 1u)brings the two atoms together near R;6a0 on the dou-bly excited autoionizing 3Su

1 potential; second, theprobability that the autoionization takes place. Bothmodels require the molecular autoionization probabilityand rely on measurements reported in the literature(Huennekens and Gallagher, 1983; Bonanno et al.,1983). The two models differ significantly in the calcula-tion of the first step, the optical excitation probability.Figure 53 shows that the local equilibrium theory over-estimates the PAI rates by about one order of magni-tude. Neither theory takes into account the extensivemolecular hyperfine structure of the excited states, andthe evidence from trap loss experiments (see Sec. V)indicates that models of collision dynamics at small de-tuning that ignore molecular excited-state hyperfinestructure usually fail.

Recently Blange et al. (1997) have performed photo-dissociation fragmentation spectroscopy on the Na2

1 ionsproduced by PAI in a conventional Na MOT. They suc-ceeded in determining the internal vibrational level dis-tribution of Na2

1 produced by the cold collision; authorsinterpreted the distribution to infer the mechanism forthis PAI process.

2. PAI and molecular hyperfine structure

Exploring the off-resonance detuning dependence ofPAI, Wagshul et al. (1993) discovered important effects

FIG. 52. Photoassociative ionization in Na collisions. FromHeather and Julienne (1993).


of molecular ground-state hyperfine structure. With aMOT loaded from a Zeeman-slowed atomic beam, to-gether with an extra probe laser, Wagshul et al. alter-nated trapping and probing with a 50% duty cycle. Trapon/off periods varied from 2 to 100 msec, and ions werecounted only during the probe period. In this scheme alarge range of probe laser detuning could be exploredwithout the complicating effects of the MOT fields (trapand repumper) and without compromising the trap per-formance. Total probe laser intensity (;3 W cm22) wasmuch greater than the trapping laser (;35 mW cm22)but had a similar geometrical configuration and polariza-tion to minimize perturbations on the MOT. Figure 54shows the spectrum obtained by Wagshul et al. (1993) inthe region between the two hyperfine ground state tran-sitions, F52→F853 and F51→F852. A broad reso-nance structure rises sharply about halfway between thetwo ground-state hyperfine transitions and extends tothe blue. The quadratic dependence of the amplitude ofthis feature with the probe laser intensity is consistentwith a two-photon process. Observing variations in thepeak intensity as the relative population between thetwo ground-state hyperfine levels were varied, Wagshulet al. concluded that the observed spectrum arises fromthe collision of two ground-state atoms, each of which isin a different ground-state hyperfine level. The interpre-tation of Fig. 54 can be understood in the following way.Atoms in different hyperfine ground states collide onthe potential labeled F511F52. The photon requiredfor a two-step excitation to the doubly excited level fallshalfway between the potentials labeled 11P and 21P . Where the attractive 21P potential crosses thisline, the transition becomes doubly resonant. The ob-served threshold for this process is within 10 MHz of theprediction. Light to the red of this frequency cannot pro-vide enough energy to reach the P1P potential startingfrom the F511F52 asymptote of the ground state.Light to the blue of the threshold frequency follows thefollowing route. First the optical field couples to the 21P curve at larger internuclear separation. The atomsthen begin to accelerate together on the 21P potentialuntil they reach a Condon point where the exciting lightbecomes resonant with a transition to the P1P curve.This broad PAI feature shows substructure consistentwith free-bound transitions to an intermediate attractivestate. Wagshul et al. (1993) carried out a model calcula-tion of the PAI feature by applying their earlier two-stepmodel (Julienne and Heather, 1991; Heather and Juli-enne, 1993) and incorporating the ground-state hyper-fine levels. The calculated structure resembles the ex-perimental feature, but appears to be less congested;This is probably because the model neglects the multi-tudinous number of excited-state hyperfine levels.

3. Two-color PAI

The work of Wagshul et al. (1993) brought to light theimportance of ground-state hyperfine structure in earlystudies of PAI spectroscopy. They used a one-colorprobe to excite the two-step PAI process. Bagnato, Mar-


cassa, Tsao, et al. (1993) investigated the same PAI pro-cesses, using two different laser frequencies, one ofwhich could be independently varied, and starting withdifferent initial states for the colliding atoms. Atomicsodium was cooled and trapped in a MOT loaded fromthe low-velocity tail of background vapor. The trappingfrequency v1 was red detuned as usual about one line-width from cycling transition 3s2S1/2(F52)→3p2P3/2(F853), and the repumper v2 was tunedclose to 3s2S1/2(F51)→3p2P3/2(F852). Depending ontheir relative intensity, these two frequencies couldpump different relative populations into the two ground-state hyperfine levels F51 and F52. A channeltron de-tector measured the PAI ion signal as a function of thefrequency of a third, tunable probe laser vp . Figures55(a) and 55(b) show the PAI profiles as vp scans to theblue of v1 and v2 . In Fig. 55(a) the intensity of v1 isstronger than v2 by a factor of three, and the insetabove the line profiles shows the interpretation of theexcitation routes. Both atomic hyperfine levels have sig-nificant population so v1 and v2 can couple all threemolecular entrance channels (111, 112, 212) to the11P and 21P attractive excited-state levels. Then vpscanning to the blue couples these two singly excitedlevels to the doubly excited autoionizing P1P curve. Totest this interpretation Bagnato, Marcassa, Tsao, et al.(1993) reversed the relative intensities in v1 and v2 soas to pump most of the atomic population to the F52level. The effect is to depopulate the 111 molecularentrance channel, and panel (b) of Fig. 55 shows that theamplitude of the PAI profile degrading to the blue of v2is greatly attenuated, consistent with this entrance chan-nel depopulation. A common feature of all the measure-ments is the asymmetry in the PAI profile. As the probelaser scans to the blue, starting from the atomic transi-tions near v1 and v2 , the PAI signal rises rapidly to apeak ;400 MHz detuning from the atomic limit, thenexhibits a slow decay over a range of about 2 GHz. Theasymmetric shape of this two-color PAI spectrum wasmodeled by Gallagher (1991) and arises from the actionof four factors: (1) first excitation from the ground stateat the initial Condon point Ri , (2) spontaneous emissionas the atoms approach on the first attractive excitedstate until they reach the second Condon Point R2 , (3)

FIG. 53. Photoassociative ionization rate constant as a func-tion of MOT light intensity. MOT laser detuned to the red ofresonance about one natural linewidth. From Bagnato, Mar-cassa, Wang, et al. (1993).


second excitation from the first attractive state to thedoubly excited state at R2 , and (4) spontaneous emis-sion at the atoms approach the autoionization zone onthe doubly excited state. Once the atoms are withinRPAI , the zone where the quasimolecule can autoionize,the four excitation-decay factors must be multiplied bythe autoionizing probability in order to calculate anoverall ion production rate. Detuning of vp to the blueeffectively increases the time between the first and sec-ond optical excitations. For short times (small blue de-tunings) the initially excited population does not havetime to undergo appreciable radiative decay before ex-citation to the doubly excited level, yet the strong accel-erating influence of the attractive C3 /R3 potential short-ens the travel time between R2 and RPAI where theatoms enter the autoionizing region. These two factorsaccount for the rapid rise in PAI rate at small blue de-tunings. As vp continues to scan to the blue, radiativedepopulation of the first excited state starts to becomesignificant and the overall PAI probability diminishesslowly. For atoms surviving to R2 on the first excitedstate, the increased radial acceleration tends to minimizeradiative decay on the doubly excited state once thepopulation undergoes the second excitation at R2 . How-ever, the probability of that excitation begins to diminishwith increased acceleration because the atoms passthrough the optical resonance around the Condon pointR2 with greater velocity. The overall effect is a slowdiminution of the PAI production rate as vp scans fur-ther to the blue. Although the asymmetric shape calcu-lated by Gallagher is in satisfactory accord with the mea-surements of Bagnato, Marcassa, Tsao, et al. (1993) theabsolute values of the rate constants differ by about anorder of magnitude. Furthermore the position of thepeak in the measured PAI profile is blue-shifted fromGallagher’s model by about 200 MHz. These discrepan-cies could be accounted for by effects discussed previ-ously in the section on trap loss, specifically (1) neglectof hyperfine structure and (2) semiclassical treatment atsmall detuning.

FIG. 54. Two-photon PAI spectrum arising from collisions ondifferent ground-state hyperfine levels. From Wagshul et al.(1993).


Bagnato, Marcassa, Tsao, et al. (1993) also investi-gated two-color PAI by prepumping all the atoms intoeither the F51 or the F52 and probing with only twofrequencies: vp ,v1 or vp ,v2 . Figures 56(a) and 56(b)show the results with insets of the excitation routes.Panel (a) shows the spectrum with two principal featureswhen only F51 is populated so that incoming scatteringflux appears only on the 111 asymptote. The onset ofthe right-most features occurs at about twice theground-state hyperfine splitting to the blue of v1 . Theasymmetric shape degraded to the blue is characteristicof initial excitation to an attractive state at long range byfixed frequency v1 followed by a second excitation tothe doubly excited level by scanning frequency vp . Theright inset above the feature shows the excitation path-way. It appears that the profile has a structure that mustcome from bound states of the doubly excited manifold.This structure remains to be investigated thoroughly.The central peak, immediately to the blue of v2 , is qua-dratic in the probe laser intensity, suggesting a two-photon single frequency process similar to that observedby Wagshul et al. (1993). The inset shows that vp carriespopulation to some mixed state of 11P and 21P, and asecond photon takes the population to the doubly ex-cited level. Panel b shows the spectrum obtained whenonly F52 is populated. Incoming scattering flux appearsonly on the 212 asymptote, and the onset of the featurebegins about 250 MHz to the red of v1 and degrades tothe red over a range of about 2 GHz. Structure is clearlyvisible and the inset above the feature shows that thescanning frequency vp excites vibration-rotation levelsassociated with a bound singly excited states while v2transfers population to the doubly excited level. Notethat the left and right insets show how the fixed andscanning frequencies exchange roles. A PAI profile de-grading to the blue means vp couples the strongly at-tractive, singly excited state (or states) to the relativelyflat, doubly excited level above it. A profile degrading tothe red implies vp coupling from the strongly attractivemanifold to the flat 212 entrance channel below it. Thefixed frequency v1 or v2 determines where on the abso-lute frequency scale these two-step excitation profilesappear. The red-degraded, structured profile revealbound states of the singly excited manifold very close tothe dissociation limit where molecular hyperfine cou-pling produces a complicated and congested spectrum.This structure was analyzed by Bagnato et al. (1994) tobe associated with a 0g

2 bound vibrational level progres-sion, but extensive hyperfine mixing near dissociationrender these symmetry labels ambiguous. Further prob-ing to the red of the hyperfine region (Ratliff et al.,1994) reveals a much simpler, highly resolved spectrum,opening the very fruitful field of precision PAI spectros-copy, reviewed in the next section.

In very recent work, Jones et al. (1997) have measuredtwo-color spectra in which the first step populates a spe-cific state in the 0g

2 or the 1g Na2 molecular electronicpotentials. Then a second high-resolution light beam


scans transitions either upward leading to autoionizingpotentials near the 3P13P doubly excited asymptote ordownward back to the manifold of potentials associatedwith the Na2 ground electronic state. This study obtainsspectroscopic constants for several levels of a doubly ex-cited 1u state, demonstrates the appearance of a newform of Condon internal diffraction from bound→freetransitions just above the 3P3/213P3/2 dissociation limit,and discusses the use of photoassociation as a source ofcold molecules. This work pioneers the high-resolutionphotoassociation spectroscopy of the doubly excitedstates of Na2 and begins to unravel the similarities anddifferences between the mechanisms of associative ion-ization at high temperature and photoassociative ioniza-tion in the ultracold regime.

In a parallel theoretical study, Dulieu et al. (1997)have shown that five molecular symmetries, 1Sg

1 , 3Su1 ,

1Pu , 3Pu , 1Dg of the doubly excited states are likely tocontribute to the molecular autoionization. They iden-tify the peaks observed by Jones et al. (1997) and attrib-uted to a doubly excited 1u state (Hund’s case (c) cou-pling), as belonging to the 1Pu member (Hund’s case(a) coupling) of the five molecular symmetries. Exactlyhow the short range 1Pu joins to the long range 1u isstill the subject of continuing investigation (Huynh et al.,1997). It appears, however, that the autoionization rateof this 1Pu state precludes it from significantly contrib-uting to the associative ionization process at thermal en-ergies.

FIG. 55. Two-color photoassociative ionization. Inset showsMOT (v1) and repumper (v2) fixed frequencies and sweepingprobe (vp) frequency. From Bagnato, Marcassa, Tsao et al.(1993).


D. Photoassociation spectroscopy in MOTs and FORTs

Although early cold alkali photoassociative ionizationexperiments explored different excitation and ionizationpathways and confirmed the essential mechanisms of theprocess, the full potential of photoassociation as a ver-satile and powerful spectroscopic technique of unprec-edented precision could not be realized until scanningprobe fields could explore beyond the hyperfine cou-pling zone of irregular and congested spectra. Two back-to-back articles appearing in Physical Review Letterssignaled a breakthrough in measuring photoassociationspectra well beyond this region. The NIST group contin-ued to investigate photoassociative ionization sodium,first with a strong probe laser and a conventional MOT(Lett et al., 1993), then with a ‘‘dark spot’’ MOT whichincreased their ionization signal about a hundred fold(Ratliff et al., 1994). In contrast the University of Texasgroup, led by D. J. Heinzen, used a far-off-resonanceoptical trap (FORT) to capture rubidium atoms (Milleret al., 1993a) and to measure photoassociation spectraby measuring fluorescence loss from the trap as a func-tion of trap frequency (Miller et al., 1993b). These twoexperiments opened a new domain of precision spectros-copy from which molecular potentials, scattering dynam-ics, atomic lifetimes, and scattering lengths have all beendetermined with an accuracy unattainable by conven-tional methods. An excellent review of PAI has recentlyappeared (Lett et al., 1996), lucidly explaining develop-ments up to about mid-1995.

1. Sodium

The NIST group first reported a PAI spectrum (Lettet al., 1993) showing regular vibrational progressions byscanning a probe laser to the red of the MOT cyclingtransition. The setup used the same conventional MOTand dedicated, retroreflected probe laser source thatWagshul et al. (1993) employed to study hyperfine ef-

FIG. 56. Two-color photoassociative ionization. Insets abovefeatures describe excitation routes. Frequencies are labeled asin the previous figure. From Bagnato, Marcassa, Tsao, et al.(1993).


fects. The PAI spectrum showed two regular progres-sions, one strong series ranging from about 5 to 100 GHzto the red of the MOT cycling transition and a secondweaker series discernible from 15 to 40 GHz red detun-ing. These regular features are the vibrational series cor-responding to long-range molecular potentials of the sin-gly excited Na2 molecule. The C3 /R3 form of theselong-range potentials arises from the resonant dipole-dipole interaction between an excited Na* (3p) atomand a ground state Na(3s) atom. Four optically activeexcited molecular potentials supporting vibrational lev-els and correlating to the 2S1/21

2P3/2 asymptote are can-didates for the observed regular progressions. In Hund’scase (c) coupling these states are labeled 1u , 0g

2 , 1g ,and 0u

1 . They will have different effective C3 coeffi-cients and their vibrational progressions will have differ-ent hyperfine substructure. Both of these characteristicsserve to identify states to which the spectra belong.

LeRoy and Bernstein (1970) and Stwalley (1970) haveshown that in this long-range regime the spacing be-tween vibrational levels can be described analytically by

dE~v !

dv5\S 2p

m D G~4/3!

G~5/6!

3

C31/3 @D2E~v !#5/6, (117)

which integrates to

Ev2D5S a

C32D ~v2vd!6, (118)

where v is the vibrational quantum number countedfrom the top of the well, and vd is the effective quantumnumber at the apparent dissociation. The number vd is afraction between 0 and 1. The first bound vibrationallevel below dissociation is always labeled v51. The termD denotes the dissociation energy, and a is a collectionof constants including the reduced mass, m. The NISTgroup fit a plot of (Ev2D)1/6 vs (v2vd) to the centroidsof the peaks for the strong and weak series. From theslope they determined C3510.3760.07 a.u. for thestrong series and C3510.860.7 a.u. for the weak series.The calculated Hund’s case (c) C3 constants for the op-tically active excited states clearly narrowed the possibleidentification of the strong and weak series to 1g and0u

1 . The measured peak widths, due to hyperfine sub-structure, were ;1 GHz, consistent with calculations for1g but not for 0u

1 . According to Williams and Julienne(1994), the 0u

1 state should show about one third thehyperfine substructure as the 1g state. Therefore, boththe strong and weak series must belong to the 1g . Sincethe apparent dissociation energy determined from Eq.(118) for the two series differs by about the ground-statehyperfine splitting, Lett et al. (1993) concluded that theorigin of the strong and weak progressions arose from212 collisions and 211 collisions, respectively. How-ever, one may ask, Why is it that the average measuredC3 lies near 10.6 a.u. with uncertainty of about 0.7%while the best C3 value for the 1g state is 9.55 a.u., adifference of about 8%? The reason is that the 1g po-tential originates from diagonalizing a 333 potentialmatrix, and C3 is only independent of R in the region of


internuclear separation where the binding energies ofthe states is much less than the atomic fine structuresplitting, i.e., C3 /R3!DEFS . In this range of R the C3term is independent of R . At shorter internuclear sepa-ration the diagonalized adiabatic states will fit to C3 con-stants that depend on the range of R over which thepotential is sampled. Williams and Julienne (1994) re-port that fitting vibrational eigenvalues obtained fromadiabatic electronic-rotational-hyperfine molecular po-tentials to Eq. (118) yields a C3 in agreement with themeasured value of the strong series within experimentalerror as long as the measured and calculated values spanthe same range in the progression. The C3 extracted fromsuch fits can be used to obtain a precision measurementof the excited atom lifetime. Section VI.E reviews pre-cision measurement of excited state lifetimes.

Ratliff et al. (1994) improved the NIST experiment bysubstituting a ‘‘dark spot’’ MOT (Ketterle et al., 1993)for the conventional MOT, increasing the ion signal bytwo orders of magnitude and permitting a probe scanover 172 cm21 to the red of the trapping transition. Fig-ure 57 shows the full range of this photoassociative ion-ization spectrum. These new results revealed the 1g ,0u

1 , and 0g2 states, identified by the energy range over

which they were observed, vibrational and rotationalspacing, and the extent of the hyperfine structure. Fur-thermore, alternately chopping the trap and probe lasersat 50 KHz while slowly scanning the probe permittedtwo kinds of measurements: (1) trap-loss spectra duringthe trap phase, as measured by ions produced by thetrapping lasers, and (2) ionization production during theprobe phase. Figure 58 shows how these two comple-mentary spectra helped distinguish the various progres-sions. The long progression of the 1g state shows clearlythe change of coupling regime from Hund’s case (c) toHund’s case (a) at closer range. Hund’s case (a) labelsthe state 1Pg and the vibrational lines split into rota-tional progressions. The molecular transition giving riseto this progression is 3Su→1g (or 1Pg at close range);however, the triplet character diminishes as the couplingscheme transforms from Hund’s case (c) to (a), and thenatural line width diminishes to less than 0.1 MHz.Therefore at closer range (detunings>50 cm21) thespectral resolution is limited only by the probe laserwidth (;2 MHz) permitting analysis of the line profilesof the resolved rotational progressions.

The individual rotational line profiles show structureand asymmetries due to the finite temperature of theMOT, residual hyperfine splitting and quantum thresh-old effects. Figure 59 shows the measured rotational lineshapes obtained by Ratliff et al. (1994) for the v548vibrational level of the 1g state. Evidence of hyperfinecontributions can be seen in the incompletely resolvedsubstructure of J51 and J52 profiles. An analysis ofthese rotational line shapes have been carried out byNapolitano et al. (1994), the results of which are shownin Fig. 60. The intensity of the photoassociation spec-trum is proportional to the thermally averaged rate co-efficient. The quantum scattering expression for the rate


coefficient in terms of the scattering matrix S and thescattering partial waves l is written as

Kp~T ,v!5kB

hQT(l50

`

~2l11 !

3E0

`

uSp~E ,l ,v!u2e2E/kBTdE

kBT, (119)

where Sp(E ,l ,v) is the S-matrix element for productchannel p , QT5(2p mkBT/h2)3/2 is the translationalpartition function, kB is Boltzmann’s constant, E5\k2/2m is the asymptotic kinetic energy of the collid-ing ground state 2S atoms, and the other terms in theexpression have their usual meanings. At relatively largedetunings where the vibration-rotation lines are isolatedfrom each other, Napolitano et al. (1994) used quantumscattering calculations with a complex potential to showthat the form of Sp(E ,l ,v) around a scattering reso-nance can be modeled as

uSp~E ,l ,v!u25GpGs~E ,l !

~E2Db!21~G/2!2 , (120)

where Db is the probe laser detuning from the boundrovibronic state of the excited level of the free-boundtransition, Gp is the decay width to the collision product(whatever that decay process might be) and Gs(E ,l) isthe stimulated rate connecting the ground and excitedlevels. The G term in the denominator is the sum of Gpand Gs , plus any other decay processes that affect thelevel in question. Since Gs52pPge(E), where Pge(E) isdefined by Eq. (28), Gs exhibits the Wigner-law thresh-old behavior due to the the quantum properties of theground-state wave function as described in Sec. III.B.This threshold behavior has the form of Gs'Al E(2l11)/2

as E→0. Three partial waves, s ,p ,d contribute to thescattering amplitude of the J-levels of Fig. 60; their in-fluence can most easily be seen on the blue (right-hand)slope of each profile. Note especially in the J52 panelof Fig. 60 the very sharp onset to the profile, illustratingthe influence of the Wigner threshold law for the s-wavecontribution (close to infinite slope as E→0). On thered side of the profile, the intensity falls off as e2Db /kBT

and at first thought one might expect to extract a colli-sion temperature from the slope of a semilog plot. How-ever, the s ,p ,d contributions to the line shapes all yieldsomewhat different effective temperatures, again due tothe influence of the Wigner threshold law operating onGs . In order to extract a meaningful temperature fromthe experimental line shape therefore the fractional par-tial wave contributions to the profile must be known. Itis also worth noting that the particular admixture of par-tial waves at a particular temperature will shift the peakpositions of the lines, and the line ‘‘position’’ must befound by modeling the onset on the blue side of theprofile (Jones, Julienne, et al., 1996).

Recently, a combination of high resolution photoasso-ciative spectroscopy of cold Na together with conven-tional molecular spectroscopy using atomic beams andheat pipe, permitted Jones, Maleki, et al. (1996) to per-


form a direct measurement of the ground-state dissocia-tion energy (D0) of Na2 with unprecedented precision.Combining three spectroscopic measurements, Jones,Maleki, et al. (1996) obtained D0 for the X 1(g

1 groundstate as D055942.6880(49) cm21, making this the mostaccurately known dissociation energy of any chemicallybound neutral molecule. This result demonstrates howcold photoassociation spectroscopy can complementconventional spectroscopy with a considerable increasein resolution.

Additional two-color experiments have revealed otherinteresting details about the PAI process. Molenaaret al. (1996) have reported a two-color PAI spectrum ina Na MOT in which they determine high-lying vibra-tional levels of the 0g

2 and observe a predissociation ofthis state at a curve-crossing interaction between the 0g

2

and the ground-state 2S1/2 (F51) hyperfine level. Simi-lar spectra near the 0g

2 dissociation have been reportedby Bagnato et al. (1994).

2. Rubidium

Although the atom trap most commonly used for col-lision studies has been the MOT, Miller et al. (1993a)pioneered the use of the far-off-resonance trap (FORT)to study photoassociation experiments in Rb. The ad-vantages of the FORT over the MOT are (1) relativesimplicity (no magnetic fields), (2) very low excited statepopulation and therefore little diffusional heating, (3)high density (1012 cm23) and (4) a well-defined polariza-tion axis along which atoms can be aligned or oriented.The disadvantage is the relatively low number of atoms(;104) trapped in the small volume (;531029 cm3).In the experiment of Miller et al. (1993b), the FORT wasloaded from a MOT by spatially superposing the fo-cused FORT laser on the MOT and alternately chop-ping the two kinds of traps. At the end of the load se-quence fluorescence detection probes the number ofatoms in the FORT. In the first version of the experi-ment, the FORT laser itself was swept in frequency from50 cm21 to 980 cm21 to the red of the atomic resonance.As the laser scanned through the bound state reso-

FIG. 57. Photoassociative ionization spectrum of Na collisionsfrom the NIST dark-spot MOT. From Ratliff et al. (1994).


nances, the free-bound association peaks were detectedby measuring the resulting loss of atoms from theFORT. Figure 61 shows the results of the trap-loss spec-tra. The increase of the background intensity as the dis-sociation limit is approached represents an increase inthe Franck-Condon overlap and is also a consequence ofthe increase in the atomic density in the trap. The solidand dashed vertical lines on the top of the spectra of Fig.61 identify two vibrational series. Comparison of theseseries with energy eigenvalues of model potentials byMovre and Pichler (1977), Krauss and Stevens (1990),and Bussery and Aubert-Frecon (1985a, 1985b) identifythem as the 1g and 0g

2 states. A noticeable feature in thespectrum obtained by Miller et al. (1993b) is the slowoscillation in the intensity of the lines. Thorsheim et al.(1987) predicted these Condon modulations. They arisefrom the Franck-Condon overlap of the ground- andexcited-state wave functions, and the oscillations reflectthe nodes and antinodes of the scattering wave functionon the entrance ground-state channel. This is a conse-quence of the reflection principle in Eq. (32). Changing

FIG. 58. Top spectrum shows trap-loss dips as probe laserscans through molecular rotational progressions. Bottom ionspectrum shows complementary photoassociative ionizationsignals. From Ratliff et al. (1994).

FIG. 59. High resolution photoassociative ionization spectrumof rotational levels associated with v548 of the long-range 1g

state of Na2 . Note residual hyperfine splitting in the J52 inset.From Ratliff et al. (1994).


the excitation frequency changes the Condon point RCso that the spectrum tracks the shape of the ground-statewave function.

In a second version of this experiment Cline et al.(1994a, 1994b) separated the trapping and scanningfunctions with two different lasers: one laser at fixed fre-quency produced the FORT while a probe laser scannedthrough the resonances. The two lasers, probe andFORT, were alternately chopped to avoid spectral dis-tortion from light shifts and power broadening. Milleret al. (1993b) restricted themselves to a spectral regionfrom 50 cm21 to 980 cm21 below the 2S1/21

2P1/2 disso-ciation limit of Rb2. The two-color experiment (Clineet al., 1994a, 1994b) enabled a greatly increased spectralrange and the identification of the 0g

2 ‘‘pure-long-range’’state, the 1g , and the 0u

1 . The high-resolution photoas-sociation spectrum is shown in Fig. 62. From this high-resolution spectra Cline et al. (1994a, 1994b) determinedthe C3 coefficient for two molecular excited states ofRb2. The values are: C3(0u

1)5(14.6460.7) a.u. andC3(1g)5(14.2960.7) a.u. The spectra also show thatthe 0u

1 lines are broadened by predissociation to the52S1/2152P1/2 asymptote from spin-orbit mixing of the0u

1 components of A1(u1 and b3Pu at short range. This

broadening is evidence of the fine-structure-chargingcollisions trap-loss process discussed in Sec. V. Recentlythe Texas group, in collaboration with the Eindhoventheory group, extended their experiments to a two-colorphotoassociation spectroscopy in which the second colorcouples population from the initially excited 0g

2 to thelast 12 vibrational levels of the 85Rb2 ground state (Tsaiet al., 1997). This high-precision spectroscopic data per-mits accurate determination of interaction parametersand collision properties of Rb ultracold collisions includ-ing the scattering length and the position of Feshbachresonances (see Sec. VIII.B), both of which are very im-portant for Bose-Einstein condensation in ultracold Rb.

FIG. 60. High resolution line profile measurements and theoryon rotational progression associated with v548 of the long-range excited 1g state of Na2 . From Napolitano et al. (1994).


As an alternative to trap-loss spectroscopy Leonhardtand Weiner (1995, 1996) investigated Rb2 photoassocia-tion spectra by direct photoassociative ionization. Figure63(b) illustrates the main steps in this experiment. At-oms held in a vapor-cell-loaded MOT are illuminatedwith two light beams: a probe beam vp scans throughthe resonances of the singly excited Rb2 long-range po-tential while a second, transfer laser at fixed frequencyv t directly photoionizes the bound excited molecule tohigh-lying vibrational states of Rb2

1 . This direct two-color photoassociative ionization process does not com-pete with an open autoionizing channel, as in sodium[Fig. 63(a)], because the manifold of doubly excitedstates P1P lies below the minimum in the dimer ionpotential. Sweeping the probe laser to the red of theatomic resonance and counting the ions, Leonhardt andWeiner (1995, 1996) produced the photoassociativespectrum shown in Fig. 64 where the FORT trap-lossspectrum obtained by Cline et al. (1994a, 1994b) hasbeen added for comparison. The direct Rb2 PAI spec-trum appears to show only one molecular state progres-sion. In fact, comparison with the FORT experimentshows that the 0g

2 progression is absent in the photoas-sociative ionization spectrum. The measured series is as-signed to the 1g state and the determination of the po-tential constant resulted in C3(1g)5(14.4360.23) a.u.in good agreement with the value determined by Clineet al. (1994a, 1994b). Direct PAI should be widely appli-cable to many trappable species, but requires the densi-ties of a dark spot MOT or a FORT to compete effec-tively with fluorescence trap-loss spectroscopy.

3. Lithium

The Rice University group, lead by R. G. Hulet, hasperformed a series of experiments in an atomic-beam-loaded MOT of 6Li and 7Li. Photoassociation trap-lossfluorescence spectroscopy has been carried out on bothisotopes and the resulting data has been used to deter-mine C3 coefficients, atomic lifetimes, and scatteringlengths. The Rice group reported the basic spectroscopy,experimental technique, and analysis late in 1995 (Abra-ham, Ritchie, et al., 1995). Earlier in the year the samegroup determined a precise atomic radiative lifetime forthe Li (2p) level using the same photoassociation spec-tral data (McAlexander et al., 1995). A sample of the6Li2 spectrum reported by McAlexander et al. (1995) isshown in Fig. 65. Indicated in the figure are the vibronicseries for the 11Su

1 and 13Sg1 states which converge on

the 2s2S1/212p2P1/2 asymptotic limit. Three rotationallines (N50, 1, and 2) are resolvable in the 13Sg

1 seriesas shown in the inset of Fig. 65. The N50 line is used forthe series analysis to construct part of the 13Sg

1 interac-tion potential. The basic approach is to fit together amolecular potential at short, medium, and long rangefrom various sources of calculated potentials and spec-tral data. At long range beyond 20a0 the potential isapproximated analytically by the form


V.2C3

R3 2C6

R6 2C8

R8 , (121)

where C3 is due to the dominant resonant dipole inter-action, and the other two dispersion terms are quitesmall compared to the dipole term. At very short range,less than 4.7a0 , and then from 7.8a0 to 20a0 the poten-tial uses ab initio calculations. In the region between4.7a0 and 7.8a0 conventional spectroscopy yields an ac-curate Rydberg-Klein-Rees (RKR) potential. The dif-ferent potential segments are then pieced together witha cubic spline and the spectral positions are calculatedwhile C3 is varied as a free parameter until the best fit isobtained. McAlexander et al. (1995) finally report aC3(13Sg

1)511.04860.066 a.u. This C3 is then used tocalculate a lifetime for the Li (2p) level. Deriving life-times from photoassociation spectra will be discussed inSec. VI.E. In more recent work, Abraham, McAlex-ander, Stoof, and Hulet (1996) reported spectra of hy-perfine resolved vibrational levels of the 1Su

1 and 3Sg1

states of both lithium isotopes, obtained via photoasso-ciation. A simple first-order perturbation theory analysisaccounts for the frequency splittings and relative transi-tion strengths of the observed hyperfine features.

4. Potassium

The research groups of Stwalley and Gould at theUniversity of Connecticut have joined forces to mount acomprehensive study of the photoassociation spectros-copy in ultracold 39K collisions. Implementation of aMOT in potassium presents an unusual situation amongthe alkalis. All the excited-state hyperfine levels arespaced within 34 MHz, making it impossible to identifywell-defined cycling and repumping transitions to cooland maintain the MOT. Nevertheless, following Will-iamson and Walker (1995), the Connecticut group wasable to realize a working potassium MOT by essentiallytreating the whole group of excited hyperfine levels as asingle ‘‘line’’ and tuning the cooling laser 40 MHz to the

FIG. 61. Photoassociation FORT trap-loss fluorescence spec-trum of Rb2 . From Miller et al. (1993b).


red of the 39K 4s2S1/2(F52)→4p2P3/2(F853) transi-tion. The ‘‘repumping’’ beam was set 462 MHz to thered of the same transition, equal to the splitting betweenthe ground-state hyperfine levels. A potassium MOT re-quires relatively high intensity: 90 mW cm22 for the‘‘cooling’’ laser and 40 mW cm22 for the ‘‘repumper,’’although both beams undoubtedly fulfill both coolingand repumping functions. Wang et al. (1996a) report aMOT performance of ;23107 captured atoms, a den-sity ;331010 cm23, at a temperature ;500 mK. Inser-tion of a dark-spot disk (Ketterle et al., 1993) in the hori-zontal components of the ‘‘repumper’’ beam enhancedthe photoassociation rate by about one order of magni-tude. With this potassium MOT, the Connecticut groupcarried out cold photoassociation spectroscopy by mea-suring trap-loss fluorescence as an intense(1 –2 W cm22) probe laser scanned about 7 cm21 to thered of the S1/2(F52)→P3/2(F853) reference. Figure 66shows an example of the regular progressions observed.In their initial report Wang et al. (1996a) identified the0g

2 , 0u1 , and 1g states associated with the S1/21P3/2 as-

ymptote. In a followup study on the ‘‘pure long-range’’states, Wang et al. (1996b) determined the molecularconstants of the 0g

2 and the C3 constants for the 0u1 and

1g states. They also pointed out the utility of the 0g2 as

an intermediate ‘‘window’’ state to access even higher-lying states K2 by optical-optical double resonance.Very recently Wang, Wang, Gould, and Stwalley (1997)have observed optical-optical double resonance PAIspectroscopy near the 39K(4s)139K(4d ,5d ,6d ,7s) as-ymptotes, using the v850 level of the 0g

2 as the inter-mediate ‘‘window.’’ The Connecticut team has also re-ported a comprehensive study of the long-rangeinteractions in potassium binary atom collisions (Wang,Gould, Stwalley, 1998b). They have observed the sixlong-range states of 0u

1 , 1g , and 0g2 symmetry, three

dissociating to the 4s2S1/214p2P3/2 asymptote and threeto the 4s2S1/214p2P1/2 asymptote. Furthermore theiranalysis has determined the C3

P and C3S constants from

FIG. 62. Cline et al. (1994a, 1994b) separated the trapping andscanning functions with two different lasers: one laser at fixedfrequency produced the FORT while a probe laser scannedthrough the resonances.


the 0g2 ‘‘pure long-range’’ state, from which they extract

a precise measure of the K(4p) radiative lifetime, aswell as the C6

P and C6S constants. More recently, they

have identified for the first time in an alkali dimer spec-trum the pure long range 1u state (X. Wang, H. Wanget al., 1998). Finally Wang, Gould, and Stwalley (1998a)have very recently reported direct observation of predis-sociation due to fine-structure interaction in the photo-association spectra of K2 .

E. Atomic lifetimes from photoassociation spectroscopy

Because cold atoms collide at temperatures corre-sponding to kinetic energy distributions of order equal

FIG. 63. PAI in sodium and rubidium collisions: (a) Potentialdiagram of Na2 and Na2

1 showing doubly excited asymptoteabove Na2

1 ionization threshold. (b) Potential diagram of Rb2

and Rb21 showing doubly excited asymptote below Rb2

1 thresh-old.

FIG. 64. Direct photoassociative ionization of 85Rb collisionsin a MOT. The upper trace is the trap-loss spectrum fromTexas FORT, and the lower trace is the Rb2

1 ion signal fromthe direct PAI. From Leonhardt and Weiner (1995).


to or less than the spontaneous emission linewidth oftheir cooling transition, they are particularly sensitive tobinary interactions when the partners are far apart. Thedominant long-range interaction between identical at-oms, one resonantly excited and the other in the groundstate, is the dipole-dipole potential discussed many yearsago by King and van Vleck (1939). They showed thatthis interaction potential between atom A in excitedstate P and atom B in ground state S is given by

V~R !561

R3 @^dx&21^dy&

222^dz&2# , (122)

where ^dq&, q5x ,y ,z are the transition dipole matrixelements between the two states. Normally the z axis istaken as the line joining the two atom centers, and C3constants are then defined in terms of the transition di-poles as a ‘‘transverse’’ component,

C3P5^dx&

25^dy&2 (123)

and an ‘‘axial’’ component,

C3S52^dz&

252C3P . (124)

The transition moments themselves are related to theatomic excited state lifetime t through the average rateof spontaneous photon emission G, written in mks units,

G5t21516p3n3

3e0hc3 ^dq&2.

With C3P also written in mks units,

C3P5

14pe0

^dq&2, (125)

the atomic lifetime can be written in terms of C3 as

FIG. 65. Photoassociation spectrum from a 6Li MOT. Insetshows rotational substructure on vibrational lines. From McAl-exander et al. (1995).


FIG. 66. Trap-loss fluorescence spectrum from photoassociation of 39K. From Wang et al. (1996a).

t53\

4C3P S l

2pD 3

(126)

Therefore, a precise determination of C3P from cold pho-

toassociation spectroscopy will determine a precise mea-sure of the atomic excited-state lifetime.

For most molecular states that exhibit strong chemicalbinding, however, the long-range, dipole-dipole interac-tion is only one of several R-dependent interactionterms in the Hamiltonian. At close range the electronclouds from each atom interpenetrate and give rise toelectron-nuclear attraction, electron-electron repulsionterms, as well as attractive quantum mechanical electronexchange terms which decrease exponentially with in-creasing R . In this near zone the electron orbital angularmomentum, with a molecular axis projection denoted byL, strongly couples to the internuclear axis, and the elec-tron spin S , orbital angular momentum, and axis rota-tion angular momenta are coupled in a scheme calledHund’s case (a), with states designated by the notationSL . At greater internuclear separation, beyond the ex-change interaction, the interatomic interaction is de-scribed by resonant dipole-dipole and dispersion termsvarying with inverse powers of R , V(R);6C3 /R3

2C6 /R62C8 /R8. At sufficiently large distances, wherethe energy separations between the different Hund’scase (a) states become small compared to the atomicfine-structure splitting, electron orbital and spin angularmomentum couple strongly together to form a totalelectronic angular momentum, whose projection on themolecular axis is designated V. The total electronic an-gular momentum couples with nuclear rotation to forma total angular momentum called Hund’s case (c), withstates labeled by V. This scheme is used to label thepotentials in Fig. 47.

In order to extract C3 from observed spectral line po-sitions, it is necessary to understand all the terms in themolecular Hamiltonian, which have contributions fromthe interatomic interactions, molecular rotation, and fineand hyperfine structure. Fortunately, these can all betaken into account with great precision and detail in thecase of photoassociation, where the spectra are due tosingle molecular states. Although a complete quantita-tive theory of the rotating molecule with hyperfine struc-


ture can be developed (Tiesinga et al., 1997) it is mucheasier to use simpler models, which are also highly quan-titative (Jones, Julienne, et al., 1996). The Hamiltonianof Movre and Pichler provides such a model. It includesthe interatomic and fine-structure interactions (althoughrotation and hyperfine terms could be readily added).Only states with the same projection V of total elec-tronic angular momentum on the interatomic axis cancouple to one another (V5L1S , where L and S arethe respective projections of orbital and spin angularmomenta on the axis). At long range the off-diagonalinteratomic interaction due to the resonant dipole cou-pling, proportional to C3 /R3, mixes different states ofthe same V. Diagonalizing the Movre-Pichler Hamil-tonian gives the adiabatic molecular potential curvesshown in Fig. 47. Therefore each individual adiabaticcurve, such as 1g or 0g

2 , has its own characteristic C3coefficient at long range.

A good example of the Movre-Pichler analysis is pro-vided by the two pure long-range 0g

2 states dissociatingto S1P . These result from a spin-orbit avoided crossingbetween two Hund’s case (a) basis states: a repulsive3P0g potential which goes as 1C3 /R3 and an attractive3S0g potential which goes as 22C3 /R3. The two adia-batic 0g

2 potentials are found by diagonalizing the po-tential matrix (Jones, Julienne et al., 1996):

P S

VMP5S C3

R322DFS

3&DFS

3

&DFS

32

2C3

R3 2DFS

3

D P

S

, (127)

where DFS is the atomic spin-orbit splitting (DFS5515.520 GHz for Na) and the zero of energy is the2S1/21

2P3/2 asymptote. Diagonalizing this matrix givesthe two 0g

2 states in Fig. 47, the upper going to 2P3/2

12S1/2 , and the lower to 2P1/212S1/2 . The upper 0g

2

potential has a shallow well and is shown in Fig. 15.Although both curves are characterized by only a singleatomic transition dipole matrix element, the diagonaliza-tion is R dependent, and the effective C3 value associ-ated with a particular adiabatic curve is R dependent,


changing from a long-range value characteristic ofHund’s case (c) coupling to a short range value charac-teristic of Hund’s case (a) coupling. For example, theattractive Hund’s case (c) 1g state from 2P3/21

2S1/2changes to a Hund’s case (a) 1Pg state at short range.We have already seen Sec. VI how the C3 value foundfrom fitting spectra for this state changes with the rangeof levels fitted. At short range the potentials are alsoaffected by other terms in the electronic Hamiltonian.As R decreases, these terms can be ordered qualita-tively: the van der Waals dispersion terms C6 /R6 firstbegin to play a role, then the exchange-overlap interac-tion due to overlapping atomic orbitals on the two at-oms, and finally the strong chemical bonding at verysmall R. The effects of all of these terms must be takeninto account, the specific details depending on the par-ticular state being analyzed. The 0g

2 state is especiallyuseful in this regard, since it depends mainly on C3 andto a lesser extent on the dispersion energy.

The Rice group used the lower 3Sg1 state of Li2

(McAlexander et al., 1995). Since this state is chemicallybound with a large binding energy, the potential must bevery accurately known over a wide range of R in orderto use C3 as a free-variable fitting parameter to mini-mize differences between a constructed model potentialand measured data. The accuracy of the lifetime deter-mination was limited to 0.6% by the use of ab initiocalculated potentials over a range of R (from 7.8a0 to20a0) where spectroscopic data was unavailable. In anupdate to this experiment (McAlexander et al., 1996)the Rice group was able to shrink the error bars evenmore by applying essentially the same approach but us-ing an experimentally derived potential over the rangeof R where they had previously relied on calculations;moreover, by analyzing the hyperfine spectrum (Abra-ham, McAlexander, Stoof, and Hulet, 1996), they re-duced the uncertainty in the vibrational energies. Thereported uncertainty in the Li (2P) lifetime from thesemeasurements is 0.03%.

The ‘‘pure’’ long-range states, first proposed byStwalley et al. (1978), support bound vibrational levelswith turning points entirely outside the region where ex-change potential makes significant contributions. Themolecular potential therefore includes only the powerseries terms with atomic coefficients C3,C6,C8 . . . andthe constant atomic spin-orbit term. Transition momentsand the atomic lifetimes can then be calculated fromEqs. (125) and (126). The NIST (Jones, Julienne, et al.,1996) and Connecticut (Wang et al., 1996a, 1996b)groups have followed this approach to study sodium andpotassium respectively. Both groups used the ‘‘pure’’long-range 0g

2 state described by the Movre-PichlerHamiltonian, Eq. (127), to obtain P-state lifetimes withuncertainties around the 0.1% level. As in the case oflithium, at this level of precision, retardation effects (Ca-simir and Polder, 1948; McLone and Power, 1964;Meath, 1968) have to be taken into account. This meansthat, in Eq. (127), the C3 term has to be replaced byfP/S

•C3 for the respective P and S states where fP/S arethe retardation correction factors,


fP5cosS 2pR

l D1S 2pR

l D sinS 2pR

l D2S 2pR

l D 2

cosS 2pR

l D.12~2pR/l!2

2(128)

and

fS5cosS 2pR

l D1S 2pR

l D sinS 2pR

l D.11

S 2pR

l D 2

2.

(129)

The significance of this factor can be gauged by compar-ing the magnitude of the retardation correction (121MHz) to the experimental precision (65 MHz) for thecase of Na2 0g

2 (Jones, Julienne, et al., 1996).Table II summarizes the results on precision lifetime

measurements. The current best lifetimes for Li, Na, K,and Rb are from photoassociation spectroscopy. Onlythe Cs lifetime is based on conventional methods (Rafacet al., 1994). The C3 coefficients for the five attractivestates from 2P3/21

2S1/2 can be found by using the d2

from Table II with the formulas in Table I of Julienneand Vigue (1991).

F. Precision measurement of scattering lengthby photoassociative spectroscopy

As the kinetic energy of a binary collision approacheszero, the number of partial waves contributing to theelastic collision reduces to the s wave, and the informa-tion inherent in the scattering event can be character-ized by the scattering length, as defined by Eq. (20),

A052 limk→0

S 1k cot h0

D (130)

where k52p/ldB is the wave vector, ldB is the de Bro-glie wavelength of the reduced mass particle, and h0 isthe s-wave phase shift due to the scattering potential.The role of the scattering length in ultracold groundstate collisions is examined in detail in Sec. III.B. Thesign and magnitude of A0 is crucially important to theattainment of Bose-Einstein condensation in a dilute gas(Cornell, 1996). Binary collisions with positive scatteringlength lead to large, stable condensates while negativescattering lengths do not, and the rate of evaporativecooling of the gas—necessary to achieve critical densi-ties and temperatures of the BEC transition—dependson the square of the scattering length (Ketterle and VanDruten, 1996).

Photoassociation spectroscopy has played a pivotalrole in determining accurate and precise values for thescattering length in collisions of Li, Na, and Rb, thethree gases in which the quantum degenerate phase hasbeen observed. The essential idea is to use photoassocia-tive spectroscopy to probe the phase of the ground-statewave function with great precision. Two approacheshave been most fruitful: (1) determination of the nodalpositions of the ground-state wave function from the


relative intensities of spectral features, and (2) determi-nation of the spectral position of the last ground boundstate below the dissociation limit. The first approachtakes advantage of the fact that photoassociative spec-troscopy line intensities are sensitive to the amplitude ofthe ground-state wave function for each contributingpartial wave near the Condon point of the transition, asindicated by Eq. (32). In particular, the slowly varying‘‘Condon modulations’’ (Thorsheim et al., 1987; Juli-enne, 1996) have minima that reveal the position of theground-state wave-function nodes. A quantum scatter-ing calculation tunes the model ground-state potentialsuntil the relative spectral intensities in the calculationmatch those determined by the photoassociative spec-troscopy experiment. The calculated phase shift then de-termines the scattering length A0 for the model. Al-though all three alkalis yielded excellent results to thisline of attack, each one presented unique features ofphotoassociative spectroscopy analysis. The second ap-proach uses a two-photon photoassociatiive spectros-copy to locate the exact position of the last bound states.After the initial free-bound photoassociation transitiona second probe laser stimulates the excited populationdownward to the last bound vibrational level of theground state. The detunings of the two probe lasers de-termine the spectral position of the last bound state,which in turn fixes the threshold phase of the ground-state potential within highly constrained limits. A quan-tum scattering calculation of the s-wave elastic phaseshift then yields the scattering length. Both of thesestrategies have been used to good advantage.

1. Lithium

The Rice group led by R. G. Hulet determined thescattering length of collisions between two 7Li atoms inthe F52, MF52 state by using the two-photon tech-nique to determine the last bound vibrational level ofthe 7Li2(a3Su

1) triplet ground state (Abraham, McAlex-ander, et al., 1995; Cote, Dalgarno, Sun, and Hulet,1995). They set up a conventional lithium MOT andtuned a photoassociating laser to the v564 level of the 13Sg

1 bound excited state, using trap-loss fluorescence tofind the free-bound transitions. Then a second probeconnected this bound excited state back to high-lyingbound states of the a 3Su

1 ground state. Resonanceswith the second probe were detected by an increase inthe fluorescence base line, representing an anti-trap-lossfluorescence peak. Figure 67(a) shows the schematic ofthe stimulated excitation and deexcitation scheme, andFig. 67(b) shows the peak in the MOT fluorescence asthe second probe sweeps over the last bound vibrationallevel of the ground state. The difference frequency be-tween the two probes, measured with a spectrum ana-lyzer, permitted precise determination of the a3Su

1 lastbound-state energy with respect to the dissociation as-ymptote. Comparison of this energy with an accuratemodel potential identified the vibrational level with thev510 quantum number. Although the model potentialsused to identify the last bound state (Moerdijk and Ver-


haar, 1994; Moerdijk, Stwalley, et al., 1994; Cote et al.,1994) were sufficient for that purpose, they both pre-dicted a v510 binding energy of 11.4 GHz whereas theRice group directly measured 12.4760.04 GHz. Thenext step therefore was to modify the dissociation en-ergy De of the potential of Cote et al. so that v510 levelmatched the measurement. The newly modified, fine-tuned model potential was then input to a quantum scat-tering calculation to determine the s-wave phase shift inthe limit of zero temperature. Equation (130) then de-termined the scattering length. The final result, takinginto account uncertainties in the long-range coefficientsof the model potential, was found to be AT(2,2)5227.360.8 a.u. where AT(2,2) signifies the scatteringlength of the triplet potential starting from the F52,MF52 doubly polarized atomic state.

In more recent work the Rice group used the Condonmodulations in X 1Sg

1→A 1Su1 photoassociation line

intensities to determine the scattering lengths in 6Li and7Li collisions (Abraham, McAlexander, Gerton, andHulet, 1996). Figure 68 shows a plot of the photoasso-ciation line intensities for the three possible combina-tions of atomic hyperfine levels (111, 112, 212) in thecase of 7Li collisions. The minima, corresponding tonodes in the ground-state wave function, occur atslightly different positions for each of the hyperfine as-ymptotes. Varying the model potential of Cote et al.(1994), until the scattering lengths give the best fit to thenode positions, resulted in AS(1,1)53963 a.u.,AS(2,1)53365 a.u., and AS(2,2)53163 a.u. for 7Licollisions. In the case of 6Li the ground-state splittingswere too close to resolve minima from the different hy-perfine levels so the final result was reported as AS54763 a.u. for 6Li collisions.

Finally in the most recent work (Abraham et al., 1997)the Rice group completed the cycle in the lithium spec-troscopy by determining the s-wave triplet scatteringlength in 6Li. Once again, as in the case of the bosonisotope 7Li, they used the two-photon photoassociationtechnique to determine the precise energy of a boundstate (v59) of the a3Su

1 triplet ground state of 6Li2 .The result is also once again a negative scattering length,but of very large amplitude (AT5221606250 a.u.)

TABLE II. Atomic lifetimes and transition moments.

Atom Lifetime, t (ns) d2 (a.u.)

7Lia (2P1/2) 27.102 (2)b (7)c 11.0028 (8)b (28)c

Nad (3P3/2) 16.230 (16)b 6.219 (6)b

39Ke (4P3/2) 26.34 (5)b 8.445 (14)b

Rbf (5P3/2) 26.23 (6)b 8.938 (20)b

Csg (6P1/2) 34.934 (94)b 10.115 (27)b

a McAlexander et al. (1996).b One standard deviation.c Systematic uncertainty.d Jones, Julienne et al. (1996).e Wang, Li, et al. (1997).f Freeland et al. (1997).g Rafac et al. (1994).


which indicates a near-threshold resonance lying justabove the dissociation limit of the a3Su

1 state.

2. Sodium

The NIST group also used the relative intensities oftheir photoassociative ionization spectra to determinethe scattering length in Na collisions (Tiesinga et al.,1996), but they were able to take advantage of a particu-larly fortuitous circumstance. One of the nodes of theground state p wave (the ‘‘last’’ node before the waveassumes an oscillation characteristic of the 500 mK tem-perature) appears directly below v50 vibrational levelof the pure long-range 0g

2 state. This p-wave node sup-presses the intensity of photoassociation transitions toodd rotational levels of this v50 state and produceslarge variations in the rotational progression intensitiesfor higher vibrational levels of the 0g

2 state as well. Fig-ure 69 shows the relative positions of the ground-state pwave, the 0g

2 state, and the ionization pathway. Figure70 shows how the J51 and J53 intensities are stronglysuppressed in the v50 rotational progression becausethey arise from the p-wave contribution to the collisionbut then reappear in the v51 and v55 spectra. By vary-ing parameters in the singlet X 1Sg

1 and triplet a 3Suground states, the NIST group was able to reproduce therelative intensities and line shapes of the PAI spectra.The fine-tuning of the potentials shifted the p-wave ands-wave nodes to positions of best-fit to the spectra, andfrom these nodal positions the scattering lengths for col-lisions between atoms in F51, MF521 could be deter-mined within narrow constraints. The result is A1,2155265 a.u.

3. Rubidium

The Texas group led by D. J. Heinzen, in collabora-tion with the Eindhoven theory group led by B. J. Ver-haar, has carried out measurements of the scatteringlength in 85Rb collisions using a FORT in which the at-oms were doubly spin polarized (Gardner et al., 1995).The atoms approach only on the ground state a 3Su

1 ,and the photoassociation trap-loss fluorescence spectros-copy measures vibration-rotation progressions in thea 3Su

1→0g2 transition. Here the 0g

2 state is not the purelong-range state but at short range becomes a compo-nent of the 3Sg

1 molecular state dissociating to 2S1/212P1/2 . Figure 71 shows how the electron and nuclearspin polarization restrict the populated 0g

2 rotationalstates to even-numbered levels. Unlike the sodium case,with the 0g

2 state from 2S1/212P3/2 , in which more than

one partial wave contributed to various members of therotational progression, the 0g

2 state from 2S1/212P1/2 has

a one-to-one correspondence between the J level andscattering partial wave. The intensity of each J peak ef-fectively measures the amplitude of each partial wave.The energies of the J50 levels of five vibrational stateswere used to determine the product of the transitiondipoles, d(P1/2)d(P3/2), from a fit to the C3 /R3 poten-tial in the range of R (;41–47 a.u.), where the photo-


association spectra are measured. A line-shape analysissimilar to Napolitano et al. (1994) was then carried outon the J50 and J52 features from which a triplet scat-tering length 21000a0,aT(85Rb),260a0 was ex-tracted. Using a Am scaling procedure and a model trip-let potential, the Texas group concluded that the tripletscattering length for 87Rb fell in the range, 185a0,aT(87Rb),1140a0 . These measurements showedthat Bose-Einstein condensation was not favorable forthe triplet state of 85Rb but was possible for 87Rb. In afollowup experiment on shape resonances in 85Rb colli-sions (Boesten et al., 1996) the same group was able tonarrow the range of triplet scattering length to 2500a0,aT(85Rb),2300a0 , confirming that 85Rb would notproduce large, stable condensates.

Table III summarizes the s-wave scattering length de-terminations for various isotopes and states of the threealkalis measured to date,

VII. OPTICAL SHIELDING AND SUPPRESSION

A. Introduction

Photoassociation uses optical fields to produce boundmolecules from free atoms. Optical fields can also pre-vent atoms from closely approaching, thereby shieldingthem from short-range inelastic or reactive interactions

FIG. 67. Determination of the last bound state in 7Li2(a3Su1):

(a) Schematic of the stimulated excitation and deexcitationscheme. (b) Peak in the MOT fluorescence as the secondprobe sweeps over the last vibrational level of the groundstate. From Abraham, McAlexander, et al. (1995).


and suppressing the rates of these processes. Recentlyseveral groups have demonstrated shielding and sup-pression by shining an optical field on a cold atomsample. Figure 72(a) shows how a simple semiclassicalpicture can be used to interpret the shielding effect asthe rerouting of a ground-state entrance channel scatter-ing flux to an excited repulsive curve at an internucleardistance localized around a Condon point. An opticalfield, blue detuned with respect to the asymptotic atomictransition, resonantly couples the ground and excitedstates. In the cold and ultracold regime particles ap-proach on the ground state with very little kinetic en-ergy. Excitation to the repulsive state effectively haltstheir approach in the immediate vicinity of the Condonpoint, and the scattering flux then exits either on therepulsive excited state or on the ground state. Figure72(b) shows how this picture can be represented as aLandau-Zener avoided crossing of field-dressed poten-tials. As the blue-detuned suppressor laser intensity in-creases, the avoided crossing gap around the Condonpoint widens, and the semiclassical particle movesthrough the optical coupling region adiabatically. Theflux effectively enters and exits on the ground state, andthe collision becomes elastic. The Landau-Zener prob-ability that the particle remain on the ground state as itpasses once through the interaction region is

Pg5PS5expS 22p\V2

av D , (131)

where a is the slope of the difference potential evalu-ated at the Condon point,

FIG. 68. 7Li collisions: (a) Plot of the photoassociation lineintensities for the three possible combinations of atomic hyper-fine levels. (b) Condon fluctuations showing nodes occurring atslightly different positions for each of the hyperfine asymp-totes. From Abraham, McAlexander, Stoof, and Hulet, 1996.


a5UdD

dRURC

5Ud@Ue~R !2Ug~R !#

dR URC

, (132)

v being the local velocity of the semiclassical particle,and V the Rabi frequency of the optical coupling,

Vge

2p517.35 MHz3AI~W cm22!3dge~a.u.!. (133)

In Eq. (133) I is the shielding optical field intensity anddge is the transition dipole moment in atomic units.Equation (131) expresses the probability of penetrationthrough the crossing region with the scattering flux re-maining on the ground state. This probability is often

FIG. 69. Relative position of the ground state p-wave, the 0g2

state, and the ionization pathway. From Tiesinga et al. (1996).

FIG. 70. Note how in Na photoassociative ionization (3Su

→0g2) the J51 and J53 intensities are strongly suppressed in

the v50 rotational progression because they arise from thep-wave contribution to the collision but then reappear in thev51 and v55 spectra. From Tiesinga et al. (1996).


termed the ‘‘shielding measure,’’ PS . The probability oftransferring to the excited state at the crossing is givenby

Pe512expS 22p\V2

av D , (134)

and the probability of exiting the collision on the excitedstate asymptote after traversing the crossing regiontwice is

Pe`5PePg5Pe~12Pe!512Pg

` . (135)

According to Eqs. (131) and (135) the probability of thescattering flux exiting the collision on the ground stateapproaches unity as I increases to a strong field. How-ever, as we shall see in Sec. VII.F, other effects enter athigh field to modify this simple behavior. In addition,such Landau-Zener models are necessarily phenomeno-logical in nature, since there are a multiplicity of poten-tial curves (see Fig. 22) and an effective value of themolecular transition dipole strength dge must be chosento fit the data.

B. Optical suppression of trap loss

Evidence for optical manipulation of trap-loss pro-cesses was first reported by Bali et al. (1994) in 85Rbcollisions in a MOT. In this experiment scattering fluxentering on the ground state transfers to excited repul-sive molecular states by means of a control or ‘‘cataly-sis’’ laser tuned 5–20 GHz to the blue of the cooling

FIG. 71. Rubidium photoassociative ionization spectra show-ing the effects of spin polarization: (a) Rb photoassociativeionization spectrum unpolarized. (b) Rb photoassociative ion-ization spectrum from doubly spin-polarized atoms. Note re-striction to even-numbered rotational levels. From Gardneret al. (1995).


frequency. When the blue detuning of the control laserbecomes greater than about 10 GHz, the scattering part-ners, receding from each other along the repulsive po-tential, gather sufficient kinetic energy to escape thetrap. The control laser thus produces a ‘‘trap-loss’’ chan-nel, the rate of which is characterized by a trap-loss de-cay constant b(I), a function of the control laser inten-sity I . Figure 73 shows the increase of b up to about6 W cm22. Above this point the trap-loss rate constantbegins to decrease, indicating a suppression of the losschannel with increasing control laser intensity. Bali et al.interpret this suppression in terms of the dressed-stateLandau-Zener crossing model depicted in Fig. 72. Thesuppression of b is a consequence of the increasinglyavoided crossing between the ground atom-field stateug ;N& and the excited atom-field state ue ;N21&. As thecontrol laser intensity increases, incoming flux on ug ;N&begins to propagate adiabatically near the crossing andrecedes along the same channel from which it entered.A followup study by this Wisconsin group (Hoffmannet al., 1996) used a blue-detuned laser to excite incomingcollision flux population from the ground level to repul-sive excited levels in a Rb MOT. The kinetic energygained by the atoms dissociating along the excited repul-sive potential probed the limits of capture velocity as afunction of MOT parameters.

The Connecticut group of Gould and coworkers hasalso studied shielding and suppression in ground-statehyperfine-changing collisions (Sanchez-Villicana et al.,1995). As discussed in Sec. V.B, exoergic hyperfine-changing collisions can also lead to loss of atoms from atrap if it is made sufficiently shallow (by decreasing thetrapping light intensity) such that the velocity gained byeach atom from the hyperfine-changing-collisionskinetic-energy release exceeds the maximum capture ve-locity of the trap. In earlier studies, Wallace et al. (1992)observed a rapidly rising branch of b, the trap-loss rateconstant, as a function of decreasing MOT laser inten-sity below about 2.5 mW cm22 in a 85Rb MOT and be-low about 4 mW cm22 in a 87Rb MOT. They interpretedthis rapid rise as due to hyperfine-changing-collisionskinetic-energy release. In order to suppress this sourceof trap loss Sanchez-Villicana et al. (1995) imposed alight field on their 87Rb MOT blue-detuned 500 MHzwith respect to the S1/21P3/2 asymptote. This suppressorlight field couples the incoming ground-state collisionflux to a repulsive excited-state potential at a Condonpoint RC;32 nm, effectively preventing the atoms fromapproaching significantly closer. The atoms recede, ei-ther along the excited-state potential or along theground-state potential, before reaching the internucleardistance where hyperfine-changing collisions takes place(<10 nm). The blue-detuned light field effectivelyshields the atomic collision from the hyperfine-changing-collisions loss process by preventing the atomsfrom approaching close enough for it to take place. Fig-ure 74 shows the effectiveness of this shielding action asa function of suppression intensity by plotting the ratioof trap loss rate constant b/b0 with and without the sup-pressor laser present. Sanchez-Villicana et al. (1995)


Rev. Mod. Phys

TABLE III. Scattering lengths (a0).

As At5AfmaxmmaxA121 A3,23 A1,21/2,2

6Li2a 145.562.5 221606250

7Li2b 13362 227.660.5

6Li7Lia 220610 140.960.223Na2

b 18563 1526539K2

c 1132↔1144 21200↔26041K2

c 180↔188 125↔16039K2

d 1278614 181.162.440K2

d 115863 11.764.441K2

d 112162 128663685Rb2

e 2500↔230085Rb2

f 14500↔1` 244061402`↔21200

87Rb2e 185↔140

87Rb2g 110365 110365 110365

133Cs2h u.260u

133Cs2i u46612u

a Abraham et al. (1997).b Tiesinga et al. (1996).c Boesten, Vogels, et al. (1996).d Cote and Dalgnaro (1997); Cote and Stwalley (1997).e Boesten, Tsai, et al. (1996).f Tsai et al. (1997).g Julienne et al. (1997).h Arndt et al. (1997).i Monroe et al. (1993).

compare their measurements to the simple Landau-Zener model of the shielding effect (see Fig. 72) and findreasonably good agreement.

The use of blue-detuned optical fields to suppresshyperfine-changing collisions has also been studied incold sodium collisions. Muniz et al. (1997) employed anovel sequential trapping technique to capture Na atomsin a MOT sufficiently shallow to observe the hyperfine-

FIG. 72. Simple one-dimensional models of optical suppres-sion: (a) Conventional semiclassical picture of optical suppres-sion. Atoms approach on the ground state and are excited in alocalized region around RC by light of frequency vS to a re-pulsive excited state. (b) Dressed-state picture whereasymptotic states are atom-field states and the optical couplingappears as a Landau-Zener avoided crossing.

., Vol. 71, No. 1, January 1999

changing-collisions loss effect. The Brazil group cap-tured 107 –108 atoms in a relatively deep MOT operatingnear the D2 line of Na. They then transferred thesecooled atoms to a much shallower MOT operating nearthe D1 line, the collisional loss from which is due prin-cipally to hyperfine-changing collisions. Then with a sup-pressor laser, blue-detuned 600 MHz with respect to theS1/21P3/2 asymptote, Muniz et al. (1997) measured thesuppression of hyperfine-changing collisions. Figure 75shows a plot of the fractional decrease in hyperfine-changing-collisions rate constant, u5(b02b)/b0 vs sup-pression laser intensity, I . Both Sanchez-Villicana et al.(1995) and Muniz et al. (1997) compared their results tothe simple Landau-Zener model, the diagram of which isshown in Fig. 72. From Fig. 74 it appears that theLandau-Zener model underestimates the suppression ef-fect somewhat in 87Rb collisions, while Muniz et al.(1997) find the model overestimates the effect in Na col-lisions. This model and its appropriateness to opticalsuppression and shielding is discussed in more detail inthe next section. Muniz et al. (1997) also compare theirresults to a more sophisticated three-dimensional, mul-tichannel close-coupling calculation described in Sec.VII.F.

C. Optical shielding and suppressionin photoassociative ionization

Although trap-loss studies have played a major role inrevealing the nature of cold and ultracold collisions, todate photoassociative ionization (PAI) has been theonly collisional process where the product is directly de-


tected, although very recent results from Wang, Wang,et al. (1998) and Fioretti et al. (1997) indicate direct ob-servation of fine-structure changing processes in K andCs ultracold collisions, respectively. As discussed in Sec.VI on photoassociation, the PAI process takes place intwo steps. Reactant partners first approach on theground-state potential. Near the outer Condon pointRC , where an optical field \v1 couples the ground andexcited states, population transfers to the long-range at-tractive state, which accelerates the two atoms together.At shorter internuclear distance, near an inner Condonpoint, a second optical field \v2 couples the populationin the long-range attractive state either to a doubly ex-cited state that subsequently autoionizes or to the ion-ization continuum directly. Optical shielding intervenesat the first step. Rather than transferring population tothe long-range attractive potential at the first Condonpoint, an optical field \v3 , tuned to the blue of theexcited-state asymptote, transfers population to a repul-sive curve from which it exits the collision on the excitedasymptote with kinetic energy approximately equal tothe blue detuning of \v3 . This process is analogous tothe weak-field, trap-loss mechanism described by Baliet al. (1994). As the intensity of the suppressor field in-creases, scattering flux begins to exit on the ground statewith negligible increase in kinetic energy, and the colli-sion partners become elastically shielded from short-range interactions. Optical suppression of photoassocia-tive ionization was first reported by Marcassa et al.(1994) in a conventional MOT setup. Four different op-tical frequencies were used for the following: (a) to pro-duce the MOT (v1 ,v2), (b) to generate the suppressorfrequency (v3), and (c) to probe the PAI inelastic chan-nel (vp). Electro-optic modulation of the repumper v2produced sideband frequencies v3 ,v4 . Although v4 wasnot necessary to observe the suppression effect, its pres-ence added new features consistent with the interpreta-tion of suppression and shielding described here. Figure76 shows the coupling and schematic routing of the in-coming flux. Figure 76(a) shows the familiar two-stepPAI process with no suppression. After excitation to theattractive 21P level, the frequency v2 transfers scatter-ing flux in a second step to the P1P level from whichassociative ionization takes place. As vp tunes to the red

FIG. 73. Trap loss b as a function of control laser intensity.From Bali et al. (1994).


of the Condon point R2 , it can no longer, in combina-tion with v2 , couple to the doubly excited level, and anabrupt cutoff of the PAI signal is predicted at that fre-quency. Figure 76(b) shows what happens when a sup-pressor frequency v3 is imposed on the trap. With vptuned to the red of v3 , scattering flux on the F52, F852 entrance channel diverts to the repulsive 21P curvearound the suppressor Condon point Rs and exits via theexcited or ground state, depending on the intensity ofthe suppressor field. Figure 76(c) depicts what happenswhen vp tunes to the blue of v3. In this case the incom-ing flux transfers to the attractive 21P curve around theCondon point Rp . However, now v3 and v4 add to v2to enhance the probability of second-step excitation tothe P1P level. Therefore PAI should be enhanced withrespect to the PAI rate represented in Fig. 76(a). Fur-thermore the cutoff to PAI should be extended to theleft of R2 because v4 is the blue sideband of v2 , appear-ing 1.1 GHz higher frequency. Figure 77 shows a plotPAI count rate vs vp detuning with and without v3 andv4 . All three predicted features—suppression, enhance-ment, and cutoff—extension, are evident and appearwhere expected.

The dependence of any shielding or suppression effecton the intensity of the optical field is obviously of greatinterest and importance. Marcassa et al. (1994) de-scribed model two-state calculations at three levels ofelaboration: (1) time-dependent Monte Carlo wave-function simulations of wave-packet dynamics, (2) quan-tum close-coupling calculations without excited-statespontaneous emission, and (3) a very simple estimatebased on a dressed-state Landau-Zener avoided crossingmodel. The result of the calculations were in closeagreement among themselves and within about 15% ofthe experiment, given that an effective value of dge inEq. (133) had to be selected in order to give the model

FIG. 74. Suppression ratio b/b0 vs suppression laser intensityI for detunings of 500 MHz (solid points) and 250 MHz (openpoints). Differently shaped symbols represent data from differ-ent runs. The solid and dashed lines show the Landau-Zenermodel calculations for the two detunings. From Sanchez-Villicana et al. (1995).


an intensity scale that could be compared with the ex-perimental one. However, laser source limitations pre-cluded intensity measurements above ;1 W cm22. In afollowup article Marcassa et al. (1995) were able to mea-sure the suppression power dependence up to 8 W cm22.Figure 78 plots the fractional intensity of the PAI signalas a function of suppressor power density. The experi-mental results are shown together with calculations fromquantum close coupling and the Landau-Zener semiclas-sical model. Agreement is good at low field, but aboveabout 5 W cm22 the data tend to lie above the theorycurve, suggesting that the intensity dependence of sup-pression ‘‘saturates’’ at an efficiency less than 100%.Calculated values do not lie outside the statistical errorbars on the data points, however, so these data do notestablish a ‘‘saturation’’ effect at high field beyond ques-tion.

Zilio et al. (1996) continued to investigate the high-field behavior of optical suppression of PAI, by measur-ing the rate dependence on the polarization (circular orlinear) of the probe beam vp . They found that as thesuppressor field increased above about 1 W cm22, circu-lar polarization became markedly more effective at sup-pressing PAI, but that neither linear nor circular polar-ization suppressed the PAI rate to the extent predictedby the simple weak-field Landau-Zener model. Figure79(a) shows the shielding measure PS as function of sup-pressor field intensity, and Fig. 79(b) shows the results ofa close-coupling, strong-field model calculation of theprocess (Napolitano et al., 1997). The model calculationshows the correct ordering of the polarization depen-dence and the magnitude of the shielding measure(which is considerably less than the Landau-Zener pre-diction) although the curvature of the experiment andtheory plots have the opposite sign. This close-couplingcalculation will be discussed in more detail in Sec. VII.Fon strong-field theories of optical shielding. In very re-cent work, Tsao et al. (1998) investigated the alignmenteffect of linear polarization by measuring PS with thesuppressor beam E-field aligned parallel and perpen-dicular to a highly collimated and velocity-selectedatomic beam. They compared their results to the predic-

FIG. 75. Fractional suppression u vs suppressor laser intensityfor a D1 trapping laser intensity of 40 mW cm22. From Munizet al. (1997).


tions of the close-coupling high-field model of Napoli-tano et al. (1997) and found reasonably good agreement.Figure 80 summarizes the experimental result and com-parison with the model calculation.

D. Optical shielding in xenon and kryptoncollisional ionization

Another example of optical shielding in ionizing col-lisions of metastable xenon has been reported by theNIST group (Walhout et al., 1995). The experimenttakes place in a MOT with the cooling transition be-tween the metastable ‘‘ground state’’ 6s@ 3

2 #2 , and the

excited state 6p@ 52 #3 . This intermediate coupling nota-

tion refers to the core total electronic angular momen-tum in square brackets, and the subscript refers to thetotal electronic angular momentum of the atom (core1outer electron). The MOT is time chopped with a 150msec period and an ‘‘on’’ duty cycle of 2

3 . Measurementswere carried out in the probe cycle (MOT off) and thetrap cycle (MOT on). Experiments in the probe cyclestart with a control (or ‘‘catalysis’’) laser sweeping overa range of detuning from about 1.5 GHz to the red ofthe cooling transition to 500 MHz to the blue. The con-trol laser induces a strong enhancement of the ionizationrate over a red detuning range of about 500 MHz, peak-ing at 20 MHz, and produces suppression at blue detun-ings over several hundred megahertz. The maximumsuppression factor (;5), occurring at 200 MHz blue,appears to ‘‘saturate’’ at the highest control laser powerof 0.5 W cm22.. With the MOT beams illuminating thesample the experiments show an even more dramaticsuppression effect. Figure 81 shows the suppression fac-tor, which is the ratio of ionization rate constant withand without the control laser, plotted against control la-ser detuning. At about 250 MHz blue detuning the sup-pression factor reaches a maximum of greater than 30.The interpretation is similar to that of the sodium case(Marcassa et al., 1994). Without the control field present,reactant scattering flux, approaching on the metastableground state, is transferred to an attractive excited stateat far internuclear separation by the trapping laser. Thecolliding partners start to accelerate toward each otherand, during their inward journey, radiatively relax backto the metastable ground state. The consequent increasein kinetic energy allows the colliding partners to sur-mount centrifugal barriers that would otherwise haveprevented higher partial waves from penetrating to in-ternuclear distances where Penning and associative ion-ization take place. The net result is an increase in theionization rate constant during the MOT ‘‘on’’ cycle.With the control laser present and tuned to the blue ofthe trapping transition, the previously accelerated in-coming scattering flux is diverted to a repulsive excitedstate before reaching the ionization region. The collidingatoms are prevented from approaching further, and theresult is the observed marked reduction in ionizationrate constant.


FIG. 76. Schematic of transi-tions showing photoassociativeionization and optical suppres-sion of photoassociative ioniza-tion: (a) Two-step PAI process;(b) suppressor frequency v3 im-posed on the collision, reroutingincoming flux to the repulsiveexcited curve; (c) with vp tunedto the right of v3 , photoasso-ciative ionization takes placewith enhanced probability dueto addition of v3 and v4 to v2 .From Marcassa et al. (1994).

An experiment similar to the xenon work has alsobeen carried out in a krypton MOT by Katori andShimizu (1994). Analogous to the xenon case, the cool-ing transition cycles between 5s@ 3

2 #2↔5p@ 52 #3 , and a

control laser sweeps from red to blue over the coolingtransition from about 2600 MHz to 1100 MHz. Sup-pression of the ionization rate was again observed withthe control laser tuned to the blue. The power in thecontrol laser was only about 4 mW cm22, and the sup-pression factor peaked at a blue detuning of about 20MHz. Although this detuning is very close to the trap-ping transition, apparently the low power in the suppres-sion laser and time chopping of the MOT cycle andprobe cycle permitted ionization rate constant measure-ments without disruption of the MOT itself. Katori andShimizu (1994) also determined the ratio of rate con-stants between associative and Penning ionization andfound it to be .1/10. The power dependence of the sup-pression effect was not investigated over a very widerange of control laser power. The maximum power den-sity was only 25 mW cm22, so no firm conclusions con-cerning strong field effects can be drawn from this re-port.

E. Optical shielding in Rb collisions

In very recent work Sukenik et al. (1997) haveoberved evidence of optical suppression in two-photon‘‘energy pooling’’ collisions in 85Rb and 87Rb MOTs.This study is a two-color experiment in which one fre-quency v1 tunes 90 MHz to the red of the 5S1/2(F51)→5P1/2(F851) 87Rb resonance transition (or the5S1/2(F52)→5P1/2(F852) 85Rb resonance transition)and populates singly excited attractive states originating


from the 5S1/215P1/2 asymptote. A second color v2 ,tunes 0–2 GHz to the blue of 5S1/2(F51)→5P1/2(F852) 87Rb transition (or 5S1/2(F52)→5P1/2(F853)85Rb transition). Violet photons are observed from the‘‘energy pooling’’ energy transfer collision,

Rb~5P1/2!1Rb~5P1/2!→Rb~5S1/2!1Rb* ~6P ! (136)

FIG. 77. Effect of optical suppression on the photoassociativeionization spectrum observed in a Na MOT. Solid curve is thephotoassociative ionization spectrum without the suppressionfield vs present. Dotted curve shows the photoassociative ion-ization spectrum vs present. Note enhancement of signal tothe right of vs , suppression to the left. Note also the cutoff ofthe photoassociative ionization signal to the left of v2 withoutthe suppression field and the extension of this cutoff to v4when vs is present. From Marcassa et al. (1994).


followed by subsequent emission from the excited Rbatom,

Rb* ~6P ! Rb(5S)1hc/~l5421 nm!. (137)

The spectrum of the violet photon emission was re-corded as a function of v2 detuning. Sukenik et al.

FIG. 78. Shielding parameter Ps as a function of suppressorfield intensity. From Marcassa et al. (1995).

FIG. 79. Comparison of experiment and theory in optical sup-pression in Na: (a) Experimental results showing increased ef-fectiveness of optical suppression by circularly polarized lightat higher field intensities. From Zilio et al. (1996). (b) Theorycalculations in good qualitative agreement with experimentalresults. Also shown is the Landau-Zener results which consid-erably overestimates suppression at high fields. From Napoli-tano et al. (1997).


(1997) report deep modulations in the fluorescence in-tensity. They attribute this detuning modulation to tran-sitions in which v2 couples singly excited hyperfine lev-els, populated by v1 excitation, to doubly excitedmolecular hyperfine levels. Studies of v1 and v2 inten-sity dependencies show saturation in both colors. The v1saturation is due to depletion of the ground-state reac-tant flux. The v2 saturation is ascribed to optical shield-ing due to the blue color coupling of the ground stateand repulsive singly excited states. Thus v2 is thought toplay a double role: (1) opening the energy-pooling chan-nel by populating doubly excited molecular states atlong range, and (2) suppressing two-step photon transi-tions by coupling incoming ground-state flux to repulsivecurves originating from the 5S15P asymptotes.

F. Theories of optical shielding

Weak-field theories applied to either Na or Xe experi-ments do not produce the experimentally measured de-pendence of ionization rate on blue-detuned optical-field power density. In particular a serious attempt tocapture all the physics of strong-field optical shieldingmust take into account the effects of light shifts, satura-tion, rapid Rabi cycling at high fields, light polarization,and spontaneous decay. Suominen et al. (1995) investi-gated these factors (except light polarization) within thesimplifying context of two-state models in which the an-gular dependence of the suppressor field Rabi couplingwas replaced by a spherical average. The essential moti-vation was to investigate the conditions under which thenaive Landau-Zener formulas—(Eqs. (131) and(134))—break down and what modifications can bemade, if any, to retain a semiclassical picture of theshielding dynamic. Suominen et al. (1995) identify threedifferent characteristic times whose relative duration de-termines the validity of various approximations. These

FIG. 80. Alignment dependence of optical suppression of Naphotoassociative ionization collisions in a beam. Results showperpendicular polarization more effective than parallel and theL-Z formula overestimates effect at higher fields. From Tsaoet al. (1998).


times follow: (1) the lifetime against spontaneous decaytg , (2) the time t tp.mv/a for a wave packet to travelfrom the excitation Condon point RC to the turningpoint R tp on the repulsive potential, with mv the localmomentum; and (3) the time it takes a wave packet totraverse half the Landau-Zener interaction region tV

. 12 \V/av , where v , V, and a have been defined in Eqs.

(131), (133), and (132). Note that tV is the only one ofthese characteristic times that depends on the shieldingfield intensity and reflects the increasing ‘‘repulsion’’ ofthe dressed potentials with increasing V at the avoidedcrossing. If tg is long compared to both t tp and tV ,spontaneous decay can be ignored and the weak-fieldLandau-Zener formulas are appropriate. If tg becomesless than t tp but still remains long compared to tV , thenspontaneous emission can be included by calculating theexcitation probability with the Landau-Zener formula,Eq. (131), followed by radiative decay. In this case theshielding probability becomes

PS5Pg1Pe@12exp~2gt tp!#512Peexp~2gt tp!.(138)

If tg becomes shorter than t tp and tV , then excitationand spontaneous emission cannot be separated becauseRabi cycling is taking place around RC . As the shieldingfield intensity increases, eventually tV become greaterthan both t tp and tg , and the semiclassical particle orwave packet never passes entirely through the interac-tion region before undergoing Rabi cycling and sponta-neous decay. In this high-field regime one would notexpect the Landau-Zener approach to be very useful.Suominen et al. (1995) investigated all these relativetime scales by comparing the Landau-Zener models toquantum Monte Carlo wave-packet calculations, the ul-timate arbiter for the usefulness of any semiclassicaltwo-level model. They found that, as expected, theLandau-Zener, Eq. (131), and modified Landau-Zener,Eq. (138), expressions fail with increasing V but thatanother modification of the Landau-Zener expression,

PS512Peexp@2g~t tp2tV!# , (139)

actually tracks the Monte Carlo calculations quite well.

FIG. 81. Suppression factor as a function of control laser de-tuning in Xe collisions. These data show the suppression factorin the ‘‘trap’’ cycle with the MOT on. From Walhout et al.(1995).


The overall conclusion is that Landau-Zener and itsmodifications remain surprisingly robust even underconditions where one would expect them to fail, at leastwithin the confines of a two-state system. Real collisionsof course involve many more than two states, and thenext question is to what extent off-resonant states caninfluence the conclusions of the two-state studies.

Suominen, Burnett, and Julienne (1996) investigatedthis issue by carrying out Monte Carlo wave-function(MCWF) calculations on a model three-state systemconsisting of a flat ground state and two excited statesvarying as 6C3 /R3. First they investigated the effect ofthe off-resonant state (the repulsive 1C3 /R3 state) onthe probability of inelastic trap-loss processes excited bya red-detuned laser resonant with the attractive(2C3 /R3) state at some Condon point RC . The resultsshowed that as the red-detuned laser Rabi frequencyincreased, the two-state and three-state MCWF calcula-tions yielded essentially the same increasing probabilityof excited-state population while the simple weak-fieldLandau-Zener model predicted a low-level saturation ofthis excited-state population. Aside from the inclusionof the off-resonant state, the essential difference be-tween the MCWF calculations and the Landau-Zenerapproach is that at high fields the MCWF method takesinto account population recycling due to successive ex-citations after spontaneous emission. The Landau-Zenerapproach makes the weak-field assumption that excita-tion and spontaneous emission are decoupled so thatafter excitation to the attractive state, followed by spon-taneous emission back to the ground state, reexcitationnever occurs. Off-resonant population of the repulsive1C3 /R3 state appears to have little effect on the results.Next, Suominen, Burnett, and Julienne (1996) switchedthe roles of the attractive and repulsive excited states byinvestigating the influence of the off-resonant state (theattractive 2C3 /R3 state) on optical shielding when ablue-detuned laser is resonant with the repulsive(1C3 /R3) state at some Condon point RC . They foundthat at weak field the results agreed with Landau-Zener,calculations, but that at strong field the shielding mea-sure PS did not approach zero but began to increase dueto the off-resonant population of the attractive excitedstate. This behavior is not observed in the experimentsof Zilio et al. (1996) and Walhout et al. (1995), where PSsaturates with increasing shielding field intensity at lev-els above those predicted by Landau-Zener but showsno evidence of a subsequent rise. The experimental re-sults cannot be explained therefore by the presence ofoff-resonant states, but must be due to some other high-field effects.

At the same time that Suominen, Burnett, and Juli-enne (1996) were carrying out the MCWF calculation toinvestigate the role of off-resonant states in a three-statemodel of shielding, they and their experimental co-authors (Suominen, Burnett, Julienne, et al., 1996) in-vestigated to what extent a two-state Landau-Zener typeof theory could explain the experimental results of Wal-hout et al. (1995). Both the alkali experiments (Mara-cassa et al., 1994; Zilio et al., 1996) and the xenon experi-


ments show that suppression and shielding are less thanpredicted by very simple two-state avoided-crossingmodels, but Fig. 82 shows that the extent of the shielding‘‘saturation’’ as a function of shielding laser intensity ismuch more dramatic in the results of Walhout et al.(1995). The results of Fig. 82 were recorded with theMOT light turned off and only the blue-detuned controllaser present. By inserting an averaged distribution ofmolecular Rabi frequencies in the Landau-Zener expo-nential argument, Suominen, Burnett, Julienne, et al.(1996) were able to get good agreement with these xe-non experimental results. At first glance it might seemreasonable to use such a distribution since the MOT ex-periment does not restrict the angle u between the col-lision axis and the control laser polarization axis, and theRabi frequency could be written as V5V0sin u. Theproblem is that there is no real justification for averagingV over u since the P ,Q ,R branches are all available foroptical coupling. Furthermore the experimental resultsshow that relative shielding is much more pronouncedwhen both the blue-detuned control laser and the red-detuned MOT lasers are turned on. With both red andblue colors present the extent of the shielding effect re-sembles much more closely the alkali results, and appli-cation of the Landau-Zener formula with a distributionof Rabi frequencies does not give good agreement. Al-though various ad hoc adaptations of the Landau-Zenerformula can yield a satisfactory comparison with the xe-non measurements, a clear, justifiable, and consistentphysical picture does not emerge from these calcula-tions.

Very recently Yurovsky and Ben-Reuven (1997) haveproposed a Landau-Zener approach in which the three-dimensional nature of the collision has been incorpo-rated into the theory. By calculating the Landau-Zenerprobability at multiple crossings, where incoming 1Sgground-state s and d waves couple through P ,Q ,Rbranches to a 1Pu repulsive excited state, several path-ways are traced out through which the incoming flux canpenetrate to the inner region, thereby rendering theshielding less efficient than predicted by the naive one-dimensional Landau-Zener model. Yurovsky and Ben-Reuven calculated the shielding measure PS as a func-tion of shielding laser intensity for three detunings andcompared to the measured results in xenon collisions(Suominen, Burnett, Julienne, et al., 1996). Although thethree-dimensional Landau-Zener theory better reflectsthe ‘‘saturation’’ behavior of PS than does the one-dimensional Landau-Zener theory, the xenon measure-ments still show a stronger saturation than either theone- or three-dimensional versions of the two-stateLandau-Zener model.

Motivated by the experimental results of Marcassaet al. (1995), Zilio et al. (1996), and Walhout et al.(1995), Napolitano et al. (1997) developed a full three-dimensional, close-coupled, quantum scattering ap-proach to optical suppression of ultracold collision rates.This work differs from the MCWF approach of Suom-inen et al. (1995) and Suominen, Burnett, and Julienne(1996) in that the calculations are not restricted to two


or three states or to only s-wave scattering. Although itdoes not take spontaneous radiation explicitly into ac-count, Suominen et al. (1995) has shown that at largeenough blue detuning the decay of the upper state canbe neglected, and Napolitano et al. (1997) carried outtheir model calculations in a detuning regime wherespontaneous emission can be safely ignored. The modelcollision examined is

A~1S !1A~1S !1P~«W q ,\vL!→A~1S !1A~1P !, (140)

where A(1S) and A(1P) are atoms in the 1S and 1Pground and excited states, respectively, and P(«W q ,\vL)represents a photon of energy \vL and unit polarizationvector «W q with q50 for linear polarization and q561for circular polarizations. In order to keep the numberof channels to a manageable size, nuclear and electronspins were excluded; however, the model does representreal scattering of the spinless alkaline earths (group IIA)and serves as a qualitative guide for understanding thebehavior of the Na and Xe experiments. The problem isset up with the atoms far apart in an asymptotic field-dressed atomic basis within a space-fixed frame definedby «W q. As the atoms approach, this space-fixed basis cor-relates to a molecular body-fixed basis through the uni-tary transformation of the symmetric top eigenfunctions;these molecular basis states are then coupled by the ra-diation field. This coupling is normally localized arounda Condon point RC defined by the blue detuning of theshielding laser field. The molecular state formed from apair of 1S ground-state atoms (1Sg

1) and the statesformed by a 1S ,1P ground-excited pair (1Su

1 ,1Pu) con-stitute the four-state molecular basis (1Pu is doubly de-generate). The goal is to calculate the probability thatscattering flux incoming on the molecular ground statewill (1) penetrate to the inner region where reactive andinelastic processes can take place, (2) transfer and exiton the 1Pu excited state, or (3) elastically scatter on theground state.

The results of the quantum close-coupling calculationsshow that the shielding parameter PS is markedly sensi-tive to both optical field intensity and to polarization,with circular polarization shielding more effectively thanlinear polarization. Napolitano et al. (1997) interpret thepolarization effect as the difference in the number ofangular momentum branches (P ,Q ,R) through whichthe ground and excited states can couple. For exampleFig. 83 shows that when two partial waves l50,2 cross inthe region of the Condon point, the angular momentumselection rules result in different coupling for linear andcircularly polarized light. For linear polarization the en-trance d wave couples only through the P and Rbranches, while for circularly polarized light the P ,Q ,Rbranches couple. The result is that circular polarizationresults in more ‘‘avoidedness’’ at the crossing than doeslinear polarization. Figure 79 demonstrates a qualitativeagreement between close-coupling calculations and theresults of Zilio et al. (1996) both in the amplitude of theshielding measure PS and in the ordering of the lightpolarization. A further result of the close-couplingtheory is the prediction of anisotropy in shielding as a


function of the linear polarization alignment with re-spect to the molecular collision axis. This behavior wasconfirmed in a beam experiment by Tsao et al. (1998).

Most shielding and suppression studies, both experi-mental and theoretical, have focused on the dependenceof PS as a function of optical field intensity at some fixedblue detuning. Band and Tuvi (1995) have predicted,however, that by varying the relative detuning of the redand blue light fields in a sodium PAI experiment, as wellas their intensities, one can produce interesting effectssuch as field-dressed ‘‘reaction barriers,’’ the height andwidth of which can be controlled by the suppression fielddetuning and intensity. These barriers would be observ-able as resonances in the PAI signal as a function of bluedetuning. The resonance positions should also shift andbroaden with increasing suppression intensity. To datewe are aware of no experimental efforts undertaken toinvestigate these interesting predictions of a model cal-culation, but shielding and suppression should providenew opportunities for optical control of inelastic pro-cesses.

FIG. 82. Penetration measure (12PS) as function of laser in-tensity for three values of blue detuning of the control orshielding laser in Walhout et al. (1995). Circles are experimen-tal points; solid line, the theory of Suominen, Burnett, Juli-enne, et al. (1996); dotted line, the standard Landau-Zener cal-culation.


VIII. GROUND-STATE COLLISIONS

Most of this review has focused on collisions of cold,trapped atoms in a light field. Understanding such colli-sions is clearly a significant issue for atoms trapped byoptical methods, and historically this subject has re-ceived much attention by the laser cooling community.However, there also is great interest in ground-state col-lisions of cold neutral atoms in the absence of light.Most of the early interest in this area was in the contextof the cryogenic hydrogen maser or the attempt toachieve Bose-Einstein condensation (BEC) of trappeddoubly spin-polarized hydrogen. More recently the in-terest has turned to new areas such as pressure shifts inatomic clocks or the achievement of BEC in alkali sys-tems. The actual realization of BEC in 87Rb (Andersonet al., 1995), 23Na (Davis et al., 1995), and 7Li (Bradleyet al., 1995; Bradley et al., 1997) has given a tremendousimpetus to the study of collisions in the ultracold regime.Collisions are important to all aspects of condensatesand condensate dynamics. The process of evaporativecooling that leads to condensate formation relies onelastic collisions to thermalize the atoms. The highly suc-cessful mean field theory of condensates depends on thesign and magnitude of the s-wave scattering length toparametrize the atom interaction energy that determinesthe mean field wave function. The success of evaporativecooling, and having a reasonably long lifetime of thecondensate, depend on having sufficiently small inelasticcollision rates that remove trapped atoms through de-structive processes. Therefore, ground-state elastic andinelastic collision rates, and their dependence on mag-netic or electromagnetic fields, is a subject of consider-able current interest.

This section will review work on ground-state colli-sions of trapped atoms in the regime below 1 mK, withparticular emphasis on the ultracold regime below 1 mK.The work on ground-state collisions could easily be thesubject of a major review in its own right, so our reviewwill be limited in scope. We do not in any way claim tobe exhaustive in our treatment. We will use a historicalapproach, as we have done for collisions in a light field,and try to cover some of the key concepts and measure-ments. The first section will review the early work in thefield, including a brief survey of the work on hydrogen.A second section will discuss the role of collisions inBEC.

A. Early work

We noted in Sec. III B that the quantum propertiesare quite well known for collisions where the de Brogliewavelength is long compared to the range of the poten-tial. We confine our interest here to the special case ofthe collision of two neutral atoms at temperatures of lessthan 1 K. Interest in this subject was stimulated in the1980’s by two developments: the possibility of achievingBose-Einstein condensation with magnetically trappedspin-polarized hydrogen (Stwalley and Nosanow, 1976;Laloe, 1980; Silvera and Walraven, 1980; Silvera and


Walraven, 1986), and the prospects of unparalleled fre-quency stability of the cryogenic hydrogen maser(Crampton et al., 1978; Hess, Kochanski, et al., 1986;Hurlimann, et al., 1986; Walsworth, et al., 1986). Theground-state hydrogen atom has a 2S electronic stateand a nuclear spin quantum number of 1/2. Coupling ofthe electron and nuclear spins gives rise to the well-known F50 and F51 hyperfine levels of the groundstate. The transition between these two levels is the hy-drogen maser transition, and the doubly spin polarizedlevel, F51, M51, with both electron and nuclear spinshaving maximum projection along the same axis, is theone for which BEC is possible in a magnetic atom trap.Both the hydrogen maser and the phenomenon of BECare strongly affected by atomic collisions of ground-statehydrogen atoms. Collisions cause pressure-dependentfrequency shifts in the maser transition frequency thatmust be understood and controlled (Kleppner et al.,1965; Crampton and Wang, 1975; Verhaar et al., 1987;Koelman, Crampton, et al., 1988), and they cause de-structive relaxation of the spin-polarized H atoms thatcan prevent the achievement of BEC (Cline et al., 1981;Sprik et al., 1982; Ahn et al., 1983; van Roijen et al.,1988).

These developments stimulated in the 1980’s theoret-ical calculations for low-temperature collision propertiesof atomic H and its isotopes. Earlier work (Dalgarno,1961; Allison, 1972) had laid the groundwork for under-standing inelastic spin exchange collisions by which twoH atoms in the F51 state undergo a transition so thatone or both of the atoms exit the collision in the F50state. Berlinsky and Shizgal (1980) extended these cal-culations of the spin exchange cross section and colli-sional frequency shift of the hyperfine transition to thelow-temperature limit. These early calculations werebased on extending the high temperature theory, based

FIG. 83. Schematic of close-coupling, strong-field model of op-tical suppression: (a) Two partial waves, s and p , opticallycoupled from the 1Sg to 1Pu , with linear polarization. Notethat the d wave couples only through P ,R branches. (b) Samecoupling but with circular polarization. Coupling is throughP ,Q ,R branches. From Napolitano et al. (1997).


on knowing the phase shifts of the ground state molecu-lar hydrogen 1Sg and 3Su potentials alone without ex-plicit inclusion of the hyperfine structure of the sepa-rated atoms. A proper quantum mechanical theorybased on numerical solution of the coupled-channelSchrodinger equation for the atoms with hyperfine struc-ture, also known as the close coupling method, was in-troduced by the Eindhoven group, and applied to fre-quency shifts in hydrogen masers (Verhaar et al., 1987;Koelman, Crampton et al., 1988; Maan et al., 1990) andrelaxation of doubly spin-polarized hydrogen in a mag-netic trap (Ahn et al., 1983; Lagendijk, 1986; Stoof et al.,1988). The close coupling method is a powerful numeri-cal tool and is the method of choice for quantitativecalculations on ground-state collisions. It is the bestmethod currently available and has been applied to avariety of species, including mixed species. A recent dis-cussion of the multichannel scattering theory for coldcollisions has been given by Gao (1997).

Collisions of species other than hydrogen have beeninvestigated. Uang and Stwalley (1980) looked at colli-sions of hydrogen and deuterium in cold magnetic trapsto assess the role of deuterium impurities in a cold spin-polarized hydrogen gas. Koelman et al. (1987) and Koel-man, Stoof, et al. (1988) calculated the lifetime of a spin-polarized deuterium gas; Tiesinga et al. (1990) andTiesinga, Crampton, et al. (1993) examined frequencyshifts in the cryogenic deuterium maser. More recentlyJamieson et al. (1996) have calculated collisional fre-quency shifts for the 1S-2S two-photon transition in hy-drogen. Tiesinga et al. (1991) calculated that the relax-ation rate coefficient for doubly spin-polarized Na wouldbe about ten times larger than for spin-polarized hydro-gen. Tiesinga, Verhaar, et al. (1992) also calculated thatthe frequency shift in a cesium atomic fountain clock(Kasevich et al., 1989; Clairon et al., 1991) might be largeenough to limit the anticipated accuracy of such a clock.The use of neutral atoms in ultra-precise atomic clocks isdiscussed by Gibble and Chu (1992). Gibble and Chu(1993) measured large collisional frequency shifts in aCs fountain. These have been verified by an experimentof Ghezali et al., 1996. Verhaar et al. (1993) argue thatother cold collision properties can be deduced fromthese clock-shift measurements, namely, that doublyspin-polarized 133Cs probably has a large negative scat-tering length; they criticize the opposite conclusiondrawn from the same data by Gribakin and Flambaum(1993), due to restrictive approximations used by the lat-ter. Gibble and Verhaar (1995) suggest that clock fre-quency shifts might be eliminated by using 137Cs.Kokkelmans et al. (1997) use accurate collisional calcu-lations for both isotopes of Rb to suggest that a Rbatomic clock will offer better performance than a Csone. Gibble, Chang, and Legere (1995) have directlymeasured the s-wave scattering cross section and angu-lar distribution in Cs atom collisions in an atomic foun-tain experiment at a temperature of T50.89 mK.

B. Bose-Einstein condensation

One of the major motivating factors in the study ofcollisions of cold ground-state neutral atoms has been


the quest to achieve Bose-Einstein condensation (BEC).This is a phase transition occurring in a gas of identicalbosons when the phase space density becomes largeenough, namely, when there is about one particle percubic thermal de Broglie wavelength. The specific crite-rion for condensation is (Lee and Yang, 1958)

nS 2p\2

mkT D 3/2

.2.612, (141)

where n represents atom density. In the condensate,there is a macroscopic occupation by a large number ofatoms of the single ground state of the many-body sys-tem, whereas in a normal thermal gas many momentumstates are occupied with very small probability of occu-pying any given one. Achieving BEC means making thedensity large enough, or the temperature low enough,that Eq. (141) is satisfied. The early work with spin-polarized hydrogen aimed at reaching high enough den-sity using conventional refrigeration techniques. Unfor-tunately, this proved to be impossible due to the lossescaused by collisions or surface recombination when thedensity was increased. An alternative approach was de-veloped by using a magnetic trap without walls (Hesset al., 1987) and evaporative cooling (Masuhara, 1988) toreach much lower temperatures. The idea is to keep thedensity sufficiently low to prevent harmful collisions.The long quest to achieve BEC for atomic hydrogen wasfinally successful in 1998 (Fried et al., 1998), three yearsafter the first demonstratin with cold, trapped alkali at-oms.

The success of laser cooling for alkali atoms gave animpetus to achieving BEC using an alkali species. Theseare similar to hydrogen in that they have 2S groundelectronic states with two hyperfine levels due to thenuclear spin (see Fig. 3). All alkali species have an iso-tope with odd nuclear spin, making the atom a compos-ite boson. Unlike hydrogen, the alkali dimer 3Su statesupports bound states; however, a metastable conden-sate is possible because of the long time scale requiredto make dimer bound states via three-body recombina-tion at low condensate density. Ordinary laser coolingmethods produce density and temperature conditionsmany orders of magnitude away from satisfying thephase-space density criterion in Eq. (141). The processof evaporative cooling was seen as a viable route toreaching BEC, and several groups set out to make itwork. This approach was spectacularly successful, andwithin a few months of each other in 1995, three groupsreached the regime of quantum degeneracy required forBEC. The first unambiguous demonstration of BEC inan evaporatively cooled atomic gas was reported by theNIST/JILA group (Anderson et al., 1995) for doublyspin polarized 87Rb, followed by evidence of BEC fordoubly spin-polarized 7Li by a group at Rice University(Bradley et al., 1995), then BEC was demonstrated atMIT (Davis et al., 1995) in the F51, M521 lower hy-perfine component of 23Na. A much clearer demonstra-tion of BEC for the 7Li system was given later (Bradleyet al., 1997). As of mid-1997, at least three additionalgroups have used similar evaporative cooling and trap


designs to achieve BEC in 87Rb and 23Na, and otherattempts remain in progress elsewhere. A large litera-ture on the subject of BEC has already been generated.Some introductory articles on the subject are given byBurnett (1996), Cornell (1996), and Townsend, Ketterle,and Stringari (1997).

Ground-state collisions play a crucial role both in theformation of a condensate and in determining its prop-erties. A crucial step in the formation of a condensate isthe achievement of critical density and temperature by‘‘evaporative cooling,’’ the process by which hot atomsare removed from the confined ensemble while the re-maining gas thermalizes to a lower temperature (Hess,1986). Elastic collisions are necessary for evaporativecooling to work, and the stability and properties of thecondensate itself depend on the sign and magnitude ofthe elastic scattering length. Two- and three-body inelas-tic collisions cause destructive processes that determinethe condensate lifetime. Therefore, these collisions havebeen of as much interest for alkali species as for hydro-gen and have been the object of numerous experimentaland theoretical studies.

First, the process of evaporation depends on elasticmomentum transfer collisions to thermalize the gas oftrapped atoms as the trapping potential is lowered.These elastic collisions represent ‘‘good’’ collisions, andthey have cross sections orders of magnitude larger foralkali species than for hydrogen. During evaporation,the rate of inelastic collisions that destroy the trappedhyperfine level, the so-called ‘‘bad’’ collisions, must re-main much less than the rate of elastic collisions. Anexcellent description of the role of these two types ofcollisions is given in the review on evaporative coolingby Ketterle and Van Druten (1996). Long before alkalievaporative cooling was achieved, Tiesinga, Moerdijk,and Verhaar, and Stoof (1992) calculated that the ratioof the ‘‘good’’ to ‘‘bad’’ collisions appears to be veryfavorable for both the F54, M54 and F53, M523trappable states of 133Cs, having a value on the order of1000. Precise predictions were not possible due to uncer-tainties in the interatomic potentials. Monroe et al.(1993) used time-dependent relaxation of trapped atomsto measure the elastic cross section for F53, M523near 30 mK to be large and independent of temperaturebetween 30 and 250 mK, 1.5(4)310212 cm2, implying ascattering length magnitude near 46(12)a0 , if the crosssection is assumed to be due to s-wave collisions. New-bury, Myatt, and Wieman (1995) similarly measured theelastic cross section for the F51, M521 state of 87Rbto be 5.4(1.3)310212 cm2, implying a scattering lengthmagnitude of 88(21)a0 , as indicated in Table III. Mea-surements on thermalization in a F51, M521 23Natrap by Davis, Mewes, Joffe et al. (1995) deduced a scat-tering length of 92(25)a0 for this level. A very recentstudy by Arndt et al. (1997) of thermalization of doublyspin-polarized 133Cs F54, M54 showed that the scat-tering length magnitude was very large, .260a0 , andthe elastic scattering cross section was near the upperbound given by the s-wave unitarity limit 8p/k2, be-tween 5 and 50 mK. All of these experimental studies


measured total cross section only, and therefore werenot able to determine the sign of the scattering length.Verhaar et al. (1993), however, calculated the sign to benegative for 133Cs F54, M54.

The second way in which atomic interactions pro-foundly affect the properties of a condensate is throughtheir effect on the energy of the condensate. The effectof atom-atom interactions in the many-body Hamil-tonian can be parametrized in the T→0 limit in terms ofthe two-body scattering length (Huang and Yang, 1956).This use of the exact two-body T matrix in an energyexpression is actually a rigorous procedure, and can befully justified as a valid approximation (Stoof, Bijlsma,and Houbiers, 1996). One simple theory that has beenvery successful in characterizing the basic properties ofactual condensates is based on a mean-field, or Hartree-Fock, description of the condensate wave function,which is found from the equation (Edwards and Burnett,1995)

S 2\2

2m¹21V trap1NU0uCu2DC5mC . (142)

Here V trap is the trapping potential that confines thecondensate,

U054p\2

mA0 (143)

represents the atom-atom interaction energy, which isproportional to the s-wave scattering length for thetrapped atomic state, and m is the chemical potential—that is, the energy needed to add one more particle to acondensate having N atoms. The condensate wave func-tion in this equation, called the Gross-Pitaevski equationor the nonlinear Schrodinger equation, can be inter-preted as the single ground-state orbital occupied byeach boson in the product many-body wave functionF(1 . . . N)5) i51

N C(i). The wave function C could alsobe interpreted as the order parameter for the phasetransition that produces the condensate.

The effect of atom interactions manifests itself in thenonlinear Schrodinger equation in the mean-field termproportional to the local condensate density, NuCu2, andthe coupling parameter U0 proportional to the s-wavescattering length A0 . In an ideal gas, with no atom in-teractions, A050, and this term vanishes. Equation(142) shows that the condensate wave function in suchan ideal gas case just becomes that for the zero-pointmotion in the trap, that is, the ground state of the trap-ping potential. For typical alkali traps, which are har-monic to a good approximation, the zero-point motiontypically has a frequency on the order of 50 to 1000 Hz,and a range on the order of 100 mm to a few mm. Forcomparison, kBT/h520 kHz at T51 mK. In actuality,as atoms are added to the condensate the atom interac-tion term becomes the dominant term affecting the con-densate wave function, and the shape of the condensatedepends strongly on the size of the U0 term, which isproportional to the product NA0 . The sign of the scat-tering length is crucial here. If A0 is positive, the inter-


action energy increases as more atoms are added (Nincreases) to the condensate. The condensate is stableand becomes larger in size as more atoms are added. Infact, a very simple approximation, called the Thomas-Fermi approximation, gives the condensate density n byneglecting the kinetic energy term in Eq. (142) in rela-tion to the other terms:

n5NuCu25m2V trap

U0. (144)

This equation is remarkably accurate except near theedge of the trap where m2V trap approaches 0 or be-comes negative, and it describes condensates of 87Rb F52, M52 and F51, M521 and 23Na F51, M521, allof which have positive scattering length. Condensates ofsuch species can be made with more than 106 atoms. Onthe other hand, if the scattering length is negative, as for7Li, increasing the number of particles in the trap makesthe interaction energy term in Eq. (142) become morenegative. The wave function contracts as more particlesare added, and, in fact, only about 1000 atoms can beadded to a 7Li condensate before it becomes unstableand can hold no more atoms (Dodd et al., 1996; Bradleyet al., 1997; Bergeman, 1997). A condensate with nega-tive scattering length is not possible in a uniform homo-geneous gas, but in an atom trap the presence of zero-point motion does permit the existence of a very smallcondensate, as is the case for 7Li.

It is perhaps not obvious why a collisional propertylike a scattering length determines the energetics of theinteracting particles. A simple motivational argument toindicate why this is the case can be given in relation toFig. 5 in Sec. III.B. The long wavelength scattering wavehas its phase shifted near R50 by the interaction poten-tial. From the perspective of the asymptotic wave theeffective origin of the oscillation near R50 is shifted bythe presence of the potential, to R5A0.0 for the caseof positive A0 and to R5A0,0 for the case of negativeA0 , as Fig. 5(b) shows. The kinetic energy associatedwith the long wavelength asymptotic wave is affected bythis shift in effective origin of oscillation. If one thinks ofthe two-particle system in terms of a single effectivemass particle in a box, the left-hand wall of the box atR50 is moved to larger or smaller R , depending on thesign of the scattering length. What is important for theenergetics is whether the change in energy is positive ornegative, relative to noninteracting atoms (the box witha wall at R50). Given that the energy of the groundstate of a reduced mass particle of mass m/2 in a box is(\2/m)(p/L)2, it is easy to work out that changing thelength of the box from L to L1A0 changes the energyby an amount proportional to A0 /m , thus lowering theenergy for the case of negative A0 and raising it for thecase of positive A0 . A rigorous analysis gives the cou-pling term in Eq. (143).

The gas thermalization studies that measure the elas-tic scattering cross section cannot determine the sign ofthe scattering length. The sign can be determined fromphotoassociation spectroscopy, which is sensitive to theshape and phase of the ground-state collisional wave


function. Much tighter constraints have also been placedon the magnitudes of scattering lengths by photoassocia-tion studies. This work was summarized in Sec. VI. Ofcourse, observation of the condensate properties deter-mines the sign as well. Even the magnitude of the scat-tering length can be found by measuring the expansionof the condensate when the trapping potential is re-moved, since the expansion rate depends on the strengthof the interaction term in the initial condensate. This hasbeen done for both the 87Rb (Holland et al., 1997) and23Na (Mewes et al., 1996; Castin and Dum, 1996; Ket-terle and Miesner, 1997) condensates, confirming themagnitude of the scattering lengths determined moreprecisely by other means. Before the determinations byphotoassociation spectroscopy, the sign and magnitudeof the scattering lengths have also been estimated fromthe best available interatomic potentials based on spec-troscopically derived potential wells and the long-rangevan der Waals potentials. This was done for 7Li (Moer-dijk and Verhaar, 1994; Moerdijk, Stwalley, et al., 1994)and 23Na (Cote and Dalgarno, 1994; Moerdijk and Ver-haar, 1994; Moerdijk and Verhaar, 1995), but the accu-racy is not as good as for later determinations by photo-association spectroscopy. Boesten et al. (1996) have usedspectroscopic data on K to calculate threshold scatteringproperties of 39K and 41K. Cote et al. (Cote and Dal-garno, 1997; Cote and Stwalley, 1997) show that revisedground-state potentials for the K2 molecule need to beused, and these give different scattering lengths than thecalculation of Boesten et al. (1996). The existence ofphotoassociation spectra for K2 should permit determi-nation of more accurate threshold scattering propertiesof both isotopes in the future.

The third way in which collisions are significant forBEC is the role of inelastic collisions, which change thestate of the trapped species. This process can produceuntrapped states, as well as cause heating by releasingkinetic energy in the inelastic process. For example, iftwo doubly spin-polarized atoms collide and produceone or two atoms in the lower hyperfine state, then oneor two units of the ground-state hyperfine splitting isgiven to the atoms, to be shared equally among them.Since this splitting is typically many hundreds of mK, theatoms easily escape the shallow traps designed to holdatoms at a few mK or less. We have already discussedhow these ‘‘bad’’ collisions can affect the process ofevaporative cooling. Fortunately, the ratio of ‘‘good’’ to‘‘bad’’ collisions is favorable in many cases, so thatevaporation actually works well and results in BEC. Onecase where evaporation appears to be unlikely due to‘‘bad’’ collisions is the case of doubly spin-polarized133Cs. Very recent experiments by (Soding et al., 1998)found that the inelastic rate coefficient for destruction ofdoubly spin-polarized 133Cs atoms is so large, aboutthree orders of magnitude larger than predicted by Ties-inga, Moerdijk, Verhaar, and Stoof (1992) that evapora-tive cooling of this level will be impossible. Leo et al.(1998) explain this unusually large inelastic collision rateas being due to a second-order spin-enhancement of theeffective spin-dipolar coupling. Evaporative cooling of


the lower hyperfine level, 133Cs F53, M523 may stillbe feasible, given the relatively low collisional destruc-tion rate for this level (Monroe et al., 1993). The sign ofthe scattering length for this level is still not certain.

Measuring an inelastic collision rate in a condensate,compared to the corresponding collision rate in a ther-mal gas, provides a way to probe the quantum mechani-cal coherence properties of the condensate. Burt et al.(1997) have shown how this can be done. Although thefirst-order coherence of a condensate can be measuredby observing the interference pattern of two overlappingcondensates (Andrews et al., 1997), collisions probehigher-order coherence properties related to the differ-ent nature of the density fluctuations in a thermal gasand in a condensate. For example, for ordinary thermalfluctuations, the average of the square of the density istwo times larger than the square of the average density,whereas, for a condensate, these two quantities areequal. The second- and third-order coherence functions,g(2)(0) and g(3)(0), respectively, measure this effect,where the argument 0 implies that two or three particlesare found at the same position; g(2)(0)52! and g(3)

3(0)53! for thermal gases, and both are 1 for a con-densate. Thus atoms are bunched in a thermal source,but not in a condensate, analogous to photons in a ther-mal source and a laser. Kagan et al. (1985) suggestedthat the three-body recombination rate might provide away to measure this property, since the rate coefficientfor three-body recombination would be 3!56 timessmaller in a condensate than in a thermal gas. Burt et al.(1997) in fact have done just this, measuring g(3)(0)57.4(2) by comparing the measured three-body recom-bination rates for thermal and condensed 87Rb in theF51, M521 level.

Condensate coherence can also be probed with two-body inelastic collisions as well as three-body ones, asdescribed by Ketterle and Miesner (1997). Stoof et al.(1989) use a collision theory viewpoint to show the dif-ference of collision rates in a thermal gas and a conden-sate, showing that the corresponding rate for a two-bodycollision in a condensate is 2 times smaller than for thethermal gas. Burnett et al. (1996) point out that a con-densate will have a two-body photoassociation spec-trum, which will also show the factor of 2 decrease inrate coefficient relative to a thermal gas. In addition, acondensate photoassociation spectrum would probe thetwo-body part of the many-body wave function in muchmore detail than an overall collision rate, since it probesthe wave function over a range of interparticle separa-tions instead of just yielding g(2)(0), as an overall colli-sion rate does. Photoassociation should be readily ob-servable in a condensate, since at the frequencies ofphotoassociation lines, the light absorption rate due totwo-body photoassociation at typical condensate densi-ties will greatly exceed the light scattering rate by freeatoms. An interesting question about photoassociationin a condensate is whether a three-body spectrum couldbe observed in which excited triatomic molecules areformed from three nearby ground-state atoms. If so, this


could provide a means for a finer grained probing ofthree-body effects in a condensate.

Inelastic collisions, including three-body recombina-tion, are important for condensates and condensate for-mation because these can lead to heating or removal ofatoms from the system. We discussed above how ‘‘bad’’collisions can affect evaporative cooling. The lifetime ofa condensate itself will be determined by collision pro-cesses. If the vacuum is not sufficiently low, hot back-ground gas species can collide with trapped or con-densed atoms, thereby transferring momentum to theatoms. Since the background gas atoms typically haveenergies on the order of 300 K, the cold atoms receiveenough momentum to be ejected from the trap. There isalso the possibility that some glancing collisions maytransfer a very slight amount of momentum, producinghot but still trapped atoms. This could be a source ofheating processes of unknown origin that have been ob-served in magnetic atom traps (Monroe, 1993; Meweset al., 1996). Even if the vacuum is good enough, inelas-tic collisions among the trapped species themselves canlimit the lifetime of the trapped gas or condensate. If thecollision rate coefficient for the destructive process isKin

(k) for a k-body collision, the trap lifetime is(Kin

(k)nk)21. For a nominal 1014 cm23 atom density in acondensate, a one-second lifetime results from a two-body rate coefficient of 10214 cm3 s21 or a three-bodyrate coefficient of 10228 cm3 s21. These rate coefficientswill be very dependent on the species and the particularhyperfine level which is trapped. The only hyperfine lev-els for which trapping and condensation are possible arethose for which the inelastic rate coefficients are suffi-ciently low.

We noted above how two-body inelastic collision ratecoefficients must be small in relation to elastic collisionrate coefficients in order for evaporative cooling towork. This is true for the F52, M52 87Rb and 7Li spe-cies and F51, M521 23Na species that have been con-densed. The inelastic rate is small for the doubly spin-polarized species for the reasons discussed in Sec. VIIIabove. It is small for F51, M521 collisions for basi-cally the same reason. An inelastic collision requires aweak spin-dipolar mechanism, since the sum of atomicM is not conserved, and, additionally, the exit channel isa d-wave for an s-wave entrance channel. The small am-plitude of the threshold d-wave leads to very small col-lisional destruction of F51, M521 for weak magneticfields. The success of sympathetic cooling, and observa-tion of a dual condensate of 87Rb F51, M521 and F52, M52 (Myatt et al., 1997), raises the obvious theo-retical question of why the inelastic collision rate coeffi-cient for the destructive collision of these two specieswas found to be so small, 2.8310214 cm3 s21. This in-elastic process, which produces two F51 hot atoms,goes by the spin-exchange mechanism, and normallywould be expected to be several orders of magnitudelarger. Three theory groups immediately answered thepuzzle by showing that this observation meant that 87Rbhad a very special property, namely the scatteringlengths of both the 1Sg and 3Su states are nearly the


same, and in fact, all scattering lengths between any twohyperfine levels of 87Rb are nearly the same (Julienneet al., 1997; Kokkelmans, Boesten, and Verhaar, 1997;Burke, Bohn, et al., 1997a, 1997b). The existence ofnearly identical scattering lengths is a sufficient condi-tion for the inelastic rate coefficient to be as small as itis. It is not a necessary condition, since a threshold scat-tering resonance could also lead to a low inelastic ratecoefficient. Such a resonance does not exist for 87Rb,however. Julienne et al. (1997) pointed out that reconcil-ing the existing data on 87Rb required that the scatteringlengths for collisions between any two pairs of F52, M52 or F51, M521 differ by no more than 4a0 andhave a value of 10365a0 . Both Julienne et al. (1997)and Kokkelmans, Boesten, and Verhaar (1997) calcu-lated that the inelastic collisional destruction rate coef-ficient for collisions of 23Na F52, M52 and F51, M521 would be 3 to 4 orders of magnitude larger thanthe one measured for 87Rb. Consequently a dual speciescondensate would be impossible for 23Na. The issue ofinelastic collision rate coefficients is crucial for the pros-pects of sympathetic cooling, which offers an attractivepath for cooling species that cannot be cooled evapora-tively, and needs to be investigated for mixed alkali spe-cies. For example, cooling is desirable for spin-polarizedfermionic species such as 6Li, which may exhibit inter-esting Cooper pairing effects in the quantum degenerateregime (Stoof et al., 1996). Spin-polarized fermionic spe-cies cannot be cooled evaporatively, since only p-wavecollisions are allowed, and these are strongly suppressedat low T (see Sec. III.B).

Two-body rate coefficients for inelastic processes tendto be small for species that can be condensed, since oth-erwise the bad collisions will limit the trap density. Butas density increases, three-body collisions will eventuallyprovide a limit on trap density and lifetime. Three-bodycollisions produce a diatomic molecule and a free atom.These two products share the kinetic energy releaseddue to the binding energy of the molecule. This is usu-ally enough energy that the particles do not remaintrapped; in any case, the molecule is unlikely to betrapped. Three-body collisions for spin-polarized hydro-gen were studied by De Goey et al. (1986, 1988). Thecollision rate coefficient is unusually small for this sys-tem, since the ground-state triplet potential does notsupport any bound states in which to recombine, andmaking ground-state singlet molecules requires a veryweak spin-dipolar transition. The rate coefficient for al-kali systems is orders of magnitude larger than for hy-drogen, since the triplet potentials support severalbound states with small binding energy. Moerdijk, Boes-ten, and Verhaar (1996) calculate the three-body ratecoefficients for doubly spin-polarized 7Li, 23Na, and87Rb to be 2.6, 2.0, and 0.04310228 cm3 s21, respec-tively. Moerdijk and Verhaar (1996) differ from the sug-gestion of Esry et al. (1996) that the three-body rate co-efficient may be strongly suppressed at low T . Fedichev,Reynolds, and Shlyapnikov (1996) give a simple formulabased on the scattering length for the case when the lastbound state in the potential is bound weakly enough.


Burt et al. (1997) note that this theory gives the magni-tude of their measured three-body rate coefficient forF51, M521 87Rb.

One of the more interesting prospects for tailoring thecollisional properties of ground-state species is to makeuse of an external field to modify the threshold collisiondynamics and consequently change either the sign ormagnitude of the scattering length or modify inelasticcollision rates. This prospect was raised by Tiesinga,Verhaar, and Stoof (1993), who proposed that thresholdscattering properties for 133Cs could be changed by amagnetic field. This is possible because of the rapidvariation in collision properties associated with a thresh-old scattering resonance. For example, a magnetic fieldcan move a molecular bound state to be located at justthe threshold energy for the collision energy of two lev-els of the lower hyperfine manifold. Moerdijk, Verhaar,and Axelsson (1995) discussed the role of resonances for6Li, 7Li, and 23Na. Although an experimental attempt tolocate predicted resonances in 87Rb was unsuccessful(Newbury et al., 1995), there is no doubt that such scat-tering resonances will exist. The question is whetherthey exist in experimentally accessible regimes of mag-netic fields. Using much more refined calculations ofthreshold scattering derived from photoassociation spec-troscopy, Vogels et al. (1997) make specific predictionsthat resonances in scattering of the F52, M522 lowerhyperfine level of 85Rb will occur in experimentally ac-cessible ranges of magnetic field. Boesten et al. (1996)calculate that 41K may offer good prospects for magneticfield tuning of the scattering length. The MIT BECgroup, led by W. Ketterle (Inouye et al., 1998) has re-cently reported an observation of magnetically inducedFeshbach resonance effects on condensate mean-fieldenergy and lifetime for Na F51, M511 confined in anoptical trap.

External fields other than magnetic ones can alsochange the scattering lengths. Fedichev, Kagan, et al.(1996) proposed that an optical field tuned near reso-nance with a photoassociation transition can be used tovary the ground-state scattering length but not cause ex-cessive inelastic scattering. Napolitano, Weiner, andJulienne (1997), while investigating optical shielding(Sec. VII), calculated that large changes in elastic scat-tering rates can be produced by light detuned by about100 natural linewidths to the blue of atomic resonance.This was partly because the optically induced mixingwith the ground state of an excited repulsive excitedstate with the 1/R3 long-range form greatly changed theground-state collision by inducing contribution from an-gular momenta other than s waves. The problem withusing light to change ground-state collision rates is thatthe light can also induce harmful inelastic processes aswell, and the free atoms also scatter off-resonant lightand experience heating due to the photon recoil. Sucheffects must be minimized in order for optical methodsto be practical. Bohn and Julienne (1997) considered theproposal of Fedichev, Kagan, et al. (1996) in more detailfor the cases of 7Li and 87Rb, giving simple formulas forestimating the light-induced changes, and showed that


there may be ranges of intensity and detuning whereuseful changes can be effected. Finally, the radiofre-quency (rf) fields used in evaporative cooling can changecollision rates. Agosta et al. (1989) showed how thestrong rf field in a microwave trap will modify collisions.Moerdijk, Verhaar, and Nagtegaal (1996) showed thatthe rf fields used in evaporative cooling would makenegligible changes in the collision rates of F51, M521 23Na or F52, M52 87Rb in a magnetic trap. Suom-inen, Tiesinga, and Julienne (1998) concur with thisanalysis for these species, but show that typical rf fieldsfor evaporative cooling can cause enhanced inelastic col-lisional relaxation of F52, M52 23Na. This is becauserf-induced nonadiabatic transitions due to motion in thetrap lead to production of other M levels, which decaywith very large spin-exchange rate coefficients (Kokkel-mans, Boesten, et al., 1997; Julienne et al., 1997).

One species that has been suggested as a viable can-didate for BEC is the metastable 3S1 state of the Heatom (Shlyapnikov et al., 1994). This long-lived speciescan be cooled and trapped (Bardou et al., 1992), andFedichev, Reynolds, Rahmanov, and Shlyapnikov(1996) calculated that the collisional ionization rate co-efficient is so small for the J51, M51 level that a po-larized gas of such a species might be stable long enoughto make trapping and condensation possible. The polar-ized gas is stable because a collision of two j51, m51atoms only occurs on the 5Sg potential of the He2 dimer,for which Penning ionization is forbidden. A gas ofmetastable atoms is only stable if complete polarizationis maintained. An unpolarized sample would rapidly de-stroy itself due to very fast Penning ionization collisions(Julienne and Mies, 1989; Bardou et al., 1992). The Pen-ning ionization rate coefficient for other spin-polarizedJ52 metastable noble gases is not known. Recent ex-periments on cold, trapped J52, M52 Xe metastableatoms indicate that collisional ionization of the polarizedgas is comparable to that for the unpolarized gas(Rolston, 1997).

IX. FUTURE DIRECTIONS

At this writing (March 1998) cold and ultracold colli-sions continue to stimulate the imagination of research-ers with new avenues of investigation opening at an ac-celerating pace. In this section we briefly sketch sometopics that appear particularly promising and significant.

Collisions in a light field present several intriguing ar-eas of investigation. For example proposals for coolingand trapping molecules have recently appeared (Bandand Julienne, 1995; Doyle et al., 1995; Bahns et al., 1996;Cote and Dalgarno, 1997) that would extend a Doppler-free precision spectroscopy to species not susceptible todirect optical cooling and may provide the starting pointfor building trapped clusters. Fioretti et al. (1998) re-cently reported the observation of translationally coldCs2 molecules that fell out of a MOT following photoas-sociation of cold Cs atoms. Julienne et al. (1998) proposethat cold molecules can be efficiently made by a stimu-lated Raman process in a Bose-Einstein condensate.


The prospects for producing cold molecules raises nu-merous issues about collisions between two cold mol-ecules, or a cold molecule and a cold atom (Balakrish-nan, Forrey et al., 1997; Balakrishnan, Kharchenko,et al., 1997). Multiple-color experiments, in whichdouble-resonance optical excitation accesses ‘‘pure long-range’’ intermediate states, have begun the spectro-scopic characterization of heretofore inaccessible highlyexcited and cold molecular Rydberg states (Wang,Gould, and Stwalley, 1998b). Two recent reports de-scribe the production of a ‘‘frozen Rydberg gas’’ afterexcitation of atomic Rydberg states in cold Cs (Mou-rachko, et al. 1998) and Rb (Anderson, et al., 1998).These experiments reveal novel many-body phenomenain a dilute gas. Collisions within and between highly col-limated, dense, bright atomic beams reveal alignmentand orientation features of ultracold collisions that areusually obscured by spatial averaging in trap collisions.A few steps have already been taken in this direction(Weiner, 1989; Thorsheim et al., 1990; Wang andWeiner, 1990; Tsao et al., 1998), but these only indicatethe potential for future studies. For example atomicbeam collisions will lead to a simplification of the analy-sis of photoassociation spectra through alignment andspin polarization (Gardner, et al., 1995). Spin orientationeffects suppressing Penning ionization in He metastableand other rare-gas metastable collisions can be also di-rectly probed in beams. Doery et al. (1998) have calcu-lated the long range-potentials of noble gas dimers,showing the existence of pure long-range states thatcould be probed by photoassociation spectroscopy. Ties-inga et al. (1998) have developed computational meth-ods that treat explicitly nuclear and electron spin andpermit accurate calculations of molecular hyperfinestructure in photassociation spectra. Atom associationbetween mixed species confined in ‘‘dual MOTs’’ willextend precision molecular spectroscopy from homo-nuclear to heteronuclear diatomic species. Initial studieswith dual MOTs are already under active investigation(Santos et al., 1995; Shaffer et al., 1997). Development ofconvenient laser sources in the blue and near-UV re-gions of the spectrum will more readily permit coolingand trapping of Group IIA atoms such as Mg, Ca, Sr,and Ba. The common isotopes of the alkaline earthshave no nuclear spin and therefore exhibit no hyperfineinteraction. Photoassociation spectra and trap-loss spec-tra should therefore be interpretable with a rigorous,unifying, quantitative theory, including effects of stronglight fields and spontaneous emission. The study of op-tical shielding and suppression in nuclear-spin-free sys-tems will also permit rigorous comparison of experimentand theory. Up to the present most cold and ultracoldcollision measurements have been carried out in the fre-quency domain. Precision molecular spectroscopy hasbeen the dominant influence. Some collision processes,however, lend themselves to the time domain; the obser-vation of ultracold collisions in real time (Boesten et al.,1996; Gensemer and Gould, 1997; Orzel et al., 1998) andwith wave packets (Varde et al., 1997) may open a fruit-


ful new avenue for studies of dynamics and moleculeformation complementary to high-precision frequencyspectroscopy.

Ground-state collisions will continue to play an in-valuable role as a probe of dilute, gas-phase quantumstatistical condensates. Output coupling of confinedBECs for the purpose of realizing beams of atoms in asingle quantum state, ‘‘atom lasers’’ (Mewes, et al., 1997)will need to address the issue of collision-limited coher-ence times. Details of scattering lengths between mixedspecies will determine the feasibility of sympatheticcooling (Myatt et al., 1997) as a useful technique for ex-tending BEC to a greater range of atomic or molecularsystems. For example, Burke et al. (1997a, 1997b) dis-cuss how 87Rb can be used for sympathetic cooling ofthe 85Rb isotope, which is difficult to cool directly be-cause of a small elastic collision cross section in the rel-evant temperature range. Control of the sign and mag-nitude of the scattering length by magnetic (Vogelset al., 1997; Burke et al., 1997a, 1997b; Inouye et al.,1998), rf, or optical field (Fedichev, Kagan, et al., 1996)manipulation will be the focus of intensive efforts be-cause a large positive scattering length provides the onlydemonstrated route to BEC via evaporative cooling.

Although we have discussed the principal future di-rections of which we are aware, the field of cold andultracold collisions is presently germinating new ideas atsuch a rate that any assessment purporting to character-ize the overall direction or emphasis even in the nearterm would be foolhardy. The only certitude one canexpress without fear of contradiction will be the need toupdate and supercede this review with another in thenear future.

ACKNOWLEDGMENTS

Financial support from the National Science Founda-tion, the Army Research Office, the Office of Naval Re-search, the FAPESP, CNPq, and CAPES is gratefullyacknowledged.

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