UBC Physics & Astronomy | UBC Physics & …qdg/resources/AMOReadingList/...of G8 and D8, using the so-called dressed-atom approach to atom-photon interactions (see for example Cohen-Tannoudji,

Manipulating atoms with photons*

Claude N. Cohen-Tannoudji

College de France et Laboratoire Kastler Brossel† de l’Ecole Normale Superieure,75231 Paris Cedex 05, France

[S0034-6861(98)01103-9]

Electromagnetic interactions play a central role in lowenergy physics. They are responsible for the cohesion ofatoms and molecules and they are at the origin of theemission and absorption of light by such systems. Thislight is not only a source of information on the structureof atoms. It can also be used to act on atoms, to manipu-late them, to control their various degrees of freedom.With the development of laser sources, this researchfield has considerably expanded during the last fewyears. Methods have been developed to trap atoms andto cool them to very low temperatures. This has openedthe way to a wealth of new investigations and applica-tions.

Two types of degrees of freedom can be consideredfor an atom: the internal degrees of freedom, such as theelectronic configuration or the spin polarization, in thecenter of mass system; the external degrees of freedom,which are essentially the position and the momentum ofthe center of mass. The manipulation of internal degreesof freedom goes back to optical pumping (Kastler,1950), which uses resonant exchanges of angular mo-mentum between atoms and circularly polarized light forpolarizing the spins of these atoms. These experimentspredate the use of lasers in atomic physics. The manipu-lation of external degrees of freedom uses the concept ofradiative forces resulting from the exchanges of linearmomentum between atoms and light. Radiative forcesexerted by the light coming from the sun were alreadyinvoked by J. Kepler to explain the tails of the comets.Although they are very small when one uses ordinarylight sources, these forces were also investigated experi-mentally in the beginning of this century by P. Lebedev,E. F. Nichols and G. F. Hull, R. Frisch. For a historicalsurvey of this research field, we refer the reader to re-view papers (Ashkin, 1980; Letokhov and Minogin,1981; Stenholm, 1986; Adams and Riis, 1997) which alsoinclude a discussion of early theoretical work dealingwith these problems by the groups of A. P. Kazantsev,V. S. Letokhov, in Russia, A. Ashkin at Bell Labs, andS. Stenholm in Helsinki.

It turns out that there is a strong interplay betweenthe dynamics of internal and external degrees of free-dom. This is at the origin of efficient laser coolingmechanisms, such as ‘‘Sisyphus cooling’’ or ‘‘Velocity

*The 1997 Nobel Prize in Physics was shared by Steven Chu,Claude N. Cohen-Tannoudji, and William D. Phillips. This lec-ture is the text of Professor Cohen-Tannoudji’s address on theoccasion of the award.

†Laboratoire Kastler Brossel is a laboratory affiliated withthe CNRS and with the Universite Pierre et Marie Curie.

Reviews of Modern Physics, Vol. 70, No. 3, July 1998 0034-6861/9

Selective Coherent Population Trapping,’’ which werediscovered at the end of the 80s (for a historical surveyof these developments, see for example Cohen-Tannoudji and Phillips, 1990). These mechanisms haveallowed laser cooling to overcome important fundamen-tal limits, such as the Doppler limit and the single pho-ton recoil limit, and to reach the microkelvin, and eventhe nanokelvin range. We devote a large part of thispaper (Sections II and III) to the discussion of thesemechanisms and to the description of a few applicationsinvestigated by our group in Paris (Section IV). There isa certain continuity between these recent developmentsin the field of laser cooling and trapping and early the-oretical and experimental work performed in the 60sand the 70s, dealing with internal degrees of freedom.We illustrate this continuity by presenting in Section I abrief survey of various physical processes, and by inter-preting them in terms of two parameters, the radiativebroadening and the light shift of the atomic groundstate.

I. BRIEF REVIEW OF PHYSICAL PROCESSES

To classify the basic physical processes which are usedfor manipulating atoms by light, it is useful to distinguishtwo large categories of effects: dissipative (or absorp-tive) effects on the one hand, reactive (or dispersive)effects on the other hand. This partition is relevant forboth internal and external degrees of freedom.

A. Existence of two types of effects in atom-photoninteractions

Consider first a light beam with frequency vL propa-gating through a medium consisting of atoms with reso-nance frequency vA . The index of refraction describingthis propagation has an imaginary part and a real partwhich are associated with two types of physical pro-cesses. The incident photons can be absorbed, more pre-cisely scattered in all directions. The corresponding at-tenuation of the light beam is maximum at resonance. Itis described by the imaginary part of the index of refrac-tion which varies with vL2vA as a Lorentz absorptioncurve. We will call such an effect a dissipative (or ab-sorptive) effect. The speed of propagation of light is alsomodified. The corresponding dispersion is described bythe real part n of the index of refraction whose differ-ence from 1, n21, varies with vL2vA as a Lorentz dis-persion curve. We will call such an effect a reactive (ordispersive) effect.

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708 Claude N. Cohen-Tannoudji: Manipulating atoms with photons

Dissipative effects and reactive effects also appear forthe atoms, as a result of their interaction with photons.They correspond to a broadening and to a shift of theatomic energy levels, respectively. Such effects alreadyappear when the atom interacts with the quantized ra-diation field in the vacuum state. It is well known thatatomic excited states get a natural width G, which is alsothe rate at which a photon is spontaneously emittedfrom such states. Atomic energy levels are also shiftedas a result of virtual emissions and reabsorptions of pho-tons by the atom. Such a radiative correction is simplythe Lamb shift (Heitler, 1954).

Similar effects are associated with the interaction withan incident light beam. Atomic ground states get a ra-diative broadening G8, which is also the rate at whichphotons are absorbed by the atom, or more preciselyscattered from the incident beam. Atomic energy levelsare also shifted as a result of virtual absorptions andreemissions of the incident photons by the atom. Suchenergy displacements \D8 are called light shifts, or acStark shifts (Barrat and Cohen-Tannoudji, 1961; Cohen-Tannoudji, 1962).

In view of their importance for the following discus-sions, we give now a brief derivation of the expressionsof G8 and D8, using the so-called dressed-atom approachto atom-photon interactions (see for example Cohen-Tannoudji, Dupont-Roc, and Grynberg, 1992, chapterVI). In the absence of coupling, the two dressed statesug ,N& (atom in the ground state g in the presence of Nphotons) and ue ,N21& (atom in the excited state e inthe presence of N21 photons) are separated by a split-ting \d, where d5vL2vA is the detuning between thelight frequency vL and the atomic frequency vA . Theatom-light interaction Hamiltonian VAL couples thesetwo states because the atom in g can absorb one photonand jump to e . The corresponding matrix element ofVAL can be written as \V/2, where the so-called Rabifrequency V is proportional to the transition dipole mo-ment and to AN . Under the effect of such a coupling,the two states repel each other, and the state ug ,N& isshifted by an amount \D8, which is the light shift of g .The contamination of ug ,N& by the unstable state ue ,N21& (having a width G) also confers to the ground statea width G8. In the limit where V!G or udu, a simpleperturbative calculation gives:

G85V2G

G214d2 , (1)

D85V2d

G214d2 . (2)

Both G8 and D8 are proportional to V2}N , i.e., to thelight intensity. They vary with the detuning d5vL2vAas Lorentz absorption and dispersion curves, respec-tively, which justifies the denominations absorptive anddispersive used for these two types of effects. For largedetunings (udu@G), G8 varies as 1/d2 and becomes neg-ligible compared to D8 which varies as 1/d . On the otherhand, for small detunings, (udu!G), G8 is much largerthan D8. In the high intensity limit, when V is large com-

Rev. Mod. Phys., Vol. 70, No. 3, July 1998

pared to G and udu, the two dressed states resulting fromthe coupling are the symmetric and antisymmetric linearcombinations of ug ,N& and ue ,N21&. Their splitting is\V and they share the instability G of e in equal parts, sothat G85G/2. One can explain in this way various physi-cal effects such as the Rabi flopping or the Autler-Townes splittings of the spectral lines connecting e or gto a third level (Autler and Townes, 1955).

B. Manipulation of internal degrees of freedom

1. Optical pumping

Optical pumping is one of the first examples of ma-nipulation of atoms by light (Kastler, 1950). It uses reso-nant excitation of atoms by circularly polarized light fortransferring to the atoms part of the angular momentumcarried by the light beam. It is based on the fact thatdifferent Zeeman sublevels in the atomic ground statehave in general different absorption rates for incomingpolarized light. For example, for a Jg51/2↔Je51/2transition, only atoms in the sublevel Mg521/2 can ab-sorb s1-polarized light. They are excited in the sublevelMe511/2 of e from which they can fall back in thesublevel Mg511/2 by spontaneous emission of a p-polarized photon. They then remain trapped in this statebecause no further s1-transition can take place. It ispossible in this way to obtain high degrees of spin orien-tation in atomic ground states.

2. Light shifts

Optical pumping is a dissipative effect because it isassociated with resonant absorption of photons by theatom. Nonresonant optical excitation produces lightshifts of the ground state Zeeman sublevels. Because ofthe polarization selection rules, light shifts depend onthe polarization of the exciting light and vary in generalfrom one Zeeman sublevel to another. Consider for ex-ample a Jg51/2↔Je51/2 transition [Fig. 1(a)]. As1-polarized excitation shifts only the Zeeman sublevelMg521/2, whereas a s2-polarized excitation shifts onlythe sublevel Mg511/2 [Fig. 1(b)]. Magnetic resonancecurves in the ground state g , which are very narrow be-cause the relaxation time in g can be very long, are thuslight shifted by a polarized nonresonant excitation, andthe sign of this shift changes when one changes the po-larization of the light beam from s1 to s2. It is in thisway that light shifts were first observed (Cohen-Tannoudji, 1961). Figure 1(c) gives an example of ex-perimental results obtained by exciting the transition61S0 , F51/2↔63P1 , F51/2 of 199Hg atoms by the non-resonant light coming from a lamp filled with anotherisotope (201Hg).

Light shifts can be considered from different points ofview. First, they can be interpreted as a radiative correc-tion, due to the interaction of the atom with an incidentfield rather than with the vacuum field. This is why Al-fred Kastler called them ‘‘Lamp shifts.’’ Secondly, theyintroduce perturbations to high precision measurementsusing optical methods, which must be taken into account

709Claude N. Cohen-Tannoudji: Manipulating atoms with photons

before extracting spectroscopic data from these mea-surements. Finally, because of their variation from onesublevel to another, the effect of light shifts can be de-scribed in terms of fictitious magnetic or electric fields(Cohen-Tannoudji and Dupont-Roc, 1972). This ex-plains why the light shifts produced by a nonresonantlaser standing wave are being more and more frequentlyused to produce spatial modulations of the Zeemansplittings in the ground state on an optical wavelengthscale. This would not be easily achieved with spatiallyvarying real magnetic fields. We will see in Section IIinteresting applications of such a situation.

C. Manipulation of external degrees of freedom

1. The two types of radiative forces

There are two types of radiative forces, associated re-spectively with dissipative and reactive effects.

Dissipative forces, also called radiation pressureforces or scattering forces, are associated with the trans-fer of linear momentum from the incident light beam tothe atom in resonant scattering processes. They are pro-portional to the scattering rate G8. Consider for examplean atom in a laser plane wave with wave vector k. Be-cause photons are scattered with equal probabilities in

FIG. 1. Experimental observation of light shifts (from Cohen-Tannoudji, 1961). For a Jg51/2↔Je51/2 transition (a), as1-polarized nonresonant excitation shifts only the sublevelg21/2 [right part of (b)] whereas a s2-polarized excitationshifts only the sublevel g11/2 [left part of (b)]. The detuning d ispositive and very large compared to the Zeeman splittings in eand g . The Zeeman splitting in the ground state is thus in-creased in the first case, decreased in the second one. (c) mag-netic resonance signal vs magnetic field, in units of the corre-sponding Larmor frequency. The central curve (circles) is theresonance curve in the absence of light shift. When the non-resonant light beam is introduced, either s1-polarized(crosses) or s2-polarized (triangles), the magnetic resonancecurve is light shifted in opposite directions.


two opposite directions, the mean momentum trans-ferred to the atom in an absorption-spontaneous emis-sion cycle is equal to the momentum \k of the absorbedphoton. The mean rate of momentum transfer, i.e., themean force, is thus equal to \kG8. Since G8 saturates toG/2 at high intensity (see Section I.A), the radiationpressure force saturates to \kG/2. The corresponding ac-celeration (or deceleration), which can be communi-cated to an atom with mass M , is equal to amax5\kG/2M5vR/2t , where vR5\k/M is the recoil veloc-ity of the atom absorbing or emitting a single photon,and t51/G is the radiative lifetime of the excited state.For sodium atoms, vR5331022 m/s and t51.631028 s, so that amax can reach values as large as106 m/s2, i.e., 105g where g is the acceleration due togravity. With such a force, one can stop a thermalatomic beam in a distance of the order of one meter,provided that one compensates for the Doppler shift ofthe decelerating atom, by using for example a spatiallyvarying Zeeman shift (Phillips and Metcalf, 1982;Prodan, Phillips, and Metcalf, 1982) or a chirped laserfrequency (Ertmer et al., 1985).

Dispersive forces, also called dipole forces or gradientforces (Askarian, 1962; Ashkin, 1980; Letokhov and Mi-nogin, 1981), can be interpreted in terms of position de-pendent light shifts \D8(r) due to a spatially varyinglight intensity (Cohen-Tannoudji, 1992). Consider forexample a laser beam well detuned from resonance, sothat one can neglect G8 (no scattering process). Theatom thus remains in the ground state and the light shift\D8(r) of this state plays the role of a potential energy,giving rise to a force which is equal and opposite to itsgradient: F52¹@\D8(r)# . Such a force can also be in-terpreted as resulting from a redistribution of photonsbetween the various plane waves forming the laser wavein absorption-stimulated emission cycles. If the detuningis not large enough to allow G8 to be neglected, sponta-neous transitions occur between dressed states havingopposite gradients, so that the instantaneous force oscil-lates back and forth between two opposite values in arandom way. Such a dressed-atom picture provides asimple interpretation of the mean value and of the fluc-tuations of dipole forces (Dalibard and Cohen-Tannoudji, 1985).

2. Applications of dissipative forces—Doppler cooling andmagneto-optical traps

We have already mentioned in the previous subsec-tion the possibility of decelerating an atomic beam bythe radiation pressure force of a laser plane wave. Inter-esting effects can also be obtained by combining the ef-fects of two counterpropagating laser waves.

A first example is Doppler cooling, first suggested forneutral atoms by T. W. Hansch and A. L. Schawlow(1975) and, independently for trapped ions, by D. Wine-land and H. Dehmelt (1975). This cooling process resultsfrom a Doppler-induced imbalance between two oppo-site radiation pressure forces. The two counterpropagat-ing laser waves have the same (weak) intensity and the


same frequency and they are slightly detuned to the redof the atomic frequency (vL,vA). For an atom at rest,the two radiation pressure forces exactly balance eachother and the net force is equal to zero. For a movingatom, the apparent frequencies of the two laser wavesare Doppler shifted. The counterpropagating wave getscloser to resonance and exerts a stronger radiation pres-sure force than the copropagating wave which gets far-ther from resonance. The net force is thus opposite tothe atomic velocity v and can be written for small v asF52av where a is a friction coefficient. By using threepairs of counterpropagating laser waves along three or-thogonal directions, one can damp the atomic velocity ina very short time, on the order of a few microseconds,achieving what is called an ‘‘optical molasses’’ (Chuet al., 1985).

The Doppler friction responsible for the cooling isnecessarily accompanied by fluctuations due to the fluo-rescence photons which are spontaneously emitted inrandom directions and at random times. These photonscommunicate to the atom a random recoil momentum\k , responsible for a momentum diffusion described bya diffusion coefficient D (Gordon and Ashkin, 1980;Letokhov and Minogin, 1981; Cohen-Tannoudji, 1992).As in usual Brownian motion, competition between fric-tion and diffusion usually leads to a steady state, with anequilibrium temperature proportional to D/a . Thetheory of Doppler cooling (Letokhov, Minogin, andPavlik, 1977; Wineland and Itano, 1979; Gordon andAshkin, 1980) predicts that the equilibrium temperatureobtained with such a scheme is always larger than a cer-tain limit TD , called the Doppler limit, and given bykBTD5\G/2 where G is the natural width of the excitedstate and kB the Boltzmann constant. This limit, which isreached for d5vL2vA52G/2, is, for alkali atoms, onthe order of 100 mK. In fact, when the measurementsbecame precise enough, it appeared that the tempera-ture in optical molasses was much lower than expected(Lett et al., 1988). This indicates that other laser coolingmechanisms, more powerful than Doppler cooling, areoperating. We will come back to this point in Section II.

The imbalance between two opposite radiation pres-sure forces can be also made position dependent thougha spatially dependent Zeeman shift produced by a mag-netic field gradient. In a one-dimensional configuration,first suggested by J. Dalibard in 1986, the two counter-propagating waves, which are detuned to the red (vL,vA) and which have opposite circular polarizations,are in resonance with the atom at different places. Thisresults in a restoring force towards the point where themagnetic field vanishes. Furthermore the nonzero valueof the detuning provides a Doppler cooling. In fact, sucha scheme can be extended to three dimensions and leadsto a robust, large and deep trap called a ‘‘magneto-optical trap’’ or ‘‘MOT’’ (Raab et al., 1987). It combinestrapping and cooling, it has a large velocity capturerange and it can be used for trapping atoms in a smallcell filled with a low pressure vapour (Monroe et al.,1990).


3. Applications of dispersive forces: Laser traps and atomicmirrors

When the detuning is negative (vL2vA,0), lightshifts are negative. If the laser beam is focused, the focalzone where the intensity is maximum appears as a mini-mum of potential energy, forming a potential well wheresufficiently cold atoms can be trapped. This is a lasertrap. Laser traps using a single focused laser beam (Chuet al., 1986) or two crossed focused laser beams (Adamset al., 1995; Kuhn et al., 1997) have been realized. Earlyproposals (Letokhov, 1968) were considering trappingatoms at the antinodes or nodes of a nonresonant laserstanding wave. Channeling of atoms in a laser standingwave has been observed experimentally (Salomon et al.,1987).

If the detuning is positive, light shifts are positive andcan thus be used to produce potential barriers. For ex-ample, an evanescent blue detuned wave at the surfaceof a piece of glass can prevent slow atoms impinging onthe glass surface from touching the wall, making thembounce off a ‘‘carpet of light’’ (Cook and Hill, 1982).This is the principle of mirrors for atoms. Plane atomicmirrors (Balykin et al., 1988; Kasevich, Weiss, and Chu,1990) have been realized as well as concave mirrors(Aminoff et al., 1993).

II. SUB-DOPPLER COOLING

In the previous section, we discussed separately themanipulation of internal and external degrees of free-dom, and we have described physical mechanisms in-volving only one type of physical effect, either dispersiveor dissipative. In fact, there exist cooling mechanismsresulting from an interplay between spin and externaldegrees of freedom, and between dispersive and dissipa-tive effects. We discuss in this section one of them, theso-called ‘‘Sisyphus cooling’’ or ‘‘polarization-gradientcooling’’ mechanism (Dalibard and Cohen-Tannoudji,1989; Ungar et al., 1989) (see also Dalibard and Cohen-Tannoudji, 1985), which leads to temperatures muchlower than Doppler cooling. One can understand in thisway the sub-Doppler temperatures observed in opticalmolasses and mentioned above in Section I.C.2.

A. Sisyphus effect

Most atoms, in particular alkali atoms, have a Zeemanstructure in the ground state. Since the detuning used inlaser cooling experiments is not too large compared toG, both differential light shifts and optical pumping tran-sitions exist for the various ground state Zeeman sublev-els. Furthermore, the laser polarization varies in generalin space so that light shifts and optical pumping rates areposition dependent. We show now, with a simple one-dimensional example, how the combination of thesevarious effects can lead to a very efficient coolingmechanism.

Consider the laser configuration of Fig. 2(a), consist-ing of two counterpropagating plane waves along the zaxis, with orthogonal linear polarizations and with the


same frequency and the same intensity. Because thephase shift between the two waves increases linearlywith z , the polarization of the total field changes froms1 to s2 and vice versa every l/4. In between, it iselliptical or linear.

Consider now the simple case where the atomicground state has an angular momentum Jg51/2. Asshown in Section I.B, the two Zeeman sublevelsMg561/2 undergo different light shifts, depending onthe laser polarization, so that the Zeeman degeneracy inzero magnetic field is removed. This gives the energydiagram of Fig. 2(b) showing spatial modulations of theZeeman splitting between the two sublevels with a pe-riod l/2.

If the detuning d is not too large compared to G, thereare also real absorptions of photons by the atom fol-lowed by spontaneous emission, which give rise to opti-cal pumping transfers between the two sublevels, whosedirection depends on the polarization: Mg521/2→Mg511/2 for a s1 polarization, Mg511/2→Mg521/2for a s2 polarization. Here also, the spatial modulationof the laser polarization results in a spatial modulationof the optical pumping rates with a period l/2 [verticalarrows of Fig. 2(b)].

The two spatial modulations of light shifts and opticalpumping rates are of course correlated because they aredue to the same cause, the spatial modulation of thelight polarization. These correlations clearly appear inFig. 2(b). With the proper sign of the detuning, opticalpumping always transfers atoms from the higher Zee-man sublevel to the lower one. Suppose now that theatom is moving to the right, starting from the bottom ofa valley, for example in the state Mg511/2 at a place

FIG. 2. Sisyphus cooling. Laser configuration formed by twocounterpropagating plane waves along the z axis with orthogo-nal linear polarizations (a). The polarization of the resultingelectric field is spatially modulated with a period l/2. Everyl/4, it changes from s1 to s2 and vice versa. For an atom withtwo ground state Zeeman sublevels Mg561/2, the spatialmodulation of the laser polarization results in correlated spa-tial modulations of the light shifts of these two sublevels and ofthe optical pumping rates between them (b). Because of thesecorrelations, a moving atom runs up potential hills more fre-quently than down [double arrows of (b)].


where the polarization is s1. Because of the finite valueof the optical pumping time, there is a time lag betweeninternal and external variables and the atom can climbup the potential hill before absorbing a photon andreach the top of the hill where it has the maximum prob-ability to be optically pumped in the other sublevel, i.e.,in the bottom of a valley, and so on [double arrows ofFig. 2(b)]. Like Sisyphus in the Greek mythology, whowas always rolling a stone up the slope, the atom is run-ning up potential hills more frequently than down.When it climbs a potential hill, its kinetic energy is trans-formed into potential energy. Dissipation then occurs bylight, since the spontaneously emitted photon has an en-ergy higher than the absorbed laser photon [anti-StokesRaman processes of Fig. 2(b)]. After each Sisyphuscycle, the total energy E of the atom decreases by anamount of the order of U0 , where U0 is the depth of theoptical potential wells of Fig. 2(b), until E becomessmaller than U0 , in which case the atom remainstrapped in the potential wells.

The previous discussion shows that Sisyphus coolingleads to temperatures TSis such that kBTSis.U0 . Ac-cording to Eq. (2), the light shift U0 is proportional to\V2/d when 4udu.G . Such a dependence of TSis on theRabi frequency V and on the detuning d has beenchecked experimentally with cesium atoms (Salomonet al., 1990). Figure 3 presents the variations of the mea-sured temperature T with the dimensionless parameterV2/Gudu. Measurements of T versus intensity for differ-ent values of d show that T depends linearly, for lowenough intensities, on a single parameter which is thelight shift of the ground state Zeeman sublevels.

FIG. 3. Temperature measurements in cesium optical molasses(from Salomon et al., 1990). The left part of the figure showsthe fluorescence light emitted by the molasses observedthrough a window of the vacuum chamber. The horizontalbright line is the fluorescence light emitted by the atomic beamwhich feeds the molasses and which is slowed down by a fre-quency chirped laser beam. Right part of the figure: tempera-ture of the atoms measured by a time-of-flight technique vs thedimensionless parameter V2/uduG proportional to the lightshift (V is the optical Rabi frequency, d the detuning and G thenatural width of the excited state).


B. The limits of Sisyphus cooling

At low intensity, the light shift U0}\V2/d is muchsmaller than \G. This explains why Sisyphus coolingleads to temperatures much lower than those achievablewith Doppler cooling. One cannot, however, decreaseindefinitely the laser intensity. The previous discussionignores the recoil due to the spontaneously emitted pho-tons which increases the kinetic energy of the atom byan amount on the order of ER , where

ER5\2k2/2M (3)

is the recoil energy of an atom absorbing or emitting asingle photon. When U0 becomes on the order of orsmaller than ER , the cooling due to Sisyphus coolingbecomes weaker than the heating due to the recoil, andSisyphus cooling no longer works. This shows that thelowest temperatures which can be achieved with such ascheme are on the order of a few ER /kB . This result isconfirmed by a full quantum theory of Sisyphus cooling(Castin, 1991; Castin and Mo” lmer, 1995) and is in goodagreement with experimental results. The minimumtemperature in Fig. 3 is on the order of 10ER /kB .

C. Optical lattices

For the optimal conditions of Sisyphus cooling, atomsbecome so cold that they get trapped in the quantumvibrational levels of a potential well [see Fig. 3(b)].More precisely, one must consider energy bands in thisperodic structure (Castin and Dalibard, 1991). Experi-mental observation of such a quantization of atomic mo-tion in an optical potential was first achieved in one di-mension (see Fig. 4 and Verkerk et al., 1992; Jessenet al., 1992). Atoms are trapped in a spatial periodic ar-ray of potential wells, called a ‘‘1D-optical lattice,’’ withan antiferromagnetic order, since two adjacent potentialwells correspond to opposite spin polarizations. 2D and3D optical lattices have been realized subsequently (seethe review papers by Grynberg and Triche, 1996; Hem-merich, Weidemuller, and Hansch, 1996; and Jessen andDeutsch, 1996).

III. SUBRECOIL LASER COOLING

A. The single photon recoil limit. How to circumvent it.

In most laser cooling schemes, fluorescence cyclesnever cease. Since the random recoil \k communicatedto the atom by the spontaneously emitted photons can-not be controlled, it seems impossible to reduce theatomic momentum spread dp below a value correspond-ing to the photon momentum \k . Condition dp5\kdefines the ‘‘single photon recoil limit.’’ It is usual inlaser cooling to define an effective temperature T interms of the half-width dp (at 1/Ae) of the momentumdistribution by kBT/25dp2/2M . In the temperaturescale, condition dp5\k defines a ‘‘recoil temperature’’TR by


kBTR

25

\2k2

2M5ER . (4)

The value of TR ranges from a few hundred nanokelvinfor alkalis to a few microkelvin for helium.

It is in fact possible to circumvent this limit and toreach temperatures T lower than TR , a regime which iscalled ‘‘subrecoil’’ laser cooling. The basic idea is to cre-ate a situation where the photon absorption rate G8,which is also the jump rate R of the atomic random walkin velocity space, depends on the atomic velocity v5p/M and vanishes for v50 [Fig. 5(a)]. Consider thenan atom with v50. For such an atom, the absorption oflight is quenched. Consequently, there is no spontaneousreemission and no associated random recoil. One pro-

FIG. 4. Probe absorption spectrum of a 1D optical lattice(from Verkerk et al., 1992). The upper part of the figure showsthe two counterpropagating laser beams with frequency v andorthogonal linear polarizations forming the 1D optical lattice,and the probe beam with frequency vp whose absorption ismeasured by a detector. The lower part of the figure shows theprobe transmission vs vp2v . The two lateral resonances cor-responding to amplification or absorption of the probe are dueto stimulated Raman processes between vibrational levels ofthe atoms trapped in the light field (see the two insets). Thecentral narrow structure is a Rayleigh line due to the antifer-romagnetic spatial order of the atoms.

FIG. 5. Subrecoil laser cooling. (a) the random walk of theatom in velocity space is supposed to be characterized by ajump rate R which vanishes for v50. (b) as a result of thisinhomogeneous random walk, atoms which fall in a small in-terval around v50 remain trapped there for a long time, onthe order of @R(v)#21, and accumulate.


tects in this way ultracold atoms (with v.0) from the‘‘bad’’ effects of the light. On the other hand, atoms withvÞ0 can absorb and reemit light. In such absorption-spontaneous emission cycles, their velocities change in arandom way and the corresponding random walk in vspace can transfer atoms from the vÞ0 absorbing statesinto the v.0 dark states where they remain trapped andaccumulate [see Fig. 5(b)]. This reminds us of what hap-pens in a Kundt’s tube where sand grains vibrate in anacoustic standing wave and accumulate at the nodes ofthis wave where they no longer move. Note, however,that the random walk takes place in velocity space forthe situation considered in Fig. 5(b), whereas it takesplace in position space in a Kundt’s tube.

Up to now, two subrecoil cooling schemes have beenproposed and demonstrated. In the first one, called ‘‘Ve-locity Selective Coherent Population Trapping’’(VSCPT), the vanishing of R(v) for v50 is achieved byusing destructive quantum interference between differ-ent absorption amplitudes (Aspect et al., 1988). The sec-ond one, called Raman cooling, uses appropriate se-quences of stimulated Raman and optical pumpingpulses for tailoring the appropriate shape of R(v)(Kasevich and Chu, 1992).

B. Brief Survey of VSCPT

We first recall the principle of the quenching of ab-sorption by ‘‘coherent population trapping,’’ an effectwhich was discovered and studied in 1976 (Alzetta et al.,1976; Arimondo and Orriols, 1976). Consider the 3-levelsystem of Fig. 6, with two ground state sublevels g1 andg2 and one excited sublevel e0 , driven by two laser fieldswith frequencies vL1 and vL2 , exciting the transitionsg1↔e0 and g2↔e0 , respectively. Let \D be the detuningfrom resonance for the stimulated Raman process con-sisting of the absorption of one vL1 photon and thestimulated emission of one vL2 photon, the atom goingfrom g1 to g2 . One observes that the fluorescence rate Rvanishes for D50. Plotted versus D, the variations of Rare similar to those of Fig. 5(a) with v replaced by D.

FIG. 6. Coherent population trapping. A three-level atomg1 ,g2 ,e0 is driven by two laser fields with frequencies vL1 andvL2 exciting the transitions g1↔e0 and g2↔e0 , respectively.\D is the detuning from resonance for the stimulated Ramanprocess induced between g1 and g2 by the two laser fields vL1and vL2 . When D50, atoms are optically pumped in a linearsuperposition of g1 and g2 which no longer absorbs light be-cause of a destructive interference between the two absorptionamplitudes g1→e0 and g2→e0 .


The interpretation of this effect is that atoms are opti-cally pumped into a linear superposition of g1 and g2which is not coupled to e0 because of a destructive in-terference between the two absorption amplitudes g1→e0 and g2→e0 .

The basic idea of VSCPT is to use the Doppler effectfor making the detuning D of the stimulated Raman pro-cess of Fig. 6 proportional to the atomic velocity v . Thequenching of absorption by coherent population trap-ping is thus made velocity dependent and one achievesthe situation of Fig. 5(a). This is obtained by taking thetwo laser waves vL1 and vL2 counterpropagating alongthe z axis and by choosing their frequencies in such away that D50 for an atom at rest. Then, for an atommoving with a velocity v along the z axis, the oppositeDoppler shifts of the two laser waves result in a Ramandetuning D5(k11k2)v proportional to v .

A more quantitative analysis of the cooling process(Aspect et al., 1989) shows that the dark state, for whichR50, is a linear superposition of two states which differnot only by the internal state (g1 or g2) but also by themomentum along the z axis:

ucD&5c1ug1 ,2\k1&1c2ug2 ,1\k2&. (5)

This is due to the fact that g1 and g2 must be associatedwith different momenta, 2\k1 and 1\k2 , in order tobe coupled to the same excited state ue0 ,p50& by ab-sorption of photons with different momenta 1\k1 and2\k2 . Furthermore, when D50, the state (5) is a sta-tionary state of the total atom1laser photons system. Asa result of the cooling by VSCPT, the atomic momen-tum distribution thus exhibits two sharp peaks, centeredat 2\k1 and 1\k2 , with a width dp which tends to zerowhen the interaction time u tends to infinity.

The first VSCPT experiment (Aspect et al., 1988) wasperformed on the 23S1 metastable state of helium at-oms. The two lower states g1 and g2 were the M521and M511 Zeeman sublevels of the 23S1 metastablestate; e0 was the M50 Zeeman sublevel of the excited23P1 state. The two counterpropagating laser waves hadthe same frequency vL15vL25vL and opposite circu-lar polarizations. The two peaks of the momentum dis-tribution were centered at 6\k , with a width corre-sponding to T.TR/2. The interaction time was thenincreased by starting from a cloud of trapped precooledhelium atoms instead of using an atomic beam as in thefirst experiment (Bardou, Saubamea et al., 1994). Thisled to much lower temperatures (see Fig. 7). Very re-cently, temperatures as low as TR/800 have been ob-served (Saubamea et al., 1997).

In fact, it is not easy to measure such low tempera-tures by the usual time-of-flight techniques, because theresolution is then limited by the spatial extent of theinitial atomic cloud. A new method has been developed(Saubamea et al., 1997) which consists of measuring di-rectly the spatial correlation function of the atomsG(a)5*2`

1`dzf* (z1a)f(z), where f(z) is the wavefunction of the atomic wave packet. This correlationfunction, which describes the degree of spatial coher-ence between two points separated by a distance a , is


simply the Fourier transform of the momentum distribu-tion uf(p)u2. This method is analogous to Fourier spec-troscopy in optics, where a narrow spectral line I(v) ismore easily inferred from the correlation function of theemitted electric field G(t)5*2`

1`dtE* (t1t)E(t), whichis the Fourier transform of I(v). Experimentally, themeasurement of G(a) is achieved by letting the two co-herent VSCPT wave packets fly apart with a relativevelocity 2vR52\k/m during a dark period tD , duringwhich the VSCPT beams are switched off. During thisdark period, the two wave packets get separated by adistance a52vRtD , and one then measures with a probepulse a signal proportional to their overlap. Figure 8(a)shows the variations with tD of such a signal S (which isin fact equal to @11G(a)#/2). From such a curve, onededuces a temperature T.TR/625, corresponding todp.\k/25. Figure 8(b) shows the variations of TR /Twith the VSCPT interaction time u. As predicted bytheory (see next section), TR /T varies linearly with uand can reach values as large as 800.

VSCPT has been extended to two (Lawall et al., 1994)and three (Lawall et al., 1995) dimensions. For a Jg51↔Je51 transition, it has been shown (Ol’shanii andMinogin, 1992) that there is a dark state which is de-scribed by the same vector field as the laser field. Moreprecisely, if the laser field is formed by a linear superpo-sition of N plane waves with wave vectors ki (i51,2, . . . N) having the same modulus k , one finds thatatoms are cooled in a coherent superposition of N wavepackets with mean momenta \ki and with a momentumspread dp which becomes smaller and smaller as theinteraction time u increases. Furthermore, because ofthe isomorphism between the de Broglie dark state andthe laser field, one can adiabatically change the laser

FIG. 7. One-dimensional Velocity Selective Coherent Popula-tion Trapping (VSCPT) experiment. The left part of the figureshows the experimental scheme. The cloud of precooledtrapped atoms is released while the two counterpropagatingVSCPT beams with orthogonal circular polarizations are ap-plied during a time u51 ms. The atoms then fall freely andtheir positions are detected 6.5 cm below on a microchannelplate. The double band pattern is a signature of the 1D coolingprocess which accumulates the atoms in a state which is a lin-ear superposition of two different momenta. The right part ofthe figure gives the velocity distribution of the atoms detectedby the microchannel plate. The width dv of the two peaks isclearly smaller than their separation 2vR , where vR59.2 cm/sis the recoil velocity. This is a clear signature of subrecoil cool-ing.


configuration and transfer the whole atomic populationinto a single wave packet or two coherent wave packetschosen at will (Kulin et al., 1997). Figure 9 shows anexample of such a coherent manipulation of atomicwave packets in two dimensions. In Fig. 9(a), one seesthe transverse velocity distribution associated with thefour wave packets obtained with two pairs of counter-propagating laser waves along the x and y axis in a hori-zontal plane; Figure 9(b) shows the single wave packet

FIG. 8. Measurement of the spatial correlation function of at-oms cooled by VSCPT (from Saubamea et al., 1997). After acooling period of duration u, the two VSCPT beams areswitched off during a dark period of duration tD . The twocoherent wave packets into which atoms are pumped fly apartwith a relative velocity 2vR and get separated by a distance a52vRtD . Reapplying the two VSCPT beams during a shortprobe pulse, one measures a signal S which can be shown to beequal to @11G(a)#/2 where G(a) is the spatial overlap be-tween the two identical wave packets separated by a . FromG(a), which is the spatial correlation function of each wavepacket, one determines the atomic momentum distributionwhich is the Fourier transform of G(a). The left part of thefigure gives S vs a . The right part of the figure gives TR /T vsthe cooling time u, where TR is the recoil temperature and Tthe temperature of the cooled atoms determined from thewidth of G(a). The straight line is a linear fit in agreementwith the theoretical predictions of Levy statistics. The lowesttemperature, on the order of TR/800, is equal to 5 nK.

FIG. 9. Two-dimensional VSCPT experiment (from Kulinet al., 1997). The experimental scheme is the same as in Fig. 7,but one uses now four VSCPT beams in a horizontal plane andatoms are pumped into a linear superposition of four differentmomentum states, giving rise to four peaks in the two-dimensional velocity distribution (a). When three of the fourVSCPT laser beams are adiabatically switched off, the wholeatomic population is transferred into a single wave packet (b).


into which the whole atomic population is transferred byswitching off adiabatically three of the four VSCPTbeams. Similar results have been obtained in three di-mensions.

C. Subrecoil laser cooling and Levy statistics

Quantum Monte Carlo simulations using the delayfunction (Cohen-Tannoudji and Dalibard, 1986; Zoller,Marte, and Walls, 1987) have provided new physical in-sight into subrecoil laser cooling (Bardou, Bouchaudet al., 1994). Figure 10 shows, for example, the randomevolution of the momentum p of an atom in a 1D-VSCPT experiment. Each vertical discontinuity corre-sponds to a spontaneous emission process during whichp changes in a random way. Between two successivejumps, p remains constant. It clearly appears that therandom walk of the atom in velocity space is anomalousand dominated by a few rare events whose duration is asignificant fraction of the total interaction time. Asimple analysis shows that the distribution P(t) of thetrapping times t in a small trapping zone near v50 is abroad distribution which falls as a power law in thewings. These wings decrease so slowly that the averagevalue ^t& of t (or the variance) can diverge. In suchcases, the central limit theorem (CLT) can obviously nolonger be used for studying the distribution of the totaltrapping time after N entries in the trapping zone sepa-rated by N exits.

It is possible to extend the CLT to broad distributionswith power-law wings (Gnedenko and Kolmogorov,1954; Bouchaud and Georges, 1990). We have appliedthe corresponding statistics, called ‘‘Levy statistics,’’ tosubrecoil cooling and shown that one can obtain in thisway a better understanding of the physical processes aswell as quantitative analytical predictions for theasymptotic properties of the cooled atoms in the limitwhen the interaction time u tends to infinity (Bardou,

FIG. 10. Monte Carlo wave function simulation of one-dimensional VSCPT (from Bardou, Bouchaud et al., 1994).Momentum p characterizing the cooled atoms vs time. Eachvertical discontinuity corresponds to a spontaneous emissionjump during which p changes in a random way. Between twosuccessive jumps, p remains constant. The inset shows azoomed part of the sequence.


Bouchaud et al., 1994; Bardou, 1995). For example, onepredicts in this way that the temperature decreases as1/u when u→` , and that the wings of the momentumdistribution decrease as 1/p2, which shows that theshape of the momentum distribution is closer to aLorentzian than a Gaussian. This is in agreement withthe experimental observations represented in Fig. 8.[The fit in Fig. 8(a) is an exponential, which is the Fou-rier transform of a Lorentzian.]

One important feature revealed by this theoreticalanalysis is the non-ergodicity of the cooling process. Re-gardless of the interaction time u, there are alwaysatomic evolution times [trapping times in the small zoneof Fig. 5(a) around v50] which can be longer than u.Another advantage of such a new approach is that itallows the parameters of the cooling lasers to be opti-mized for given experimental conditions. For example,by using different shapes for the laser pulses used inone-dimensional subrecoil Raman cooling, it has beenpossible to reach for cesium atoms temperatures as lowas 3 nK (Reichel et al., 1995).

IV. A FEW EXAMPLES OF APPLICATIONS

The possibility of trapping atoms and cooling them atvery low temperatures, where their velocity can be aslow as a few mm/s, has opened the way to a wealth ofapplications. Ultracold atoms can be observed duringmuch longer times, which is important for high resolu-tion spectroscopy and frequency standards. They alsohave very long de Broglie wavelengths, which has givenrise to new research fields, such as atom optics, atominterferometry and Bose-Einstein condensation of dilutegases. It is impossible to discuss here all these develop-ments. We refer the reader to recent reviews such asAdams and Riis (1997). We will just describe in this sec-tion a few examples of applications which have beenrecently investigated by our group in Paris.

A. Cesium atomic clocks

Cesium atoms cooled by Sisyphus cooling have an ef-fective temperature on the order of 1 mK, correspondingto a rms velocity of 1 cm/s. This allows them to spend alonger time T in an observation zone where a micro-wave field induces resonant transitions between the twohyperfine levels g1 and g2 of the ground state. IncreasingT decreases the width Dn;1/T of the microwave reso-nance line whose frequency is used to define the unit oftime. The stability of atomic clocks can thus be consid-erably improved by using ultracold atoms (Gibble andChu, 1992; Lea et al., 1994).

In the usual atomic clocks, atoms from a thermal ce-sium beam cross two microwave cavities fed by the sameoscillator. The average velocity of the atoms is severalhundred m/s; the distance between the two cavities is onthe order of 1 m. The microwave resonance between g1and g2 is monitored and is used to lock the frequency ofthe oscillator to the center of the atomic line. The nar-rower the resonance line, the more stable the atomic


FIG. 11. (Color) The microgravity clock prototype. The left part is the 60 cm360 cm315 cm optical bench containing the diodelaser sources and the various optical components. The right part is the clock itself (about one meter long) containing the opticalmolasses, the microwave cavity and the detection region.

clock. In fact, the microwave resonance line exhibitsRamsey interference fringes whose width Dn is deter-mined by the time of flight T of the atoms from onecavity to another. For the longest devices, T , which canbe considered as the observation time, can reach 10 ms,leading to values of Dn;1/T on the order of 100 Hz.

Much narrower Ramsey fringes, with sub-Hertz line-widths, can be obtained in the so-called ‘‘Zachariasatomic fountain’’ (Zacharias, 1954; Ramsey, 1956). At-oms are captured in a magneto-optical trap and lasercooled before being launched upwards by a laser pulsethrough a microwave cavity. Because of gravity they aredecelerated; they return and fall back, passing a secondtime through the cavity. Atoms therefore experiencetwo coherent microwave pulses when they pass throughthe cavity, the first time on their way up, the second timeon their way down. The time interval between the twopulses can now be on the order of 1 sec, i.e., about twoorders of magnitude longer than with the usual clocks.Atomic fountains have been realized for sodium(Kasevich et al., 1989) and cesium (Clairon et al., 1991).A short-term relative frequency stability of 1.3310213t21/2, where t is the integration time, has beenrecently measured for a one meter high cesium fountain(Ghezali et al., 1996; Ghezali, 1997). For t5104 s,Dn/n;1.3310215 and for t533104 s, Dn/n;8310216

has been measured. In fact such a stability is most likely


limited by the hydrogen maser which is used as a refer-ence source, and the real stability, which could be moreprecisely determined by beating the signals of two foun-tain clocks, is expected to reach Dn/n;10216 for a oneday integration time. In addition to the stability, anothervery important property of a frequency standard is itsaccuracy. Because of the very low velocities in a foun-tain device, many systematic shifts are strongly reducedand can be evaluated with great precision. With an ac-curacy of 2310215, the BNM-LPTF fountain is pres-ently the most accurate primary standard (Simon et al.,1997). A factor of 10 improvement in this accuracy isexpected in the near future.

To increase the observation time beyond one second,a possible solution consists of building a clock for opera-tion in a reduced gravity environment. Such a micro-gravity clock has been recently tested in a jet plane mak-ing parabolic flights. A resonance signal with a width of7 Hz has been recorded in a 1022g environment. Thiswidth is twice narrower than that produced on earth inthe same apparatus. This clock prototype (see Fig. 11) isa compact and transportable device which can also beused on earth for high-precision frequency comparison.

Atomic clocks working with ultracold atoms could notonly provide an improvement of positioning systemssuch as the GPS. They could also be used for fundamen-tal studies. For example, one could build two fountains


clocks, one with cesium and one with rubidium, in orderto measure with a high accuracy the ratio between thehyperfine frequencies of these two atoms. Because ofrelativistic corrections, the hyperfine splitting is a func-tion of Za where a is the fine-structure constant and Zis the atomic number (Prestage, Tjoelker, and Maleki,1995). Since Z is not the same for cesium and rubidium,the ratio of the two hyperfine structures depends on a.By making several measurements of this ratio over longperiods of time, one could check Dirac’s suggestion con-cerning a possible variation of a with time. The presentupper limit for a/a in laboratory tests (Prestage,Tjoelker, and Maleki, 1995) could be improved by twoorders of magnitude.

Another interesting test would be to measure with ahigher accuracy the gravitational red shift and the gravi-tational delay of an electromagnetic wave passing near alarge mass (Shapiro effect, Shapiro, 1964).

B. Gravitational cavities for neutral atoms

We have already mentioned in Section I.C the possi-bility of making atomic mirrors for atoms by using bluedetuned evanescent waves at the surface of a piece ofglass. Concave mirrors [Fig. 12(a)] are particularly inter-esting because the transverse atomic motion is thenstable if atoms are released from a point located belowthe focus of the mirror. It has been possible in this wayto observe several successive bounces of the atoms [Fig.12(b)], and such a system can be considered as a ‘‘tram-poline for atoms’’ (Aminoff et al., 1993). In such an ex-periment, it is a good approximation to consider atomsas classical particles bouncing off a concave mirror. In aquantum mechanical description of the experiment, onemust consider the reflection of the atomic de Brogliewaves by the mirror. Standing de Broglie waves can thenbe introduced for such a ‘‘gravitational cavity,’’ whichare quite analogous to the light standing waves for aFabry-Perot cavity (Wallis, Dalibard, and Cohen-Tannoudji, 1992). By modulating at frequency V/2p theintensity of the evanescent wave which forms the atomicmirror, one can produce the equivalent of a vibratingmirror for de Broglie waves. The reflected waves thushave a modulated Doppler shift. The corresponding fre-quency modulation of these waves has been recentlydemonstrated (Steane et al., 1995) by measuring the en-ergy change DE of the bouncing atom, which is found tobe equal to n\V , where n50,61,62, . . . [Figs. 12(c)and (d)]. The discrete nature of this energy spectrum is apure quantum effect. For classical particles bouncing offa vibrating mirror, DE would vary continuously in a cer-tain range.

C. Bloch oscillations

In the subrecoil regime where dp becomes smallerthan \k , the atomic coherence length h/dp becomeslarger than the optical wavelength l5h/\k52p/k ofthe lasers used to cool the atom. Consider then such anultracold atom in the periodic light shift potential pro-


duced by a nonresonant laser standing wave. The atomicde Broglie wave is delocalized over several periods ofthe periodic potential, which means that one can pre-pare in this way quasi-Bloch states. By chirping the fre-quency of the two counterpropagating laser waves form-ing the standing wave, one can produce an acceleratedstanding wave. In the rest frame of this wave, atoms thusfeel a constant inertial force in addition to the periodicpotential. They are accelerated and the de Broglie wave-length ldB5h/M^v& decreases. When ldB5lLaser , thede Broglie wave is Bragg reflected by the periodic opti-cal potential. Instead of increasing linearly with time,the mean velocity ^v& of the atoms oscillates back andforth. Such Bloch oscillations, which are a textbook ef-fect of solid-state physics, are more easily observed withultracold atoms than with electrons in condensed matterbecause the Bloch period can be much shorter than the

FIG. 12. Gravitational cavity for neutral atoms (from Aminoffet al., 1993, and Steane et al., 1995). (a) trampoline for atoms.Atoms released from a magneto-optical trap bounce off a con-cave mirror formed by a blue detuned evanescent wave at thesurface of a curved glass prism. (b) number of atoms at theinitial position of the trap vs time after the trap has beenswitched off. Ten successive bounces are visible in the figure.(c) principle of the experiment demonstrating the frequencymodulation of de Broglie waves. The upper trace gives theatomic trajectories (vertical position z vs time). The lowertrace gives the time dependence of the intensity I of the eva-nescent wave. The first pulse is used for making a velocityselection. The second pulse is modulated in intensity. This pro-duces a vibrating mirror, giving rise to a frequency-modulatedreflected de Broglie wave which consists in a carrier and side-bands at the modulation frequency. The energy spectrum ofthe reflected particles is thus discrete so that the trajectories ofthe reflected particles form a discrete set. (d) this effect is de-tected by looking at the time dependence of the absorption ofa probe beam located above the prism.


relaxation time for the coherence of de Broglie waves(in condensed matter, the relaxation processes due tocollisions are very strong). Figure 13 shows an exampleof Bloch oscillations (Ben Dahan et al., 1996) observedon cesium atoms cooled by the improved subrecoil Ra-man cooling technique described in Reichel et al. (1995).

V. CONCLUSION

We have described in this paper a few physical mecha-nisms allowing one to manipulate neutral atoms withlaser light. Several of these mechanisms can be simplyinterpreted in terms of resonant exchanges of energy,angular and linear momentum between atoms and pho-tons. A few of them, among the most efficient ones, re-sult from a new way of combining well-known physicaleffects such as optical pumping, light shifts, and coher-ent population trapping. We have given two examples ofsuch cooling mechanisms, Sisyphus cooling and subre-coil cooling, which allow atoms to be cooled in the mi-crokelvin and nanokelvin ranges. A few possible appli-cations of ultracold atoms have been also reviewed.They take advantage of the long interaction times andlong de Broglie wavelengths which are now availablewith laser cooling and trapping techniques.

One can reasonably expect that further progress inthis field will be made in the near future and that newapplications will be found. Concerning fundamentalproblems, two directions of research at least look prom-ising. First, a better control of ‘‘pure’’ situations involv-ing a small number of atoms in well-defined states ex-hibiting quantum features such as very long spatialcoherence lengths or entanglement. In that perspective,atomic, molecular and optical physics will continue toplay an important role by providing a ‘‘testing bench’’for improving our understanding of quantum phenom-ena. A second interesting direction is the investigationof new systems, such as Bose condensates involving amacroscopic number of atoms in the same quantumstate. One can reasonably hope that new types of coher-ent atomic sources (sometimes called ‘‘atom lasers’’) willbe realized, opening the way to interesting new possibili-ties.

It is clear finally that all the developments which haveoccurred in the field of laser cooling and trapping arestrengthening the connections which can be established

FIG. 13. Bloch oscillations of atoms in a periodic optical po-tential (from Ben Dahan et al., 1996). Mean velocity (in unitsof the recoil velocity) vs time (in units of half the Bloch pe-riod) for ultracold cesium atoms moving in a periodic opticalpotential and submitted in addition to a constant force.


between atomic physics and other branches of physics,such as condensed matter or statistical physics. The useof Levy statistics for analyzing subrecoil cooling is anexample of such a fruitful dialogue. The interdisciplinarycharacter of the present researches on the properties ofBose condensates is also a clear sign of the increase ofthese exchanges.

REFERENCES

Adams, C. S., H. J. Lee, N. Davidson, M. Kasevich, and S.Chu, 1995, Phys. Rev. Lett. 74, 3577.

Adams, C. S., and E. Riis, 1997, Prog. Quantum Electron. 21,1.

Alzetta, G. A., A. Gozzini, L. Moi, and G. Orriols, 1976,Nuovo Cimento B 36B, 5.

Aminoff, C. G., A. M. Steane, P. Bouyer, P. Desbiolles, J.Dalibard, and C. Cohen-Tannoudji, 1993, Phys. Rev. Lett. 71,3083.

Arimondo, E., and G. Orriols, 1976, Lett. Nuovo Cimento 17,333.

Ashkin, A., 1980, Science 210, 1081.Askarian, G. A., 1962, Zh. Eksp. Teor. Fiz. 42, 1567 [Sov.

Phys. JETP 15, 1088 (1962)].Aspect, A., E. Arimondo, R. Kaiser, N. Vansteenkiste, and C.

Cohen-Tannoudji, 1988, Phys. Rev. Lett. 61, 826.Aspect, A., E. Arimondo, R. Kaiser, N. Vansteenkiste, and C.

Cohen-Tannoudji, 1989, J. Opt. Soc. Am. B 6, 2112.Autler, S. H., and C. H. Townes, 1955, Phys. Rev. 100, 703.Balykin, V. I., V. S. Letokhov, Yu. B. Ovchinnikov, and A. I.

Sidorov, 1988, Phys. Rev. Lett. 60, 2137.Bardou, F., 1995, Ph.D. thesis (University of Paris XI, Orsay).Bardou, F., J. P. Bouchaud, O. Emile, A. Aspect, and C.

Cohen-Tannoudji, 1994, Phys. Rev. Lett. 72, 203.Bardou, F., B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C.

Westbrook, A. Aspect, and C. Cohen-Tannoudji, 1994, C. R.Acad. Sci. 318, 877.

Barrat, J. P., and C. Cohen-Tannoudji, 1961, J. Phys. Radium22, 329 and 423.

Ben Dahan, M., E. Peik, J. Reichel, Y. Castin, and C. Salomon,1996, Phys. Rev. Lett. 76, 4508.

Bouchaud, J. P., and A. Georges, 1990, Phys. Rep. 195, 127.Castin, Y., 1991, Ph.D. thesis (University of Paris, Paris).Castin, Y., and J. Dalibard, 1991, Europhys. Lett. 14, 761.Castin, Y., and K. Mo” lmer, 1995, Phys. Rev. Lett. 74, 3772.Chu, S., J. E. Bjorkholm, A. Ashkin, and A. Cable, 1986, Phys.

Rev. Lett. 57, 314.Chu, S., L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ash-

kin, 1985, Phys. Rev. Lett. 55, 48.Clairon, A., C. Salomon, S. Guellati, and W. D. Phillips, 1991,

Europhys. Lett. 16, 165.Cohen-Tannoudji, C., 1961, C. R. Acad. Sci. 252, 394.Cohen-Tannoudji, C., 1962, Ann. Phys. (Paris) 7, 423 and 469.Cohen-Tannoudji, C., 1992, ‘‘Atomic motion in laser light,’’ in

Fundamental Systems in Quantum Optics, Les Houches ses-sion LIII, edited by J. Dalibard, J. M. Raimond, and J. Zinn-Justin (Elsevier Science, Amsterdam), p. 1.

Cohen-Tannoudji, C., and J. Dalibard, 1986, Europhys. Lett. 1,441.


Cohen-Tannoudji, C., and J. Dupont-Roc, 1972, Phys. Rev. A5, 968.

Cohen-Tannoudji, C., J. Dupont-Roc, and G. Grynberg, 1992,Atom-Photon Interactions: Basic Processes and Applications(Wiley, New York).

Cohen-Tannoudji, C., and W. Phillips, 1990, Phys. Today 43,No. 10, 33.

Cook, R. J., and R. K. Hill, 1982, Opt. Commun. 43, 258.Dalibard, J., and C. Cohen-Tannoudji, 1985, J. Opt. Soc. Am.

B 2, 1707.Dalibard, J., and C. Cohen-Tannoudji, 1989, J. Opt. Soc. Am.

B 6, 2023.Ertmer, W., R. Blatt, J. L. Hall, and M. Zhu, 1985, Phys. Rev.

Lett. 54, 996.Ghezali, S., 1997, Ph.D. thesis (University of Paris, Paris).Ghezali, S., Ph. Laurent, S. N. Lea, and A. Clairon, 1996, Eu-

rophys. Lett. 36, 25.Gibble, K., and S. Chu, 1992, Metrologia 29, 201.Gnedenko, B. V., and A. N. Kolmogorov, 1954, Limit Distri-

butions for Sum of Independent Random Variables (AddisonWesley, Reading, MA).

Gordon, J. P., and A. Ashkin, 1980, Phys. Rev. A 21, 1606.Grynberg, G., and C. Triche, 1996, in Coherent and Collective

Interactions of Particles and Radiation Beams, Proceedings ofthe International School of Physics ‘‘Enrico Fermi,’’ editedby A. Aspect, W. Barletta, and R. Bonifacio (IOS, Amster-dam), Vol. 131, p. 243.

Hansch, T. W., and A. L. Schawlow, 1975, Opt. Commun. 13,68.

Heitler, W., 1954, The Quantum Theory of Radiation (Claren-don, Oxford).

Hemmerich, A., M. Weidemuller, and T. W. Hansch, 1996, inCoherent and Collective Interactions of Particles and Radia-tion Beams, Proceedings of the International School of Phys-ics ‘‘Enrico Fermi,’’ edited by A. Aspect, W. Barletta, and R.Bonifacio (IOS, Amsterdam), Vol. 131, p. 503.

Jessen, P. S., and I. H. Deutsch, 1996, in Advances in Atomic,Molecular, and Optical Physics, edited by B. Bederson and H.Walther (Academic Press), Vol. 37, p. 95.

Jessen, P. S., C. Gerz, P. D. Lett, W. Phillips, S. L. Rolston, R.J. C. Spreeuw, and C. I. Westbrook, 1992, Phys. Rev. Lett. 69,49.

Kasevich, M., and S. Chu, 1992, Phys. Rev. Lett. 69, 1741.Kasevich, M., E. Riis, S. Chu, and R. de Voe, 1989, Phys. Rev.

Lett. 63, 612.Kasevich, M. A., D. S. Weiss, and S. Chu, 1990, Opt. Lett. 15,

607.Kastler, A., 1950, J. Phys. Radium 11, 255.Kuhn, A., H. Perrin, W. Hansel, and C. Salomon, 1997, in OSA

TOPS on Ultracold Atoms and BEC, edited by K. Burnett(Optical Society of America, Washington, DC), Vol. 7, p. 58.

Kulin, S., B. Saubamea, E. Peik, J. Lawall, T. W. Hijmans, M.Leduc, and C. Cohen-Tannoudji, 1997, Phys. Rev. Lett. 78,4185.


Lawall, J., F. Bardou, B. Saubamea, K. Shimizu, M. Leduc, A.Aspect, and C. Cohen-Tannoudji, 1994, Phys. Rev. Lett. 73,1915.

Lawall, J., S. Kulin, B. Saubamea, N. Bigelow, M. Leduc, andC. Cohen-Tannoudji, 1995, Phys. Rev. Lett. 75, 4194.

Lea, S. N., A. Clairon, C. Salomon, P. Laurent, B. Lounis, J.Reichel, A. Nadir, G. Santarelli, 1994, Phys. Scr. Vol. T51, 78.

Letokhov, V. S., 1968, Pis’ma. Zh. Eksp. Teor. Fiz. 7, 348[JETP Lett. 7, 272 (1968)].

Letokhov, V. S., and V. G. Minogin, 1981, Phys. Rep. 73, 1.Letokhov, V. S., V. G. Minogin, and B. D. Pavlik, 1977, Zh.

Eksp. Teor. Fiz. 72, 1328 [Sov. Phys. JETP 45, 698 (1977)].Lett, P. D., R. N. Watts, C. I. Westbrook, W. Phillips, P. L.

Gould, and H. Metcalf, 1988, Phys. Rev. Lett. 61, 169.Monroe, C., W. Swann, H. Robinson, and C. E. Wieman, 1990,

Phys. Rev. Lett. 65, 1571.Ol’shanii, M. A., and V. G. Minogin, 1992, Opt. Commun. 89,

393.Phillips, W. D., and H. Metcalf, 1982, Phys. Rev. Lett. 48, 596.Prestage, J., R. Tjoelker, and L. Maleki, 1995, Phys. Rev. Lett.

74, 3511.Prodan, J. V., W. Phillips, and H. Metcalf, 1982, Phys. Rev.

Lett. 49, 1149.Raab, E. L., M. Prentiss, A. Cable, S. Chu, and D. E. Prit-

chard, 1987, Phys. Rev. Lett. 59, 2631.Ramsey, N., 1956, Molecular Beams (Oxford University, Ox-

ford).Reichel, J., F. Bardou, M. Ben Dahan, E. Peik, S. Rand, C.

Salomon, and C. Cohen-Tannoudji, 1995, Phys. Rev. Lett. 75,4575.

Salomon, C., J. Dalibard, A. Aspect, H. Metcalf, and C.Cohen-Tannoudji, 1987, Phys. Rev. Lett. 59, 1659.

Salomon, C., J. Dalibard, W. Phillips, A. Clairon, and S. Guel-lati, 1990, Europhys. Lett. 12, 683.

Saubamea, B., T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M.Leduc, and C. Cohen-Tannoudji, 1997, Phys. Rev. Lett. 79,3146.

Shapiro, I. I., 1964, Phys. Rev. Lett. 13, 789.Simon, E., P. Laurent, C. Mandache, A. Clairon, 1997, in Pro-

ceedings of the 11th European Frequency and Time Forum(Neuchatel, Switzerland), p. 646.

Steane, A., P. Szriftgiser, P. Desbiolles, and J. Dalibard, 1995,Phys. Rev. Lett. 74, 4972.

Stenholm, S., 1986, Rev. Mod. Phys. 58, 699.Ungar, P. J., D. S. Weiss, E. Riis, and S. Chu, 1989, J. Opt. Soc.

Am. B 6, 2058.Verkerk, P., B. Lounis, C. Salomon, C. Cohen-Tannoudji, J. Y.

Courtois, and G. Grynberg, 1992, Phys. Rev. Lett. 68, 3861.Wallis, H., J. Dalibard, and C. Cohen-Tannoudji, 1992, Appl.

Phys. B: Photophys. Laser Chem. 54, 407.Wineland, D., and H. Dehmelt, 1975, Bull. Am. Phys. Soc. 20,

637.Wineland, D. J., and W. Itano, 1979, Phys. Rev. A 20, 1521.Zacharias, J., 1954, Phys. Rev. 94, 751.Zoller, P., M. Marte, and D. F. Walls, 1987, Phys. Rev. A 35,

198.

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UBC Physics & Astronomy | UBC Physics & …qdg/resources/AMOReadingList/...of G8 and D8, using the so-called dressed-atom approach to atom-photon interactions (see for example Cohen-Tannoudji,