Background Information for lectures on High …indico.ictp.it/event/a0111/material/1/17.pdfThese are preliminary lecture notes, intended only for distribution to participants. strada

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SMR1302- 19

WINTER SCHOOL ON LASER SPECTROSCOPY AND APPLICATIONS

19 February - 2 March 2001

Background Informationfor lectures on

High Resolution Spectroscopy and Laser Cooling

M. INGUSCIOL.E.N.S. - Lab. Europeo di Spettroscopie Non Lineari

Largo Enrico Fermi, 2 - Firenze, Italy

These are preliminary lecture notes, intended only for distribution to participants.

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SPECTROSCOPY, LASERMASSIMO INGUSCIO, Department of Physics of the University, Firenze, Italy

ANTONIO SASSO, Department of Physics of the University, Napoli, Italy

Introduction 371

1. Widths of Spectral Lines 374

1.1 Mechanisms of LineBroadening 374

1.1.1 Homogeneous Broadening 374

1.1.2 Inhomogeneous Broadening ... 3752. High-Resolution

Spectroscopy 375

2.1 Saturation Spectroscopy 377

2.2 IntermodulatedSpectroscopy 380

2.3 Polarization Spectroscopy 381

2.4 Velocity-Selective Optical-Pumping Spectroscopy 383

2.5 Polarization-IntermodulatedExcitation Spectroscopy(POLINEX) 384

2.6 Multiphoton Spectroscopy 386

2.6.1 Selection Rules for Two-Photon Transitions 387

2.6.2 Transition Probabilities andLine Shapes for Two-PhotonTransitions 387

2.6.3 Doppler-Free MultiphotonTransitions 390

2.6.4 Applications of High-Resolution SpectroscopicTechniques to the HydrogenAtom: Lamb-Shift andRydberg-ConstantMeasurements 390

3. High-Sensitivity LaserSpectroscopy 391

3.1 Excitation Spectroscopy 3943.2 Optoacoustic Spectroscopy .... 3953.3 Optogalvanic Spectroscopy .... 3953.4 Intracavity Spectroscopy 3983.5 Fast Modulation

Spectroscopy 4004. Time-Resolved

Spectroscopy 4024.1 Lifetime Measurements 4024.1.1 Phase-Shift Method 4024.1.2 Pulse Excitation 4034.2 Quantum-Beat Spectroscopy .. 4035. Ultrahigh-Resolution

Spectroscopy 4055.1 Laser Cooling 4075.2 Atom Traps 4085.2.1 Magneto-optical Traps 408

Glossary 410Works Cited 411Further Reading 412

INTRODUCTION

Most of what we know of atomic and mo-lecular structure has been derived from spec-troscopic analysis of the radiation emittedwhen atoms or molecules are excited. Frommeasurements of the wavelengths and inten-sities of observed spectral components, pio-neering spectroscopists, with diligent and pa-tient work, laid the basis upon which theexploration of the physics of microsystemswould begin.

The foundation of spectral analysis dates

back to about 1860, with Kirchhoff and Bun-sen. The first instruments for wavelengthmeasurement were spectrographs, which, intheir modern version, continue to occupyan important position in the spectroscopylaboratory. The sample under investigationwould be suitably excited, generally bymeans of an electrical discharge, and theemitted light dispersed by prisms or diffrac-tion gratings. Finally, the emergent light wassent to a photographic plate upon which thespectra were recorded [excitation spectros-copy, Fig. l(a)]. Alternatively, light from a

Encyclopedia of Applied Physics, Vol. 19 © 1997 VCH Publishers, Inc.3-527-29300-0/97/$5.00 + .50

371

372 Spectroscopy, Laser

lenslens

sample

dispersingelement screen

(photoplate)

lens

(a)

lens

lens

dispersingelement

sample

slit

(b)

detector

FIG. 1. Principle of (a) excitationand (b) absorption spectroscopy. In(a), the sample is excited, for in-stance, by an electrical discharge,and the emitted light sent into aprism spectrograph. In (b), lightfrom a lamp is dispersed and thesample then investigated using onlywavelengths that are resonant witha transition of the atomic or molec-ular species.

lamp could be selected in wavelength andused to record the absorption of the sample[absorption spectroscopy, Fig. l(b)].

The first approach to treating the largeamount of information accumulated duringthe early decades of these investigations con-sisted of finding some regularity in the se-quences of spectral lines. Liveing and Dewar,in around 1880, emphasized physical simi-larities between the spectral lines of alkalielements and overtones in acoustics. In1883, Hartley found an important numericalrelationship that allowed, from the largenumber of lines in any given spectrum, theisolation of those groups of lines (multiplets)that were undoubtedly related. It soon be-came evident that the frequency v was morerepresentative than the wavelength A. Never-theless, because we measure directly not fre-quency but wavelength, it is customary inspectroscopy to use, instead of the frequencyitself, the wave number v = I/A.

Our knowledge of spectral series formulasdates from a discovery by Balmer, in 1885.He reproduced very closely the wavelengthsof the nine then-known lines in the visiblespectrum of hydrogen (Balmer series) usinga simple formula:

n being a variable integer number assumingthe values 3, 4, 5, . . . for, respectively, thefirst, second, third, and so on lines in thespectrum. Other series of hydrogen were in-vestigated in the ultraviolet by Lyman in1906 and, two years later, in the infrared byPaschen.

A more general formula applicable toother series and elements was found by Ryd-berg. Using the comparatively large mass ofwavelength data then available for alkali ele-ments, Rydberg found that the wave num-bers of the observed spectral sequence weredescribed by the formula

= v - RJ(n + /x)2(2)

An = 3645.6n2/(n2 - 4) A, (1)

where /JL and v are constants depending uponthe series, v representing the high-frequencylimit to which the lines in the series con-verge. The constant R^ in Eq. (2), now calledthe Rydberg constant, was found to be inde-pendent of the series considered and showedslight variations from one atom to another.The level scheme of the hydrogen atom isshown in Fig. 2, and a photograph of theBalmer spectral series is shown in Plate 1.

As is well known, at the beginning of thiscentury many experimental observations notexplicable in terms of classical physics led to

Spectroscopy, Laser 373

"1

0 n

.Confini

itinuutn

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

90,000

100,000

110,000

so

3

Paschen

3S

inBalmer

n00

frequency of the emitted radiation is deter-mined by the condition that

- 4

-3

ft t001(0

Lyman

FIG. 2. Energy-level diagram of the hydrogen atom,showing some of the lines of the principal series.

the formulation of quantum mechanics. Inparticular, Niels Bohr was to apply the ideaof quantum mechanics to the hydrogenatom, arriving at a theoretical formula for itsspectrum that agreed with observations andwith the empirical formulas of Balmer andRydberg. According to Bohr's model, elec-trons move around the nucleus with definiteenergies. When an electron makes a transi-tion from one allowed stationary state withenergy Eu to another of lower energy Eh the

hv = Eu- Eh (3)

where h is the Planck constant (h = 6.63 X10~34 J s). Conversely, it is necessary to ab-sorb a quantum of electromagnetic energy—a photon—with energy hv in order to inducea transition from a lower- to a higher-energylevel. Measurements of the wavelengths ofspectral lines are therefore related to the sep-arations between energy levels, while themeasured intensities are proportional to thetransition probabilities.

From this basic idea of Bohr, experimen-tal and theoretical investigations of atomicand molecular structure have evolved consid-erably over the intervening decades. Newerand more accurate experimental observa-tions have stimulated the development ofdeeper and more sophisticated theories, andtheoretical predictions have, in their turn,challenged experimentalists to seek ever finerstructure in the spectra of atoms and mole-cules. This process gained a stimulating im-petus from the discovery of the laser, whichopened a new era in spectroscopy. On onehand, laser sources have allowed a return toclassical spectroscopic techniques with an in-crease in spectral resolution and sensitivityof several orders of magnitude. More impor-tantly, though, the many peculiarities of la-ser radiation (high intensity and spectral pu-rity, short light pulses, etc.) have made itpossible to explore the interaction betweenradiation and matter in a completely newlight. Many experiments that could not beperformed with classical radiation are nowfeasible, and new techniques based upon co-herent light have reached, in the spectra ofatoms and molecules, the ultimate limits ofresolution.

Besides fundamental physics, spectro-scopic investigations are of direct impor-tance in fields such as astrophysics, combus-tion, and plasma physics and in anincreasing variety of applications. Optical ab-sorption and emission spectroscopy, for ex-ample, are used for the remote sensing of at-mospheric pollutants and for medical tests.

In the following pages, the main tech-niques of modern laser spectroscopy are ex-amined. For discussions in detail, the readeris referred to more specific and exhaustive


textbooks as, for instance, Corney (1977),Demtroeder (1981), Svanberg (1992), and Le-tokhov and Chebotayev (1977). In this workthe physics of the laser is not treated; for areview of this subject, we suggest the follow-ing excellent books: Siegman (1971), Svelto(1976), and Yariv (1975).

1. WIDTHS OF SPECTRAL LINES

The more deeply we understand the struc-ture of an atom or molecule, the higher isthe resolution with which we need to exam-ine its absorption and emission spectra. Thelimit of resolution is determined by thewidth of the spectral lines. According tothe Rayleigh criterion, two close lines withequal intensities are resolved if the dip pro-duced in the overlapping profile drops to 0.8of the maximum intensity. In general, thespectral resolving power R of any dispersinginstrument is defined by

R = \/A\ = vlAv (4)

where AX = A2 — Xx is the minimum sepa-ration of two closely spaced lines that arejust resolved.

The primary limit to the spectral resolu-tion is usually the optically dispersing ele-ment used in the spectroscopic apparatus.Even if the experimental resolution is ex-tremely high, however, spectral lines will stillexhibit an intrinsic breadth. In other words,the lines in discrete spectra are never strictlymonochromatic but show a spectral distri-bution I{v) around the central frequency v0

= (Eu — Ei)/h, Eu and £/ being the energiesof the upper and lower levels of the transi-tion under investigation. The width of such aline profile is usually expressed in terms ofthe so-called full width at half maximum(FWHM), which corresponds to the fre-quency interval Av where

I(v0 - Avll) = I(v0 + Avll) =

1.1 Mechanisms of Line Broadening

(5)

There are several mechanisms that causea broadening of the emission or absorptionlines of atoms or molecules. These are usu-ally divided into two categories, homoge-neous and inhomogeneous broadening, de-

pending upon whether the probability ofabsorption or emission of radiation with fre-quency v is equal (homogeneous) or is notequal (inhomogeneous) for all the atoms ofthe sample.

1.1.1 Homogeneous Broadening Thenatural (also named radiative or spontane-ous) width is an example of homogeneousbroadening and represents the intrinsicbreadth of a spectral line. The finite lifetimeT of an excited level gives rise, as describedby the uncertainty principle of Heisenbergthat AEAt ^ /z, to an uncertainty in the en-ergy of the level. As a consequence, the ra-diation emitted when an electron jumpsfrom that level to the ground state is com-posed of different frequencies, and the rela-tive frequency distribution gives rise to a lineshape described by a Lorentzian curve,

= I(vo)L(v -- v0) + (y/2)2], (6)

where 70 is the total intensity of the line andy — 2irAvnat = \ITU is the homogeneouswidth (FWHM). The line shape L{y - v0) isnormalized; i.e., the area under the curve isunity.

If the lower level of the transition is alsoradiative with a lifetime 77, then the radiativewidth becomes

Avn3Lt — 1/2TTTW + 1/2TTT/. (7)

Typically, the lifetime of radiative levels is ofthe order of tens of nanoseconds, and hencetypical natural widths are of a few MHz.

Transitions do not necessarily occur be-tween any arbitrary pair of energy levels be-cause of general selection rules derivingfrom the conservation of energy, angularmomentum, parity, etc. In general, such lev-els are said to be metastable and the lineshape of the transition connecting them isextremely narrow. Such atoms usually decaythrough nonradiative processes, such as col-lisions with other atoms or with the walls ofthe gas container. For this reason, collisionsalso play an important role in the shape ofspectral profiles. Collisions can be inelasticor elastic, depending upon whether the inter-nal energy of the colliding partners is modi-fied or not after the collision. Elastic colli-sions affect the phase but not the amplitude


of the wave function of the atomic state. Thetotal effect of the collisions is to produce ashift and a broadening of the line shape. Col-lisional (or pressure) broadening is propor-tional to the pressure p of the collision part-ner species, and the total homogeneouswidth is given by the sum of two terms:

7hom = Trad + Tcoll = Trad + (8)

In some cases, the interaction time of atomswith the radiation field can be shorter thanthe spontaneous lifetime, giving rise to time-of-flight broadening. In such circumstancesthe linewidth is determined by the interac-tion time r, and in the limiting case thewidth becomes

y = \I2ITT. (9)

1.1.2 Inhomogeneous BroadeningThe main cause of inhomogeneous broaden-ing for atoms and molecules in the gas phaseis the Doppler effect. The central frequencyvQ emitted by an excited atom moving withvelocity v relative to the rest frame of the ob-server will be shifted by a quantity Av givenby

— V — y0 = (10)

where the wave vector k (Ikl = 2TT/A) indi-cates the direction of the emitted photon. Atthermal equilibrium the molecules of a gasfollow a Maxwellian velocity distribution.The whole line shape is hence given by thesuperposition of differently Doppler-shiftedemitters, and the observed line shape is aGaussian:

where the FWHM is given by

Avo = 7.16 X lO-7vo(T/M)1'2

(11)

(12)

with T the gas temperature (expressed in kel-vins) and M the molar mass (expressed ingrams). For the sodium Dx line, the Dopplerwidth is, at T = 500 K, AvD = 1.7 GHz,about two orders of magnitude broader thanthe radiative width. Since the Doppler widthis proportional to the frequency v, line

shapes of transitions in the infrared and mi-crowave spectral regions are, in practice, lit-tle affected, while in the optical domain (thevisible and ultraviolet) the Doppler effect isthe dominant broadening mechanism.

From the previous discussion we can con-clude that homogeneous and inhomogeneousbroadening are governed by two line profiles:the Lorentzian and the Gaussian. Althoughthe two curves appear similar, significant dif-ferences appear if we compare two curveshaving the same half-width, as shown inFig. 3.

In our discussion of Doppler broadening,we have neglected the homogeneous broad-ening of each velocity class of atoms. Thismeans that we should properly write thewhole line shape as a superposition of differ-ent Lorentzian curves, each centered at theDoppler-shifted frequency and weighted by aGaussian curve. The result of such a convo-lution of a Lorentzian and a Gaussian iscalled a Voigt profile (see Fig. 4). In practice,when AVQ > Avhorn> the line shape is well de-scribed by a Gaussian curve.

2. HIGH-RESOLUTION SPECTROSCOPY

The intrinsic width of a spectral line isdetermined by spontaneous emission. It maybe desirable to reach such a limit experimen-tally for two main reasons: simply because itrepresents the highest resolution possibleand to be able to investigate properties ofisolated atoms in the absence of externalperturbations (other than the laser radiationused for the experiment itself). Especiallywhen the gaseous sample under investigationis rarefied in order to render collisions neg-ligible, the Doppler effect will be the first ob-stacle to overcome.

At first sight, the easiest approach to re-ducing Doppler broadening might appear tobe by cooling the sample. However, the slowdependence of AvD upon temperature [seeEq. (12)] means that, for instance, to reduceAvD by two orders of magnitude, the gastemperature must be decreased by four or-ders of magnitude. This is practically feasibleonly for atomic and molecular species withhigh vapor pressures.

Another approach that has been used toreduce Doppler broadening is the use ofatomic beams. If a well-collimated beam of


FIG. 3. Comparison between Lorentzian (homogeneous broadening) and Gaussian (inhomogeneous broaden-ing) line shapes having equal maxima.

particles is irradiated perpendicularly by res-onant radiation, then the first-order Dopplereffect is removed (k-v = 0). Because of thefinite collimation of the beam, of course, theDoppler effect will still be present due tothe small velocity component perpendicularto the atomic beam. If 6 is the beam diver-gence, the residual Doppler effect is given by

It is worth noting that the relativistic sec-ond-order Doppler effect, depending upon (v/c)2, is not eliminated even with a perfectly

collimated beam. This quadratic Doppler ef-fect can play an important role in ultrahigh-resolution spectroscopy, and we shall seelater some novel techniques based upon lasercooling of the atoms that aim to overcomethis limit. It should also be noted that, be-cause the atoms interact with the laser radi-ation downstream of the nozzle, the methodof atomic beam spectroscopy is not appro-priate for the study of species in excitedstates with short lifetimes.

Doppler-free resolution is also possible us-

FIG. 4. Voigt profile obtained fromthe convolution of Gaussian andLorentzian profiles (from Demtroe-der, 1981).


ing classical light sources. This has beenachieved, for instance, using double-reso-nance schemes in which two successive tran-sitions, one in the visible and another in theradio-frequency region, are excited simulta-neously (see Fig. 5). The radio-frequencytransition will be between two closely spacedenergy levels, such as fine-structure or hyper-fine levels; the Doppler width of these tran-sitions, as explained previously, will be neg-ligible, and the levels' spacing, if not theirabsolute positions, may thus be measuredwith high resolution.

Level splittings may also be investigatedwith the methods of level-crossing (Hanle ef-fect) or quantum-beat spectroscopy. Thesetechniques are based on the interference be-tween different atomic radiative transitionchannels, which lead to a spatial distributionof the intensity and polarization of the emit-ted radiation. The need for radiation of highspectral purity is avoided, and these tech-niques are particularly valuable for accuratemeasurements of fine and hyperfine struc-ture and of the Zeeman and Stark splittingsof atomic and molecular levels.

The advent of tunable laser sources hasallowed the development of nonlinear tech-

niques that eliminate Doppler broadeningwithout suffering from these restrictions.They are therefore competitive with andcomplementary to the older spectroscopicmethods and have led to a number of novelspectroscopic studies that would otherwisehardly have been possible. These techniquesare applicable to atoms and molecules incells and hence have an immediate practicaladvantage with respect to beam methods,which require complex and expensive appa-ratus.

Doppler-free laser spectroscopic tech-niques are mainly based on two differentschemes: saturation spectroscopy and Dopp-ler-free two-photon spectroscopy. In the for-mer, narrow-band cw laser radiation "marks"a small group of atoms within a narrowrange of axial velocities. In two-photon spec-troscopy an atom undergoes a transition be-tween two quantum states of the same paritythrough the absorption of two photons,whose energies add up to the required en-ergy. If the photons are absorbed from coun-terpropagating beams, the Doppler effect iscancelled. The basic principles of these sub-Doppler techniques are outlined using arather simple approach in the following par-agraphs.

radio-frequency

visible radiation

1

FIG. 5. Level scheme for double resonance involvingoptical and radio-frequency radiation.

2.1 Saturation Spectroscopy

We start with a single optical transitionbetween two energy levels 1 and 2. Initially,we suppose that such a transition is homo-geneously broadened. We recall that in linear(low intensity) absorption spectroscopy, theabsorption of the intensity / as a function ofthe optical thickness z of the sample is givenby the well-known Beer-Lambert law

I{z) =/(O)exp(-oz), (13)

where a is the absorption coefficient, a quan-tity depending upon microscopic propertiesof the medium under investigation. For lowlight intensities, the population densities ofthe upper (N2) and lower (Nx) levels of theinvestigated transition are only slightlychanged, and the coefficient a is thus inde-pendent of intensity:

a = cr^N, - (g^)^]. (14)

where cr12 represents the absorption cross


section and gx and g2 the degeneracies of theupper and lower levels, respectively. Athigher intensities, N1 may decrease signifi-cantly while the upper-state density N2 in-creases, ultimately rendering the atomicsample completely transparent (AN = N2 -Ni = 0). Since N2 and Nx are both functionsof intensity, the absorption coefficient itselfbecomes nonlinearly intensity dependent. Inthis case, it can be shown that the popula-tion difference AN can be written as

AN = ANM + S(v)l (15)

where AN0 is the population difference in theabsence of laser radiation and S(v) is the so-called saturation parameter,

S(v) = So(y/2)2

- <o0f + (y/2)2 (16)

where y is the homogeneous breadth of thetransition and OJ = 2-irv the optical angularfrequency. The saturated absorption coeffi-cient as(v) thus becomes

- a>0)2 2 > (17)

where C is a constant. Comparing Eq. (17)with the unsaturated absorption profile ofEq. (14), we see that saturation decreases theabsorption by a factor 1 + S(v) and broad-ens the line profile by a factor (1 + S0)

1/2, a s

shown in Fig. 6.We now consider the effect of the motion

of the atoms. If the laser beam is a mono-chromatic wave of angular frequency co andwave vector k, the Doppler shift will mean

FIG. 6. Homogeneous line shape affected by satu-ration (from Demtroeder, 1981).

that only atoms having a velocity componenttoward the laser in the interval v ± Av givenby the relation

(v ± = a) — (o0 ± y (18)

will absorb. This causes a hole in the veloc-ity distribution of the lower level and a cor-responding bump in the upper-level distribu-tion. As we have seen, this group of atomswill undergo saturation, and hence the hole(or dip) will exhibit a saturated profile. If thelaser beam, after crossing the sample, is re-flected back into the cell, then two symmet-ric holes (or dips) will be produced in the ve-locity distribution. As the laser frequencyapproaches the resonant frequency v0 of thetransition, the two holes will collapse intoone, and the two counterpropagating beamswill simultaneously interact with the samegroup of atoms; see Fig. 7(a). Tuning the la-ser frequency across the Doppler width, itcan be shown (in the weak-field approxima-tion) that the absorption coefficient will begiven by

as((o) = aD((o)

- ^ 1(a> - (o0y +

(19)

FIG. 7. (a) Holes in the velocity distribution of thelower level produced by two counterpropagating laserbeams when the laser frequency is off resonanceand in resonance, (b) The absorption profile as afunction of the laser tuning.


which represents a Doppler profile modifiedby the presence of a minimum at the centerof the line [the Lorentzian term in Eq. (19)]as shown in Fig. 7(b).

In the previous analysis we have consid-ered an atomic system at thermal equilib-rium where Nx < N2. The same discussioncan be extended to systems where a popula-tion inversion has been produced. This is,for instance, the case of a laser medium withan inhomogeneously broadened gain profile.If the length of the single-mode laser cavityis tuned so that the laser frequency vx = v0,the output power will show a minimumaround the central frequency v0.

Historically, saturation spectroscopy atoptical frequencies dates from 1963 (threeyears after the discovery of the laser) whenMcFarlane et at (1963) and, independently,Szoke and Javan (1963) demonstrated thesaturation minimum at the center of theemission curve of a single-mode He-Ne la-ser. Nevertheless, saturation spectroscopy be-came extensively used as a spectroscopictechnique only with the advent of tunabledye lasers. Hansch et al (1971) developed anexperimental arrangement that provided aDoppler-free profile with a good signal-to-noise ratio. This scheme is shown in Fig. 8.The primary beam is divided in two using apartially reflecting mirror (beam splitter).One of the two beams has a higher intensitythan the other, by about a factor of 10, andis usually named the pump or saturatingbeam, while the weaker is called the probe

or analysis beam. The two beams are sent inopposite directions through the cell, and theintensity of the transmitted probe is detectedusing, for example, a photodiode. When thelaser is tuned across the absorption line, theprobe beam sees the hole burned by the sat-urating beam, and the recorded line shapeconsists of a Gaussian with a dip at the cen-ter. In order to isolate the saturating contri-bution, a phase-sensitive detection schemecan be used. For this purpose, the intensityof the pump beam is mechanically choppedat a given frequency—a few hundred Hz—and the signal processed using a lock-in am-plifier.

A typical saturation spectrum is shown inFig. 9 for the neon transition ls4(J = 1)-2p4(J = 0) at 607.4 nm, obtained with a sin-gle-mode dye laser. The lower level of thistransition is metastable and is populated byan electrical glow discharge. Figure 9(a)shows the Doppler-broadened absorptionprofile obtained with a single laser beam,from which a 2-GHz Doppler width may beestimated, corresponding to a gas tempera-ture of about 500 K. A slight asymmetry isapparent, due to the presence of two closelyspaced transitions corresponding to the twoisotopes 20Ne and 22Ne present in the dis-charge with natural abundances of 91% and9%, respectively. When counterpropagatingbeams are used, Lamb dips are produced atthe resonances of the two isotopes [Fig.9(b)]. Finally, a Doppler-free recording maybe obtained using a phase-sensitive detection

tock-inamplifier

signal

-saturating team probe*detector

FIG. 8. Typical experimental ar-rangement for saturation spectros-copy (from Hansch, 1975).


(a)

FIG. 9. Recording of the neon transition at 607.4 nmusing a tunable single-mode dye laser. In (a), the lineis recorded in presence of a single beam (Dopplerbroadened), while in (b), the effect of the saturationproduced by the pump beam is visible within theDoppler profile as a homogeneous dip. (c) Doppler-free recording obtained by means of a phase-sensi-tive detection scheme.

of magnitude more sensitivity than absorp-tion spectroscopy.

Intermodulated spectroscopy is a tech-nique derived from saturation spectroscopy(Sorem and Schawlow, 1972) in which oneof these transverse detection schemes isused. The basic principle is illustrated in Fig.10. Here, the intensities of the pump (/pump)and probe (/probe) beams are of the same or-der of magnitude, and the beams are modu-lated at frequencies fx and f2, respectively;i.e.

*pump(20)

The intensity of the fluorescent (or optogal-

vanic) signal Ifi induced simultaneously by

the two beams is

Ifl - CAfsat(/pump (21)

where C is a constant that takes into accountthe transition probability and the detectionefficiency, while Afsat is the saturated popula-tion density of the lower level and can bewritten as

- a(/p u m p

/ p r o b e ) ] ,

(22)

No being the unsaturated population density.By use of Eqs. (20) and (22), and assuming

scheme [Fig. 9(c)]; in this case, the linesfrom two isotopes are completely resolved,and their separation is about 1.7 GHz.

2.2 Intermodulated Spectroscopy

As we have seen in the preceding section,saturation spectroscopy monitors changes inthe absorption of the probe beam caused bya pump beam. This can significantly limitthe sensitivity when weak transitions are tobe studied. In such cases it is better to use"transverse" detection schemes where differ-ent observables are detected, such as fluores-cent, optogalvanic, and optoacoustic signals,which we treat in the following sections. Forall these techniques the detected signal isover a zero ground, and this can offer orders

chopper

LASER

^ 1

<

lock-inamplifier

/

f., , cathode

rO r1 • \

J L—^=y—

signal | "r L?FIG. 10. Scheme for intermodulated spectroscopy.The sample consists of an electrical discharge, andthe measurement is made by recording the fluores-cence or optogalvanic signal at the sum or differencefrequency produced by the interference of the modu-lation frequencies of the two laser beams.


/pump = Io, it may readily be shownpp

that Eq. (21) becomes

- a)0)

X [1 + 1/2 cos(27rfxt) + 1/2 cos(27if2f)]+ C7gao(fi> - wo)[l + 1/4 cos2(2<irfxt)+ 1/4 cos2(27rf2t) + cos(27rfxt)+ cos(2irf2t) + cos{2Trf1t) cos(27jf2f)]X L(a> - a\>). (23)

This shows that the fluorescence signal iscomposed of several terms modulated at dif-ferent frequencies. The first two terms de-pend linearly upon intensity: they are modu-lated at the frequencies fx and f2,respectively, and originate from the pure in-teraction of each laser beam. Their depen-dence upon the laser frequency is, therefore,given by the Doppler-broadened absorptiona0(co — <o0). The term cos{2Trflt) cos(27rf2t)t

quadratically dependent upon intensity, isthe most interesting because it gives rise totwo terms modulated at the beat frequencies(fi + fi) a n d l/i — fi\- These quadratic termscorrespond to the simultaneous interactionof the two counterpropagating laser beamswith the same velocity class of atoms, whichoccurs when the laser frequency is tunednear the center of the line. Therefore, if thedetected fluorescence signal is processed bya lock-in amplifier set to one of the two beatfrequencies, a Doppler-free signal will be ob-tained.

It is interesting to note that both satura-tion and intermodulated spectroscopies arebased upon the simultaneous interaction ofatoms with two counterpropagating laserbeams, along whose axis the atoms havezero velocity; the interacting atoms may thusbe compared to an atomic beam moving per-pendicular to the laser beams. Nevertheless,there is an important difference with respectto a true atomic beam because, in a cell, at-oms undergo collisions that, besides givingrise to pressure broadening, are also respon-sible for the broad residual pedestal shownin Fig. 10(c).

An unexpected signal is also observedwhen the laser frequency is far enough awayfrom line center but still within the Dopplerwidth. This effect is due to so-called velocity-changing collisions, which may be describedsimply as follows. When vx ¥^ vOf the probeand pump beams interact with two symmet-

rically displaced velocity classes of atoms.Nevertheless, collisions tend to thermalizethe velocity distribution, and atoms with avelocity v "marked" by the pump beam can,after one or more collisions, acquire a veloc-ity —v. If this occurs within the detectiontime, these atoms may thereby interact withboth the pump and probe beams.

The Doppler-broadened pedestal dependsupon several collisional parameters and thelifetimes of the levels involved. This contri-bution to the Doppler-free spectrum can beuseful in learning about collisional physicsitself but prevents full resolution of closelyspaced spectral lines. In order to improvethe resolution and sensitivity, other tech-niques are required.

2.3 Polarization Spectroscopy

Polarization spectroscopy is based on thelight-induced birefringence and dichroism ofan absorbing gas. These effects were first ob-served by Fornaca et at (1963) using classi-cal radiation sources; polarization spectros-copy using single-mode tunable lasers wasdemonstrated by Wieman and Hansch(1976).

In a polarization spectroscopy experi-ment, the laser beam is split into two beamsof different intensities, the pump beam beingcircularly polarized by means of a quarter-wave plate while the probe beam is polarizedlinearly. As with the previous Doppler-freetechniques, the two beams pass in oppositedirections through the sample cell. Thetransmitted probe beam passes through ahigh-extinction-ratio polarizer that is set tomaximum extinction, before falling upon thedetector.

The basic idea of polarization spectros-copy can be understood with the help of Fig.11, which shows a transition between twolevels having angular momenta of Jx = 2and J2 = 1. Both levels are degenerate with2 7 + 1 Zeeman sublevels labeled with quan-tum number m = —7 . . . +/ , describing theprojection of / along a quantization axisthat, in our case, is the laser beam direction.Transitions m" -> m' follow the selection ruleAm = ±1 for right (cr+) and left (cr~) circu-larly polarized light. Thus, if the pump beamis a+ polarized only, some of the lower sub-levels are partially depleted (the sublevelswith m = - 2 , - 1 , and 0). The angular mo-


m1 -1 0

FIG. 11. The cr+ pump beam in-duces transitions with Am} = +1creating a birefringence in the gase-ous sample. The birefrengence ismonitored with a second counter-propagating beam (probe beam)normally extinguished between twocrossed polarizers.

mentum of the light is hence transferred tothe atomic sample, producing a nonuniformpopulation of the m sublevels, which corre-sponds to a macroscopic anisotropy of thesample. Such induced birefringence can be"read" by the linearly polarized probe beam,which may be considered to be a superposi-tion of a+ and a~ circular polarizations. Inthe absence of the pump beam, and whenthe laser frequency is resonant with the tran-sition under investigation, the intensitytransmitted by the crossed polarizer is

/(pump off) = /o(* + & + b2) exp(-o/), (24)

where /0 is the probe beam intensity, x is thefinite extinction coefficient of the polarizer, 8is the angular deviation from orthogonality,and L is the thickness of the sample. Finally,b is a term that takes into account the resid-ual birefringence introduced by stress in-duced in the windows of the cell.

When the laser frequency is tuned towithin the homogeneous width of the line,the cr+ and a" components experience dif-ferent absorption coefficients a+ and a_ andrefractive indices 77+ and 77 _ because of ve-locity-selective pumping by the <J+ pump

beam. As a consequence, after passingthrough the sample, the resulting probe po-larization will generally be elliptical, with theminor axis rotated with respect to the trans-mission axis of the analyzer (x axis in Fig.12). In the limit of low saturation and a thinsample, it can be shown that the transmittedintensity / after the crossed polarizer is

LbAa \ ( {AaL)2

.2(1 + x2)) + Vl6(l + x2),(25)

where x = (co - a)0)/y is the laser detuningrelative to the homogeneous width and Aa isthe difference between the absorption coeffi-cients of the cr+ and cr~ probe components.In Eq. (25), only Aa appears, because At) isrelated to Aa through the Kramers-Kronigdispersion relations.

Equation (25) shows the spectral profileof the polarization signal; it is given by aconstant term I0(x + fl2 + b2), a dispersion-shaped term, and a Lorentzian. By an appro-priate choice of the experimental conditions,a pure Lorentzian or dispersive shape can beobtained.

to lock-in

laser beam BS chopper

r 1 spatial""filter

PL

FIG. 12. Experimental arrange-ment for polarization spectroscopy.


In Fig. 13 are shown typical line shapesfor the 5S2-

5JP3 oxygen transition at A =

777.4 nm, obtained by polarization spectros-copy using a diode laser. Atomic oxygen wasproduced by dissociation in a radio-fre-quency discharge where molecular oxygenwas present at trace levels. The two extremeregimes of symmetrical Lorentzian [Fig.13(a)] and dispersive [Fig. 13(b)] shapeswere achieved by setting the angle 0 betweenthe two polarizers to two different values.Narrower line shapes, down to one-half ofthe homogeneous width, can be achieved byrecording the first derivative of the disper-sion-shaped signal, as shown in Fig. 13(c)(the derivative of a line profile is simply ob-tained by modulating the laser frequency in-stead of the intensity). The "artificial" sub-natural resolution reached in this way canprovide some advantages for high-resolutionspectroscopy or can be used to lock laser fre-quencies.

As we have already discussed, in satura-

75MHz

FIG. 13. Typical recording of the 5S2-5P3 oxygen

transition obtained with polarization spectroscopy byusing a diode laser tunable around 777.4 nm. (a), (b)The extreme cases of Lorentzian and dispersive lineshapes, respectively, (c) By modulating the laser fre-quency instead of the amplitude, the derivative of aLorentzian shape is obtained. (From Gianfrani et ai,1991.)

tion spectroscopy the atomic populations areprobed and line shapes are affected by veloc-ity-changing collisions that rethermalize thevelocity distribution altered by the pump la-ser beam. In contrast, polarization spectros-copy probes the dichroism and birefringenceoriginating from velocity-selective orientationof the atomic sample. Collisions do not usu-ally conserve atomic angular orientation, andthe recorded line shapes are unaffected bythe residual Doppler background, so that thespectral resolution is further improved.

2,4 Velocity-Selective Optical-PumpingSpectroscopy

Polarization spectroscopy is a particularcase of a more general class of Doppler-freetechniques based on velocity-selective opticalpumping (Pinard et al, 1979). The popula-tion distribution among the Zeeman sublev-els can be expressed in terms of a power se-ries in m (m = — I/I . . . + I/I):

nmk, (26)

where Nm is the probability of occupation ofthe sublevel m. The zero-order term (k = 0)is the population P and is given by

P = ^Nmf (27)

while the first (k = 1) and second (k — 2)orders are the orientation O and alignment Aexpressed by

-\J{J + 1).

(28)

(29)

Generally, the population distribution will bea superposition of orientation and alignment.Nevertheless, for particular light polariza-tions, a pure orientation or alignment can beachieved. For instance, pure orientation iscreated with circularly polarized light [atomsaccumulate in the sublevels with m = J(a+)or m = —/(o-~)], while pure alignment maybe induced using linearly polarized light.

If the polarization rather than the ampli-tude of the pump beam is modulated, theorientation or alignment may be monitored.


In the orientation modulation technique, forinstance, an electro-optic modulator (EOM)is square-wave modulated to provide an al-ternating a+lcr~ polarization of the pumpbeam while the probe beam is circularly po-larized. Alternatively, for alignment detec-tion, the EOM provides a polarization thatalternates between vertical and horizontal(see Fig. 14).

Typical recordings obtained using orien-tation and alignment detection are shown inFig. 15 for the 5S2-5P3 oxygen transition (deAngelis et ai, 1991). It may be seen that, in

recorder

FIG. 14. Experimental setup to monitor alignmentand orientation. A and B are optical components ac-cording to the observable to be detected (see follow-ing description). The laser diode (LD) is mounted inan extended cavity controlled by a piezoelectric crys-tal (PZT), and its frequency is monitored by a Fabry-Perot interferometer (FPI). The polarization of thepump beam is modulated by means of an electro-op-tic modulator (EOM). In order to monitor orientation,the EOM produces an alternating a+-a~ polarizationof the pump, while A is a linear polarizer followed bya quarter-wave retarder, and B is absent. For align-ment detection, the EOM renders the polarization al-ternately vertical and horizontal. A is a linear polar-izer, and B is again absent. To monitor thepopulation (saturation spectroscopy), the EOM maybe used like a chopper to modulate the intensity, withlinear polarizers at A and B. (From de Angelis et al.,1991.)

the case of alignment detection, atoms un-dergoing velocity-changing collisions becomedepolarized and do not contribute to the re-sidual Doppler pedestal.

2.5 Polarization-IntermodulatedExcitation Spectroscopy (POLINEX)

The intermodulated technique was im-proved by Hansch and co-workers by modu-lating the polarizations, rather than the am-plitudes, of the counterpropagating beams.This method is referred to as the polariza-tion-intermodulated excitation, or POLINEX(Hansch and Toschek, 1970). When the laseris tuned to within the homogeneous width,the total excitation rate is still modulated atthe sum or difference frequency, but thecombined absorption of the two beams ingeneral depends upon their relative polariza-tion. When the two laser beams share thesame polarization, they will be preferentiallyabsorbed by atoms of the same orientation,with a resultant increase in saturation; whenthe polarizations are different, the twobeams will tend to interact with atoms ofdifferent orientation and, as a consequence,the total saturation decreases.

In addition to the density-matrix formal-ism used by Hansch and Toschek (1970),POLINEX may be described using a simplerrate equation approach (Teets et at, 1977). Ifoptical pumping effects are neglected, and inthe approximation of a Doppler width muchlarger than the homogeneous width, it ispossible to obtain an analytical expressionfor the Doppler-free line shape.

There is a substantial difference betweenPOLINEX and polarization spectroscopy, al-though in both schemes the signal is derivedfrom the orientation and alignment. Indeed,in polarization spectroscopy, the signal de-pends upon both dichroism and birefrin-gence induced by the absorbed light, and asa consequence the line shapes are usuallyasymmetric. In contrast, POLINEX spectros-copy detects only the dichroism, and the lineshapes are symmetric.

One advantage of POLINEX over inter-modulation spectroscopy is that the cross-over signals are frequently of opposite signsand can therefore be easily distinguishedeven in complicated spectra. It should benoted, however, that if amplitude modula-tion is unintentionally present together with


-400 -200 0 200Av(MHz)

(a)

400

-200 -100 0Av(MHz)

100

(b)

b)

200

FIG. 15. Comparison between ve-locity-selective (a) orientation and(b) alignment detection of the J =2-J' = 3 transition of oxygen at777.4 nm. The line shape (a) isgiven by the superposition of a Lor-entzian curve and a Gaussiancurve due to the velocity-changingcollisions.

the polarization modulation, spurious signalscan be produced.

A comparison between intermodulationand POLINEX spectroscopy is shown in Fig.16 for the copper transition at 578.2 nm.Copper vapor was produced by sputtering ina hollow-cathode discharge. It is interestingto note that in POLINEX spectroscopy theresidual Doppler background is absent. Thisis a consequence of the fact that atomic ori-entation and alignment are not conservedduring collisions. Another difference betweenthe two spectra is the absence in the POLI-NEX spectrum of transitions starting fromlevels with F = 0, which cannot be oriented.

The various techniques previously dis-

cussed, combining saturation and optical ac-tivity, have been extensively used in manyspectroscopic applications. The ability to de-plete selectively a specified rotational-vibra-tional level by optical pumping can be usedto simplify complex molecular spectra. Thismethod, named polarization labeling spec-troscopy, uses two lasers. The first laserbeam is again split into two counterpropa-gating beams (pump and probe) and lockedto the center of the polarization signal of atransition (j/'J") -» (v1J'). The second laseracts as a probe laser: it passes collinearlywith the probe beam of the first laserthrough the sample cell, after which the twoprobe beams are separated and individually


F' 2I

F 3

I

2I3

i: i i ; II : 11 iI10

frequency laser

(a)

(b)

1 2 I 21 2 I I 2 II I I N II I I I2 2 2 21 10 1 10

i iI

IS

FIG. 16. Comparison between (a) intermodulatedspectroscopy and (b) POLINEX detection of the Cutransition at 578.2 nm. Lines in the spectrum (a) arebroader than in the spectrum (b) because they areaffected by velocity-changing collisions.

detected. When the frequency of the probelaser is tuned, a polarization signal is pro-duced every time the probe laser hits a tran-sition that shares a common level with thepump transition (*/',/') -> (vf,J').

2.6 Multiphoton Spectroscopy

In many spectroscopic techniques, two ormore photons are involved. As a simple ex-ample, we consider the case where a level bis selectively excited by a first laser of fre-quency a)x. This level represents a platformfrom which a second photon can be ab-sorbed to reach a higher excited level c.Such a two-step excitation approach hasbeen extensively used for many double-reso-nance schemes where the two photons liein a wide spectral region (radio frequency,microwave, infrared, and visible). We now,however, focus our attention on a class ofmultiphoton spectroscopic techniques (Bruz-zese et al, 1989) in which the jump from thelower level a to the excited level c occursthrough the absorption of two or more pho-tons (see Fig. 17) without intermediate lev-els. This possibility was predicted upon theintroduction of quantum mechanics, and oneof the first examples was represented by theRaman process, in which a photon of fre-quency a) is absorbed and another photon oflower (Stokes process) or higher (anti-Stokesprocess) frequency u! is emitted, with con-sequent excitation of a different state of theatomic or molecular system. In the low-in-tensity (spontaneous emission) regime, thisprocess is linear.

The first theoretical study considering thepossibility of inducing transitions throughtwo-photon absorption was performed in1931 by Goeppert-Mayer (1931). Her ideawas, at the time, experimentally unfeasible

////S//N |

i1

2

/ / / / / / /> / / / / / / / / / / / /

k11

(a) (b) (C)

FIG. 17. Schematic representation of multiphoton processes: (a) nonresonant, (b) quasiresonant, and (c) mul-tistep excitation. Dashed lines represent virtual states.


because of the low intensity of the availableradiation sources: The higher-order terms ofthe perturbation theory, which describe mul-tiphoton processes, become relevant onlywhen the intensity of the electromagneticfield is sufficiently large with respect to theCoulombian atomic field. Since then, how-ever, the development of maser and, succes-sively, laser sources has offered the possibil-ity of carrying out two-photon and moregenerally multiphoton processes, therebyproviding powerful tools to investigate a newclass of phenomena.

The first experimental evidence of multi-photon processes was produced by Brossel etal. (1954), who used an intense radio-fre-quency field, and by Battaglia et al (1959)with microwaves. In 1962, just two years af-ter the operation of the first laser, Abella(1962) observed two-photon absorption be-tween the 62S1/2 ground state and the 92D3/2

state of cesium by tuning a Q-switched rubylaser at A = 693.5 nm by temperature con-trol of the active medium. However, mostspectroscopic applications based on multi-photon absorption followed the later devel-opment of pulsed tunable dye lasers.

The main advantages offered by two- (ormore) photon processes are the following:

1. The selection rules for transitions inducedby multiphoton absorption are differentfrom those for single-photon processes.For instance, in the electric dipole ap-proximation, two-photon transitions canoccur only between levels having thesame parity. As a consequence, it is pos-sible to induce transitions from theground state not otherwise feasible withsingle-photon absorption.

2. Two-photon excitation can be achieved byusing visible or ultraviolet radiation in-stead of the higher-frequency radiationrequired for an equivalent single-photontransition.

3. If the two photons are absorbed from op-posite directions, the first-order Dopplereffect can be removed, and the spectralresolution thus improved.

2.6,1 Selection Rules for Two-PhotonTransitions In the electric dipole approxi-mation, the interaction Hamiltonian actsonly on the orbital quantum numbers of theelectrons. If we consider polarized light (<r+,

a , and TT), the interaction components aretransformed as the component of an irreduc-ible tensor of rank one 7 ^ , with q = +1 fora* polarization, q = — 1 for a~ polarization,and q = 0 for ir polarization. Applying suchan operator changes the angular momentumby 1. As a consequence, applying the opera-tor twice, we have

AL = 0, 2,AmL = qx + q2. (30)

In the presence of a strong external magneticfield, L and S are completely uncoupled (thePaschen-Back effect), and the quantumnumbers of spin cannot change:

Ams = Amt ~ 0> (31)

while in the contrasting case in which thespin-orbit interaction and hyperfine interac-tion are not negligible, it can be shown fromthe Wigner-Eckart theorem that

\AF\ < 2; AmF = qx + q2. (32)

We consider two cases of particular impor-tance to many spectroscopic applications.The first concerns transitions between two sstates: the electronic and nuclear spins canbe modified only through coupling with theorbital angular momentum and, since L = 0,we have AF = AmF = 0. The second caseconcerns two states with the same total an-gular momentum / . If / = 0 or 2> only a sca-lar operator can couple these two states, andwe again have AF = AmF = 0.

2.6.2 Transition Probabilities and LineShapes for Two-Photon Transitions Wesuppose that an atom having velocity v inter-acts with two laser beams of the same fre-quency oj and wave vectors kj and k2, re-spectively. If the intensities of the twocounterpropagating beams are the same (Ix

= I2 = /), then the two-photon transitionrate from a state la) to a state \b) can be cal-culated from second-order perturbation the-ory:


l(a>ab - 2co - ktv + k2v)2 + (yJ2)2

Jab

(a>ab -2a>- 2klV)2 + (yabl2)2

_._ Jab

(a>ab - 2(o + 2k2vf + (yabl2)'

XCO - 0)n

(33)

where yab is the homogeneous linewidth ofthe transition a-b and where R{j are the ma-trix elements

(34)

For the particular case of two counter-propagating beams (k2 = -k2), the firstterm of the transition probability [Eq. (33)]is not affected by the velocity distribution ofthe atoms. Indeed, since the laser frequencyo) will be shifted by +kv and —kv, respec-tively, for the two beams, the two Dopplershifts will exactly compensate, and we have

Eb - Ea = h((o + kv) + h(a) - kv) = 2ha)ab.(35)

In other words, when the frequency of thelaser is resonant [hco = (Ea — Eb)/2], all ofthe atoms will undergo the two-photon tran-sition with the same probability, indepen-dent of their velocities.

The second term of Eq. (34) representsthe absorption of two photons from thesame laser beam. In this case, for a given la-

ser frequency o>, there is a group of atomicvelocities that, because of the Doppler effect,will see the frequency resonant with thetransition a -* b:

Eh - Ea - 2h(o = ±hkv. (36)

The line shape of a two-photon transition istherefore given by the superposition of twocurves: a Lorentzian curve, representing theDoppler-free term, and a Gaussian curve thattakes into account the velocity dependenceof the second term of Eq. (34) (see Fig. 18).If the polarizations of the two counterpropa-gating laser beams are the same, it can beshown that the area under the Lorentziancurve is twice that under the Gaussian curve.In order to improve the resolution, the Gaus-sian background can be removed with a suit-able choice of polarization of the two laserbeams.

Doppler-free two-photon spectroscopy wasfirst proposed by Vasilenko et al (1970) atNovisibirsk and demonstrated experimentallyby Cagnac et al (1973) at Paris and Bloem-bergen et al (1974) at Harvard. The investi-gated atom was chosen to be sodium for itsfavorable energy level scheme, the two-pho-ton transitions 3S-5S at 602.23 nm and 3S-AD at 578.73 nm both being in a spectralrange where tunable dye lasers are easilyavailable. Moreover, for both transitions, theintermediate 3P level makes the transitionprobabilities relatively large. Figure 19 showsa typical recording for the transition 3S1/2-5S1/2, where the two-photon absorption was

0) - JCV

1

2r

2(u - ky) = a) 122(» = at12

FIG. 18. Typical Doppler-free two-photon spectrum. The residual Doppler background is due to the absorptionof two photons traveling in the same direction.


1.772 GHi

|*—1.772 GHi

60 MHz

FIG. 19. The excitation spectrum for two-photon ex-citation of the AD level in sodium. The peaks corre-spond to the hyperfine transitions between theground state 3S1/2(F = 1, 2) and the final state403/2,5/2- The 1 772-GHz splittings are due to the hy-perfine splitting of the ground state, while the 1.027-GHz splitting reflects the AD fine structure. (FromBjorkholm and Liao, 1974).

monitored by observing cascading fluores-cence from the AP to the 3>S state at 330.3nm. This demonstrates another advantage ofthis technique, whereby an induced transi-tion can be studied by detecting fluorescenceat a wavelength quite different from that ofthe excitation radiation. Both the lower 3S1/2

and upper 5S1/2 levels consist of two hyper-fine levels (F = 3 and 4). Because of thetwo-photon selection rules, only hyperfinetransitions with AF = 0 can be induced, giv-ing rise to two spectral components sepa-rated by the combined hyperfine splittings ofthe lower and upper levels. As is shown inFig. 20(a), the two hyperfine components,separated by less than the Doppler width,are not resolved if copropagating beams areused. A higher resolution is achieved usingtwo counterpropagating beams, as shown inFig. 20(b).

Like saturation spectroscopy, Doppler-freetwo-photon spectroscopy has been used toinvestigate fine effects in the structure of at-oms and molecules, such as isotope shifts,hyperfine splittings, Zeeman and Stark ef-fects in weak fields, and collisional broaden-ings and shifts. It is interesting to note thattwo-photon signals are not significantlylower than those obtained in saturation spec-troscopy, especially if an intermediate levelis present midway between the initial and fi-nal states. This is easily explicable if we re-call that in saturation spectroscopy the sig-

(a)

(b)

16.3 MHz

FIG. 20. Typical two-photon spectra in sodium withone laser resonant with the 3S-3P1/2 transition andthe second swept around the 3P-AD transition. Thetwo laser beams copropagate in (a) and are counter-propagating in (b). (From Bjorkholm and Liao, 1976).

nal is produced by atoms having zerovelocity along the laser-beam direction, whilein two-photon spectroscopy all atoms con-tribute to the signal. This feature is impor-tant, for it means that two-photon transi-tions can be investigated using cw lasers ofrelatively low intensity.

A generalization of the opposing-beamtechnique to demonstrate resonant enhance-ment of the two-photon transition probabil-ity has been carried out by Bjorkholm andLiao (1976) in sodium vapor. Their experi-ment was performed with use of two single-mode dye lasers operating at different fre-quencies. One laser produced a fixedfrequency matched to the 3S-3P transition,while the second laser was tuned around the3P-AD transition. Because the frequencies ofthe two lasers were not equal, Doppler ef-fects were not in this case completely re-moved, and the residual Doppler effect madeit impossible to be simultaneously resonantwith all the velocities of the Maxwellian ther-mal distribution. Consequently, the use oftwo unequal frequencies results in a decreasein the excitation efficiency. To demonstrateresonant enhancement, Bjorkholm and Liaoused different wavelengths for the first laserand tuned the second around the two-photonresonance (Eb — Ea = ho)x 4- hco2). As isshown in Fig. 21, when the first laser ap-


30 40 5850 60 70 80 90 5900 10 20 30 40 5950

FIG. 21. Normalized two-photon transition rates for the 3S(F = 2)-4D5/2 and 3S(F = 2)~4D3/2 transitions as afunction of the laser detuning (from Bjorkholm and Liao, 1976).

proaches the 3S-3P resonance, the signal in-creases by seven orders of magnitude. It isalso interesting to note that for frequenciesbetween those of the two fine structure com-ponents 3JP1/2 and 3P3/2, the signal decreasesbecause of quantum interference effects.

2.6.3 Doppler-Free Multiphoton Tran-sitions The Doppler effect can also be can-celed when three or more photons are ab-sorbed if a suitable geometry of laser beamsis used. Consider the case of an atom irradi-ated by different beams having wave vectorski. The Doppler shift for each interaction iski-v, and hence, if the atom absorbs n pho-tons, the total Doppler shift is given bySiki-v, where the sum extends to all n ab-sorbed photons. So, if

2 ^ = 0, (37)

the n-photon transition is unaffected by theDoppler effect. In the particular case of pho-tons of the same frequency, if n = 2, thetwo laser beams are counterpropagating (k2

= — k2), while for n = 3, the laser beamsare directed along the bisectors of an equi-lateral triangle.

The first Doppler-free three-photon exper-iment, in which two photons were absorbedand one emitted, was made by Grynberg,Biraben, Bassini, and Cagnac.

2.6.4 Applications of High-ResolutionSpectroscopic Techniques to the Hydro-gen Atom: Lamb-Shift and Rydberg-Con-stant Measurements Spectroscopic inves-tigations of atomic hydrogen allow thetesting of fundamental physical theory andthe measurement of physical constants, withhigh accuracy. Doppler-free two-photonspectroscopy has been used by severalgroups to measure the Rydberg constant andto determine the Lamb shifts of the hydro-gen energy levels accounted for by quantumelectrodynamics.

The Rydberg constant is of particular im-portance because it is composed of only fun-damental constants (Roo = rae4/8€o/z3c), sothat a very precise measurement of R^ al-lows the consistency of other constants to be


checked. The Rydberg constant can be deter-mined by measuring the energy separationbetween two levels having different principalquantum numbers. R*, is related to theseseparations through the Bohr formula, withcorrections for the isotope shift, Dirac fine-structure effects, and quantum-electrodyn-amical radiative effects.

Figure 22 shows a saturation spectrum ofthe Balmer-a line of atomic hydrogen pro-duced in a gas discharge. Thanks to the highresolution offered by saturation spectros-copy, the separation between the 2S1/2 and2P1/2 levels (Lamb shift) was thus deter-mined. Furthermore, absolute wavelengthmeasurements of the strong 2P3/2-3>D5/2 com-ponent provided a new and more accuratevalue of the Rydberg constant. Nevertheless,the accuracy achievable in such experimentsis generally dependent upon the characteris-tics of the atomic sample; indeed, a gas dis-charge introduces further broadenings andshifts of atomic lines due to collisions withother atoms and electrons.

Atomic beams would, on the other hand,present an ideal atomic sample if collisionsand other perturbations could be kept tonegligible levels and if first-order Dopplerbroadening could be eliminated. To this end,the group of Lichten at Yale performed anoptical study of hydrogen very similar to thehistoric experiment by Lamb at radio fre-quencies (Amin et al, 1981). Molecular hy-drogen was dissociated by means of a hotwire, and the metastable 2S level was popu-lated through collisions with a transverseelectron beam. The atomic beam then inter-acted with a laser beam of low intensity. Ab-sorption of the laser beam was monitored bydetecting a variation in the flux of metasta-ble atoms after the interaction with the laserbeam. From an absolute wavelength mea-surement, Lichten derived one of the mostaccurate values of the Rydberg constant, R^= 109 737.315 73(3) cm"1.

Atoms with a single optical electron havealso been used to determine the Rydbergconstant. The idea is to investigate highly ex-cited levels (Rydberg states) in which elec-trons experience a hydrogen-type Coulombfield. The advantage offered by this approachis that lasers with high spectral purity areunnecessary because the energy differencebetween closely spaced Rydberg levels corre-sponds to radiation wavelengths in the sub-millimeter range.

The first excitation of the 2s level by two-photon absorption was performed by Hanschand co-workers (1975, 1977) at Stanford.The two-photon 1S-2S transition is particu-larly difficult because, in contrast to the 3S-5S transition in sodium, no intermediate lev-els are present. Furthermore, the 243-nm la-ser radiation required for this experiment isin a difficult spectral region and could beproduced only by frequency doubling a 486-nm pulsed dye laser by means of a lithiumfluoride nonlinear optical crystal. The two-photon transition was observed through col-lision-induced fluorescence on the Lyman-a(2P-1S) transition at 121 nm. Frequencycalibration was performed by sending part ofthe laser at 486 nm into a second dischargecell and observing the saturated spectrum ofthe Balmer-j8 line. In absence of the Lambshift, the frequency of the 1S-2S transitionshould be four times that of the 2S-AP line.The frequency difference between the tworesonances (see Fig. 23) is thus determinedby the Lamb shifts of the IS and 2S states.Using a previously measured value of the 2SLamb shift, the Lamb shift of the groundstate was determined to be 8151(30) MHzfor hydrogen and 8177(30) MHz for deute-rium.

The first version of the 1S-2S two-photonmeasurement used pulsed lasers, and thelinewidth of the observed transition was thuslimited by the broad spectral width of theselasers. More recently, Hansch et al have in-vestigated the same transition with much im-proved resolution by combining cw laserswith Fabiy-Perot built-up cavities.

3, HIGH-SENSITIVITY LASERSPECTROSCOPY

Laser radiation has been used extensivelyin spectroscopy to achieve not only high res-olution but also high sensitivity, in order toinvestigate very weak transitions or to moni-tor species present at trace level.

The absorption coefficient a is related tomicroscopic parameters of the investigatedatomic or molecular line by the relation

a = [27r2lAtl2^2(7V1 - N2)/37^phjg(Av)9 (38)

where vl2 is the frequency of the transition,/JL is the dipole moment of the transition, 77 is


(a)

(b)

(c)

—A U— LAMB SHIFT

-5 0 5 10LASER TUNING [GIGAHERTZ]

15

FIG. 22. Balmer-a line of atomic hydrogen: (a) energy levels with fine-structure transitions, (b) Doppler-limitedemission line profile, and (c) saturation spectrum with optically resolved Lamb shift (from Hansch, et al., 1975b).


A HYOROGEN/ \ iSi-2Sj.I 1 ' *

I If 1

-J V_JI

\

I

0

1

DEUTERIUMIS. - 2S.

0 25

I

I

as : _LASER TUNING [GIGAHERTZ]

168.0

l

169.25

I2 1 l 672

EFFECTIVE FREQUENCY OF ABSORBED RADIATION (GIGAHERTZ]673

FIG. 23. Doppler-free two-photon spectrum of the 1 s-2s transition of (a) atomic hydrogen and (b) deuterium(from Hansch, et ai, 1975b).

the refractive index, and g(Av) is the lineshape. To determine a, we use the well-known Beer law, which describes the radia-tion absorbed by a length L of the sample:

= 7(0)exp[-oL]. (39)

In the case of weak absorption, a L < 1, Eq.(39) can be approximated by the first-orderpower series

- oL). (40)

Hence, the absorption coefficient can be de-termined from the fractional loss of the in-cident intensity:

cJL = [1(0) - 7(L)]//(0). (41)

For weak absorption, however, this methodcannot be very accurate because it measuressmall differences above a large background,and the minimum detectable absorption isthus limited by the amplitude fluctuations ofthe radiation source.

The advent of lasers has unleashed a wide

variety of spectroscopic techniques that haveincreased the sensitivity enormously, allow-ing the detection of even a single atom. Inthe following section, we discuss some ofthese techniques.

The first approach is based upon the mod-ulation of the laser frequency at a frequencyf. If the amplitude of the modulation is suf-ficiently small (with respect to the width ofthe investigated line) and the transmitted ra-diation is detected with a lock-in amplifiertuned to the same modulation frequency f,the recorded signal is proportional to the de-rivative of the absorption spectrum. The ad-vantage of this phase-sensitive method of de-tection is to restrict the noise to a narrowband centered around the modulation fre-quency f. In terms of the signal-to-noise ra-tio, frequency modulation is more sensitivethan amplitude modulation because the fre-quency-independent background due to lightscattered from cell windows etc. is blocked.We note that in some applications it may bemore practical to modulate the frequency ofthe absorption curve rather than the laser, asis the case, for example, with intracavitymagnetic resonance spectroscopy.


Other more sensitive methods make useof observations related to the energy ab-sorbed rather than the difference in thattransmitted [1(0) - /(L)]. Induced fluores-cence, for instance, is proportional to thenumber of photons absorbed and provides avery sensitive detection scheme (excitationspectroscopy). In some cases, the absorbedenergy can be transformed into thermal en-ergy, giving rise to acoustic waves that arereadily detected by a sensitive microphone(photoacoustic spectroscopy). Alternatively, ifthe atomic or molecular sample is preparedin a discharge or flame, absorption can bemonitored by detecting the variation in thedischarge current or ionization in the weakplasma represented by the flame (optogal-vanic spectroscopy). This method can begeneralized to the case of neutral samples, inwhich an atom or molecule is ionizedthrough the resonant absorption of two ormore photons. The electron-ion pairs thusproduced are detected with unitary effi-ciency, and this method (resonance ioniza-tion spectroscopy) has been demonstrated todetect a single atom. A further techniqueachieves "amplification" of the absorption byincluding the sample within a laser cavity(intracavity spectroscopy).

3.1 Excitation Spectroscopy

Consider a spectroscopic arrangement inwhich the laser radiation is resonant with anatomic transition 1 -» 2. The number of pho-tons absorbed per unit time and unit pathlength, nabs, is proportional to the incidentphoton flux, the absorption cross section ai2,and the atomic density Nx in the lower level.The number of photons emitted per unittime from the excited level, nfl, is propor-tional to the number of photons absorbed,

(42)

where A2 = 2mA2m is the total spontaneoustransition probability from the level 2 to allthe levels with Em < E2. The quantum effi-ciency r] is

v = A2/(A2 + R), (43)

R being the collisional de-excitation rate. In

the absence of collisions, all the atoms inlevel 2 decay emitting photons (77 = 1) thatcan be collected with a given geometrical ef-ficiency 8 by a photomultiplier. Photons col-lected on the photocathode surface give riseto the emission of electrons at a rate

(44)

where r)e is the quantum efficiency of thephotocathode—i.e., the ratio between thenumbers of emitted electrons and incidentphotons. Modern photocathodes reach quan-tum efficiencies in excess of 20%, and, byuse of photon-counting techniques, it is pos-sible to measure absorption rates of 103 pho-tons per second. With a flux of photons Nphot

= 1018 photons s"1 (corresponding to a laserbeam of 1 W at 500 nm), this implies that itis possible to detect a relative absorption ofAll I = 10 ~14. This impressive sensitivity canbe further enhanced by placing the sampleinside a laser cavity where the light flux isone or two orders of magnitude larger.

When the laser radiation is tuned acrossan absorption line, the total fluorescence in-tensity 7fl = NphotaacNi reproduces the ab-sorption spectrum. Although the positions ofthe lines in the excitation spectrum are co-incident with those of the absorption spec-trum, the relative intensities can be different.Indeed, the quantum efficiency of both thetransitions and the photocathode are depen-dent upon the laser wavelength. The geomet-rical efficiency may also vary: indeed, excitedatoms with relatively long lifetimes may dif-fuse out of the interaction volume beforeemitting any fluorescence photons.

Excitation spectroscopy has been used inatomic- and molecular-beam spectroscopywhere the density of particles is rather low.As an example, in Fig. 24 we show a typicalspectrum of molecular sodium, excited withradiation at 604 nm. The molecular densityin the beam was N = 108 mol cm"3 whilethe absorption length was 10 mm. In thecase of atomic sodium, absolute atomic den-sities down to 102 atoms cm"3 have beenmeasured. The technique has also been suc-cessfully used to investigate radicals andshort-lived intermediate products in chemi-cal reactions.

Excitation spectroscopy becomes less sen-sitive in the infrared region, where the pho-tocathode efficiency decreases. In addition,


R3

NA2(AZ — XI)v' = 17 — v"=0

R5

R1R2

P1I

604392 604.413 nm

FIG. 24. Excitation spectrum of molecular sodiumobtained in a molecular beam with radiation at 604nm (from Demtroeder, 1981).

vibrational-rotational levels excited by infra-red photons have lifetimes several orders ofmagnitude longer than do electronic states.As a consequence, excited molecules escapefrom the detection volume before they radi-ate. For such investigations, optoacousticspectroscopy can be more advantageous.

3.2 Optoacoustic Spectroscopy

Although the optoacoustic effect was ob-served for the first time more than a centuryago (Bell, 1881), spectroscopic applicationsbegan only with the development of lasersources (Tarn, 1986). The technique is basedon the fact that energy absorbed by mole-cules can be converted into rotational, vibra-tional, and translational energy. At thermalequilibrium, this energy is distributed overall the degrees of freedom, leading to heatingof the sample. If the laser intensity is modu-lated at audio frequencies, sound waves ofthe same frequencies are induced inside theirradiated cell where they can be detectedwith sensitive microphones.

Optoacoustic spectroscopy is an uncon-ventional detection scheme (Ernst and Ingus-cio, 1988) in the sense that microscopic pro-cesses, in the form of atom-photon ormolecule-photon interactions, are detectedthrough macroscopic changes of the systemas a whole (in this case, an acoustic wave).Several advantages are offered with respectto conventional spectroscopic techniqueswhere usually light is monitored. In thiscase, the detector is only sensitive to the ra-diation absorbed, and the problem of stray

light is completely removed. Moreover, inthe infrared, where the sensitivity of com-mon light detectors is very poor, optoacous-tic detection can prove to be the only option.

Applications of optoacoustic spectroscopyare numerous and have concerned the mon-itoring of species at low concentrations ingases, liquids, and solids.

The sensitivity of optoacoustic spectros-copy can be enhanced if the cell is shaped toform a resonant acoustic cavity at the mod-ulation frequency, allowing the detection ofconcentrations as low as one part in 109.This has also allowed the study of moleculesat low pressures, where collisional broaden-ing is small in comparison with Dopplerbroadening. In this case, Doppler-free tech-niques have been combined with thismethod, and, as a demonstration, we showin Fig. 25 a typical two-photon Doppler-freespectrum of the 0 -> v2 vibrational transitionof NH3 obtained using two counterpropagat-ing beams from a CO2 laser, one being tunedto the 10P(18) line at 945.9802 cm"1 and thesecond to the 10P(34) line at 931.0014 cm"1.The energy difference between the final andinitial states of the investigated transitiondiffered slightly from the sum of the twophoton energies, and resonance wasachieved by shifting the energy levels usingan external electric field (Stark effect). Bychanging the intensity of the electric field,transitions between different Zeeman sublev-els m are brought into resonance with thefixed laser frequencies, giving rise to thepeaks shown in Fig. 26.

3.3 Optogalvanic Spectroscopy

The absorption of photons can alter theelectrical equilibrium of an irradiated sam-ple. In such cases, charge variations can bemonitored, instead of the absorption oremission of light. As with the optoacoustictechnique discussed above, the methodsbased on this principle may be consideredunconventional in the sense that a macro-scopic response of the system is monitored.Moreover, because of the high efficiencywith which electrical charges may be col-lected, very high sensitivities can beachieved. We discuss here two of the morerelevant techniques: resonant ionizationspectroscopy (RIS) and optogalvanic (OG)spectroscopy.


6 7 8 9

electric field (kV/cm)

FIG. 25. Optoacoustic recording of the two-photon transitions in NH3 observed using the P(18) and P(34) CO2

laser lines (from Minguzzi et al., 1982a).

20electric field (kV/cm)

FIG. 26. Optoacoustic laser Starkspectrum of NH3, recorded with intermodulated detection (from Min-guzzi et al., 1982b).


RIS (Hurst et al, 1979) is a particularcase of the more general multiphoton spec-troscopy already discussed. In RIS, the finalstate is the continuum, and two or morephotons resonant with intermediate levelsare absorbed. Essentially, schemes have beensuggested for all the elements of the periodictable by which the element may be ionizedthrough the absorption of two or more pho-tons of suitable wavelengths. The main ad-vantages are that a species can be ionized se-lectively and that the electron-ion pairsproduced may be collected with 100% prob-ability. For this purpose, standard systemsemployed in nuclear physics can be used asionization chambers or proportional cham-bers. The high sensitivity of RIS has beenproved by detecting a single atom of caesiumin a background of 1019 atoms of argon. Adetection sensitivity of just a few atoms isnecessary for studies of very rare events orig-inating in such nuclear reactions as those in-duced by solar neutrinos, 37Cl(i>,e)37Ar.

Even when the atoms or molecules underinvestigation are in a discharge—a weakplasma—resonant laser radiation can still af-fect the electrical equilibrium of the dis-charge itself. This change is observed as anincrease or decrease in the conductivity ofthe discharge and is known as the optogal-vanic effect (OGE), first described by Pen-ning (1928), who noted a variation in the im-pedance of a neon discharge when it wasirradiated with emission from an adjacentneon discharge. Extensive and practical ap-plications of the OGE had to await the intro-duction of tunable lasers (Barbieri et al,1990).

The mechanisms that translate the OGEinto current perturbation are rather complexbecause of the large number of processes oc-curring within the discharge, but as a firstapproximation the effect can be explained byconsidering the difference between the ioni-zation cross sections at for the two states in-volved in the optical transition. Such a sim-ple model explains the fact that opticalexcitation leads to a change in the numberof charges (electrons and ions), which inturn causes a change in the discharge impe-dance. In a typical OG detection scheme, thelaser beam is amplitude modulated by achopper, and variations in current are de-tected by means of a lock-in amplifier. Theexperimentally observed perturbation of the

discharge characteristics induced by laser ra-diation is usually sufficiently small that theOGE can be considered linearly proportionalto the number of photons absorbed.

Many features make OG spectroscopy at-tractive when compared with other conven-tional spectroscopic techniques in the ultra-violet, visible, and near-infrared regions ofthe spectrum. First, it is a relatively econom-ical technique, since it does not require theuse of such devices as a monochromator orphotomultiplier tube. A glow discharge is aninexpensive way of obtaining quite large den-sities of excited states in volatile elements,especially in metastable states, and gaseousstates of refractory elements are easily pro-duced by sputtering in hollow-cathode dis-charges. A remarkable population of atomsin excited levels is present in the dischargeas a result of electron-neutral-atom colli-sions, allowing transitions between excitedlevels to be investigated. OG spectroscopypermits one to record atomic and molecularlines that, otherwise, would be measurableonly in an atomic or molecular beam. In mo-lecular spectroscopy, OG techniques are themost natural way to study atomic and mo-lecular species and radicals produced fromparent molecular compounds that are pres-ent in the discharge.

OG spectroscopy is intrinsically more sen-sitive than that based upon absorption, theformer yielding a signal against a zero back-ground, the latter recording a small variationsuperimposed on a large signal. In compari-son with fluorescence techniques, OG spec-troscopy offers the advantage of being unaf-fected by either the luminosity of thedischarge or the stray scatter of excitationradiation.

To underline the diverse mechanisms cor-responding to different spectroscopic tech-niques, we show in Fig. 27 a comparison be-tween various measurements of the samespectrum of neon: by fluorescence, optogal-vanic, and optoacoustic spectroscopy. Theneon was excited in a positive column at afew Torr and irradiated by a tunable dye la-ser. The dramatic differences in the relativeintensities of the recorded lines are evidenceof the various mechanisms governing the dif-ferent spectroscopic techniques.

The numerous methods developed forsub-Doppler spectroscopy, such as satura-tion, intermodulated, and two-photon spec-


FL ' ( a )a)

h)

b)

oc

OA'1 (C)

LJL JL_JL

FIG. 27. Transitions in neon from 560 to 620 nm. The three spectra are obtained using (a) fluorescence, (b)optogalvanic, and (c) optoacoustic detection. (From Ernst and Inguscio, 1988).

troscopy, can also be applied using the OGtechnique. The OG signal is actually propor-tional to the absorption of radiation by theatoms of the discharge, so that the same sat-uration methods used for other detectionschemes, such as optoacoustic or fluores-cence spectroscopy, can be used. Transitionsbetween highly excited states of atomic oxy-gen populated through the dissociation of O2

molecules in a discharge are shown in Fig.

28. The Doppler-broadened spectrum wasobtained by using a tunable dye laser around616 nm, while higher resolution wasachieved by means of optogalvanic intermo-dulated spectroscopy.

3.4 Intracavity Spectroscopy

The spectroscopic sensitivity can be en-hanced if the sample is placed within the la-


(a)

(b)

3p5P2 4d5D,(c)

CO

(d)

I 1 i i i i i i t i i i » i i

300 MHz — ^

FIG. 28. Spectrum of an O2-Ne discharge around616 nm: (a) fluorescence spectrum; (b) optogalvanicsignal obtained by frequency scanning a multimodedye laser; (c) Doppler-limited line profile; (d) inter-modulated optogalvanic spectrum of the 5P2-

503,2,itransitions (from Barbieri et ai, 1990).

ser cavity. Indeed, at a laser output powerPouV the power inside the laser cavity is

inside = (lOTout = qPoiiV (45)

T being the transmissivity of the output cou-

pling mirror. By suitably shaping the lasercavity, a region can be created where thebeam is focused (the waist) and the intensitythus further increased. For aL < 1 the ab-sorbed intensity is

AI = (\IT)oLIout, (46)

i.e., \IT times the absorption rate observedoutside the cavity. For a typical transmissiv-ity of 0.02 (R = 98%), the amplification fac-tor is 50.

If the cell cannot be placed inside the ac-tive resonator, an external passive resonatormay be used. By matching the laser outputto a fundamental mode of the passive reso-nator and making as low as possible thelosses g, the power circulating inside the res-onator can be \lq times the laser outputpower. Passive resonators represent an im-provement over multipass cells and, sincethe atoms or molecules experience two coun-terpropagating beams, a Doppler-free schememay be adopted.

Intracavity absorption can be monitoredthrough the laser-induced fluorescence or us-ing other detection schemes already dis-cussed (optoacoustic, optogalvanic, etc.). Afurther improvement in the sensitivity can beobtained if the output power is monitoredand the laser is running close to threshold.Indeed, because of the nonlinear response ofthe laser near threshold, small changes inthe losses Ag caused by the absorbing intra-cavity sample lead to a dramatic variation ofthe laser output. If Go is the unsaturatedgain and g the total loss, it can be shownthat the amplification factor q is

q = G0/(G0 - g)(g + Ag) - Go/g(Go - g),(47)

where it has been supposed that Ag < g.Equation (47) shows that the sensitivity canbe greatly enhanced when the threshold con-dition is approached (Go -• g). Nevertheless,an upper limit is imposed by the huge insta-bilities that a laser exhibits when such a con-dition is reached. Amplification factors of 105

have been obtained, allowing the study ofvery weak transitions having oscillatorstrengths below 10"12 or with very low con-centrations below 108 atoms cm"3.

Intracavity spectroscopy has been exten-sively used in the infrared and far infrared


where the alternative of fluorescence spec-troscopy does not exist. The CO2 laser, withtens of lines around 10 /mi, is commonlyused, and, because of the narrow tunabilityof the laser, the resonance condition is usu-ally achieved by shifting the atomic or mo-lecular levels by means of the Stark and Zee-man effects. The Stark effect is used forpolar molecules, while Zeeman shifting af-fects paramagnetic species. By extrapolatingthe measured frequency to zero field, infor-mation on the electric dipole moment or gy-romagnetic (g) factors can be obtained.

Most rotational molecular lines lie in thefar infrared (50 /mi < A < 1 mm) where co-herent radiation is usually generated by op-tically pumped lasers. An apparatus for in-tracavity spectroscopy in this spectral regionhas been developed by Evenson (1981) to in-vestigate the rotational structures of mole-cules and the fine structure of atoms. In thequasiconfocal laser cavity is placed the sam-ple cell, which is separated from the activemedium by a window at Brewster's angle.The population inversion in the active me-dium is created by a CO2 laser, and the far-infrared output is detected by means of a liq-uid-helium-cooled bolometer; such detectorsshow very low noise and hence enhancedsensitivity. Absorption coefficients of 10"10

cm"1 have been measured in studies of tran-sient species and free radicals produced indischarges, and of intermediate products inchemical reactions.

An interesting demonstration of the highsensitivity reached by this technique isshown in Fig. 29, which shows the fine-struc-ture 3P0~

3î component of the ground stateof silicon. Atoms of silicon were produced bydissociation of silane (SiH4) in a microwavedischarge, and the resonance condition wasachieved by applying a magnetic field of1121.2 Gauss. It should be noted that thereis an excellent signal-to-noise ratio in spiteof the extremely low transition probability A= 8 X 10"6 s"1.

3.5 Fast Modulation Spectroscopy

We have already seen that a significantimprovement in sensitivity over direct ab-sorption techniques may be achieved bymodulating the laser amplitude and using aphase-sensitive detection scheme. This prin-ciple can be extended to modulating the la-

b

S i ^

I1Iif 11

ia

MAGNETIC FIELD

FIG. 29. Laser magnetic resonance measurement ofthe silicon ground-state fine-structure component3P0-3Pi of the 129.5-mm line CH3OH. In (a) and (b),the far-infrared cavity was slightly detuned to fre-quencies respectively higher and lower than the tran-sition to the ground state. (From Evenson, 1981).

ser frequency (wavelength modulation) at arelatively high frequency (kilohertz) and byan amount typically several times smallerthan the width of the absorption line of thespecies under investigation. Scanning the la-ser wavelength and using ac detection at themodulation frequency, or twice the modula-tion frequency, provides a detected signalthat is the first or second derivative of theabsorption line shape. When the modulationfrequency is comparable with or larger thanthe linewidth (megahertz or gigahertz range),the two modulation sidebands may be re-solved, and the technique is known asfrequency-modulation spectroscopy. Suchtechniques are particularly suited to semi-conductor diode lasers, whose wavelengthsmay be modulated at very high frequenciessimply by modulating the injection current.


— - 30

S-«o= -50J -60Q.

2 -ro< -eo

O-IOOr . , , o

-120

CARRIER (dc)

CALC. SHOT NOISE LEVEL

I _L

AMPLIFIER LEVEL

JOkHl IOOkH2 IMH2FOURIER NOISE FREQUENCY

10MHzFIG. 30. Dye-laser amplitude noisespectral density (300 /xW lightpower, 10 kHz analysis bandwidth).

In this fashion, absorptions down to 10 9

cm"1 can be measured. The advantages ofthese techniques derive essentially from thehigh modulation frequencies, which allowthe predominantly low-frequency \lf noise tobe rejected. Figure 30 shows the noise spec-trum for a diode laser in the megahertz re-gion where the amplitude fluctuations aredominated by shot noise.

In Fig. 31, the same transition in methanehas been observed using absorption, wave-length-modulation, and frequency-modula-tion techniques. The signal-to-noise ratio inthe FM case clearly represents an improve-ment of several orders of magnitude. More-over, the sloping baseline is removed, andhigh discrimination against signals that donot show wavelength dependence is intro-duced.

Still higher sensitivity can be achievedwith heterodyne detection, which is an ex-tension to optical and infrared wavelengthsof the superheterodyne radio receiver. Abeam of radiation of frequency vs is detectedby a photodiode that absorbs photons andreleases charge carriers, the photocurrent be-ing proportional to the square of the totalelectric field Es of the incident radiation.With direct detection, this means that thephotocurrent is proportional to the power ofthe optical or IR field.

In heterodyne detection, however, the in-coming radiation is mixed in the detectorwith radiation from a local oscillator (Et) offrequency vh which could be a diode laser or

(a)

30 GHz

'600 MHz(c)

- 600 MHz

FIG. 31 . (a) Pure absorption signal on the third over-tone of methane (CH4) at 100 Torr. Derivative lineshapes recorded at 500 mTorr with (b) WM and (c)two-tone FM spectroscopy.


a CO2 laser, producing a photocurrent con-taining the terms E*, Ef, Est and Eb The firsttwo terms correspond to the optical and lo-cal oscillator powers. More interesting is themixing term EsEt that oscillates at the beatfrequency vh = \vs — vt\ between the twofields. This frequency difference vb can bechosen to be in the radio-frequency regionwhere amplification can readily be per-formed. The heterodyne receiver can thusdetect radiation in a narrow spectral bandaround the local oscillator frequency, andthe bandwidth is determined by electronicfilters instead of optical filters or dispersiveoptics. The power signal-to-noise ratio of aheterodyne receiver is

SNR = >qPs/hvB, (48)

where rj is the heterodyne quantum effi-ciency, Ps is the signal power, and B is thebandwidth of the receiver. It is sometimesalso convenient to use phase-sensitive detec-tion at the frequency of a beam chopper toincrease further the signal-to-noise ratio.

4. TIME-RESOLVED SPECTROSCOPY

In the preceding sections we have dis-cussed a variety of spectroscopic techniquesin which information about atomic or mo-lecular structures is obtained by analyzingthe response of the system as a function ofthe optical frequency. In this section, by con-trast, we consider the temporal response of asystem when it is irradiated by short laserpulses. We will see that spectroscopic tech-niques based on the temporal and frequencydomains can provide complementary infor-mation.

The availability of short and intense laserpulses allows the transfer of a large fractionof an irradiated sample of ground-state at-oms or molecules into excited states. By useof step-wise excitations with synchronized la-sers, highly excited levels can be reached. Bymonitoring of the temporal decay of fluores-cence from a given level, the lifetime of thatlevel can be measured. If two closely lyingstates are excited coherently, a new class ofphenomena can be studied, such as quantumbeats, whereby atomic interference is ob-served as a temporal modulation of the fluo-rescent intensity. The recent development of

ultrashort laser pulses, of picosecond or sub-picosecond duration (Shapiro, 1977), makespossible the investigation with extremelyhigh time resolution of ultrafast relaxationprocesses occurring during the excitationand deactivation of molecular states.

The interaction between atoms and veryshort, intense laser pulses also gives rise tostrong nonlinear processes. Atoms in very in-tense laser fields can be ionized by absorbingtens of photons. Furthermore, within thesame laser pulse, the ions produced can befurther ionized several times (multichargeionization), while quasifree photoelectronsmay absorb further photons (above-ioniza-tion spectroscopy). In such strongly nonlin-ear regimes, high-order harmonics reachinginto the x-ray region can be generated.

4.1 Lifetime Measurements

Lifetime measurements are of particularinterest in spectroscopy, for the lifetime isdirectly related to the transition probability,and hence, such measurements make possi-ble the testing of quantum mechanical cal-culations. Anomalies among the lifetimeswithin a molecular band can also provide in-formation about perturbations due to cou-pling with other series. Finally, lifetimes areof more applied interest—for instance, in la-ser physics.

In Sec. 1 we showed that the radiativewidth of a line is related to the lifetimes ofthe lower and upper levels of the transition.We also mentioned that the homogeneouswidth is usually affected by such otherbroadening effects as saturation, time offlight, collisions, etc. Even if these furtherbroadening mechanisms are removed, to de-termine from a transition linewidth the life-time of a given level requires knowledge ofthe lifetime of the other level (unless it isthe ground state). Unlike frequency-domainspectroscopy, time-resolved measurement al-lows the lifetime to be evaluated directly.Several experimental approaches have beendeveloped.

4.1.1 Phase-Shift Method This methodis based on the use of a cw laser, sinusoidal-ly modulated in amplitude at a frequency fand resonant with a transition i -* k of fre-quency ojik. The rate equation for the finalstate k is


dNkldt = aF(N{ - Nk) - Nk/r, (49)

where a is the absorption cross section, Fthe photon flux, and r the lifetime of thelevel k. The difference N{ - Nk in Eq. (49)takes into account absorption and stimulatedemission, while Nhlr represents the sponta-neous emission decay. The photon flux is

F = (I0/hcoik)(l + a s cos<oikt. (50)

The induced resonant fluorescence is modu-lated at the same frequency but is shifted inphase with respect to the forcing excitationfield by an amount #, related to the levellifetime T by

tan<f> = (51)

By measuring the phase shift cf> with a lock-in amplifier, the lifetime r can thus be esti-mated.

This method fails when two or more lev-els with different lifetimes are simulta-neously populated. In this case the inducedfluorescence for each level presents differentphase shifts, which are not easily determina-ble. The method also suffers in the event ofstimulated emission [see Eq. (49)], which isespecially likely to be present when lasersources are used.

4.1.2 Pulse Excitation Another ap-proach to lifetime measurement makes useof pulsed lasers able to excite the atoms ormolecules in a time short compared with thelifetime of the investigated level. The fluores-cence decay can be observed in the absenceof stimulated emission in two different ways.

With the first method the fluorescence ismonitored directly by means of a transientdigitizer or a boxcar. To reconstruct thewhole exponential curve using either tech-nique requires many fluorescence photonsper excitation pulse, and nonlinearities in thephotomultiplier response present a furtherlimitation to this technique. These difficultiescan be overcome by a delayed-coincidencemethod, which operates in the regime of ex-tremely low intensity. In this technique, sin-gle-photon counts are recorded while therepetition rate of the excitation laser pulse ischosen to be as high as possible. At the heartof the experimental scheme is a time-to-am-plitude converter. A trigger signal synchro-

nized to the laser pulse starts a voltage rampat time tQy which is stopped by the first fluo-rescence photon after the excitation pulse,detected at time tx. The ramp height is thusproportional to the time interval tx — t0 andis stored by a multichannel analyzer. Aftereach excitation pulse, a given channel will beincreased by one unit, and the voltage distri-bution recorded by the multichannel ana-lyzer thus yields the decay curve directly (seeFig. 32).

It is important that the detection proba-bility be kept below one fluorescence photonper excitation pulse to avoid so-called pileup.If two or more photons were generated, thephotons with short delays would be over-rep-resented, and the decay curve would be al-tered.

Although a high repetition rate is requiredin order to provide a reasonable measure-ment time, the pulse frequency also has anupper limit, for the time interval betweentwo pulses must be longer than the lifetimeof the investigated level. Moreover, the pulsefrequencies must not exceed the reciprocalof the dead time of the electronic chain,which is typically 100 ns.

The decay of an excited state can also bemonitored with a second laser (probe) tunedto a transition sharing a common level withthe investigated transition (pump-and-probetechnique). When the pump pulse alters thepopulation of this level, the probe pulse canmonitor how quickly the population is re-stored through relaxation processes if a con-trolled delay is introduced between pumpand probe pulses. The pump-and-probe tech-nique is particularly useful for investigatingultrafast processes (picosecond and femto-second range) where conventional techniquesfail.

4.2 Quantum-Beat Spectroscopy

If two or more closely spaced atomic ormolecular levels are simultaneously excitedby a short laser pulse at t = t0, the totalwave function will be given by the superpo-sition of the wave functions of the excitedstates:

(52)

where ak is the probability amplitude that


end prism

modulator

end mirror C 3

beam splitterlens

No cell

photodiode

Tt excittr

mode lockedargon ion laser

photodiode

sampling

oscilloscope

lens

monochromotoi

photomultiplitr

discriminator

discriminatorfrequency

divider 2 : 1

stop!

ratemeter

start

time to ampli-tude converter

multi channelanalyzer

(a)

FIG. 32, (a) Schematic diagram of the delayed-coincidence technique, (b) Fluorescence decay, reconstructedby a multichannel analyzer. (From Demtroeder, 1981.)

the light pulse has prepared the atom inlevel \k). The time-dependent intensity of thefluorescence emitted when the excited levelsdecay into the final state If) is given in termsof the matrix element by

= CI\{®f\€r\V{t))\2, (53)

where e is the polarization vector and *P(t) isthe temporal evolution of the total wavefunction, given by


exp [-(iEkf/h + yk/2)t].

(54)

Inserting (54) into (53) and assuming equalthe decay rates of levels 1 and 2 (y1 = y2 =y), we have

= Cexp(-yt)(A + B cosa)21t), (55)

where A is the sum of the transition proba-bilities 1 -> f and 2 -> f, 5 is the interferenceterm, and co2i = (£2 ~ £i)/&. The time-de-pendent fluorescence is hence given by thesuperposition of an exponential curve and anoscillation with frequency co21. The measure-ment of such an oscillation allows the deter-mination of very closely spaced energy levels.Quantum-beat spectroscopy therefore allowsDoppler-free resolution.

In Fig. 33 a typical example of a Zeemanquantum beat for a resonance line in ytter-bium is shown. The three Zeeman sublevelsof the upper level 3P1 are excited simulta-neously, but only the levels with m; = +1and m; = — 1 can decay into the groundstate ô (transitions with Arrij = 0 for m; =0 -> nij• = 0 are forbidden by the selectionrules). The modulation observed in the time-

dependent fluorescence is thus related to theseparation of these two levels.

If further levels are excited simulta-neously, modulations appear in the time-dependent fluorescence at a different fre-quency, and their measurement in the timedomain becomes rather difficult. In this casethe Fourier transform of the time-dependentfluorescence intensity yields its spectral dis-tribution I((o) with sub-Doppler resolution.Figure 34 illustrates as an example quantumbeats measured by Andra et ai (1975) in thefluorescence following excitation of three hy-perfine levels in the 6p 2P3/2 state of the137Ba+ ion.

5. ULTRAHIGH-RESOLUTIONSPECTROSCOPY

Many of the processes by which spectrallines are broadened are due to the randommotion of the atoms. Let us consider anatom of mass M and velocity v. On accountof its motion, the atom absorbs photons at afrequency (or given by the relation

O)' = O)0 + kv — (O0V2/2c2, (56)

where <u0 = (Eb — Ea)/h is the resonance fre-

1000 2000 3000 4000

time/ns

FIG. 33. Typical example of Zeeman quantum beat for the resonance line in ytterbium (from Svanberg, 1992).


10s

104

103

102

10'

' • • k

466.7

F=3

625.1

158.310

A = 125.0 (5) MHzB= 91.6 (5) MHz

FIG. 34. Quantum beats in fluorescence from the 6p 2P3 /2 state of the 137Ba+ ion (from Andra, 1975). The up-per curve represents the time evolution of the emitted fluorescence following excitation, and the lower curve isthe Fourier-transform spectrum.

quency and the second and third terms rep-resent, respectively, the first- and second-or-der Doppler shifts.

In the preceding sections we have seenthat the linear (first-order) Doppler shift canbe canceled by use of atomic beams or bymeans of one of the numerous Doppler-freespectroscopic techniques, but such methodsfail to eliminate the second-order Dopplershift, which, as shown in Eq. (56), dependsquadratically upon the atomic velocity. An-other effect strictly connected to the atomicmotion is the previously discussed transit-time (or time-of-flight) broadening. In manycases these two broadenings determine theultimate limit of spectral resolution. This isthe case, for instance, for transitions involv-ing long-lived levels for which the radiativewidth is very narrow. Full resolution of suchnarrow transitions is of importance in test-ing the fundamental laws of physics and formany metrological applications such asatomic clocks. In order to achieve maximumresolution, it is necessary to reduce theatomic velocities, since this reduces the sec-ond-order Doppler shift and increases the in-

teraction time between the atoms and the la-ser field.

When an atom interacts with a lightbeam, the emitted and absorbed photonscarry much information about the atomicstructure, this being the essence of spectros-copy. But the interaction of photons with at-oms can also be used to manipulate the ki-netic status of the atoms, i.e., their velocity.This phenomenon, usually manifested as so-called laser cooling, was first suggested in-dependently by Hansch and Schawlow(1975) for neutral species and by Winelandand Dehmelt (1975) for trapped ions. A moredetailed discussion of this subject is reportedin the article ATOMIC COOLING AND TRAPPING.

In the last few years, there has been in-creasing interest in the use of near-resonantphoton scattering to cool and trap atoms andions. This interest is motivated in part by thephysics governing such phenomena and inpart by the wide variety of atomic and mo-lecular physics experiments to which it maybe applied. The production of ultracoldatomic samples alone, having temperaturesof a few tens of microkelvins, has opened up


a r ich area for experiments in atomic andmolecular physics. One of the main applica-tions is in ultrahigh-resolution spectroscopy,since for very slow atoms the first- and sec-ond-order Doppler shifts and the pressureshift are eliminated, and Doppler-free tech-niques are no longer required.

The production of cold atoms moreoveroffers the possibility of revisiting the physicsof collisions in a new light. There are twoprincipal reasons for investigating collisionsin atomic traps: The first is to understand,and perhaps to minimize, the role of suchcollisions because they represent a signifi-cant loss channel; the other is to explore thefundamental physics of ultracold collisionsbetween atoms with large de Broglie wave-lengths, for which hitherto unseen effectscan be expected.

The recoil depends on the energy of thephoton (h v) and the mass M of the atom (Av= hvlcM). For instance, for sodium (M = 23amu) and A = 589 nm, the velocity changeper photon absorbed is about 3 cm s"1. Ifthe initial atomic velocity is typical of room-temperature distributions (about 103 m s"1),the number of scattered photons required tostop the atoms will be nphot = vlAv = 3 X104. The cooling transition a -> b must there-fore be closed, in the sense that atoms in up-per level b may decay radiatively only to theinitial state a. For sodium, the 3S1/2-3P3/2transition is suitable for such a coolingscheme, even though the presence of hyper-fine structure renders the transition incom-pletely closed (optical pumping).

During laser cooling, atoms undergo amaximum deceleration given by

5.1 Laser Cooling

The principle of laser cooling is best illus-trated by a two-level atomic system (Meystreand Stenholm, 1985; Chu and Wieman,1989). We consider an atom of mass M mov-ing along the z axis with a velocity v. A pho-ton of frequency co traveling in the oppositedirection can be absorbed if co is appropri-ately Doppler shifted with respect to the res-onance frequency coo.

The photon momentum hvlc changes thevelocity of the atom by

a =

Av = hvlMc. (57)

We suppose that the excited atom can decayradiatively only back to the initial state. Dur-ing this process, the atom also receives a ve-locity kick when it radiates a photon byspontaneous or stimulated emission. In thepresence of a single laser beam, stimulatedphotons will be emitted in the same direc-tion as the incoming photons, and as a con-sequence, the momentum transferred by theabsorbed photon will be canceled. In con-trast, the spontaneous photons will be emit-ted in random directions; the average mo-mentum transfer is in this case zero, andthere is thus overall a net momentum trans-fer from the radiation to the atoms. Becausethis process depends upon spontaneousemission, the radiation pressure force thatresults is often known as the spontaneousforce.

where r is the lifetime of the upper level ofthe cycling transition and the factor 2 takesinto account the fact that atoms spend onlyhalf of their time in the upper level. Againfor sodium, a = 106 m s"2, which corre-sponds to 105 times g. This strong accelera-tion is sufficient to bring to rest a thermalsodium atom in 1 ms and over a distance of0.5 m, which is quite reasonable on the lab-oratory scale.

An experimental apparatus suitable for la-ser cooling is shown in Fig. 35. A laserbeam, appropriately detuned with respect tothe atomic resonance, travels against an ef-fusive atomic beam. A second laser beam,acting as an analysis beam, is roughly collin-ear with the cooling laser. When the laserfrequency is tuned onto resonance, atomsabsorb and re-emit photons that can be de-tected by a photomultiplier. The analysisbeam is swept in order to probe the velocitydistribution modified by the cooling beam.Part of the analysis beam is sent perpendic-ular to the atomic beam to provide a zero-velocity marker.

The magnetic coils in Fig. 35 produce avarying magnetic field along the atomicbeam direction, which shifts the Zeemansublevels so that the transition frequency be-tween two Zeeman levels changes with posi-tion. With a suitable dependence of B(z), thetransition frequency for moving atoms can


FIG. 35. Schematic diagram of theapparatus used for the laser coolingof neutral atoms (from Beverini etai, 1989).

be resonant with the fixed laser-cooling fre-quency.

The early experiments on laser cooling ofneutral atoms used a sodium beam chosenboth because the resonance transition isclosed and accessible to tunable single-modedye lasers and because of its short radiativelifetime. Many atoms have since been manip-ulated with laser light, and in Fig. 36 weshow the result of laser-cooling atoms of cal-cium using the transition 1S0-

1P1 at 422 nm.

5.2 Atom TrapsIn the preceding section we have dis-

cussed a method of stopping the atoms in anatomic beam. Another aspect of atomic ma-

nipulation is confinement to a region ofspace where the atoms can then be investi-gated.

In 1985, Chu et al demonstrated the pos-sibility of cooling neutral sodium atoms inso-called optical molasses. Magneto-opticaltraps represent a refinement of this ideawhereby atoms are localized at higher densi-ties and cooled to temperatures limited bythe atomic recoil energy.

5,2.1 Magneto-optical Traps The mag-neto-optical trap (MOT) (Raab et at, 1986)depends upon two features: the Zeeman in-teraction with an inhomogeneous magneticfield, and the radiative selection rules that

Ca

Laser cooling on

Laser cooling off

v(mfc) 900 600 300FIG. 36- Laser cooling of calcium ob-served in presence of magnetic field.


transitions between magnetic levelsin a n atom.

T h e basic principle of the MOT can be il-lustrated by considering a hypothetical atomwith- a spin S = 0 (ms = 0) ground state anda sp in S = 1 (ms = - 1 , 0, +1) excitedstate. In a weak, inhomogeneous magneticfield B(z) = bzt the energy levels will be splitthrough the Zeeman effect by an amount AE= /uLmsbz- In the presence of two counter-propagating waves with a+ and cr~ circularpolarizations (see Fig. 37), with a frequencytuned below the atomic resonance, an atomat z > 0 wiH absorb more cT photons thana+ photons since the laser frequency iscloser to the Ams = - 1 transition fre-quency. Similarly, for an atom at z < 0, theZeeman shift is reversed, and the atom willabsorb more a+ photons. Because of the mo-

mentum carried by these photons, the posi-tion-dependent differential absorption leadsto the presence of a force, which pushes at-oms into the region z = 0. The scheme canbe readily extended to three dimensions byusing three pairs of counterpropagatingbeams along the x, y, and z directions and aspherical quadrupole magnetic field, asshown in Fig. 37.

This type of trap can also be used for at-oms with a more complicated hyperfinestructure such as, for instance, sodium or ce-sium. Atomic samples with densities of 1010

atoms cm"3 and at temperatures of a fewtens of millikelvins have been obtained. Atsuch low temperatures, atoms move with ve-locities of a few centimeters per second,which allows direct Doppler-free spectro-scopic measurements. In Fig. 38 is shown a

(a)

FIG. 37. Arrangement for a mag-neto-optical trap, (a) The energy-level scheme for an atom havingspin S = 0 in the ground stateand S = 1 in the excited state,(b) Experimental arrangement of athree-dimensional trap. (FromRaab et ai, 1986.)


FIG. 38. (a) Resonant fluorescence from cold atomiccesium produced in a magneto-optic trap, (b) Forcomparison, the saturated absorption spectrum isshown.

spectrum of the cesium 6S1/2(F = 3, 4)-bPmiF = 3, 4) transition obtained by laserirradiation of a cloud of trapped atoms [Fig.38(a)] and a conventional saturated absorp-tion spectrum [Fig. 38(b)]. The linewidth ob-served with the cold atoms is about 30 MHz,somewhat wider than the natural width ofthe transition (5.3 MHz). This is essentiallydue to the saturating effect of the trappinglaser. In order to improve the resolution, theatoms should be measured after switchingoff the trapping and repumping laser beams.

GLOSSARY

Absorption Spectroscopy: The study ofradiant energy that is characteristically ab-sorbed by a particular atom or molecule.

Bennet Hole: The hole that is producedin the velocity distribution when narrow-band laser light is absorbed.

Coherence: One of the most importantproperties of the laser radiation. It is essen-tially related to the high spectral purity ofthe laser light. Coherence takes part also inthe excitation of atomic or molecular transi-tion when this is realized with laser sources.

Doppler Linewidth: The width of a spec-tral line caused by the Doppler effect.

Doppler-Free Spectroscopy: Spectro-scopic techniques that allow elimination ofthe Doppler broadening. They are essentiallybased on two main approaches: velocity-se-lective saturation or counterpropagating two-photon absorption.

Emission Spectroscopy: The study of ra-diant energy that is characteristically emittedby atoms or molecules after a suitable exci-tation.

Intracavity Spectroscopy: High-sensitiv-ity spectroscopic techniques where theatomic or molecular sample under investiga-tion is inserted into a laser cavity or a pas-sive cavity having a high quality factor.

Laser: An acronym for light amplificationby stimulated emission of radiation. A devicethat emits a high-intensity, narrow-spectral-width, and highly directional beam of lightby stimulated emission in an atomic or mo-lecular system where a population inversionbetween two levels has been produced.

Laser Beam: The bright stream of lightemitted by a laser. The near-zero divergencemakes this beam extremely interesting for awide variety of applications (surgery, com-munications, printing, physics, chemistry,etc.).

Laser Cooling: A method based on thescattering of near-resonant photons to slowatoms in an atomic beam to very low veloc-ity.

Laser Trapping: A method based preva-lently on laser radiation to trap neutral andionized atoms in a small volume at ex-tremely low kinetic energy and, hence, lowtemperatures.

Linewidth: The spread in wavelength of aspectral line. Usually it is measured as thefull width at half maximum (FWHM).

Multiphoton Spectroscopy: Spectro-scopic techniques based on the simultaneousabsorption of two or more photons.

Natural Linewidth: The width of a spec-tral line caused by the finite lifetime of thelevels.

Optogalvanic Spectroscopy: Spectro-scopic techniques where atomic or moleculartransitions are monitored through the varia-tion of current of an electrical dischargewhen this is irradiated by resonant radiation.

Optoacoustic Spectroscopy: Spectro-scopic techniques where atomic or moleculartransitions are monitored through the detec-tion of an acoustic wave produced by ab-sorption of radiation.

Saturation: The condition reached in atwo-level system in presence of resonant ra-diation when the populations of the two lev-els become equal.


Spectral Line: The response of a gas-phase sample obtained from the emission orabsorption of radiation when the frequencyis tuned around an atomic or moleculartransition.

Spectroscopy: The branch of physicsconcerning the measurement of the emissionand absorption spectra of atoms and mole-cules. According to the frequency range ofthe electromagnetic radiation, we speak ofvacuum ultraviolet, ultraviolet, visible, infra-red, or microwave spectroscopy.

Spectrum: The set of lines recorded afteremission or absorption of radiation.

Time-Resolved Spectroscopy: Spectro-scopic techniques that allow one to investi-gate fast relaxation processes involving thelevels excited with a short-pulse laser.

Two-Photon Spectroscopy: Spectroscopic technique where a transition forbid-den in the dipole approximation can beinduced by the absorption of two counter-propagating photons.

Works Cited

Abella, I. D. (1962), Phys. Rev. Lett. 9, 453-455.Amin, S. R., Caldwell, C. D., Lichten, W.

(1981), Phys. Rev. Lett. 47, 1234-1238.Andra, H. J. (1975), in G. zu Putlitz, E. W. We-

ber, A. Winnacker (Eds.), Atomic Physics, PlenumPress, New York.

Barbieri, B., Beverini, N., Sasso, A. (1990), Rev.Mod. Phys. 62, 603-644.

Battaglia, A., Gozzini, A., Polacco, E. (1959),Nuovo dm. 14, 1076-1081.

Bell, A. G. (1881), Philos. Mag. 11, 308.Beverini, N., Giammanco, F., Maccioni, E.,

Strumia, F., Vissani, G. (1989), /. Opt. Soc. Am. B6, 2188-2193.

Bjorkholm, J. E., Liao, P. F. (1974), Phys. Rev.Lett. 33, 128-131.

Bjorkholm, J. E., Liao, P. F. (1976), Phys. Rev.A 14, 751-760.

Bloembergen, N., Levenson, M. D., Salour, M.M. (1974), Phys. Rev. Lett. 32, 867-869.

Brossel, J., Cagnac, B., Kastler, A. (1954) /.Phys. Radium 15, 6-8.

Bruzzese, R., Sasso, A., Solimeno, S. (1989),Riv. Nuovo Cimento 12 (7).

Cagnac, B., Grynberg G., Biraben, F. (1973), /.Phys. (Paris) 34, 845-858.

Chu, S., Wieman, C. (Eds.) (1989), Special is-sue on Laser Cooling and Trapping of Atoms, /.Opt. Soc. Am. B 6, 2018-2278.

Corney, A. (1977), Atomic and Laser Spectros-copy, Clarendon Press, Oxford.

de Angelis, M., Inguscio, M., Julien, L., Marin,F., Sasso, A., Tino, G. M. (1991), Phys. Rev. A 44,5811-5819.

Demtroeder, W. (1981), Laser Spectroscopy,Springer Series in Chemical Physics, Springer-Ver-lag, Berlin Heidelberg.

Ernst, K, Inguscio, M. (1988), Riv. Nuovo dm.11 (2).

Evenson, K. M. (1981), Faraday Disc. R. Soc.Chem. 71, 7-14.

Fornaca, G., Gozzini, A., Strumia, F. (1963), inR. Servant, A. Charm (Eds.), Electronic MagneticResonance and Solid Dielectric, North HollandPubl. Comp., Amsterdam, p. 554.

Gianfrani, L., Sasso, A., Tino, G. M., Marin, F.(1991), Nuovo Cim. D 13, 1221-1234.

Goeppert-Mayer, M. (1931), Ann. Phys. (Leipzig)9, 273-294.

Hansch, T. W., Toschek, P. (1970), Z. Phys.236, 213-244.

Hansch, T. W., Levenson, M. D., Schawlow, A.L. (1971), Phys. Rev. Lett. 26, 946-949.

Hansch, T. W., Schawlow, A. (1975), Opt. Com-mun. 13, 68-69.

Hansch, T. W. (1977), in: N. Bloembergen(Ed.), Proceedings of the International School ofPhysics "Enrico Fermi," Course LXFV, NonlinearSpectroscopy, Amsterdam: North-Holland, p. 17.

Hansch, T. W., Lee, S. A., Wallenstein, R., Wie-man, C. (1975a), Phys. Rev. Lett 34, 307-309.

Hansch, T. W., Schawlow, A., Series, G. W.(1975b). Sci. Am. 240 (3), 94-110.

Hurst, G. S., Payne, M. G., Kramer, S. D.,Young, J. P. (1979), Rev. Mod. Phys. 51, 767-819.

Letokhov, V. S., Chebotaev, V. P. (1977), Non-linear Laser Spectroscopy, Springer Series in Opti-cal Science, Springer Verlag, Berlin, Heidelberg,New York.

McFarlane, R. A., Jr., Bennet, W. R., Jr., Lamb,W. E. (1963), Appl. Phys. Lett 2, 189-190.

Meystre, P., Stenholms, S., (Eds.) (1985), spe-cial issue on Mechanical Effects of Light, /. OptSoc. Am. B 2, 1706-1860.

Minguzzi, P., Profeti, S., Tonelli, M., di Lieto,A. (1982a), Opt Commun. 42, 237-240.

Minguzzi, P., Tonelli, M., Carrosi, A. (1982b), 7.Mol. Sped. 96, 294-305.

Penning, F. (1928), Physica (The Hague) 8, 13-23.

Pinard, M., Aminoff, C. G., Laloe, F. (1979),Phys. Rev. A 19, 2366-2370.

Raab, E., Prentiss, M., Cable, A., Chu, S., Prit-chard, D. (1986), Phys. Rev. Lett 59, 2631-2634.

Shapiro, S. L. (1977), Ultrafast Light Pulse, Top-ics in Applied Physics, Vol. 18, Springer, Berlin,Heidelberg, New York.

Siegman, A. E. (1971), An Introduction to La-sers and Masers, McGraw-Hill, New York.

Sorem, M. S., Schawlow, A. L. (1972), OptCommun. 5, 148-151.


Svanberg, S. (1992), Atomic and MolecularSpectroscopy, Springer Series on Atoms andPlasma, Springer-Verlag, Berlin, Heidelberg.

Svelto, O. (1976), Principles of Lasers, PlenumPress, Heyden, London, New York.

Szoke, A., Javan, A. (1963), Phys. Rev. Lett. 10,521-524.

Tarn, A. C. (1986), Rev. Mod. Phys. 58, 381-431.Teets, R. E., Kowalski, F. V., Hill, W. T., Carl-

son, N. W., Hansch, T. W. (1977), in: A. H. Zewail(Ed.), Advances in Laser Spectroscopy I, SPIE Pro-ceedings No. 113, Bellingham, WA: SPIE, p. 409.

Vasilenko, L. S., Chebotaev, V. P., Shishaev, A.V. (1970), Zh. Eksp. Teor. Fiz. Pis'ma Red. 12, 161-165 [JETP Lett. 12, 113-116].

Yariv, A. (1975), Quantum Electronics, Wiley,New York.

Wieman, C, Hansch, T. W. (1976), Phys. Rev.Lett. 36, 1170-1173.

Wineland, D., Dehmelt, H. (1975), Bull. Am.Phys. Soc. 20, 637.

Further Reading

Corney, A. (1977), Atomic and Laser Spectros-copy, Clarendon Press, Oxford.

Demtroeder, W. (1981) Laser Spectroscopy,Springer Series in Chemical Physics, Springer-Ver-lag, Berlin, Heidelberg.

Svanberg, S. (1992), Atomic and MolecularSpectroscopy, Springer Series on Atoms andPlasma, Springer-Verlag, Berlin, Heidelberg.

Svelto, O. (1976), Principles of Lasers, PlenumPress, Heyden, London, New York.

Yariv, A. (1975), Quantum Electronics, Wiley,New York.

SPECTROSCOPY, MOLECULARSee MOLECULAR SPECTROSCOPY

HIGH SENSITIVITY TRACE GAS MONITORINGUSING SEMICONDUCTOR DIODE LASERS

C Corsif, and M. Ingusciotf

^niversita degli Studi di FirenzeDipartimento di NeuroscienzeViale Pieraccini 650139 Firenze, Italy

Êuropean Laboratory for Nonlinear Spectroscopy (LENS) andINFM-Istituto Nazionale Fisica della MateriaLargo E. Fermi 250125 Firenze, Italy

1. INTRODUCTION

In recent years semiconductor diode lasers in the visible near-infrared have been applied tohigh sensitivity gas detection for a variety of environmental, medical and industrial applications.

The advantage of laser-based gas sensor with respect to conventional electro-chemical andsemiconductor point sensors resides in their characteristics of non-intrusiveness, high gasselectivity, high detection speed and low cost.

In particular, great attention has been attracted by InGaAs-InP Distributed Feed-Back(DFB) diode lasers, operating at room temperature. Thanks to the knowledge and technologydeveloped for telecommunication diode lasers emitting around 1.3|im and 1.5|im, these lasers canbe easily designed to emit single mode almost anywhere in the region between l[im and 2(im.

This spectral region is of particular interest because of the presence of molecular overtonevibrational bands for many important gases, like for instance, CO2, CO, H2S, HC1, HF, NH3,CH4, H2O, NO, N2O.

Although the transition strengths are at least one order of magnitude weaker than those forthe fundamental bands in the mid-infrared, this problem is compensated, in real applications, bythe advantages coming from the reduced opacity of the atmosphere and from the possibility ofconnecting the lasers with optical fiber systems, developed for telecommunication purposes.

Furthermore diode lasers are particularly suited for high-sensitivity absorption spectroscopybecause they show much smaller amplitude noise than most other laser sources. In addition,diode lasers can easily be modulated at frequencies up to several GHz.

Optical Sensors and Microsystems: New Concepts, Materials, TechnologiesEdited by Martellucci et al., Kluwer Academic / Plenum Publishers, New York, 2000. 193

This allows to apply various detection techniques1'2'3 such as low-wavelength modulationspectroscopy (LWM) and frequency modulation spectroscopy (FM)4*5 depending on thesensitivity level necessary to be reached.

In principle, sensitivities near to the fundamental quantum limit (shot-noise limit) arepossible. In practice, quantum-limited sensitivity has proven to be difficult to obtain, due to anumber of technical noise sources.

The noise sources can be divided in detector noise, noise due to amplitude fluctuations ofthe laser field and optical noise due to interference fringes.

1.1 Detector noise

There are three major components in a photodetector noise: Johnson (thermal) noise,detector shot noise, and 1/f-noise.

Johnson noise is due to the thermal fluctuations of the charge carrier density within theresistor itself. The thermal noise current can be expressed as:

R ) <»

where K is the Boltzmann constant, T the temperature, A v the detection bandwidth and R thedetector system resistance. The Johnson noise has a white frequency spectrum and can bereduced by cooling the detector.

Shot noise is due to quantum fluctuations of the radiation field. They give rise tofluctuations of the detected current in a photodetector and can be expressed by:

where e is the electronic charge, t] is the quantum efficiency of the detector, P is the incidentpower and v is the photon energy. Shot noise is also white and independent of the modulationfrequency but is proportional to the square root of the laser power. For optical powers in the mWrange the output noise-voltage of the photodetector is dominated by shot noise rather than byJohnson noise.

Often devices show various sources of other noise mechanisms. In many cases thisadditional noise shows a 1/f dependence, but unfortunately no theoretical analysis are available.An empirical expression for the 1/f noise current is given by:6

(3)

where C is a proportional factor, a and b are constants close to unity. The 1/f noise depends onthe manufacturing processes, in particular on electric contacts and surfaces. It dominates thedetector noise for frequencies below 1 kHz and drops below the Johnson and shot noise levels athigher frequencies. The rms 1/f noise current shows an approximately linear dependence on thephotocurrent and therefore on the light intensity.Since all three detector noise sources depend onthe detection bandwidth Av a reduction of the bandwidth results in a noise-reduction. It is alsoconvenient to work at higher frequencies where the 1/f-detector noise has become lower than theshot noise.

194

"5

500-

400-

300-

200-

100-

0.00 0.25 0.50 0.75 1.00

1/SQRT(N)

Figure 1. Improving the signal to noise ratio decreasing the detection bandwidth, by means of signal averaging. Infigure the signal noise is plot versus the inverse of the square root of the number of averages, which is proportionalto the detection bandwidth, for a DFB laser at 1578 nm.

1.2 Laser Excess Noise

In practice, the sensitivity of absorption measurements is often limited by excess-noisewhich is due to fluctuations of the laser power. The fluctuations are generated by external effectslike current and temperature instabilities, mechanical vibrations, or optical feedback as well as byintrinsic noise sources, such like photon and carrier density fluctuations and partition noise. Theexternal noise sources can be minimized by battery-driven or highly stabilized current sourcestogether with proper alignment, ar-coatings, and optical isolators. The intrinsic noise dependsmostly on the manufacturing process and design and on the operating conditions, liketemperature and current.

Temperaturecontroller

Diodelaser

Sample cell Photodiode

Rampgenerator

Currentdriver

Reference Photoc liode

to the frequency analysis•

Computer Digitaloscillos.

- Direct absorption- LWM Technique- FM Technique- TTFM Technique

«—DifferentialAmplifier

Figure 2. Schematic view of a laboratory set-up for a semiconductor diode laser spectrometer.

195

Several investigations of amplitude-noise characteristics of lead-salt lasers7 and DFB andDBR-lasers8 have shown that in most cases the laser excess-noise exhibit a 1/f dependence withcut-off frequencies ranging between 1 and 100 Mhz. The rms current at the detector caused bythe laser-excess noise can be expressed as:

where a defines the frequency dependence of the laser-excess noise and ranges between 0.8 and1.5, Pex defines the magnitude of the laser power fluctuations at 1 Hz in a 1 Hz detectionbandwidth. Pex is approximately proportional to the laser power and depends on the intrinsicnoise of the diode laser and on the external effects of the particular measurement system. As inthe case of detector noise, laser excess noise is detection bandwidth dependent and can bereduced with appropriate techniques.

1.3 Residual Amplitude Modulation

Frequency modulation of diode lasers results in a simultaneous amplitude modulation, sincenot only the wavelength depends on the current but also the laser power. This residual amplitudemodulation (RAM) becomes often the main noise source in high-sensitive absorptionmeasurements. The RAM can be reduced using a dual beam subtraction method, in which thelaser beam is split in a reference beam and a probe beam, which are detected by the same kind ofphotodiodes.

It is important that the amplifier have the same quantum efficiency, the same gain and thesame frequency response. Therefore all electronic components have to be selected with great careto avoid manufactural differences of photodiodes and electric components, such as resistance,operational amplifiers and capacities.

1.4 Interference Fringes

Every transmitting element in the optical path, such as beamsplitters, lenses, absorption cellwindows or the laser collimator itself can create Fabry-Perot fringes. These optical fringes oftenexhibit a free spectral range (FSR) comparable to the linewidth of absorption lines when the laseris scanned across the lines. They appear as periodic oscillations with sufficient amplitudes toobscure weak absorptions signals. Interference fringes arise from reflections between parallelsurfaces in the optical path and the transmission depend on the laser wavelength. Any scanning ofthe wavelength results in an amplitude variation if resonant structures appear in the path. The freespectral range between two fringes can be expressed as:

where n is the refractive index of air and / is the distance between the optical surfaces. Thedistance /ranges typically between 1 m and 5 mm, which corresponds to resonances at a distanceof 150 MHz and 30 GHz, respectively.

There are several possibilities to minimize these etalon effects. For example, if the freespectral range of the fringes is very different compared to the width of the absorption feature theycan be removed by filtering the detected signal. However, for spacings comparable to thelinewidths the fringes can only be removed by a careful design of the experimental set-up.

196

§

on<

Detection limit in air: 7ppm

Partial Pressure of NH- in air (Torr) 500mTorr of NH3 in 760 Torr of air

iOOmTorr of NH3 in 760 Torr of air

Frequency (arb)

Figure 3. Second derivative profiles of different pressures of NH3 in air: a detection limit of 7 ppmxm in air isextrapolated. The combination band of ammonia at 1.5jam is investigated, with a laser at 1494 nm.

Whenever possible, transmissive optics should be avoided or anti-reflection coated opticsshould be used.

It is important to note that only detector 1/f noise and laser excess noise depend on thedetection frequency, while all noise sources discussed so far depend on the detection bandwidthAv. Therefore, high sensitivity detection can be achieved by increasing the detection frequency(to a value for which the laser excess noise is lower than the shot- and Johnson noise) and byreducing the detection bandwidth.

1.5 Detection Techniques

In general, in a direct absorption measurements, the small changes in the transmittedamplitude, arising from gas traces, have to be distinguished on a large background, resulting in apoor sensitivity.

Frequency modulation techniques are based on a fast modulation of the laser emission in thefrequency domain resulting in an amplitude modulation of the ligth intensity. By means of phasesensitive detection is then possible to extract only the absorbed signal from the background of thetransmitted power. An additional increase in sensitivity is due to the fact that the detectionfrequency can be moved to larger values, in order to reduce the laser amplitude fluctuations.

The modulation techniques are divided according to the modulation frequency.In low wavelength modulation spectroscopy (LWM), the laser frequency is modulated at a

relatively low frequency (hundreds of kHz), which is small compared to the width of the line tobe probed. The observed signal arises from the difference in the absorption of different sidebandswhich probe simultaneously the absorption line. The signal is then demodulated, at n times themodulation frequency. Usually, first (n=l) and second (n=2) derivative detection are used.

197

n

oQ.

O

JQ

Frequency (arb)

Figure 4. Second derivative absorption signal of CO at the exit of a catalytic exhaust with a DFB laser at 1578 nm.

In frequency modulation spectroscopy (FM), the laser is modulated at much higherfrequencies, that range usually between 100 MHz and several GHz, comparable with the width ofthe absorption features, and produce sidebands which are widely spaced in frequency. In thistechnique, only one sideband is absorbed at a time, giving rise to a heterodyne beat signal at themodulation frequency. The detection frequency is moved in a region where the laser excess noisepresents its minimum value and, in addition, the selective absorption of the sidebands results in alarger detected signal.

In order to retain a similar sensitivity for atmospheric pressure broadened absorptionprofiles, the laser must be modulated at frequencies in excess of one Ghz. Electronics forheterodyne detection at these frequencies can be complicated. The problem can be simplified byusing two-tone frequency modulation spectroscopy (TTFM).9 The laser emission issimultaneously modulated at two distinct but closely spaced frequencies, Vj+v2 and v,-v2, oncemore comparable to the line-width of interest (Vi~a few GHz for pressure broadened profiles andv2~few MHz). The heterodyne beat signal is then obtained at the much lower frequency 2v2,eliminating the need of high speed detectors and electronics. Such detection frequency can beanyway maintained sufficiently large (a few MHz) to avoid the 1/f excess noise and reach thesame noise level as in the FM technique.

It has to be noted that such high modulation frequency can be obtained only insemiconductor diode laser by means of the injection current. Electro-Optical modulators (EOM)could be a possible solution for other laser sources, but efficient modulation at frequencies above1 GHz are not very easy to be reached.

Further improvement in sensitivity can be obtained by means of a double beamconfiguration, in which amplitude fluctuations can be canceled out, achieving a shot-noise limiteddetection.

1.6 Bandwidth reduction

Another important parameter that greatly influences the sensitivity of the apparatus is theelectronic detection bandwidth of the signal. Indeed, many noise sources (shot-noise, Johnsonnoise, laser excess noise) depend on it, in particular they decrease proportionally to the squareroot of the bandwidth.

For example, an appropriate filtering of the signal can result in an enhancement of thesensitivity. In addition, by means of a signal averaging, the noise can be further reduced, as if thebandwidth was decrease according to the relation:7

2T(0

"5c'55LL

-

-

1

' ^ ^ \ \ \ \

- V

To

Frequency (GHz)-2 0 2

Frequency (GHz)

Figure 5. Two-tone signal from 90 mTorr of pure H2S and b) in an atmosphere of air (lHz detection bandwidth),with a DFB laser at 1578 nm.

(6)

where n is the number of averages. This effect has been verified experimentally with a DFB diodelaser at 1578 nm, as can be seen in Figure 1.

2. EXPERIMENTAL SET-UP

In Figure 2. an experimental laboratory setup of a typical tunable diode laser sensor isshown. Different diode lasers are used, depending on the gas under study. The emissionfrequency is selected with an appropriate control of laser temperature and current in order tomatch the chosen molecular line. A current ramp of about 10 mA amplitude is added to theinjection current, leading to continuous sweep of the laser frequency of several GHz, whichallows to record the whole selected line with one current scan.

The output beam of the laser is collimated and splitted into three parts: one is used for thefrequency control, the second one is used to detect the unabsorbed light intensity and the thirdone is used to detect the transmitted light intensity.

The absorbed and reference signals are amplified and subtracted from each other to avoidthe background slope due to the amplitude modulation produced by the current scan.

The sample cell, a 1.5 m long Pyrex tube, is pumped out to 10"4 Torr and filled with the gassample.

This simple experimental set-up is used to extract atmospheric relevant broadeningparameters with the advantage of a straightforward interpretation of the broadened absorptionprofile.10

For high sensitivities measurements LWM and TTFM techniques are used.In the low wavelength modulation apparatus an ac component at coo=3.5 kHz is added to

the injection current, the absorbed signal is demodulated at a frequency 2coo by means of a lock-inamplifier.

In the two-tone frequency modulation scheme the signal from a synthesizer at Vi=2.04 GHzis mixed with that at V2=5.35 MHz of a function generator.

199

0.0 0.5 1.0 1.5

Photocurrent0-5 (mA0-5)

Figure 6. Noise power as a function of the optical power incident on the photo-detector. The linear dependenceindicates that the sensitivity is shot-noise limited, since other noisesources depend in a dififerent way on the opticalpower.

The output, at vmOd=2 Ghz±5.35 MHz is coupled to the laser diode current by means of ahigh pass filter. The signal of the function generator is frequency doubled and is mixed in a phasedetector with the filtered and amplified absorption signal coming from an InGaAs-PIN photo-diode. The output of the phase-detector, which is proportional to the Fourier component of thesignal from the photo-diode at vdet, is low pass filtered (100 Hz), amplified and recorded on adigital oscilloscope.

To perform a quantum limited detection a careful attention to experimental details likeamplifiers has to be paid. In particular, for the balanced dual-beam set-up used in our application,it is of fundamental importance that the probe signal and the reference signal are detected with thesame kind of photo-diodes and amplified with the same kind of amplifiers.

Table 1. Comparison between minimum detectable concentrations in air obtained with dififerent detectiontechniques.

NH3

CO

CO2

H2S

o2

HC1

A

nm

1494.35

1579.74

1579.57

1577.32

760.89

1742.3

Detection limit in air

ppmxm

500 (AD)

800 (AD)

400 (AD)

7 (LWM)

170 (LWM)

260 (LWM)

130 (LWM)

1000 (LWM)

0.5 (LWM)

7 (TTFM)

10 (TTFM)

4 (TTFM)

200

To avoid manufactural differences of resistances or capacitance they have to be tested andchosen carefully. For that reason a low-noise pre-amplifier circuit for the photo-diodes wasespecially designed to work for detection frequencies between 1 and 40 MHz.

The preamplified output signals of the photo-detector are combined in a 180° rf hybrid,which shifts one signal by 180° before adding it to the other one, eliminating in this way allcoherent fluctuations of the two beams. A quantitative description of our quantum limited dualbeam spectrometer can be found elsewhere.11

With all the detection techniques listed before a series of signal to noise ratio measurementsat different total and partial gas pressures are performed. The minimum detectable gas pressure isthen obtained by a nonlinear fitting procedure, extrapolating the value at which the signal to noiseratio becomes unity.

3. RESULTS AND DISCUSSION

The sensitivity limits from the three detection techniques are reported using the absorptionlines of some interesting molecules in the l|Lim-2|nm region (Table 1.).

3.1 Ammonia NBb

The combination band Vi+2v4 at 1.5|im has been investigated. Low wavelength modulationspectroscopy has been performed on the strongest component at X = 149435 nm using a DFBdiode laser (Fig.3). A minimum detectable concentration of 7 ppmxm of ammonia in air has beenachieved.This low detection limit is important for monitoring the NH3 concentration in variousapplications. Selective Catalytic Reduction (SCR) of NOX with ammonia represents the mosteffective technology currently available for deep NOX removal. Furthermore, alteration in NH3

concentration in the breath can indicate severe epatic failure.

3.2 Carbon Monoxide CO, Carbon Dioxide CO2 and Hydrogen Sulphide H2S

These three molecules have, in the region around 1578 nm, overlapping overtone bands: 3v(CO), 2vi+2v2+V3 (CO2)12 and V1+V2+V3 (H2S)13. By using one single semiconductor diode laserand a TTFM scheme, a minimum detectable absorption of 5xlO~7 is achieved. This corresponds ata minimum detectable concentration in air of 7 ppmxm for CO, 10 ppmxm for CO2 and 4ppmxm for H2S.

CO and CO2 are important in combustion processes and exhaust gases control (Figure 4.),while H2S detection is fundamental for security on sour-oil-rigs (Figure 5.).

Furthermore a TTFM spectrometer is a powerful tool for non-invasive diagnostic of humanbreath, where detection of the isotopic ratio 12CO2/

13CO2 can give information of the presence ofHelicobacter Pylori in the stomach.

3.3 Molecular Oxygen O2

A magnetic dipole transition of the b % + (v'=0)<-X3£g~ (v"=0)-band at 761nrn is studied.A low wavelength modulation scheme is used, but a sensitivity of only 1000 ppmxm is achieved,because the transitions' line-strengths in this forbidden band are more than one order ofmagnitude weaker than those of the other molecules investigated.

201

Table 2. Self-broadening and Air-broadening coefficients for the gases under study (* measured at 1.65 jum).

NH3

CO

CO2

H2S

o2

HC1

A

(nm)

1494.35

1579.74

1579.57

1577.32

760.89

1742.3

Self-broadening

(MHz/Torr)

37(3)

3.95(9)

4.1(3)

6.7(2)

2.04(2)

8.6(4)

Air-broadening

(MHz/Torr)

4.0(2)*

3.67(8)

3.42(7)

3.09(9)

1.92(5)

2.9(5)

3.4 Hydrogen Chloride HC1

The first overtone band (2v) at 1.77|im is investigated14 with a low wavelength modulationand two-tone frequency modulation apparatus. Sensitivity of 0.5 ppmxm is reached. Monitoringof HC1 is very important in the emitted gases of waste incinerator.

For the shot-noise limited detection the absorption spectrometer has been tested using FfeSas a target gas, once the noise sources are minimized and the optimal working conditions arefound. We have measured the rejection ratio using the difference method.

We have demonstrated that the used dual beam configuration can suppress laser noise bymore than 25 dB, in the region of interest near 10 MHz. Then we have measured the differenceof the two channels, using the 180° combiner, and the sum using the 0° combiner, that is only 5dB over the difference. So, taking in account the rejection ratio we concluded that we havereached the shot-noise leveL The detection limit of H2S in air was found to be approximately 500ppb over 1 meter path-length.

As an additional check to the reached shot-noise limit, the light noise was measured as afunction of the square root of the optical power incident on the photo-detector (see Figure 6.). Inaddition, pressure broadening parameter are measured for all these molecules. Results are listed inTable 2.

4. CONCLUSION

The achieved quantum limited sensitivity demonstrate that using distributed feedback diodelasers in combination with two-tone frequency modulation spectroscopy is a powerful techniquefor gas detection. It is possible to detect traces gases in air with a sensitivity on a ppm-level over a1-meter path-length for some molecules of industrial, medical and environmental interest.

The natural combination with fiber optics technology make these sensors attractive for alarge variety of "in situ" measurements.

ACKNOWLEDGMENTS. Many collegues and visitors have contributed to the progress indiode laser spectroscopy at Lens. We are indebted to them all: R. Benedetti, F. D'Amato, P. De

202

Natale, M. De Rosa, IC Ernst, ML Gabrysch, IC Giulietti, L. Lorini, F. Marin, G. Modugno, RS.Pavone, N. Pique, M. Prevedelli, M. Snels.

We want to thank Prof. M. Rosa-Clot and SIT (Science Industry and Technlogy) forstimulating help in the transfer of our knowledge to industry.

This work has been supported by ASI contract ARS96107, ECC contract ERB FMGE CT950017 and the Fire & Gas project N. OG-0269-95.

REFERENCES

1. F.S. Pavone, "Diode lasers and their applications in spectroscopy", Rivista del Nuovo Cimento, Vol 19: 1-42(1996)

2. F.S. Pavone, M. Inguscio, "Frequency- and Wavelength-Modulation Spectroscopies: Comparison ofExperimental Methods Using an AlGaAs Diode Laser", Appl. Phys. B, 56:118-122 (1993)

3. M. Gabrysch, "High-Sensitivity Spectroscopy Using Semiconductor Diode Laser in the Visible and NearInfrared Spectral Region", Ph.D. Thesis, Heidelberg 1997

4. G.C. Bjorklund, "Frequency-modulation spectroscopy. a new method for measuring weak absorptions anddispersions", Opt. Lett. 5, no.l: 15-17 (1980)

5. J.A. Silver, "Frequency-modulation spectroscopy for trace species detection: theory and comparison amongexperimental methods", App. Opt., 31 no. 6: 707-717 (1992)

6. Hi. Schiff, G.I. Mackay, J. Bechara "Air Monitoring by Spectroscopic Techniques", John Wiley & Sons (1994)7. P. Werle, F. Slemr, M. Gehrtz and C. Brauchle, "Quantum limited FM-spectroscopy with a lead salt diode

laser", Appl Phys. B 49:99 (1989)8. W.H. Richardson, Y. Yamamoto, "Quantum correction between the junction-voltage fluctuation and the

photon-number fluctuation in a semiconductor laser", Phys. Rev. Lett. 66,1963 (1991)9. G.R. Janik, C.B. Carlisle, T.F. Gallagher, "Two-tone frequency-modulation spectroscopy", J. Opt Soc. Am.,

B3: 1070-1074 (1986)10. C. Corsi, M. Gabrysch, M. Inguscio, "Detection of molecular oxygen at high temperature using a DFB-diode-

laser at 761 nm", Opt. Comm., 128: 35-40 (1996)11. C. Corsi, M. Gabrysch, F. Marin, G. Modugno "Quantum noise limited detection with semiconductor diode

laser", App. Phys. B, special issue on "Environmental Trace Gas Detection Using Laser Spectroscopy" (1998)12. M. Gabrysch, C. Corsi, F.S. Pavone, M. Inguscio, "Simultaneous detection of CO and CO2 using a

semiconductor DFB diode laser at 1.578 Jim", App. Phys. B, B65:75-79 (1997)13. G. Modugno, C. Corsi, M. Gabrysch and M. Inguscio, "Detection of H2S at the ppm level using a

telecommunication diode laser", Opt. Comm. in press14. C. Corsi, S. Czudzynsky, F. D'Amato, M. De Rosa, K. Ernst, M. Inguscio, "Detection of HC1 an the first and

second overtones using semiconductor diode lasers at 1.7 pm and 1.2 jmn", in press.

203

La Rivista del Nuovo Cimentodella Societa Italiana di Fisica1999

G. M. TINO and M. INGUSCIO

condensation

Editrice Compositori Bologna

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La RIVISTA DEL NUOVO CIMENTO 1999

G. M. TINO and M. INGUSCIO

Experiments on Bose-Einstein condensation

RIVISTA DEL NUOVO CIMENTO VOL. 22, N. 4 1999


G. M. TINO( 1 ) and M. INGUSCIO(2)

(1) Dipartimento di Scienze Fisiche delVUniversita di Napoli and Istituto Nazionale di Fisicadella Materia, Complesso Universitario di Monte S. Angelovia Cintia, 1-80126 Napoli (Italy)

(2) Istituto Nazionale di Fisica della Materia, European Laboratory for Nonlinear Spectroscopyand Dipartimento di Fisica deirUniversitd di FirenzeLargo E. Fermi 2, 1-50125 Firenze {Italy)

(ricevuto il 27 Maggio 1998)

1 1. Introduction2 2. Cooling and trapping of atoms3 2*1. Laser cooling and trapping of atoms3 2*1.1. Slowing of an atomic beam5 2*1.2. Optical molasses6 2*1.3. Sub-Doppler temperatures in optical molasses8 2*1.4. Magneto-optical trapping of atoms11 2*1.5. Other schemes for laser cooling and trapping of atoms12 2*2. Magnetic trapping13 2*2.2. Quadrupole traps14 2*2.2. Losses due to Majorana spin-flips14 2*2.3. The time-averaged orbiting potential (TOP) trap15 2*2.4. The optical-plug trap16 2*2.5. The Ioffe-Pritchard-type traps17 2*2.6. Permanent-magnets traps17 2*2.7. Comparison of magnetic traps18 2*3. Evaporative cooling of trapped atoms22 3. Studies of Bose-Einstein condensates22 3*1. Experimental procedure to achieve BEC24 3*2. Observation of a Bose condensate27 3*3. Measurement of energy and ground-state occupation as a function of temperature29 3*4. Study of collective excitations and propagation of sound in a Bose-Einstein condensate31 3*5. Condensation of atoms with attractive interactions33 3*6. Production of two condensates by sympathetic cooling34 3*7. The coherence properties of the condensate and the "atom laser"38 4. Conclusions and future prospects

1. - Introduction

Bose-Einstein condensation (BEC) is a purely quantum phenomenum first predictedby A. Einstein in 1924 [1,2]. A gas of non-interacting identical particles described by a

© Societa Italiana di Fisica 1

2 G. M. TINO and M. INGUSCIO

wave function symmetric under the exchange of any two particles, should undergo a phasetransition when the de Broglie wavelength becomes comparable to the spacing betweenparticles. Under these conditions, the wave packets overlap and the indistinguishabilityof the particles becomes important. Particles described by a symmetric wave functionare called bosons from S.N. Bose who was the first to study their statistical behaviourin the case of photons [3]. Bosons obey Bose-Einstein statistics. As is well known, thesymmetrization postulate of quantum mechanics admits only another class of particles,those described by an antisymmetric wave function, which are called fermions and followFermi-Dirac statistics [4,5]. If the temperature of a gas of bosons is lowered, below atransition temperature a macroscopic fraction of the particles should suddenly occupythe single lowest-energy quantum mechanical state. The condition for the transition isthat nA^B = 2.612 [6], where n is the density of particles, and A^B = h/\/2nmk&T isthe thermal de Broglie wavelength, with h Planck's constant, m the mass of particles, feeBoltzmann's constant, and T the gas temperature (the quantity nA^B is usually calledphase-space density and a system of identical particles for which nA^B ~ 1 is referred toas a quantum-degenerate system). In fact, this prediction had never been tested directlyalthough quantum degeneracy is at the origin of the behaviour of superfluid helium andexciton gases in semiconductors. A comprehensive review of experimental and theoreticalwork on BEC before 1995 can be found in [7].

In 1995, BEC was first observed directly by E. A. Cornell and collaborators in a gasof 87Rb atoms [8] and by W. Ketterle's group in a gas of Na atoms [9]. Evidence ofBEC in a gas of 7Li was also reported in [10,11]. These important results were achievedafter several years of experimental efforts initially focused on atomic hydrogen. Theobservation of BEC in hydrogen was prevented by the presence of collisional processesleading to losses of atoms from the sample and by the difficulty in the detection of theatoms. In fact, the success of experiments of BEC on alkali atoms was the result ofthe combination of methods of magnetic confinement and evaporative cooling, mainlydeveloped for hydrogen, with the techniques of laser cooling and trapping which areparticularly efficient for alkali atoms. Also important was the occurrence of favourablecollisional parameters and the possibility of easy optical diagnostics of the atomic sample.

This paper describes these recent experiments of BEC in dilute vapors of alkali atoms.Since the first observation of BEC, an amazing number of experimental and theoreticalpapers have been published. Because of this rapid development, it is impossible to givea complete bibliography and some of the results discussed here will necessarily becomeobsolete soon. An updated bibliography and information on recent results can be foundat the Internet site http://amo.phy.gasou.edu:80/bec.html/. The purpose of this paper isto summarize the experimental methods that allowed the achievement of BEC, in severallaboratories now, and to describe the main results obtained so far in the study of whatcan be considered as a new state of matter.

2. — Cooling and trapping of atoms

In recent years there was a dramatic progress in cooling and confinement of neutralatoms using electromagnetic fields. Cooling was achieved by using optical fields from lasersources (laser cooling) or by selective ejection of more energetic atoms from a trappingregion using microwave radiation (evaporative cooling). Wall-free confinement of the coldatoms was accomplished by using laser beams (optical traps), by static or slowly varyingmagnetic fields (magnetic traps), or by a combination of the two (magneto-optical traps).

In this section we describe these cooling and trapping techniques with particular

EXPERIMENTS ON BOSE-EINSTEIN CONDENSATION 3

emphasis on the methods which have been used so far to achieve Bose-Einstein conden-sation. The important "ingredients" of the experiments which have been successful sofar are magneto-optical trapping, cooling in optical molasses, magnetic trapping, andevaporative cooling.

2*1. Laser cooling and trapping of atoms, - The field of laser cooling and trapping ofatoms has been one of the most active fields of research in physics in the past decade.Starting from the initial proposals that dated back to the seventies, several methodswere developed which allowed to reach lower and lower temperatures. The study of thecooling processes became of interest in its own right as new mechanisms were discovered.In 1997, the Nobel Prize in Physics was awarded to C. Cohen-Tannoudji, S. Chu andW. D. Phillips for their important contributions in the development of methods to cooland trap atoms with laser light. It is beyond the scope of this review to describe indetail the different methods developed and the results obtained by the many groups inthe world that have been involved in the research in this field. In this section, we onlybriefly describe the techniques of laser cooling and trapping that were important in theexperiments in which Bose-Einstein condensation was achieved. A comprehensive reviewon both experimental and theoretical work in this field can be found in [12-16].

2*1.1. Slowing of an a tomic beam. When an atom with mass m absorbs oremits a photon with frequency z/, because of conservation of total momentum its velocitychanges by the recoil velocity vr = hv/(me) = h/(m\). For the D2 transition in sodiumatoms, A = 589 nm, m = 23 a.m.u., vT — 3 cm/s. For cesium, A = 852 nm, m = 133a.m.u., vT = 3.5 mm/s. This change in velocity is small but if several photons arescattered, a considerable change in the atomic velocity can result. For example, a sodiumatom moving with a velocity o ~ 1000 m/s interacting with a laser beam propagatingin the opposite direction would stop after the absorption of VQ/VV ~ 33000 photons. Inthe case of a thermal cesium beam (vo ~ 300 m/s) about 86000 photons are required tostop the atoms.

The absorption of each photon is followed by spontaneous emission. Because theemission direction is random, spontaneous emission has no effect, on the average, onthe atomic velocity. The acceleration of the atom depends on the rate of scattering ofphotons. For high intensity of the light, an absorption-emission cycle requires a time ofthe order of the excited-state lifetime r. For the transitions considered above, r ~ 20 ns.Therefore atoms can be stopped in a few milliseconds.

In order to describe the process in more detail, a simple model can be consideredof a two-level atom interacting with a plane light wave propagating in the direction nwith frequency v^ and intensity I (fig. 1). The lower and upper states of the atom haveenergies E% and Ee, respectively, separated by the quantity Ee — Eg = hi/A-

The number of absorption-emission cycles in one second is given by the rate of spon-taneous emission 1/r times the probability of occupation of the excited level, given by

( 1 ) Vf [v) ==

[5 - (isL/c)v • n]2 '

where 6 — v\, — Z A, IO is the saturation intensity, and F = 1/(2TTT) is the natural widthof the atomic resonance.

The effect of the exchange of momentum between the atom and the radiation fieldis that the atom experiences a force F s p called spontaneous force or radiation pressure

a)

• M M

atomssource

-1*|

atomicbeam

<cG. M. TINO and M. INGUSCIO

V <VAL A

laserbeam

Fig. 1. - Slowing of an atomic beam using laser light, a) Simplified scheme of the apparatus;b) velocity distribution of the atoms with no laser (dashed line) and in the presence of afixed-frequency counterpropagating laser beam (continuous line).

force. This force is given by the change of momentum of the atom in the unit of time,that is by the momentum of a photon times the number of absorption-emission cycles inthe unit of time:

(2)huL 1 1

Fsp(v) = n—- — - I/Ioc 2T 2 1 + (7/Jo) + (4/r2)[<5 - (VL/C)V . n]2

Equation (2) gives the average force on the atom over several absorption cycles. Thisexpression is valid if changes of the quantity 5eff = [<$ — {VL/C)V • n]2 can be neglected inthe time interval considered.

Slowing of an atomic beam with a counterpropagating laser beam is efficient if thelaser frequency is smaller than the atomic resonance frequency in order to compensate forthe Doppler shift. As can be seen from eq. (2), the spontaneous force is most substantialwhen [8 + v^v/c)] <C F. For a given laser frequency, only atoms with a velocity withinan interval of width Av — Tc/vi,y called the capture range, will be slowed down. Figure 1shows the change in the atomic velocity distribution produced by a counterpropagatinglaser beam with frequency v\, <VK- Atoms in the velocity capture range are slowed andtheir velocity distribution is narrower. Therefore, this scheme produces not only slowingbut also cooling of the atoms. The problem of this scheme is that only a small fractionof the atoms interact with the laser. Also, the process stops when the atoms, because ofthe change in velocity, are no longer in resonance.

EXPERIMENTS ON BOSE-EINSTEIN CONDENSATION

V L VL

Laboratory reference frame

c> • <Atomic reference frame

Fig. 2. - Scheme for 1-dimensional cooling of atoms in optical molasses.

Two methods have been devised to solve these problems. The first method is calledlaser chirping [17]: The laser frequency v\, is changed in time, typically in the form of asuccession of ramps, so that groups of atoms see the laser in resonance until they reachthe final velocity. The second method exploits a static space-varying magnetic field totune the frequency uA [18]. The advantage of the laser chirping method is its simplicitythat does not require particular changes in the atomic beam apparatus. It was furthersimplified by the recent advent of diode lasers whose frequency can be easily varied at ahigh rate. However, with this method, only "bunches" of atoms interact with the laserand eventually the slowed atoms are not localized but they are spread along the beamlength. In the second scheme, instead, the nonuniform magnetic field can be designedso that almost every atom will see the fixed-frequency laser in resonance starting from aposition which depends on the initial velocity. If the magnetic field is properly designed,a noticeable fraction of the atoms can be slowed down with this method and brought ata given velocity at the end of the magnet. The drawback of this scheme is that a largesolenoid is required, whose shape must be accurately optimized. Also the presence of anintense magnetic field can represent a limit in some experiments.

The above methods are both used in present beam-slowing experiments and the choiceof one method or the other depends on the particular requirements of the experiment.

2*1.2. Opt ical molasses. The possibility of cooling an atomic gas using laser radia-tion was first proposed by T. W. Hansch and A. L. Schawlow in 1975 [19]. They pointedout that if a moving atom is irradiated with counterpropagating laser beams that aretuned slightly below the atomic resonance frequency, the Doppler shift will cause theatom to absorb preferentially photons moving opposite to its velocity (fig. 2). As dis-cussed above, the atom will slow down. In a gas, this process will produce a cooling ofthe gas. The kinetic energy of the atoms is dissipated and converted into energy of theelectromagnetic field.

If stimulated emission of photons is neglected, using eq. (2) the force on the atoms


can be written as the sum of the forces produced by each of the laser beams:

J

In the limit of small atomic velocity (v < Tc/2vi), the force can be written as

For low velocities, the force on the atoms depends linearly on velocity as in a viscousmedium. This suggested the name optical molasses for this cooling configuration. Thesame scheme can be extended to three dimensions using three pairs of counterpropagatinglaser beams in three orthogonal directions. An optical molasses was first demonstratedexperimentally by S. Chu and coworkers in 1985 with sodium atoms [20].

The minimum temperature expected for the atoms in optical molasses can be esti-mated, with the model considered above, taking into account two effects: The first is thecooling effect produced by the viscous force. The second is a heating of the atoms dueto the random direction in which photons are absorbed and emitted. The temperatureof the atomic sample at which these two effects balance is

The minimum temperature is obtained for 5 = —T/2 and is given by

(6) kBTD = ^ .

TD is called Doppler limit. Typical values of TD are 240 JJK for sodium and 120 jiK forcesium.

An important reference temperature for laser-cooled atomic gases is the so-calledrecoil temperature. Because the exchange of momentum between atoms and radiationtakes place by discrete amounts, corresponding to the single photon momentum, theminimum spread of the atomic energies cannot be smaller than the energy correspondingto the single photon recoil. The recoil temperature Tr is therefore defined by

Tr is 2.4 jiK for sodium and 197 nK for cesium.

2*1.3. Sub-Doppler t empera tu re s in opt ica l molasses. Accurate measure-ments of the temperature of the atoms in optical molasses, first performed by W. Phillipsand coworkers by time-of-flight methods [21], showed that the temperature was muchsmaller than the expected Doppler limit temperature and not far from the recoil limit.This unexpected result was explained by J. Dalibard and C. Cohen-Tannoudji [22] andby S. Chu and coworkers [23]. They developed a theory of sub-Doppler laser coolingtaking into account light-induced shifts of the atomic levels and optical-pumping effects.


Energy

0 A/8 A/4 3A/8 A/2 5A/8 z

Fig. 3. - Mechanism leading to sub-Doppler cooling of atoms in 1-D optical molasses for twocounterpropagating laser beams with orthogonal linear polarizations. The so-called Sisyphuscooling is an effect of the position-dependent light shift of the levels and of optical pumping.

A mechanism leading to sub-Doppler cooling can be qualitatively understood consid-ering a 1-D model in which atoms having, for example, an angular momentum Jg = 1/2in the lower state and Je = 3/2 in the upper state interact with two counterpropagat-ing laser beams with orthogonal linear polarizations. The polarization of the resultingfield varies with position. Figure 3 shows the light-shifted atomic levels. The shift ofa given Zeeman sublevel depends on the quantity 1/5 and on the excitation probabilityfor atoms in that level. Since this probability depends on the light polarization, theposition-dependent polarization of the light leads to a position-dependent shift of theatomic levels. The combined effect of the spatially dependent light shift of the levels andof optical pumping is at the origin of the cooling mechanism. An atom starting, for ex-ample, at z = A/8 in the g-1/2 state and moving to the right, will climb the potential hillso that part of its kinetic energy will be converted into potential energy. If the velocityis sufficiently small, near the top of the hill the atom will be optically pumped into the5+1/2 state. The potential energy gained at the expense of kinetic energy is dissipatedin this spontaneous Raman anti-Stokes process. Optical pumping is then the mechanismallowing dissipation of energy and cooling of the atoms. This cooling effect was calledSysiphus cooling because, as in the Greek myth, the atoms are forced to continuouslyclimb potential hills. The minimum temperature predicted by this model is of the orderof the light-induced splitting of the levels. A linear dependence on 1/5 is then expected.

Optical molasses are often realized with a light polarization scheme different from theone considered above. The two counterpropagating laser beams have opposite circularpolarizations. Also in this case, sub-Doppler temperatures can be reached. The mech-anism can be understood as follows: for two counterpropagating beams with oppositecircular polarization and same amplitude, the resulting field is linearly polarized with thepolarization vector rotating along the standing wave thus forming a helix with a pitch A.It was shown in [22] that with this polarization configuration, the combined effect ofthe polarization rotation, light shift, and optical pumping produces a motion-induced


difference in the population of the ground-state Zeeman sublevels. This gives rise to animbalance between the radiation pressures of the two beams and therefore to a frictionforce. This mechanism only works for J > 1 in the ground state.

The cooling mechanisms described above both depend on the presence of a light po-larization gradient. However, since different processes lead to cooling in the two cases,different values for the friction and diffusion coefficients result. The steady-state tempera-tures are instead found to be about the same and to be proportional to the quantity 1/5.This behaviour was confirmed experimentally for three-dimensional molasses down totemperatures of a few /xK [24], corresponding to about ten times the recoil temperature.

2*1.4. Magneto-opt ica l t r app ing of atoms. The magneto-optical trap (MOT)is nowadays a common method to confine and cool neutral atoms. In particular, it playeda crucial role in BEC experiments. It was first proposed by J. Dalibard and demonstratedexperimentally in [25].

A MOT consists of a combination of a set of laser beams and a quadrupole magneticfield. It is usually realized with six laser beams, which form three orthogonal pairs eachmade of two counterpropagating beams with opposite circular polarizations, intersectingin the zero point of the magnetic field generated by two parallel coils with currents flowingin opposite directions.

The basic mechanism can be understood, in the frame of a model similar to the one forDoppler cooling, by considering the 1-dimensional scheme shown in fig. 4. For simplicity,we consider an atom with a J = 0 ground state and a J = 1 excited state. The magneticfield, that is null at the center and has a magnitude proportional to the distance fromthe trap center, produces a position-dependent Zeeman splitting of the levels, as shownin the figure. If the laser frequency v^ is smaller than the atomic resonance UA andthe polarizations of the laser beams are chosen as indicated in the figure, atoms willexperience a force towards the center of the trap. In fact, an atom displaced to one sidewith respect to the position where the magnetic field is null, will preferentially absorbphotons from the laser beam coming from that direction and will be pushed towards theorigin. As for optical molasses, the force on an atom can be written, in the low intensitylimit, as the sum of the forces due to each of the beams. In this case, however, theZeeman shift of the levels must be taken into account. In the limit of a small atomicvelocity v and a small Zeeman frequency shift KZ, the force can be written as

The motion of an atom is therefore that of a damped harmonic oscillator. The im-portant characteristic of the MOT is that the atoms are not only confined but theirkinetic energy is also dissipated as in atomic molasses. In fact, it was found [26] thatsub-Doppler mechanisms due to polarization-gradient effects also play an important rolein a MOT, leading to an increase of the confining force and to temperatures below theDoppler cooling limit. Atoms can be collected in a MOT from a slowed atomic beam asin [25] or from the room temperature vapor in a cell, as first demonstrated in [27].

The rate of change in the number N of trapped atoms is given by

(9) —--R-'iN-Bl n2dV


a)

c)

TO = - 1j

^ ^

m = + 1

a*

E

T

Fig. 4. - Magneto-optical trapping, a) 1-dimensional scheme of the experimental set-up for amagneto-optical trap, b) Transition scheme, c) 1-dimensional model of the MOT.


where R is the collection rate, 7 is the loss rate due to collisions of the trapped atomswith background gas, and the last term gives the loss rate associated with collisionsbetween trapped atoms with a density n and loss coefficient /3. In the case of a vapor-celltrap, if the last term in eq. (9) is neglected, the steady-state number of trapped atoms isNs = jR/7. The rate R at which atoms with a velocity smaller than the capture velocityof the trap vc enter the volume V defined by the intersection of the laser beams is [27]

(10) R =v* / m( V

\2KTJ

where n is the density of the atoms in the vapor, T is the temperature of the cell, and mis the atomic mass. The capture velocity of the trap can be calculated as the maximumvelocity for which an atom can be stopped within a distance equal to the laser beamsdiameter [28]. A rough estimate can be obtained considering a constant photon scatteringrate r = l /2rn where rn is the excited-state lifetime. The stopping distance for an atomwith initial velocity v is /stOp = v2/2rvr, where vT is the recoil velocity. By imposing thatthe stopping distance equals the trapping beams diameter d, one gets

(11) ^ Vc

Equations (10) and (11) show that the loading rate R is proportional to d4. The lossrate 7 due to collisions of the trapped atoms with the hot atoms in the background vaporis given by 7 = nafiKT/m)1/2, where a is the collisional cross-section. The number Nof trapped atoms is then given by

(12)1 n d4 V; / m \ :

6 T-2 a \2KTJ' n

In the case of a Cs MOT, rn = 30 ns, a = 2 x 10"13 cm2 [27], vr = 3.5 mm/s; if the laserbeams have a diameter of 1 cm and the MOT is loaded from a room temperature vapor,one gets N « 5 x 109 atoms which is of the right order of magnitude compared to whatcan be obtained experimentally in these conditions.

An important quantity, especially for BEC experiments, is the density of the trappedatoms. The maximum density of atoms that can be achieved in a MOT is limitedby two main factors: first, the collisions between ground-state and excited-state atomscontributing to the (3n2 loss term in eq. (9) become important at the atomic densitiestypically obtained in MOTs; the second limit is due to repulsive forces between trappedatoms caused by reabsorption of scattered photons. Above a critical value of density,an increase of the number of trapped atoms gives a larger cloud with no increase ofdensity. This represents also a limiting factor for the maximum number of atoms thatcan be confined in a MOT. In [29], a scheme called dark-spot trap was demonstratedthat allowed a noticeable increase in the density of trapped atoms. The key idea is toisolate the cold atoms from the trapping light. This is obtained by optically pumping theatoms in the center of the trap into a state where they do not interact with the trappinglaser. This scheme allows an increase of both the number of trapped atoms and theirdensity. More than 1010 atoms can be confined in a dark-spot trap at densities of about1012 atoms/cm3.


The maximum phase-space density obtained with the methods of laser cooling andtrapping described above is in the range 10~5-10~~4.

2*1.5. Other schemes for laser cooling and t r app ing of a toms. In the efforttowards the achievement of BEC in atomic gases, several methods have been developedto increase the phase-space density in a sample of atoms. Here, we only focus on themethods that were used in the experiments in which condensation was observed so far. Inthese experiments, a combination of optical cooling and trapping methods and of othertechniques based on magnetic trapping and evaporative cooling was used. However,the possibility of reaching quantum degeneracy conditions with purely optical schemescannot be excluded. Amongst the methods which look more promising in this prospect,are trapping by optical dipole traps and subrecoil cooling schemes such as Raman coolingand velocity-selective coherent population trapping.

The dipole trap is an optical trap based on the dipole force acting on an atom in thepresence of nearly resonant light with a nonuniform intensity. This force is not caused bythe scattering of photons due to spontaneous emission but can be considered as the effectof processes of absorption-stimulated emission of photons. In the dressed-atom picture,the dipole force arises from the spatially varying shift of atomic levels produced by alight field with a nonuniform intensity profile. For a laser frequency smaller than theatomic resonance frequency, the atoms will be attracted towards high-intensity regions.In contrast, for a positive detuning the atoms will be repelled from the high-intensityregions. A dipole trap was first realized in [30] using a single, red-detuned, focused laserbeam. At the focus, the intensity has an absolute maximum such that three-dimensionalconfinement is possible.

Since no cooling mechanism is present in this trap, only atoms colder than the trapdepth can be loaded into it. Also, the lifetime of the atoms in the trap is limited by theheating due to absorption and spontaneous emission of photons. The heating rate canbe reduced by using large values of detuning for the trapping beam. Indeed, the photonscattering rate is proportional to I/S2 while, for a large detuning, the trap depth dependson 1/5. With the so-called far-off-resonance trap (FORT), trap depths of the order ofmK can be obtained with lifetimes of several seconds [31]. Recently, a hybrid electro-optical dipole trap was demonstrated [32]; a static electric field was used to increasethe confinement along the axial direction of the focused laser beam, that is a weak-confinement direction in the single-beam dipole trap. Blue-detuned dipole traps are alsointeresting and have been demonstrated [33]. In these traps, atoms can be confined bythe repulsive walls created with the blue-detuned light and kept in regions where light isvirtually absent.

Optical dipole traps have several positive features: they allow confinement of atomsat very high densities and in extremely small volumes. Contrary to the case of magnetictraps, the atoms in a dipole trap are not polarized. This can be of particular importancein experiments where mixed-spin systems are studied. Light traps can be quickly switchedon and off. They can allow the confinement of atoms for which magnetic traps cannotbe used. The main drawbacks of these traps are the high intensity required in order touse large values of detuning, and the need for good spatial stability of the trapping laserbeam.

Of particular interest is the possibility of combining trapping in optical dipole trapsand subrecoil cooling methods. Two cooling methods have been demonstrated leading toatomic samples where the velocity spread is smaller than the atom's recoil velocity afteremission of a photon: velocity-selective coherent population trapping (VSCPT) [34-36]


0.30

-0.30

1000 2000B(G)

500 1000B(G)

Fig. 5. - Energy of the Zeeman components of the hyperfine levels in the ground state of 87Rband 85Rb as a function of the magnetic field.

and Raman cooling [37-40]. These cooling methods are both based on the idea of let-ting the atoms randomly scatter photons until they fall by spontaneous emission into avelocity-selective state which is no longer coupled to the laser field. In VSCPT, a c.w.light field tuned to a Jg = 1 -> Je = 1 transition produces a velocity-selective darkstate. This state is a linear superposition of magnetic substates having different linearmomenta. In Raman cooling, on the other hand, the light field is pulsed. Atoms arepushed towards v = 0 with a sequence of laser pulses which induce velocity-selective Ra-man transitions followed by resonant excitation and spontaneous emission. Appropriatechoice of laser pulse shape and detuning prevents atoms with v ~ 0 from being furtherexcited. While VSCPT requires a specific type of atomic transition, Raman cooling canbe applied to any three-level system with two long-lived states.

Raman cooling of atoms in a dipole trap was demonstrated in [41]. Sodium atomswere cooled down to a temperature of about 1 fiK with a density of 4 x 1011 atoms/cm3,corresponding to a factor ~ 400 in phase space density from BEC.

2*2. Magnetic trapping. - Magnetic traps consist of an inhomogeneous magnetic fieldwith a local minimum. Atoms with a non-zero magnetic moment /i in an inhomogeneousmagnetic field B have an interaction energy given by W = - / i • B and experience a forceF = V(/x • B). The direction of the force depends on the orientation of the magneticmoment with respect to the field. If the atomic motion is adiabatic (the atom does notchange Zeeman sublevel), it can be described by a local potential given by the atomicmagnetic moment times | B |. In the presence of a minimum of the magnetic field | B |,atoms which are in a state whose energy increases with increasing magnetic field experi-ence a restoring force towards the minimum region and can be trapped. Since for a staticmagnetic field only minima can exist in free space [42], only atoms in low-field-seekingstates can be trapped. Figure 5 shows the energy of the Zeeman components of theground state of 87Rb and 85Rb atoms as a function of the magnetic field. Magnetostatictraps work for 87Rb atoms in. | F = l,ra = -1) and | F = 2,m = 1,2) states and for85Rb atoms in | F = 2, m = - 1 , -2) and | F = 3, m = 1,2,3) states.

A magnetic trap was first used by W. Paul to confine neutrons [43]. Magnetic trappingof neutral atoms was first demonstrated for Na [44]. More recently, magnetic traps were


a)

13

Fig. 6. - (a) Scheme of a quadrupole trap, (b) Scheme of a IofFe-Pritchard trap.

used to confine H [45,46], Li [11], Na [47,48], K [49], Rb [50,51], Cs [27,52], and Eu [53].Compared to magneto-optical traps, magnetic traps allow the confinement of the

atoms at much lower temperatures because the effects due to radiation are absent (forexample, the photon recoil limit). Magnetic traps are, however, much shallower thanMOTs. The typical depth can be estimated considering a magnetic trap made of twoparallel coils with currents flowing in opposite directions. As discussed in the following,such a configuration produces a field which is zero at the trap center and increases linearlyaround it. For a system of ~ 2 cm radius coils separated by ~ 3 cm carrying a current of~ 500 At, the axial field gradient at the trap center is of the order of 100 G/cm and thelowest threshold is A 5 ~ 100 G. Since the magnetic moment of a ground-state alkaliatom is /x « /iB, atoms can be trapped only if k&T < /J>BAB « 10~25 J, correspondingto a temperature of about 10 mK. Compared with atoms emerging from a thermal beamor a vapor at 300 K, it is clear that atoms must be pre-cooled before being loaded intoa magnetic trap.

It must be noted that since magnetic traps are conservative traps, an independentmechanism is required to cool the atoms once they are in the trap. The methods demon-strated so far are evaporative cooling, which is described in detail in the next section,and sympathetic cooling via elastic collisions with another cold species in the trap (seesect. 3*6).

2*2.1. Quadrupole t r aps . The simplest scheme to realize a magnetic trap is theanti-Helmholtz configuration schematically shown in fig. 6a. This configuration producesa spherical-quadrupole magnetic field crossing a zero at the center and varying linearlyaround it. For two coils with radius R separated by a distance A and a current / flowing,the field components in cylindrical coordinates are [54]

(13) Bz = 2cz, Bp = -cp, = 0,


where c = ZÎAR2/2{A2 + R2f/2.The potential energy for an atom in the trap is given by

(14) W = /i | B |=

This type of trap was used in the initial experiments of atom trapping. In addition tothe simplicity of the construction, it offers the advantage of a tight confinement of theatoms compared to other traps, discussed in the following, which are characterized by aparabolic minimum around a finite bias field.

A serious drawback of this trap becomes apparent, however, when it is used in com-bination with evaporative cooling. As the temperature of the atoms is lowered and theirdensity increases, a loss of atoms from the trap is observed. This loss mechanism is dueto the so-called Majorana spin-flips.

2*2.2. Losses due to Majorana spin-flips. As the atoms move in a magnetictrap, they experience a magnetic field changing direction in a complicated way. In orderfor the atoms to be trapped, the atomic magnetic moments must be oriented so that theyare attracted towards the field minimum region. If the magnetic field changes slowly, theatomic magnetic moment precesses around the field and follows it adiabatically. Theadiabaticity condition is violated if the Larmor frequency is smaller than the rate ofchange of the field direction. This is in fact the case for atoms passing close to the zero-field point in the center of a quadrupole trap. The probability of nonadiabatic transitionsfor a beam of oriented atoms crossing a region where the magnetic field goes to zero wascalculated by E. Majorana [55]. Atoms in a quadrupole trap can undergo a spin-fliptransition from one Zeeman sublevel to another and be lost from the trap [56]. The lossrate I/TO caused by this effect can be estimated as follows [50]. If an atom with velocity vand mass m passes near the center of the trap with a minimum distance 6, it can undergoa nonadiabatic spin-flip if the Larmor frequency is smaller than the rate of change of themagnetic field direction v/b. For a radial gradient of the field dBr/dr = B'q, the Larmorfrequency is ~ /j,bB'q/h. Loss then occurs within an ellipsoid of radius bo ~ (vh/iiBq)1/2.The loss rate is given by the flux through this ellipsoid, that is, the density of atoms N/l3

times the area of the ellipsoid ~ 6Q times the velocity t>, where TV is the number of trappedatoms and I is the radius of the atom cloud. The mean velocity and cloud size are relatedby the virial theorem: mv2 ~ jilB'q. It follows that To ~ (m/h)l2. For 87Rb atoms, forexample, m/h is about 105s/cm2. The loss rate increases as the atoms are cooled andspend more time at the bottom of the trap near the zero-field point. In order to reducelosses due to spin-flips, different schemes were invented that prevent the atoms fromcrossing a zero-field region. They are briefly described in the following.

2*2.3. The t ime-averaged orbi t ing po ten t i a l (TOP) t r ap . In the first ex-periment in which Bose-Einstein condensation was observed [8], a magnetic trap, namedTOP trap, was used which had been invented trying to overcome the storage time lim-itation due to spin-flip losses while keeping the advantage of tight confinement of aquadrupole trap [50].

The idea of the TOP trap is to add to the quadrupole magnetic field a continuouslychanging bias field that moves the location of the field zero around much faster than theatoms can follow. In ref. [50] this was realized by adding to the static quadrupole field,produced by two horizontal coils in anti-Helmholtz configuration, a small magnetic field

10 G, rotating in the horizontal plane with a frequency O;TOP of the order of


(a) | (b)

•

ITOP TRAP (Bb * 0)

(C) , (d)m = -1

^ m = +1

Fig. 7. - Comparison of magnetic-field configuration and resulting potential for a quadrupoletrap (a, b) and for a time-averaged orbiting potential (TOP) trap (c, d). (Prom W. Petrich etal. (1995), with permission.)

10 kHz (fig. 7). The value of U;TOP must be much smaller than the Larmor frequencyfor atoms in the bias field and much larger than the oscillation frequency of the atomsin the trap:

(15) vL oc STOP > ^TOP > fcMb •

The motion of the atoms is then governed by the time average of the instantaneouspotential. It can be shown [50] that, using cylindrical coordinates with z indicating theaxis of the quadrupole trap, the resulting potential energy has the form

(16) W?Op{p9z) ~ //STOP + -—-(P2 + 8*2) -4-£>TOP

The trap is therefore harmonic to lowest order and the magnetic field in the center ofthe trap is STOP ¥" 0-

2*2.4. The opt ical -plug t r ap . Another successful method to "plug the hole" lead-ing to losses in quadrupole traps makes use of the optical dipole force to repel the atomsfrom the central region of the trap. This method was used in the first experiment inwhich Bose-Einstein condensation was observed in a gas of sodium atoms [9]. The opti-cal plug was created by an Ar+ laser beam tightly focused at the point of zero magneticfield. The blue-detuned far-off-resonance light produced a light-shift repulsive barrierthat prevented the atoms from crossing the zero-field region in the trap. As for theother trapping schemes, the atoms were cooled by rf-induced evaporation. The potentialexperienced by the atoms is then a combination of the magnetic trapping potential, therepulsive potential due to the laser beam and the effect due to the rf. The resultingpotential is shown in fig. 8.


Optical Plug

Magnetic FieldRfInducedEvaporation

-30

-100

Fig. 8. - Adiabatic potential for an optical-plug trap. The total potential is a combination ofthe magnetic quadrupole trapping potential, the repulsive potential of the laser plug, and theenergy shifts due to the rf. (Prom K. B. Davis et al. (1995), with permission.)

2*2.5. The Io f fe -Pr i t chard- type t r aps . The trap called Ioffe-Pritchard (IP) trapwas one of the first schemes to be suggested for trapping atoms [57]. It is similar to theIoffe configuration introduced earlier for plasma confinement [58]. The basic scheme ofthis trap is shown in fig. 6b. It is made of two coils with parallel current, producing aso-called bottle field, and four straight conductors with current in alternating directionsproducing a quadrupole field for transverse confinement. If we use cylindrical coordinateswith z indicating the axial direction, in proximity of the center the field componentsare [54]

(17)

= c1+c3{z2-p2/2) +

p + c2pcos(2</>)

<f) = -c2psin(2(f)) + . . . ,

Bp = —

where c\, c2 and c$ are coefficients which can be calculated using the expressions for thefield produced by a coil and by a straight conductor.

The resulting field | B | is given by

(18) I B |2= c\ [c2 -

With proper choice of the configuration, the origin can be made a minimum for motionin any direction. The resulting potential is harmonic with a nonzero bias field at theorigin:

(19)2 Vci

In this case, an atom passing near the center of the trap never finds a vanishing field andspin-flips can be suppressed. For this purpose, the Larmor frequency for atoms in thebias field must be larger than the vibrational frequency of the atoms in the trap:

(20)


Fig. 9. - Magnetic trap made of six permanent magnet cylinders. (Figure courtesy of R. Hulet.)

where JE?IP is the bias field at the center of the trap (given by the c\ term in eq. (17)). Abias field B\? of a few Gauss is typically sufficient.

In fact, in recent experiments of BEC, trap schemes that can be considered as variantsof an IP trap were used. One is called "baseball" trap because it is made of coils followingthe pattern of the seams on a baseball [51]. Another scheme is called "cloverleaP trap,again because of the shape of the coils [48]. These schemes will not be described in detailhere for brevity.

2*2.6. Pe rmanen t -magne t s t r aps . Intense fields and tight field curvatures can beeasily obtained using permanent magnets. A trap made of permanent magnets was usedin the experiment of BEC of 7Li [10,11] (fig. 9). In this experiment, the trap oscillationfrequencies were of the order of 150 Hz with a bias field of about 0.1 T. As discussedin subsect. 2*3, a large value of the field curvature, that is of oscillation frequency,makes evaporative cooling more efficient. On the other hand, permanent magnets havethe disadvantage that the magnetic fields cannot be changed. This can be a seriouslimitation, for example, for the observation of the condensate.

2*2.7. Comparison of magnet ic t r aps . Several factors must be taken into ac-count for a comparison of the different schemes of magnetic traps for atoms. In experi-ments aiming to achieve BEC by evaporative cooling, it is important to reach the highestcollisional rate after the atoms have been compressed in the magnetic trap' while keepingthe Majorana spin-flips to a negligible level. Important issues are also the simplicity ofconstruction and the possibility of optical access. This latter considerations limited theuse of traps made of superconducting magnets to hydrogen experiments where cryogenicsystems are used.


If the geometric mean of the field curvatures is considered as a relevant figure of merit,a quantitative comparison can be made of TOP and IP traps [8,48]. The magnetic fieldgradient Bf and curvature B" produced by a coil of radius Rc at a distance ~ Rc are

(21) (B1) ~ — (B") ~?±~ — ~ —

where Bo is the field at the center of the coil. This gives a good estimate of the axialcurvature in the IP trap. From eq. (18) it can be seen that the curvature of the radialfield is given by

(22) (B")x%y ~ | ^

where B\p is the bias field at the center of the trap.For the TOP trap, eq. (16) gives a curvature

(23) (B")TOP *

The quantities to be compared are therefore

(24) (B'WB'fa ~ ^ - ,

Equations (15) and (20) indicate that in order to avoid spin-flips, the minimumof a TOP trap must be much larger than BJP of an IP trap. Therefore for comparablefield gradient B', the radial curvature of the TOP trap is smaller than that of an IP trap(typically by two orders of magnitude). In the axial direction, instead, the curvature ofthe TOP trap is typically hundred times larger than that of an IP trap. The geometricmean of the curvatures is then usually better in an IP trap than in a TOP trap.

2*3. Evaporative cooling of trapped atoms, - Evaporative cooling of a gas of trappedatoms is based on the selective removal of atoms which have an energy higher thanthe average energy per atom and on rethermalization of the sample by collisions. Sincethe average energy of the atoms remaining in the trap is reduced in this process, thenew equilibrium state of the gas corresponds to a lower temperature. As described inthe following, methods have been demonstrated to force the cooling to proceed at agiven rate. With optimized evaporative procedures, it is possible to achieve a dramaticreduction of the gas temperature and, more important for BEC experiments, an increaseof the phase-space density by 5-6 orders of magnitude.

Evaporative cooling was first proposed [59] and demonstrated [60] in the effort toachieve BEC of atomic hydrogen. The technique was later extended to alkali atoms [50,47] and, in combination with laser cooling, played a key role in the recent experimentson BEC.

Different methods of evaporative cooling have been demonstrated [61]. In the ex-periments on alkali atoms in which BEC was observed so far, rf-induced evaporationwas used. An rf field was used to induce spin-flip transitions from trapped to untrappedstates. The resonance frequency depends on the local magnetic field. Excitation can then


e>W"eg

CDz0-

i ,

i

\

y

* • • * *. * *

^ * ^ x ^

^ weak rf ^ ^diabatic picture

RF

v — " " " " ^ t r o n g r r ^ âdiabatic potential curves

Position

Fig. 10. - Rf-induced evaporative cooling. (Figure courtesy of W. Ketterle.)

be energy-selective taking advantage of the connection between position and potentialenergy for the atoms in the trap (fig. 10).

In this subsection, we discuss some general features of the process and give the generalscaling laws and the relevant parameters following essentially the presentation in [61].Several detailed studies of evaporative cooling of trapped atoms have been published.An accurate overview on the subject can be found in [61,62].

We consider a sample initially made of N trapped atoms at a temperature T. Theatoms are confined by a three-dimensional potential U(r) oc r3/5. Cases of particularinterest for the present discussion are the linear potential (S = 3) and the parabolicpotential (S — 3/2) (x). The average energy per atom is given by

(25)

As mentioned above, evaporative cooling works by "cutting the edge" of the confiningpotential so that high-energy atoms can escape. We indicate the depth of the resultingpotential with et = rfk^T and the average energy of the removed atoms with (rj + k)k^T.If dN atoms escape from the trap, the energy of the Nf = N — dN atoms remaining in

C) For an anisotropic potential, U(x,y,z) = aSi + $2 + 3 •

\1/Sl + 6 | y \1/S2 + c | z \1/Sz with S =


the sample is reduced by

(26) diS = d/V L + A;) - Q

Rethermalization by collisions leads to a new temperature T' = T - dT, given by thecondition

(27) TV' (1 + s\ kBT -dE = N'(^+ s\ kBTf.

From eqs. (26),(27), neglecting second-order terms in diVdT, it follows that

dT

where

77 + k(29)

2 ~r°

a is a crucial parameter in the evaporative cooling process, giving the temperature changeper particle lost. In particular, if a is kept constant during the process, eq. (28) is easilyintegrated and gives

(30) ^

Note that 77 3> 1 —> a > 1; eq. (30) shows then how efficient evaporative cooling canbe. Equation (30) also shows that in the evaporation process T is no longer independentof N. The scaling laws for the relevant physical quantities can then be expressed in termsof the number of atoms in the sample, by taking the dependence on N and T of thesequantities for atoms in equilibrium in a confining potential and replacing T by Na. Theresults are shown in table I. From the scaling laws in the table, it is possible to determinefor which values of a, and consequently of 77, evaporative cooling works efficiently. A firstconsideration is that in order to have an increase of the phase-space density D when thenumber of atoms in the trap is reduced, the exponent of N in the expression of D mustbe negative, that is

(31) a >

Another condition to be fulfilled is that the evaporation does not lead to a reduction ofthe collisional rate nae\v, which would slow down the process. The reduction of v mustbe compensated by an increase of the density n. Table I shows that this is the case if

(32) a >7TT-0 2

This regime is called "runaway evaporation". Conditions (31) and (32) indicate thatefficient evaporation requires a value of 77 as large as possible. This is in agreement


TABLE I. - Dependence of relevant physical parameters on the number of trapped atoms N duringevaporation.

Temperature, T oc Na

Volume, V oc NaS

Density, n oc JV(1"a<5)

Average velocity, v oc Na'2

Elastic collision rate, nav oc N[1~a{S~^)]

Phase space density, D oc JV [1-a (*+i ) ]

with the intuitive argument that the larger the energy of the evaporating particles, thelarger the temperature change. On the other hand, a large value of 77 implies a lowrate of evaporation. If the evaporation time is too long, other mechanisms such ascollisions or Majorana spin-flips can produce losses of atoms from the trap. Optimizationof the evaporative cooling then requires a compromise between efficiency and speed of theevaporation process taking into account the unavoidable loss mechanisms. It is beyondthe scope of this review to analyze this problem in detail. An analytical expression forthe rate of evaporation can be easily obtained, however, following an argument givenin [61]. This rate can then be compared with the rate of losses of atoms from the trap.The argument is the following: For a large value of 77, the rate of evaporation of theatoms is the same as the rate of production of atoms with energy larger than r)k&T. Theevaporation rate can be written as the number of atoms with energy higher than r]k^Ttimes the elastic collision rate l/rei. The number of atoms with E > r)k&T in theuntruncated Boltzmann distribution can be approximated with 2N^/rf/7êxp[—rj\. Theelastic collision rate is given by UQGV, where rip is the density of the particles, a is theelastic collision cross-section, and v = y/2rjkBT/m — ^/7rrjv/2 is the velocity of atomswith energy r}k&T. The rate of evaporation can therefore be written as

(33) AT = -Nnoavrje'11 = - — .7~

Equation (33) gives the rate of change of the number of atoms due to evaporation.The loss of atoms due to background gas collisions or other loss mechanisms can be takeninto account by considering an exponential decay of the number of atoms with a timeconstant TIOSS- The rate of change of the collisional rate can then be written as

1 d , x I"1-<*(<*-!)(34) Mnav dt I rev rioss

In order to have a constant or increasing collisional rate during evaporation, the term inthe square bracket must be null or negative, that is

(35)Tel Tei a [0 — 2 j — -L

The quantity rev/re\ can be easily calculated in the limit of large rj considered above,


giving

(36) Tev

Te\ V

For arbitrary values of 77, this ratio can be written in terms of generalized gamma func-tions.

Equation (35) shows that efficient evaporative cooling requires the rate of elasticcollisions to be much larger than the rate of collisions causing losses of atoms from thetrap (typical numbers in the experiments are TIOSS & 200 — 500 times re\).

Two kinds of collisions are responsible for atom losses from magnetic traps: collisionswith background gas and inelastic collisions. Losses due to background gas collisions canbe reduced by reducing the pressure of background gas. Ultra-high vacuum chambersare indeed used in the experiments. Inelastic collisional processes can be either binarycollisions, such as dipole relaxation and spin relaxation, or three-body recombination.Since evaporative cooling is based on collisions, a high density is required; inelasticcollisions are then unavoidable. The situation is however quite different for differentatoms depending on their collisional parameters. The success of BEC experiments onalkali atoms, compared with the experiments on hydrogen, is due to the fact that alkaliatoms have a much larger elastic cross-section. Since the rate of inelastic collisions isroughly the same as in hydrogen, the ratio of "good" to "bad" collisions is much largerfor alkali atoms. An analysis of the relevance of different kinds of collisions for differentatoms is beyond the scope of this review. A comprehensive treatment of the subject canbe found in [61] and references therein.

3. — Studies of Bose-Einstein condensates

3*1. Experimental procedure to achieve BEC. - In the previous section, the ingredientsof BEC experiments were presented, that is the methods to cool and trap the atoms.Here, we describe the way these ingredients were put together in the first experimentsin which BEC was observed. Table II gives a synthetic view of the general scheme of aBEC experiment with the possible variations demonstrated in the different laboratories.The procedure to observe BEC can in fact be divided into four main steps.

The first step is laser cooling and trapping of the atoms. This was accomplished byusing a single vapor-cell MOT, or a laser-slowed atomic beam, or a double-MOT appa-ratus in which the atoms are loaded in a first MOT from the vapor and then transferredinto a second, high-vacuum, chamber. In the first experiments, a dark-spot MOT wasused to increase the atoms density. A molasses phase can be used to reduce the temper-ature of the atoms before loading them into the magnetic trap. In this first step a largegain in phase-space density is obtained. For example, the phase-space density in a roomtemperature vapor of Rb at a pressure of 10~10 Torr is of the order of 10~20. After lasercooling and confinement, a value in the range 10~6-10~5 can be reached.

The second step is the magnetic trapping of the atoms: This is the aspect in whichthe different experiments have more differentiated. As described in subsect. 2*2, severalschemes have been developed to achieve a tight confinement of the atoms and to avoidlosses due to Majorana spin-flips. The trap configuration must also be suitable forevaporative cooling of the atoms in the trap. An important stage in this phase is theadiabatic compression of the trapped atoms. After loading, the atoms in the trap areadiabatically compressed by increasing the trap curvature. This leads to an increase


TABLE II. - Experimental procedures used to study Bose-Einstein condensation in atomic gases.

Single vapor-cell (JILA) [8]1) Laser cooling and trapping Slowed atomic beam (MIT, Rice) [9,48,11]

Double-MOT (JILA) [51]

TOP (JILA) [8]Optical plug (MIT) [9]

2) Magnetic trapping Permanent magnets (Rice) [11]Cloverleaf (MIT) [48]Baseball (JILA) [51]

3) rf-induced evaporative cooling (JILA, MIT, Rice)

Absorption imaging (JILA, MIT) [8,9,48,51]4) Optical imaging Dark-ground imaging (MIT) [63]

Polarization imaging (Rice) [11]

of the temperature and density of the atoms, with the phase-space density remainingconstant. The increased rate of elastic collisions obtained in this way is important toachieve the necessary conditions of rethermalization in the evaporative cooling process.

The third important stage is indeed evaporative cooling. In all of the experimentswhere BEC was obtained so far, atoms in the magnetic trap were cooled with evaporativecooling by using rf-induced evaporation. As mentioned in subsect. 2*3, the magneticallytrapped atoms are irradiated with rf radiation. The rf frequency is tuned near theresonance frequency for transitions between nearby Zeeman sublevels. This producesspin-flip transitions from trapped to untrapped states. Since higher-energy atoms canreach higher-field regions of the trap, spin-flip resonance frequencies are more shiftedthan for colder atoms. By varying the frequency of the rf field, colder and colder atomsare eliminated from the trap. As discussed in subsect. 2*3, if the right procedure isfollowed, this process can give an increase of the phase-space density of several orders ofmagnitude and eventually allows to reach the condition for BEC (phase-space density ~

!)•Once conditions for BEC have been reached, a proper observation method must be

used. For this fourth stage, the methods demonstrated so far are based on absorptionimaging and phase-contrast imaging. The basic idea and experimental realization ofthese detection methods are described in the next subsection.

In order to give a specific example of an experimental procedure to reach BEC, we candescribe the way followed in the first, now almost "historical", realization at JILA [8].About 107 87Rb atoms were first collected in a dark-spot MOT from the room tempera-ture vapor in a glass cell. Because of the very low pressure of Rb in the cell (10~n Torr),the loading of the MOT took about 300 s. The trapped atoms were compressed andcooled to 20 fjK by adjusting the field gradient and laser frequency. The laser beamswere then switched off and the magnetic trap (a TOP trap in this experiment), wasquickly switched on. The magnetically trapped atoms were compressed by adiabaticallyincreasing the quadrupole field. This produced a sample of 4 x 106 atoms in the trap with


Fig. 11. - Absorption imaging method. (Figure courtesy of S. Wieman.)

a temperature of about 90 /iK and an average density of about 2 x 1010 atoms/cm3. Thetrap had an axial oscillation frequency of 120 Hz. Evaporative cooling was then appliedfor about 70 s. During this time, the rf frequency was ramped down from the initial valueto the final value that determined the final temperature. At the end of the evaporativecooling, the magnetic trap was switched off, letting the atoms expand freely. After 60ms of expansion, the cloud was observed using an absorption imaging technique. Theatoms were irradiated with a resonant laser beam for 20 / s and the shadow producedby the atoms was imaged on a charge-coupled-device (CCD) array, as described in thefollowing.

3*2. Observation of a Bose condensate, - As mentioned above, in the experimentsperformed so far, at the end of the cooling process the atoms distribution was observedby imaging with an optical probe. Two major approaches have been followed: The firstis absorption imaging of the sample, the second is based on the detection of refractiveeffects caused by the atoms.

Absorption imaging was used, for example, in [8,9]. The atoms were illuminatedwith a short pulse of resonant light and the shadow of the cloud was imaged onto aCCD detector (fig. 11). Because of the high optical density (D « 100-300) of the atomcloud, however, quantitative measurements could not be performed by direct imaging ofthe atoms in the trap. At the end of the evaporative cooling, the cloud was allowed toexpand by switching off the trapping fields and the absorption image was recorded after atime of several milliseconds. The larger size of the sample after the expansion also reducesspatial resolution requirements for the optical system. A result is shown in fig. 12. Thismethod of observation corresponds to a 2-dimensional time-of-flight measurement of the


Fig. 12. - False-color images showing the velocity distribution of a sample of cold 87Rb atoms.Left: before condensation, center: just after the transition, right: for a nearly pure condensate.Images were recorded by the absorption imaging method after releasing the atoms from the trapand letting them expand freely. (Figure courtesy of E. A. Cornell.)

velocity distribution. The recorded image gives the initial velocity distibution of theatoms in the trap. However, for a harmonic confining potential, the spatial distributionand the velocity distribution are the same. From a single image, information can then beextracted on the density and the temperature of the atoms in the trap. This method isobviously a destructive observation method because each atom scatters several photonswhile being probed, heating the gas. Experiments are therefore performed by repeatedcycles of load-evaporation-probing.

An alternative method to observe the condensate was demonstrated in [63]. By usinga so-called dark-ground imaging technique [64], the dispersively scattered light was ob-served. In the dark-ground imaging technique, a collimated probe beam is sent through aweakly absorbing sample (fig. 13). The coherently scattered light is collected with a lenssystem and imaged onto a camera. The probe beam is blocked after passing the sampleby inserting a small opaque object at the position where the beam comes to a focus.If Eo is the electric field of the incident probe beam, after passing through the sample itacquires a spatially dependent phase (5 — <p + ia/2 so that the transmitted electric fieldis given by E = Eo e%&. When the opaque object is inserted to block the incident beam,the resulting intensity is given by Js = El \ ei(5 - 1 |2. In the case a < | cp | < 1, theresulting signal is /s « E^p2 = /o<p2.

Dispersive imaging offers two important advantages with respect to absorption imag-ing. First, because the signal depends on the phase shift tp, it is possible to detune thefrequency of the probe light from resonance at values for which the optical density is


Imaging lens

CondensateWire

Probe laser

Scattered light

Fig. 13. - Schematic of dark-ground imaging method.

CCD

small. This allows probing of dense clouds of atoms directly in the trap. Second, dis-persive imaging can be a nondestructive detection method. In [63], it was demonstratedthat by probing the condensate with a pulse of off-resonant light, several consecutiveimages of the same condensate could be taken with no observable deterioration of thesignal (fig. 14). Indeed, in dispersive scattering the photons are elastically scattered by asmall angle. Also, if a light pulse duration longer than the oscillation period in the trapis used, no momentum is transferred to the atoms on the average.

When the phase shift cp is small, the dark-ground signal is small because it depends oncp2. A signal linear in (p can be obtained by a phase-contrast imaging method [11,65]. Insitu imaging allowed the MIT group to observe the propagation of sound in a condensate,

Fig. 14. - Nondestructive observation of a condensate. (A) Images of the same condensate taken1 s apart. (B) Two pictures taken 6 s apart during decompression of the cloud. (From M. R.Andrews et al. (1996), with permission.)


to study the formation of the condensate and to directly observe the collective excitations.In [11], the birefringence of the sample of atoms polarized in the trap magnetic field wasprobed using a linearly polarized laser beam. After passing through the atomic sample,the polarization of the probe beam was analyzed with a polarizer. The intensity of thetransmitted light was measured with a photodetector. This detection method allowed insitu imaging of condensates with only ~ 103 atoms.

Some remarks can be made concerning dispersive imaging detection methods. Asdiscussed above, they can be considered as nondestructive detection methods. In fact,the phase of the condensate is changed under the effect of light. Only the number ofatoms in the condensate can be measured in this way. Also, one could expect lightscattering to be affected by the quantum degenerate state of the atoms. Such effects arenegligible, however, under typical experimental conditions [66].

3*3. Measurement of energy and ground-state occupation as a function of temperature.Images of the ultra-cold atoms as the one shown in fig. 12, obtained by resonant absorp-tion imaging, contain a wealth of information about important thermodynamic quanti-ties. The integrated area under the distribution gives the total number, JV, of atoms inthe sample. The number of atoms in the ground state of the system, iVo, can be obtainedfrom the integrated area under the narrow feature centered on zero velocity. The meansquare radius of the expanded cloud as a function of time gives the mean square velocity,that is average energy, of the atoms. The temperature, T, can also be extracted fromthe analysis of the images. From the data on mean energy as a function of tempera-ture, the specific heat can be determined, that is an important quantity in the study ofphase transitions. At high densities of condensed atoms, the repulsive mean-field energybecomes important and can be measured.

In the work reported in ref. [67], a series of images of ultra-cold clouds of 87Rb atomswere analyzed to extract the ground-state occupation and the mean energy as a functionof temperature. The number of atoms in the ground state was defined as the number ofatoms contributing to the central feature in the optical-depth images. The condensateshape was fit with a 2-d Gaussian. The width, aspect ratio and peak height were found tobe functions only of the number of atoms in the feature. The procedure gave consistentvalues of 7V0 for temperatures down to about one half of the transition temperature.The temperature was determined by fitting a Gaussian to the high-energy tail of thevelocity distribution. This procedure is based on the assumptions that the high-energypart of the velocity distribution remains in thermal equilibrium with the central part ofthe cloud and that it can still be described by a Maxwell-Boltzmann distribution becausethe density of the atoms in the outer part of the cloud is lower. Figure 15 shows thedata obtained for the ground-state fraction No/N as a function of scaled temperatureT/To. The scale temperature, To, is the critical temperature for a noninteracting gasin the thermodynamic limit. The solid line in the figure shows the behaviour expectedfor the ideal gas of bosons in an anisotropic potential in the limit of infinite numberof atom: No/N = 1 — (T/To)3. The dotted line shows the curve calculated includingfinite-number corrections [68,69]. The dashed line is, instead, a least-squares fit of thedata with the function No/N = 1 — (T/Tc)

3, with Tc as a parameter. The fit gives avalue for the critical temperature Tc = 0.94(5)Tb. The accuracy of the data is howevertoo low to show a difference between Tc and To. An accurate measurement of the criticaltemperature would allow to test theories taking into account mean-field [70,71] or many-body [72] interaction effects.

From the same images analyzed to measure the ground-state fraction, information


• v\

12

8

4

•

0.0 0.5 1.0 1.5

0.5 1.0 1.5T/T0(N)

Fig. 15. Total number of atoms (inset) and ground-state fraction as a function of scaled tem-perature T'/To. (Prom J. R. Ensher et al. (1996), with permission.)

on the energy and specific heat was obtained. The specific heat is usually defined asthe temperature derivative of the energy per particle at constant pressure or constantvolume. In this case it is given by the slope of the scaled energy vs. temperature plotwith the confining potential held constant. The result is shown in the inset of fig. 16 andcompared with the specific heat of other relevant systems. The data indicate the presenceof a step in the specific heat near the empirical transition temperature. The observedstep is smaller than the one predicted by a finite-number, ideal-gas theory [68,69]. Thedifference can be explained as an effect of interactions.

CQ

o

3d SHO. 4He data /

A /i /\ /

i /

jy

4

2

o

•

/

0.8

'I

1.0 1

-

.2

^3d box

-

3 -

1 ,

0.6 0.8 1.0ir

1.2 1.4

0(N)

Fig. 16. - Specific heat as a function of scaled temperature T/To for various theories and ex-periments. The inset shows the experimentally determined specific heat (bold line) and thetheoretical prediction for a finite number of ideal bosons in a harmonic trap. (Prom J. R.Ensher et al. (1996), with permission.)


b)

y y ^/ /

ItV /

/

jtvdt/

A/

Fig. 17. - Equipotential contours for the two trap perturbations corresponding to the m = 0and m — 2 collective modes. (Prom D. S. Jin et al. (1996), with permission.)

A related work was performed at MIT on trapped Na atoms [48]. The dependenceof the condensate fraction on temperature and the interaction energy as a function ofthe number of atoms in the condensate were investigated. The study of the effect ofinteraction energy was made possible by the high density of atoms achieved in the new"cloverleaf" trap. The measured interaction energy per atom showed a good agreementwith the NQ'b behaviour expected from the theory.

3*4. Study of collective excitations and propagation of sound in a Bose-Einstein con-densate. - The collective excitations of the Bose-Einstein condensate have been studiedboth by the JILA group and by the MIT group. The excitation of the lowest collectivemodes of the condensates in the trap was achieved by a time-dependent perturbation ofthe magnetic trap potential. For a noninteracting gas, all modes have frequencies whichare integer multiples of the trap frequency. Deviations from this behaviour are expectedas an effect of interatomic interactions. In the mean-field picture, the interaction term inthe Gross-Pitaevskii equation depends on Nayfv, where N is the number of atoms, a isthe scattering length and v is the trap frequency. The normal modes of the condensatecan be classified by a quantum number m that gives the angular momentum projection inthe direction of the trap symmetry axis. In [73] the m = 0 and m = 2 modes of a dilutecondensate of 87Rb were investigated and their frequency measured as a function of theinteraction strength. The m = 0 mode corresponds to an oscillation in radial size. Them — 2 mode resembles an ellipse whose major axis rotates in the plane orthogonal to thetrap axis (fig. 17). The measured frequencies for the two excitation modes as a functionof the interaction strength are shown in fig. 18. In the figure, the experimental data arecompared with the theoretical predictions based on mean-field theory. The measuredfrequencies deviate from the harmonic trap values showing a fairly good agreement withtheoretical calculations.

A similar experiment was performed by the MIT group on a sodium condensate. Theystudied m — 0 oscillations. The measured frequency 30.0(2) Hz agreed with the theo-retical prediction y/b/2 vz [74]. The oscillations observed in this experiment showed adamping with a decay time, for a nearly pure condensate, of 250(40) ms for the oscillationat ~ 30 Hz. Measurements on a thermal cloud (T/Tc « 2) gave a damping time of 80ms for oscillations at a frequency of 35(4) Hz = 1.2(2)vz.

The dependence of the damping rate and of the oscillation frequencies on the sampletemperature was studied in [75,76]. In particular, temperatures were investigated forwhich both condensate and noncondensate atoms are present, a regime which is not wellunderstood theoretically. In [75], the m = 0 and m — 2 modes were studied with the


Fig. 18. - Measured frequencies for the m = 0 (triangles) and m = 2 (circles) excitation modesas a function of the interaction strength. Lines show the result of mean-field calculations. (FromD. S. Jin et al (1996), with permission.)

same technique used in [73]. The main results of this work can be summarized as follows(fig. 19): A temperature-dependent shift of the excitation frequencies of the condensatecan be observed, with the two modes shifting in opposite directions. In addition, for them = 0 mode, a sharp feature in the temperature dependence is observed at T « 0.62 Tc.This feature is not observed for the m = 2 mode. For both the m = 0 and m — 2condensate excitations the damping rate quickly decreases with decreasing temperature.The two decay rates show the same behaviour. Interestingly, at temperatures whereboth condensed and noncondensed atoms are present, the condensate excitations areobserved to decay more rapidly than the ones of noncondensed atoms. In the MITexperiment [76], collective excitations of the Bose gas at nonzero temperatures werestudied for large condensates and in conditions corresponding to the hydrodinamic limit.The hydrodynamic oscillation of the thermal cloud, corresponding to the first sound, wasobserved. Also, an antisymmetric oscillation of the thermal cloud and the condensate,analogous to the out-of-phase second sound mode in liquid helium was identified. Theresults of this work cannot be completely explained with presently existing models andwill certainly require a deeper investigation.

The propagation of sound in a dilute Bose-Einstein condensate was studied in [65].Density perturbations were excited using the dipole force of a focused, blue-detuned laserbeam. The propagation of sound waves was observed by recording rapid sequences ofnondestructive phase contrast images.

The speed of sound c(r) is given by [77-79]

(37)

where n(r) is the local density, m is the mass of the particles and U = Antfa/m gives theinteraction of bosons with scattering length a. The basic idea of the experiment reportedin [65] is schematically shown in fig. 20. A condensate was formed, as in the otherexperiments described above, with the atoms confined in a magnetic trap. At time t = 0,the blue-detuned laser beam was switched on or off. The beam was focused at the centerof the trap. The wavelength used (514 nm) was far enough from the sodium resonance


2.1

31

(0

0.4 0.6 0.8 1.0 1.2

1 0 :

Fig. 19. - Temperature-dependent excitation spectrum: (a) Frequencies for m = 0 (triangles)and m = 2 (circles) modes in the case of the condensate (solid symbols) and noncondensate(open symbols) clouds, (b) Damping rate. (From D. S. Jin et al. (1997), with permission.)

at 589 nm to make heating from spontaneous emission negligible. If the laser beamwas switched on after producing the condensate, the effect was to push the atoms outfrom the center of the trap creating two density peaks that propagated outward. If thefocused laser beam was kept on during the evaporative cooling and switched off after theformation of the condensate, negative density perturbations were generated propagatingoutward with the same speed. The speed of sound was determined by measuring theposition of the density peaks as a function of time. The resulting value was in goodagreement with the value expected from eq. (37) (fig. 21).

3*5. Condensation of atoms with attractive interactions. - The experiments that firstclearly showed BEC in dilute gases were performed on 87Rb [8] and 23Na [9]. Boththese atomic species have a positive 5-wave scattering length as, indicating that for coldatoms the interatomic interaction is repulsive. Evidence for Bose-Einstein condensationof 7Li, an atom for which as is known to be negative, was first published in [10] and laterconfirmed in [11] with more accurate values for the number of condensate atoms. In themean-field theory, the interaction energy is given by U = 47rft2asn/ra, where n is thedensity and m is the mass of the atoms. If as < 0, the energy decreases with increasing n,so the condensate is not stable and tends to collapse. For atoms in a confining potential,


(a) (b)

\

\ A

Fig. 20. - Scheme for the excitation of positive (a) or negative (b) density perturbations in acondensate using a focused, blue-detuned laser beam. The perturbations propagate at the speedof sound. (Prom M. R. Andrews et al. (1997), with permission.)

recent theories predicted, however, that the kinetic zero-point energy can balance theattractive interaction and lead to a metastable condensate with only a small number ofatoms in the condensate [80,81].

The experiment reported in [11] gave results in agreement with theoretical predictions.As discussed in subsect. 3*2, a phase-contrast technique was used to observe the atomsdirectly in the trap. In this experiment, a permanent-magnet trap was used, so that onlyin situ imaging was possible. For temperatures between 120 and 330 nK, the maximumcondensate number was found to be between 650 and 1300. This result is consistent withwhat is expected from theory for the trap used in this experiment. Work is actually in

8

f3)

Fig. 21. - Speed of sound vs. condensate peak density. The solid line is the theoretical prediction.(From M. R. Andrews et al. (1997), with permission.)


Fig. 22. - Absorption image showing two 87Rb condensates produced by evaporative cooling ofatoms in the | 1,-1) state and sympathetic cooling of atoms in the | 2,2) state. (Prom C. J.Myatt et al. (1997), with permission.)

progress to study the effect of attractive interactions on BEC also for other atoms, suchas 85Rb, for which the s-wave scattering length is predicted to be negative.

A related subject under investigation is the observation of the so-called Feshbachresonances [82] in the scattering length of alkali atoms [83]. Such resonances could allowto control the magnitude or even the sign of the scattering length by applying externalmagnetic, optical or radio-frequency fields. Feshbach resonances were recently observedin an experiment performed at MIT on Bose-Einstein condensates of Na atoms [84].A key ingredient in this experiment was the possibility of confining the condensate inan optical dipole trap [85]. It was predicted indeed that Feshbach resonances in Nawould be observed for a relatively small applied magnetic field only for atoms in high-magnetic-field seeking states. Since these atoms cannot be trapped in magnetic traps,they were confined using the optical trap. The occurrence of the resonances was detectedby two methods: The enhancement in the rate of inelastic collisions was detected as anincrease of the losses of atoms from the trap as the magnetic field was swept acrossthe resonance. Also, a clear evidence for a dispersive variation of the scattering lengtharound the resonance was obtained by measuring the interaction energy for the trappedcondensate vs. magnetic field by the time-of-flight absorption imaging method.

3*6. Production of two condensates by sympathetic cooling. - In [51], a cooling methodcalled sympathetic cooling was used to produce two overlapping condensates made of87Rb atoms in two different spin states. In sympathetic cooling, an atomic species iscooled via collisions with another species at a lower temperature. This method was firstdeveloped to cool ions in ion traps [86]. It had never been used before to cool neutralatoms, where interactions are in general weaker. In the experiment reported in [51], thetwo species corresponded to 87Rb atoms in the | F — l,ra = — 1) and | F = 2,m = 2)spin states. Trapped atoms in the | F = l ,m = -1) state were cooled, by evaporativecooling, down to Bose-Einstein condensation as in other experiments described above. Inthis case, however, atoms in the other spin state were left in the trap and observed at theend of the cooling process. Figure 22 shows the absorption image recorded by probingthe atoms in F — 2 and the atoms in F — 1. In this case, the trap axis was slightly tiltedwith respect to the horizontal plane so that gravity produced a separation of the two


!

•100^

1-

• |1,-1) state° |2,2) state (x 10)

o o °o

o o° fib ° ° \

o

1 10 100Temperature [\iK]

Fig. 23. - Number of atoms in the | 1, -1} and | 2, 2) states for different temperatures during thesympathetic-evaporative cooling process. (From C. J. Myatt et al. (1997), with permission.)

condensates. With a well-aligned trap, however, nearly perfectly overlapping condensateswere obtained. Because of the different magnetic moments, evaporative cooling actspreferentially on the atoms in the | F — l,ra = -1) state which are less tightly confinedin the trap. Therefore the evaporative cooling process leaves the initial number of atomsin the | F = 2, m = 2} state nearly unchanged (fig. 23). The study of the behaviour of thecondensates provided important information on collisional parameters. By measuring thedensities and loss rates for the atoms in the two states for various overlapping conditions,it was possible to determine the rate constant for binary inelastic collisions such as spin-exchange collisions. Another interesting result came from the observation that in thepresence of the two condensates, the | F — 2, m — 2) condensate was displaced upwardfrom its equilibrium position because of interaction with the | F = 1, m = -1) condensedatoms. This provided an indication that the interaction between the two condensates isrepulsive.

The demonstration of the possibility of cooling a sample of atoms down to quantumdegeneracy using sympathetic cooling is of great interest because of possible applicationsto other species. In particular, sympathetic cooling might allow the achievement ofquantum degeneracy conditions in a gas of fermionic atoms. For spin-polarized fermionicatoms, in fact, simple evaporative cooling is not efficient because s-wave collisions arenot possible for atoms in the same state. Also, the two fermionic atoms for whichexperiments are presently in progress, namely 6Li [87] and 40K [88], have a low naturalabundance so that it is important to use a cooling technique which does not requirea large initial number of atoms. More generally, sympathetic cooling may enable theachievement of Bose-Einstein condensation of atoms for which evaporative cooling doesnot work efficiently.

37. The coherence properties of the condensate and the "atom laser". - A gas ofbosons all occupying the same state is described by a macroscopic wave function that isthe solution of the nonlinear Schrodinger equation. Therefore, a Bose condensate shouldshow coherence properties analogous to the ones of laser light. Such properties were in-vestigated experimentally starting with the most striking effect, that is interference of the"matter waves" of two Bose condensates [89]. In this experiment, Na atoms were confinedin a trap created by focusing a blue-detuned light sheet at the center of a "cloverleaf'


0 0.5 1Absorption

Fig. 24. - Interference of two expanding condensates, for two different powers of the light sheetused to produce the double condensate. (Prom M. R. Andrews et al. (1997), with permission.)

magnetic trap. The combination of magnetic force and optical dipole force produceda double-well potential. After evaporative cooling, two separate Bose condensates wereproduced instead of the single cigar-shaped condensate usually produced in the simplemagnetic trap. The distance between the condensates could be varied by changing theintensity of the laser light. The overlapping of the two condensates was obtained byswitching off the trap and letting the condensates expand. The interference pattern wasobserved by absorption imaging. In order to increase the spatial resolution, only a thinslice of the cloud was observed. This was achieved by first optically pumping the atomsin the relevant part of the cloud into a given hyperfine state using a thin sheet of lightand then probing the atoms in this state with a pulse of resonant light. Figure 24 showsthe interference of two condensates as observed, after 40 ms of expansion, for two differ-ent intensities of the blue-detuned laser light in the trap. Nearly straight fringes can beobserved with a contrast that, after calibration, was found to be'in the range 50%-100%.The measured spacing between the fringes, ~ 20 fim and ~ 15 /im, respectively, wasconsistent with the value of the de Broglie wavelength A associated with the relativemotion of the atoms

(38)ht

md '

where h is Planck's constant, m is the atomic mass, d ~ 30-40 fj,m was the initial spatialseparation between the condensates and t = 40 ms was the expansion time before theobservation. The slight curvature of the fringes in fig. 24a was attributed to the effect of


interactions which becomes nonnegligible for small separations of the two condensates.The main result of this work was to demonstrate the first-order coherence of a Bosecondensate and long-range correlations over the extent of the condensate. The methodsdeveloped allowed for the first time to address experimentally the debated question ofthe phase of a Bose-Einstein condensate showing the existence of such a phase and thepossibility of comparing the phases of two condensates.

Higher-order coherence properties of the condensate were studied in [90]. Startingfrom the analogy between the intensity fluctuations in a light beam and the density inan atomic sample, information on statistical correlations in a condensate was obtainedby studying three-body recombination rates. Indeed, in [91] it was calculated that therate of three-body recombination in a condensate for noninteracting atoms should bereduced by a factor 3! with respect to the corresponding rate in a thermal cloud withthe same mean density. In the experiment reported in [90], collision rates were inferredfrom the loss rate of atoms from the trap. Three-body recombination processes couldbe distinguished from other loss processes by measuring the loss rate as a function ofdensity. By comparing the data obtained for noncondensed and condensed samples, itwas found that the three-body recombination rate was smaller in the case of a condensateby a factor 7.4 (2.0). The measured value is then in agreement with the theoretical valueof 3! and demonstrates that density fluctuations are reduced in a condensate relative toa thermal gas. In the language of quantum optics, condensate atoms appear to be lessbunched than thermal atoms.

Several papers have recently discussed the analogies between coherent matter andcoherent light [92-96]. One of the most interesting prospects in this field is the realizationof an "atom laser". In fact, atoms in a magnetic trap can be considered as analogous tophotons in an optical cavity. The thermal cloud of ultracold atoms plays the role of theactive medium and, in the experiments performed so far, evaporative cooling acts as the"excitation" mechanism which makes the atoms accumulate into a single state of the trap.It is obvious that, contrary to the photon case, the number of atoms cannot be amplified.However, the enhanced probability for bosons to occupy an already occupied state of thesystem can be seen as the analogue of an amplification mechanism. A key element in therealization of the "atom laser" is the possibility of coupling out particles in a coherentway. The simplest way to couple atoms out of the condensate is to switch off the trappingpotential and let the atoms fall under the effect of gravity. A more controlled method wasdemonstrated in [97]. Rf radiation was used to create Bose condensates in a superpositionof trapped and untrapped states. After producing a condensate of 5 x 106 sodium atomsin the F = 1, mp = — 1 ground state, a resonant rf magnetic field was switched onwith a polarization orthogonal to the magnetic field at the trap center. This coupled apart of the trapped atoms into the mp = 0 and into the rriF = 1 states. Atoms in thernp — 1 state were repelled from the trap while the untrapped mp = 0 atoms expandedfreely while moving downward under the effect of gravity. The amplitude of the rf pulsecould be adjusted to get an out-coupling of atoms between 0% and 100%. Interestingly,by varying the intensity of the rf pulse, Rabi oscillations were observed with the sameRabi frequency as for the single atom. Using subsequent rf pulses, multiple pulses ofatoms out of the condensate were obtained (fig. 25). In the same work, another outputcoupler scheme was demonstrated using a nonadiabatic rf sweep through resonance totransfer atoms into the untrapped states. This scheme is less sensitive than the other tochanges in the bias magnetic field of the trap. In the same work, the possibility of takingadvantage of Majorana spin-flips as an out-coupling mechanism was also shown.

In [89], a test of the coherence of the pulses of atoms was performed. Pulses from two


The atom laser at 200 Hz repetition rate

37

0 Density scale (arbitrary units) 1

{field of view 2.5 mm x 5.0 mm )

Fig. 25. - Atom laser. The figure shows successive pulses of coherent sodium atoms coupled outfrom a Bose-Einstein condensate. (Figure courtesy of W. Ketterle.)

separate condensates were combined and high-contrast interference fringes were observed.This completed the proof that a Bose condensate with an out-coupling mechanism suchas the one just described can be considered as a realization of an "atom laser".


4. — Conclusions and future prospects

At the time of this writing, almost three years have passed since the first observation ofBEC in atomic gases. The laboratories where BEC was first achieved had the possibilityof investigating a wide range of interesting phenomena some of which had been predictedtheoretically but never studied experimentally in such simple and well-controlled systems.During this time, several laboratories in the world have been trying to reproduce theexperiments and to achieve BEC.

Recently, a few laboratories announced the observation of BEC: D. Heinzen's group atthe University of Texas at Austin observed BEC of 87Rb atoms using a Zeeman-slowedatomic beam and evaporative cooling of the atoms in a TOP magnetic trap. L. Hauand collaborators at the Rowland Institute in Cambridge, MA, used a Zeeman-slowedatomic beam of Na and a Ioffe trap. The group of M. Kasevich, at Stanford University,achieved BEC of 87Rb atoms with a single vapor cell and a TOP trap. At the Universityof Konstanz in Germany, the group of G. Rempe observed BEC of 87Rb atoms in adouble MOT apparatus with a Ioffe-type magnetic trap. In the laboratory of T. Hanschat the University of Munich/MPI for Quantum Optics, BEC of 87Rb was achieved usinga double-MOT apparatus and a Ioffe-type magnetic trap. The group of W. Phillips atNIST, Gaithersburg achieved BEC with sodium using a Zeeman-cooled atomic beam anda TOP trap. It is interesting to notice that in this experiment BEC was reached alsoby evaporating only with the "circle of death" of the TOP trap. At the Ecole NormaleSuperieure in Paris, J. Dalibard and colleagues observed BEC of rubidium atoms usingan apparatus based on a double-MOT and a Ioffe-type magnetic trap.

The experimental results also stimulated a lot of theoretical work and several paperswere published interpreting the new observations and proposing new possible experi-ments.

Amongst the future challenges one can envisage the achievement of BEC with otheratoms. Of particular interest are H and He*. Work is also in progress on K [49], Cs [52,98],and even on molecules [99].

The effect of interactions needs to be further investigated. The case of atoms withattractive interactions and the possibility of modifying the strength of the interaction byexternal fields are certainly important subjects to be further explored.

There are important phenomena that have been predicted theoretically and observedin other systems. Notable examples are vortices and second sound that have been widelyinvestigated in superfluid helium. The study of these phenomena in the well-controlledconditions of experiments with dilute atomic gases will provide important information.

The possibility of manipulating the condensates should allow studies of tunnellingeffects between two condensates and the observation of Josephson-type phenomena.

The future possibilities of atom lasers can be imagined considering the major impactthat optical lasers had in several fields.

Finally, one of the most interesting prospects is to investigate the properties of anultracold gas of fermionic atoms. At phase space densities similar to the ones achievedin BEC experiments, the behaviour of a degenerate Fermi gas may be studied and, atstill lower temperatures, the superfluid phase transition could be observed [100-103].Promising candidates which are presently investigated are 6Li [87] and 40K [88].

Note added in proofs. - After the completion of this review, many interesting papersappeared reporting new results in the investigation of BEC. An extensive review of thetheory of BEC in trapped gases has been published in [104]. A discussion of recent


theoretical and experimental results can be found in the proceedings of the E. FermiSchool on "Bose-Einstein condensation in atomic gases" held in Varenna in the summer1998 [105]. During that School, the first evidence of BEC in atomic hydrogen waspresented [106].

* * *

The authors thank J. R. ENSHER for a critical reading of the manuscript and E. A.CORNELL, W. KETTERLE, R. HULET, and J. DALIBARD for providing copies of figures.

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Autorizzazione del Tribunale di Bologna n. 3382 del 24 dicembre 1968

ISSN 0393-697X

La Rivista del Nuovo Cimentovolume 22 serie 4 numero 4 1999


G. M. Tino and M. Inguscio

1. Introduction2. Cooling and trapping of atoms

2 1 . Laser cooling and trapping of atoms2 1 . 1 . Slowing of an atomic beam21.2 . Optical molasses2*1.3. Sub-Doppler temperatures in optical molasses2*1.4. Magneto-optical trapping of atoms2*1.5. Other schemes for laser cooling and trapping of atoms

2*2. Magnetic trapping22.2 . Quadrupole traps22.2 . Losses due to Majorana spin-flips2*2.3. The time-averaged orbiting potential (TOP) trap2*2.4. The optical-plug trap2*2.5. The loffe-Pritchard-type traps2*2.6. Permanent-magnets traps2'2.7. Comparison of magnetic traps

2*3. Evaporative cooling of trapped atoms3. Studies of Bose-Einstein condensates

3 1 . Experimental procedure to achieve BEC3*2. Observation of a Bose condensate3*3. Measurement of energy and ground-state occupation as a function of temperature3*4. Study of collective excitations and propagation of sound in a Bose-Einstein condensate3*5. Condensation of atoms with attractive interactions3*6. Production of two condensates by sympathetic cooling3*7. The coherence properties of the condensate and the "atom laser"

4. Conclusions and future prospects

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