Laser Cooling Below the Doppler Limit by Polarization

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Vol. 6, No. 11/November 1989/J. Opt. Soc. Am. B 2023

Laser cooling below the Doppler limit by polarizationgradients: simple theoretical modelsJ. Dalibard and C. Cohen-Tannoudji

Collage de France et Laboratoire de Spectroscopie Hertzienne de 1'EcoleNormale Sup6rieure [Laboratoireassoci6 au Centre National de la Recherche Scientifique (LA18) et i l'Universit6 Paris VI], 24, rue Lhomond,F-75231ParisCedex 05,FranceReceived April 3, 1989; accepted June 29, 1989

We present twocoolingmechanisms that lead to temperatures wellbelow the Dopplerlimit. These mechanisms arebased on laser polarization gradients and work at low laser power when the optical-pumping time betweendifferentground-state sublevels becomes long. There is then a large time lag between the internal atomic response and theatomic motion, which leads to a large cooling force. In the simple case of one-dimensional molasses, we identify twotypes of polarization gradient that occur when the two counterpropagating waves have either orthogonal linearpolarizations or orthogonal circular polarizations. In the first case, the light shifts of the ground-state Zeemansublevels are spatially modulated, and optical pumping amongthem leads to dipole forces and to a Sisyphus effectanalogous to the one that occurs in stimulated molasses. In the second case (+-ar configuration), the coolingmechanism is radically different. Even at very lowvelocity, atomic motion produces a population difference amongground-state sublevels, which gives rise to unbalanced radiation pressures. From semiclassical optical Blochequations, we derive for the two cases quantitative expressions for friction coefficients and velocity capture ranges.The friction coefficients are shown in both cases to be independent of the laser power, which produces anequilibrium temperature proportional to the laser power. The lowest achievable temperatures then approach theone-photon recoil energy. We briefly outline a full quantum treatment of such a limit.

physical mechanism that underlies the first proposalsof free atoms' or trapped ions2 is the Dopp-Consider, for example, a free atom moving in awave, slightly detuned to the red. Because ofDoppler effect, the counterpropagating wave gets closerstronger radiation pressure on theave. It follows that thevelocity is damped, as if the atom were moving in amedium (optical molasses). The velocity capturea process is obviously determined by theof the atomic excited state

kA - r, (1.1)k is the wave number of the laser wave. On the otherby studying the competition between laser coolingandintroduced by the random nature of spon-that for two-level atoms the

TD that can be achieved by such a meth-3 '4kBTD = r (1.2)

is called the Doppler limit. The first experimental dem-seemed to agree with such a5 6In 1988,it appeared that such a limit could be overcome.by the National Institute of

Technology Washington group7

showed thatmuch lower than TD, and even approachingTR given by

h2 k'kBTR = 2M' (1.3)where M is the atomic mass, can be observed on laser-cooledsodium atoms at low aser powers. Such an important resultwas confirmed soon after by other experiments on sodium8and cesium. 9A possible explanation for these low temperatures basedon new cooling mechanisms resulting from polarization gra-dients was presented independently by two groups at thelast International Conference on Atomic Physics in Paris.910We summarize below the broad outlines of the argument":

(i) The friction force experienced by an atom moving in alaser wave is due to the fact that the atomic internal statedoes not follow adiabatically the variations of the laser fieldresulting from atomic motion. 2" 3 Such effects are charac-terized bya nonadiabaticity parameter (, defined as the ratiobetween the distance VTcoveredby the atom with a velocityv during its internal relaxation time r and the laser wave-length X= 1/k, which in a standing wave is the characteristiclength for the spatial variations of the laser field

VTE = - = kvi-. (1.4)(ii) For a two-level atom, there is a single internal time,which is the radiative lifetime of the excited state

1TrR= -,r (1.5)so that0740-3224/89/112023-23$02.00 1989 Optical Society of America

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2024 J. Opt. Soc. Am. B/Vol. 6, No. 11/November 1989

e = kvrR = kv (1.6)But for atoms, such as alkali atoms, that have several Zee-man sublevels gr g', ... in the ground state g, there isanother internal time, which is the optical-pumping time rp,characterizing the mean time that it takes for an atom to betransferred by a fluorescence cycle from one sublevel gm oanother gm'. We can write

__1 (1.7)where r is the mean scattering rate of incident photons andalso can be considered the width of the ground state. Itfollows hat for multilevel atoms wemust introduce a secondnonadiabaticity parameter

e' = kvrp = kv (1.8)At lowlaser power, i.e., when the Rabi frequency Q s smallcompared with r, we have Tp >> TR and consequently r


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Vol. 6, No. 11/November 1989/J. Opt. Soc. Am. B 2025curves exhibit a very narrow structure around v0, with a width of a few recoil velocities, in agreement withpredictions.

TWO TYPES OF POLARIZATIONIN A ONE-DIMENSIONALsection, we consider two laser plane waveswith thefrequency 0L that propagate along opposite directionsthe Oz axis and we study how the polarization vector ofelectric field varies when one movesalong Oz. Letand 6o' be the amplitudes of the two waves and e and e' bepolarizations. The total electric field E(z, t) in z atwritten as

polarized along y. For z = 0, y coincides with e Whenone moves along Oz, ey rotates, and its extremity forms ahelix with a pitch X [Fig. 1(a)].B. The lin lin Configuration-Gradient of EllipticityWe suppose now that the two counterpropagating waveshave orthogonal linear polarizations

e= vC'= ey.

(2.6a)(2.6b)

If we suppose, in addition, that the two waves have equalamplitudes, we get from Eqs. (2.2) and (2.6)E(z, t) = +G(z)exp(-iWLt) + c.c., (2.1)

the positive-frequency component 6+(z) is given by6 (z) = 60 ee'z + o efe-ikz. (2.2)

a convenient choice of the origin on the Oz axis, we cantake Goand 6 0' real.The a+-a- Configuration-Pure Rotation ofconsider first the simple case in which

e = e+ =-I (ex+ iy), (2.3a)

'e'= e_ (= --if). (2.3b)two waves have opposite circular polarizations, a- forwavepropagating toward z < 0 and u+ or the other wave.Inserting Eqs. (2.3) into Eq. (2.2),we get

+(z) = -t ( -60)ex- ( 0 + O)Ey, (2.4)ex = excos kz-ey sin kz, (2.5a)ey = exsin kz + y Cos kz. (2.5b)

total electric field in z is the superposition of two fieldsquadrature, with amplitudes (' - 60 )/Fand (60' + 60)/and polarized along two orthogonal directions ex and yfrom e. and e by a rotation of angle s = -kz aroundWe conclude that the polarization of the total electricelliptical and keeps the same ellipticity, (o' - &o)/+ 0) for all z. When one moves along Oz, the axes ofellipse just rotate around Oz by an angle ( = -kz. Asthe periodicity along z is determined by the laser= 2r/k.Previous analysis shows that, for a +--a- configuration,have a pure rotation of polarization along Oz. By purethat the polarization rotates but keeps the sameOnecan show hat the a-f-- configuration is heone that gives such a result.In the simple case in which the two counterpropagatinga-+and a- have the same amplitude = o', the totalfield is, according to expression (3.4)below,linearly

&+(z) = 6oA(cos kz - i sin kz X) (2.7)The total electric field is the superposition of two fields inquadrature, with amplitudes 6J- cos kz and 62 sin kz,and polarized along two fixed orthogonal vectors (e, E)/parallel to the two bisectricesof e, and ey. It is clear that theellipticity changes now when one moves along Oz. From Eq.(2.7) we see that the polarization is linear along e = (e. + ey)/V'2n z = 0, circular (a-) inz = /8, linear along 2 = ( -y)lJ- in z = X/4,circular (-+) in z = 3X/8, is linear along -el in z= X/2, and so on... [Fig. 1(b)].If the two amplitudes 60 and ' are not equal, we stillhave the superposition of two fields in quadrature; however,now they are polarized along two fixed but nonorthogonaldirections. For 60 = 6 o' the nature of the polarization ofthetotal field changes along Oz. Such a result generally holds;i.e., forall configurations other than the o-+-a-one, there aregradients of ellipticity when one moves along Oz (excluding,of course, the case when both waves have the same polariza-tion).C. Connection with Dipole Forces and RedistributionThe two laser configurations of Figs. 1(a) and 1(b) differradically with regard to dipoleforces. Suppose that we havein z an atom with several Zeeman sublevels in the groundstate g. For example, we consider the simple case of a Jg =1/2 Je = 3/2 transition for which there are two Zeemansublevels, g_1/2 and g+11, in g and four Zeeman sublevels in e.It is easyto see that the z dependence of the light shifts of thetwoground-state sublevelsis quite different for the twolaserconfigurations of Figs. 1(a) and 1(b). For the -+-o- configu-ration, the laser polarization is always linear, and the laserintensity is the same for all z. It follows that the two light-shifted energies are equal and do not vary with z [Fig. 1(c)].On the other hand, since the Clebsch-Gordan coefficientsofthe various transitions g - em' are not the same, and sincethe nature of the polarization changes with z, one can easilyshow (see Subsection 3.A.1) that the two light-shifted ener-gies oscillatewith z for the lin I lin configuration [Fig. 1(d)]:the g2 sublevelhas the largest shift for a a-+ olarization theg- 1 2 sublevel has the largest shift for a a- polarization,whereas both sublevels are equally shifted for a linear polar-ization.The striking difference between the z dependences of thelight-shifted energies represented in Figs. 1(c) and 1(d)

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(a) (b)'C4Zi +

0y

%.,X~ ~ ___ YF-2 -6C 1 C

OX

l I I I >A18 X14 3A/8 A/2 Z

e-3/ 2 e-l/2 e1/2 e3 /2

I/

-_'i Cd)91/2 I _K 2 K 2Fig. 1. The two types ofpolarization gradient in a 1-D molasses and the corresponding light-shifted ground-state sublevels for a Jg = 1/2 Je= 3/2 atomic transition. (a) a+-a- configuration: two counterpropagating waves,a+and a- polarized, create a linear polarization that rotatesin space. (b) lin lin configuration: The two counterpropagating waves have orthogonal linear polarizations. The resulting polarization nowhas an ellipticity that varies in space: for z = 0 linear polarization along el = (eY+ e)/12; for z = X/8 a- polarization; for z = X/4 linear polariza-tion along 62 = ( - ey)/v2; for z = 3X/8 a+ circular polarization .... (c) Light-shifted ground-state sublevels for the a+-a configuration: Thelight-shifted energies do not vary with z. (d) Light-shifted ground-state sublevels for the lin lin configuration: The light-shifted energiesoscillate in space with a period X/2.means that there are dipole or gradient forces in the configu-ration of Fig. 1(b), whereas such forces do not exist in theconfiguration of Fig. 1(a). We use here the interpretation ofdipole forces in terms of gradients of dressed-state ener-gies.16 Another equivalent interpretation can be given interms of redistribution of photons between the two counter-propagating waves, when the atom absorbs a photon fromone wave and transfers it via stimulated emission into theopposite wave.' 2 "0 It is obvious that conservation of angu-lar momentum prevents such a redistribution from occur-ring in the configuration of Fig. 1(a).2 ' After it absorbs ao+photon, the atom is put into e+1/2 or e+3/2, and there are no a-transitions starting from these levels and that could be usedfor the stimulated emission of a a- photon. For more com-plex situations, such as for a Jg = 1 - J, = 2 transition (seeFig. 5 below), redistribution is not completely forbidden butis limited to a finite number of processes. Suppose, forexample, that the atom is initially in g-1. When it absorbs aa- photon, it jumps to eo. Then, by stimulated emission of aa- photon, it falls to g+,, from where it can be reexcited to e+2by absorption of a o-+photon. However, once in e+2, theatom can no longer make a stimulated emission in the a-wave, since no a- transition starts from e+2. We thus havein this case a limited redistribution, and one can show that,as in Fig. 1(c), the light-shifted energies in the ground statedo not vary with z (see Subsection 3.B.1). The situation iscompletely different for the configuration of Fig. 1(b).Then, each a-+ r a- transition can be excited by both linear

polarizations e,and ey,and an infinite number of redistribu-tion processes between the two counterpropagating wavescan take place via the same transition g - em+l or em-1.This is why the light-shifted energies vary with z in Fig. 1(d).Finally, let us note that, at first sight, one would expectdipole forces to be inefficient in the weak-intensity limitconsidered in this paper since, in general, they become largeonly at high intensity, when the splitting among dressedstates is large compared with the natural width r.'6 Actual-ly, here we consider an atom that has several sublevels in theground state. The light-shift splitting between the two os-cillating levels of Fig. 1(d) can be large compared with thewidth I' of these ground-state sublevels. Furthermore, weshow in Subsection 3.A.2 that for a moving atom, even withweak dipole forces, the combination of long pumping timesand dipole forces can produce a highly efficient new coolingmechanism.3. PHYSICAL ANALYSIS OF TWO NEWCOOLING MECHANISMSIn this section, we consider a multilevel atom moving in alaser configuration exhibiting a polarization gradient. Webegin (Subsection 3.A) by analyzing the lin I lin configura-tion of Fig. 1(b), and we show how optical pumping betweenthe two oscillating levels of Fig. 1(d) can give rise to a newcooling mechanism analogous to the Sisyphus effect occur-ring in stimulated molasses.16 "1 Such an effect cannot exist

h.,

y

(C)

J. Dalibard and C. Cohen-Tannoudji

Cy+- - t

Fy

I


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Vol. 6, No. 11/November 1989/J. Opt. Soc. Am. B 2027the configuration of Fig. 1(a) since the energy levels ofare flat. We show in Subsection 3.B that there is amechanism associated with the o+-o-o configu-but it has a completely different physical interpreta-

The emphasis in this section will be on physical ideas. Aquantitative analysis, based on optical Bloch equa-is presented in the followingsections.

Multilevel Atom Moving in a Gradient of Ellipticitylaser configuration is the lin lin configuration of Fig.As in Subsection 2.C, we take a Jg = 1/2 Je = 3/2The Clebsch-Gordan coefficientsof the variousgm em, are indicated in Fig. 2. The square ofcoefficients give the transition probabilities of thetransitions.

Equilibrium Internal State for an Atom at Restthat, for an atom at rest in z, the energies andof the two ground-state sublevelsdepend onSuppose, for example,that z = X/8 othat the polarization[Fig. 1(b)]. The atom is optically pumped in g- 112 sosteady-state populations of g-112 and g11 are equal1and 0, respectively. Wermustalso note that, since the a-starting from g- 1/2 is three times as intense as theransition starting from g,12, the light shift A-' of g-11 islarger (in modulus) than the light-shift A+' of

12. Weassume here that, as usual in Doppler-cooling ex-the detuninga=W - WA (3.1)

the laser frequency WLand the atomic frequency WAnegative so that both light shifts are negative.If the atom is at z 3X/8, where the polarization is o-+Fig.the previous conclusions are reversed. The popula-of g-1/2 and g2 are equal to 0 and 1, respectively,the atom is now optically pumped into g 2. Bothshifts are still negative, but we now have A+' = 3-'.Finally, if the atom is in a place where the polarization isfor example, if z = 0, X/4, A/2.. . [Fig. 1(b)], symmetryshowthat both sublevels are equally populat-the same (negative) light shift equal to 2/3the maximum light shift occurring for a -+or -All these results are summarized in Fig. 3, which shows. asof z the light-shifted energies of the two ground-of an atom at rest in z. The sizes ofthe blackrepresented on each sublevel are proportional to the

e 3 e_11 2 e+1/2

Energy

0 X/8 A/4 3X/8 A/2 5A/8 zLin C Lin C+ Lin Y

9-1/2 9+1/2 9_1/2Fig. 3. Light-shifted energiesand steady-state populations (repre-sented by filled circles) for a Jg = 1/2 ground state in the lin linconfiguration and for negative detuning. The lowestsublevel,hav-ing the largest negative light shift, is also the most populated one.

steady-state population of this sublevel. It clearly appearsin Fig. 3 that the energies of the ground-state sublevelsoscillate in space with a spatial period /2 and that thelowest-energysublevel is also the most populated one.2. Sisyphus Effect for a Moving AtomWe suppose now that the atom is moving along Oz, and wetry to understand how its velocity can be damped. The keypoint is that optical pumping between the two ground-statesublevels takes a finite time rp. Suppose that the atomstarts from the bottom of a potential valley,forexample, at z= X/8 (see Fig. 4), and that it moves to the right. If thevelocityv issuch that the atom travels over a distance of theorder of /4 during p, the atom will on average remain onthe same sublevel, climbthe potential hill, and reach the topof this hill before being optically pumped to the other sub-level, i.e., to the bottom of the next potential valleyat z = 3X/8. From there, the same sequence can be repeated (see thesolid lines in Fig. 4). It thus appears that, because of thetime lag T, the atom is alwaysclimbingpotential hills, trans-forming part of its kinetic energy into potential energy.Here wehave an atomic example of the Sisyphus myth thatis quite analogous to the coolingeffect that occurs in stimu-lated molasses and discussed in Ref. 16. Note, however,that the effectdiscussed in this paper appears at much lowerintensities than in Ref. 16since it involves two ground-statesublevels with spatially modulated light shifts A' muchsmaller than r; on the contrary, in Ref. 16 he modulation ofthe dressed-state energies is much larger than r.

9-1/2 9+1/2Atomic level scheme and Clebsh-Gordan coefficients for a= 1/2 Je = 3/2 transition.

3. Mechanism of Energy Dissipationand Order ofMagnitude of the Friction CoefficientThe previous physical picture clearly showshow the atomickinetic energy is converted into potential energy: The atomclimbs a potential hill. In the same way as for the usualdipole forces, one can also understand how the atomic mo-mentum changes during the climbing. There is a corre-

- -

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2028 J. Opt. Soc. Am. B/Vol. 6, No. 11/November 1989On the other hand, we can also calculate dW/dt from F:

dW Fv.dtSince we evaluate only orders of magnitude, we can keep thelinear expression (3.2) of F, even when v is given by expres-sion (3.3), so that dW 2dtEquating expressions (3.4) and (3.6) and using expression(3.3) of v, we finally obtain

a--hk 2 A.

o X/8 A/4 3A/8 A/2 5A/8 zFig. 4. Atomic Sisyphus effect in the lin lin configuration. Be-cause of the time lag r, due to optical pumping, the atom sees on theaverage more uphill parts than downhill ones. The velocity of theatom represented here is such that vrp - X, in which case the atomtravels overs X in a relaxation time TP. The cooling force is thenclose to its maximal value.

sponding change of momentum of the laser field because of acoherent redistribution ofphotons between the two counter-propagating waves. Photons are absorbed from one waveand transferred by stimulated emission to the other wave.All these processes are conservative and could occur in bothways. The atom could slide down a potential hill and trans-form its potential energy into kinetic energy. Opticalpumping is the mechanism of energy dissipation essentialfor introducing irreversibility into the process and for pro-ducing cooling. We see from Fig. 4 that when the atomreaches the top of the hill, there is a great probability that itwill absorb a laser photon hWL and emit a fluorescence pho-ton, blue-shifted by an amount corresponding to the light-shift splitting between the two ground-state sublevels. Thegain of potential energy at the expense of kinetic energy isdissipated by spontaneous Raman anti-Stokes photons thatcarry away the excess of energy. Here also we find a mecha-nism quite analogous to the one occurring in stimulatedmolasses.16 Note, however, that the energy dissipated hereis much smaller, since it corresponds to the light shift of theground state at low laser power.From the previous discussion, we can derive an order ofmagnitude of the friction coefficient a appearing in the low-velocity expression

F= -av (3.2)of the friction force. It is clear in Fig. 4 that the maximumvalue of the friction force occurs when vrp - X/4, i.e., when

kv- r, (3.3)where r' = 1/Tp. For this value of v, the energy dissipatedduring Tp is of the order of -hA' (sinceA' < 0), so the energydissipated per unit time is

dW -hA = -hA'r'. (3.4)Tp

Since all the previous considerations are restricted to thelow-intensity limit (we want to have rF, IA'I> r) than theoptimal friction coefficient for the usual Doppler cooling,which is of the order of hk 2 .3 ,4 One must not forget, howev-er, that the velocity capture range of this new friction force,which is given by expression (3.3), is much smaller than thevelocity capture range for Doppler cooling (given by kv - r).One can also understand why a is so large, despite the factthat the size hA'! of the potential hills of Fig. 4 is so small.We see in expression (3.7) that hIA'I is divided by r', which isalso very small since the optical-pumping time is very long.In other words, the weakness of dipole forces is compensatedfor by the length of the optical-pumping times.B. Multilevel Atom Moving in a Rotating LaserPolarizationThe laser configuration is now the o-+-a-aser configurationof Fig. 1(a) for which the laser polarization remains linearand rotates around Oz, forming an helix with a pitch X. Asshown in Fig. 1(c), the light shifts of the ground-state sublev-els remain constant when the atom moves along Oz, andthere is no possibility of a Sisyphus effect. Such a result iseasily extended to all values ofJg. If Jgwere larger than 1/2,we would have in Fig. 1(c) several horizontal lines instead ofa single one, since sublevels gm with different values of Imlhave different light shifts and since there are several possiblevalues for mlwhen Jg > 1/2.

Energy

III- Tl--

I IIIIl l

(3.5)

(3.6)

(3.7)

I I I I 1 2!-


II'


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9-1 9o 9+15. Atomic level scheme and Clebsh-Gordan coefficients for a= 1 Je = 2 transition.

In this subsection we describe a new coolingmechanismworks in the o-+-o--aserconfiguration for atoms with Jg1 and is quite different from the one discussed in Subsec-3.A. We show that, even at very low velocity, there is anorientation alongOz that appears in the ground statea result of atomic motion. Because of this highly sensi-motion-induced atomic orientation, the two counterpro-waves are absorbed with different efficiencies,gives rise to unbalanced radiation pressures and con-to a net friction force. We consider here thepossible atomic transition for such a scheme, theJg = 1 Je = 2 (see Fig. 5).

Internal State for an Atom at Restfirst that the atom is at rest in z = 0. Ifwe takequantization axis along the local polarization, which isz = 0 [see Fig. 1(a)], and if we note that Ig_)y, Igo)y,Ig)y,eigenstates of Jy (J: angular momentum), we see thatpumping, with a 7rpolarization along Oy, will concen-atoms in Igo)y, ince the optical-pumping rate Ig-), ->roportional to (1/,/2)2(1/)2 = 1/4 is greater than theIgo)y Ig_)y proportional to (T73)2(11/@)2 1/9. Thepopulations of g0)y, Ig_1)y, and g+,)y are equal4/17, and 4/17, respectively.We must also note that, since the r transition startingIgo)yis 4/3 as more intense as the two r transitionsfrom lg9+)y,oth sublevels Ig~,)y undergo the sameshift A,', smaller (inmodulus) than the light-shift A' of

sublevels adjust themselves to values such that the meannumber of Stokes processes from Igo), o Ig-1), balances themean number of anti-Stokes processes from Ig_)y to Igo)y.It is thus clear that in steady state, the mean number ofphotons emitted per unit time at WL + ( 0'/4) and WL - (A0 '/4) will be equal, giving rise to a symmetrical fluorescencespectrum.Sofar, we haveconsidered only an atom at rest in z = 0. Ifthe atom is in a different location but still at rest, the samecalculations can be repeated, giving rise to the same valuesfor the light shifts (since the laser intensity doesnot changewith z) and to the same steady-state populations. We mustnote, however, that the wave functions vary in space, sincethe light-shifted Zeeman sublevels are the eigenstates of thecomponent of J along the rotating laser polarization ey. Itfollows that, when the atom moves along Oz, nonadiabaticcouplings can appear among the various Zeeman sublevelsundergoing different light shifts.

2. Moving Atom-Transformation to a Moving RotatingFrameThe atom is now moving with a velocity v along Oz:z = Vt. (3.11)

In its rest frame, which moves with the same velocity v, theatom seesa linear polarization ey, whichrotates around Oz nthe plane xOy, making an angle with Oy [see Fig. 1(a) andEq. (2.5b)]= -kz = -kvt. (3.12)

It is then convenient to introduce, in the atomic restframe, a rotating frame such that in this moving rotatingframe the laser polarization keeps a fixed direction. Ofcourse,Larmor's theorem tells us that, in this moving rotat-ing frame, an inertial field will appear as a result of therotation. This inertial field looks likea (fictitious) magneticfield parallel to the rotation axis Oz and has an amplitudesuch that the corresponding Larmor frequency is equal tothe rotation speed k. More precisely, one can show (seeAppendix A) that the new Hamiltonian, which governs the

A0' = 4/,. (3.10)in the previous subsection,we take a red detuning so that0' and A' are both negative. Figure 6 represents the light-ground-state sublevels in z = 0 with their steady-populations.For subsequent discussions, it will be useful to analyzethe spectrum of the fluorescencelight emitted by anat rest in z = 0. We suppose that the laser power isweak (


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2030 J. Opt. Soc. Am. B/Vol. 6, No. 11/November 1989atomic evolution after the unitary transformation to themoving rotating frame, contains, in addition to a couplingterm with a fixed-polarization laser field, an extra inertialterm resulting from the rotation and equal to

Vrot = kvJ. (3.13)If we compare the new Hamiltonian in the moving rotatingframe with the Hamiltonian for an atom at rest in z = 0considered in Subsection 3.B.1 we see that all the new effectsthat are due to atomic motion in a rotating laser polarizationmust come from the inertial term [Eq. (3.13)]. Since J, hasnonzero matrix elements among the eigenstates of Jy, thisinertial term introduces couplings proportional to kv be-tween Igo)y and Igai)y (double arrows in Fig. 6). Thesecouplings are sometimes called nonadiabatic since they van-ish when v tends to 0. We show in the next subsection howthey can give rise to an atomic orientation in the groundstate that is parallel to Oz and sensitive to kv.

3. Motion-Induced Orientation in the Atomic GroundStateIn order to get a clear insight into the modifications intro-duced by the inertial term [Eq. (3.13)] and also to under-stand the energy exchanges between the atom and the laserfield, it is important to determine first what the new energylevels are in the ground state as well as the new steady-statedensity matrix in such an energy basis. To simplify thecalculations, we assume here that the light shift IA'I s muchlarger than F', so that the energy splitting between theground-state energy levels is much larger than their widths:r 0, 6 H+. (3.21)We show in the next subsection how this motion-inducedpopulation difference can give rise to a new much moreefficient friction mechanism than the one used in Dopplercooling.4. Remarks(i) We saw in Subsection 3.C.2 that the problem studiedhere is formally equivalent to the problem of an atom at rest,



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Vol. 6, No. 11/November 1989/J. Opt. Soc. Am. B 2031with a laser field linearly polarized along Oy anda static magnetic field Bo along Oz, with anthat the Larmor frequency in Bo is equal toChanging v is equivalent to changing IBoI; uch a formu-of the problem allows us to establish a connectionthe effects studied in this paper and other well-that were previously observed in optical-

It is well known, for example, thatBo in a direction differentthe symmetry axis of the laser polarization cangive risevery narrow structures in the variations with B01of theor emitted. Examples of such structures areresonances and Hanle resonances instates.2 '2 3 These resonances have a narrowABOuch that the Larmor frequency in AB( s equal tor' of the ground state, which can be very small atActually, the problem studied here is a little more compli-than a pure Hanle effect in the ground state since thenot only introduces an atomic alignment along

((3J,,2-J2) differs from zero) but also produces lightthe Igm),,states, which have the same symmetry asthat would be produced by a fictitious staticEo parallel to Oy. In the absence of Bo, the(3J,2._J2) produced by optical pumping doesprecess around Eo, which has the same symmetry axis.Bo is applied this alignment starts to precess aroundgivingrise to a new nonzero component of the alignment+ J,,Jx). It is the interaction of this alignment with Eogives rise to the orientation (J,) along Oz. In a certainan analogy between the motion-inducedorientation studied here and the effects described in24 and 25 and dealing with the orientation produced byof an atomic alignment with a real or ficti-electric field.(ii) One can easily understand why the new coolingstudied in this section does not work for astate Jg = 1/2. In a J = 1/2 state no alignment cantheorem). Optical pumping with alight cannot therefore introduce any an-in a Jg = 1/2 ground state, at least when v,i.e., B01,small. It is only when B0! is large enough to producedetuning comparable with r that the two counter-laser beams begin to be scattered with differentleading to usual Doppler cooling. Another waythis result is to note that the two sublevelscannot be connected to the same excited state by the+ and o--,so that no coherencecanbetween these two sublevels.Getting Unbalanced Radiation Pressures with aAtomic Orientation

1(a) and 5, one sees there is a six timesthat an atom in Ig.-), will absorb a -propagating toward z < 0 than that it will absorb a +propagating toward z > 0. The reverse conclusionsfor an atom in Igi)z.If the atom moves toward z > 0 and if the detuning 6 iswe saw above in expression (3.21) that the sublevel1), is more populated than g+1)z. It followsthat thepressures exerted by the two o--and o-+waves willThe atom will scatter more counterpropa-

gating a- photons than copropagating a+ones, and its veloc-ity will be damped. Note that here we ignore the Zeemancoherence between Ig i)zand Ig+i)z.We will see in Section5that the contribution of such a Zeemancoherencedoes notchange the previous conclusion.We must emphasize here that the fact that the two radia-tion pressures become unbalanced when the atom movesisdue not to the Doppler effect, as in Doppler cooling,but to adifferenceof populations in the ground state that is inducedby the inertial term [Eq. (3.13)]. It appears clearly in Eq.(3.20) that the dimensionless parameter characterizing thisnew cooling mechanism is kV/IA'I. At low laser power A'1


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2032 J. Opt. Soc. Am. B/Vol. 6, No. 11/November 1989can reach a steady state and get an energy resolution betterthan hlA'I for the scattered photons. (Of course, this timeshould not be too long so that we can neglect any velocitychange resulting from the friction force.) It is then clearthat the steady-state populations of Igo), and Igl), adjustthemselves to values such that the rate of Stokes processesfrom go)y to gL)_yalances the rate of anti-Stokes processesfrom Ig+)y to Igo),. As in Subsection 3.B.1, we find a fluo-rescence spectrum that remains symmetric even when theatom moves, since there are always as many Raman Stokesas Raman anti-Stokes photons emitted per unit time. Con-sequently, it does not seem appropriate, as was done in thefirst explanations of this cooling mechanism, to invoke di-rect nonadiabatic transitions between Igo)yand Ig)y con-verting kinetic energy into potential energy, this potentialenergy then being dissipated byRaman anti-Stokes process-es.Actually, for this second cooling scheme the dissipation is,as in Doppler cooling,due to the fact that in the laboratoryframe the fluorescence photons have on average a blueDoppler shift on the Rayleigh line as well as on the twoRaman lines. Consider, for example, the fluorescence pho-ton following he absorption of a - photon. If it is reemit-ted toward z < 0, it has no Doppler shift in the laboratoryframe, whereas it has a Doppler shift 2kv when it is reemit-ted toward z > 0. On average, the Doppler energy dissipat-ed by a fluorescence cycle involving a - photon is hkv. Fora a+ photon a similar argument gives -hkv. The totalDoppler energydissipated per unit time is therefore equal todW__rFT'(-+l-II-l)hkv - hkV2 (3.25)dt A'lwhich coincides with the power -Fv dissipated by the fric-tion force [expression (3.23)].

J. Dalibard and C. Cohen-Tannoudjiemphasize that we do not impose any condition on kv ndthe inverse r' of the pumping time -r,. Atomic motion cantherefore greatly affect the atomic ground-state dynamicsand then induce a large velocity-dependent force.First, we study the internal atomic evolution. Startingfrom the optical Bloch equations that describe the evolutionof the atomic density operator,2 6 we obtain two equationsgiving the evolution of the ground-state populations. Wethen calculate the average radiative force as a function of theatomic velocity, which givesthe friction coefficient and thevelocity capture range for this coolingmechanism. Finally,we evaluate the momentum diffusion coefficient and derivethe equilibrium temperature of atoms cooledin the lin 1 linconfiguration.A. Internal Atomic EvolutionAll the calculations of this paper are done in the electric-dipole and rotating-wave approximations so that the atom-laser field coupling can be written asV =-[D+ *G6(r)exp(-ioLt) + D- -(r)exp(iWLt)]. (4.1)D+ and D- are the raising and lowering parts of the atomicelectric-dipole operator, and A+and 6- = (+)* are thepositive- and negative-frequency components of the laserelectric field. The laser field for the lin 1 lin configurationwas given in Eq. (2.7). Inserting its value into the atom-field coupling and using the Clebsh-Gordan coefficients in-dicated in Fig. 2, we get

V = a sin kzi e3 /2 ) (g1/ 21'+ 1lel/2 ) (.-1/ 2ljX exp(-iWLt) + a Cos kz[le- 3 /2 ) (g.-1 21

+ 1 le.-1/2) (9l/21exp(HiWLt)+ h.c.4. THEORY OF LASER COOLING IN THEfin 1 fin CONFIGURATIONThis section is devoted to a quantitative study of the coolingmechanism presented in Subsection 3.A. Let us recall thatthis mechanism is based on a Sisyphus effect induced by aspatial modulation of the light shifts of the atomic Zeemanground sublevels. As in Subsection 3.A, we study here a Jg= 1/2 -J = 3/2 atomic transition and use a semiclassicalapproach to evaluate the friction force and the equilibriumtemperature reached by the atom.Our treatment is limited to the low-power domain (Q


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p(gi, ej) = (gjIplej)exp(-iWLt), (4.5)p is the steady-state density operator.We now need to calculate the steady-state value of thee), using optical Bloch equations.we have for p(g,12, e 3 /2)

2, e3/2) = + )p(gj, e3 2) - i cos kz(e_/21p1e/2)+ a sin kz[(g 12 jpjg, 2 ) - (e3 21ple 3/ 2 )]. (4.6)

2(t). It can be simplified in the low-powerand low-following ways:(i) The low-intensity hypothesis (


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(4.17) where the friction coefficient a is equal toAs sketched in Fig. 3, the two light-shifted ground-statesublevels oscillate in space with a period X/2. This spatialdependence then causes a state-dependent force;taking thegradient of AE1/2, we get

f12 =-d- AE+1/2 = -2 hkhso sin 2kz, (4.18)dz 3so that the average force [Eq. (4.15)] can be written as

f = fl/2]Il/2 + f- 1/2 ,1-1/2. (4.19)This force f is then just the average of the two state-depen-dent forcesf1/2weighted by the populations of these states.Such an expression is similar to the one obtained for a two-level atom in intense laser light, provided that the levelsg+1/are replaced by the two dressed levels.' 6 However, an im-portant difference arises here since each of the two levelsinvolved in Eq. (4.19) is essentially a ground-state sublevelwith a long residence time -p. By contrast, in the two-levelproblem the two dressed states were strongly contaminatedby the excited state, so that their lifetime was r-1.We now calculate the populations 111/2 in order to evalu-ate the force from Eq. (4.19). First, we take an atom at rest.These two populations are then equal to their steady-statevalue [Eq. (4.14b)], so that we obtain

2f(z, v = 0) - hkhso sin 2z cos 2kz3dU=_ , (4.20)dz

where the potential U is given byU(z) = - - h6so sin2 2kz. (4.21)6

The fact that f(z, v = 0) derives from a potential is againsimilar to the corresponding result for the dipole force actingon a two-level atom in a standing wave.We now consider a very slow atom for which the Dopplershift kv is smaller than 1/TP. Note that this condition ismuch more stringent than the condition kv


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0. 2

I-2

-0. 2

T FORCE (Uri t k r/2)0.04

VELOCITYunit /k

2

7. Variations with velocity v of the force due to polarization gradients in the lin lin configuration for a Jg = 1/2 Je = 3/2 transitionThe values of the parameters are Q2 0.3r, a = -r. The dotted curve shows sum of the two radiation pressure forces exerted in-two Doppler-shifted counterpropagating waves. The force due to polarization gradients leads to a much higher friction(slope at v = 0) but acts on a much narrower velocity range.

DkBT= - (4.30)auss the validity of the semiclassical approxi-

In order to calculate the exact value of Dp, one could2 "13we prefer to use a heuristic calculation here.There are three main contributions to Dp; the two firstare already present for a Jg = 0 Je = 1 transition, 2 'third one is specific of an atom with several ground-

(i) There are fluctuations of the momentum carriedby fluorescence photons.(ii) There are fluctuations in the difference among theof photons absorbed in each of the two laser waves.(iii) There are fluctuations of the instantaneous dipoleback and forth between f,/2(z) and f-/ 2 (z) at1/Tp.For a Jg = 0 Je = 1 transition, 2 ' the two first contribu-give for a dipole radiation pattern

(4.31)

nitude for these two contributions in the case of a Jg = 1/2Je = 3/2 transition. To evaluate the third contribution(coefficient Dp"), we start from

D dr[f(t)f(t+ T) - 2], (4.32)which must be calculated for an atom at rest in z (the label zwas omitted for simplification). The force f(t) oscillatesbetween f,/2(z) and f-.1 2 (z), and its correlation function canbe written as

f(t)f(t+ T) > Y fjP(i, t; , t + T),i=+1/2 j=1/2 (4.33)where P(i, t; j, t + r) represents the probability of being instate i at time t and in state j at time t + T. The calculationis then similar to the one done to evaluate the fluctuations ofthe dipole force for a two-level atom (Ref. 16, Subsection4B), and it leads to

Dp 4[f, 2(z)] 1/ 2St(z)II_/ 2St(z)Tp= 2h2 k2 - so sin4(2kz).

Once this is averaged over a wavelength, it givesDp"_ 3 622rDr-Th h2 -So.

(4.34)

(4.35)assume that Eq. (4.31)still gives the good order of mag-D- '' 7h 2k2rso.10

-w __ _



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2036 J. Opt. Soc. Am. B/Vol. 6, No. 11/November 1989This second coefficient exceeds the first one as soon as 161slarger than r. Therefore, neglecting Dp', we get

16 r==, BT Dp h1I 416a 4 so (4.36)or, using definition (4.11) for so,

161 > - kBT 8161* (4.37)It appears in these expressions that the residual kineticenergy is of the order of the light shift hA' of the groundstate. This simple result must be compared with the oneobtained for a two-level system [Eq. (1.2)]. It may lead to amuch lower temperature, and it is consistent with two ex-perimental observations,7 -9 as long as the other hyperfinelevels of the transition can be ignored:

(i) At a given power ( fixed), the temperature decreaseswhen the detuning increases.(ii) At a given detuning, reducing the power decreasesthe temperature.Another important remark concerns the comparison be-tween the residual kinetic energy kBT/2 and the potentialU(z) derived in Eq. (4.21). We actually find that these twoquantities are of the same order, which indicates that in thestationary state atoms are bunched around the points z =nX/4 rather than uniformly distributed. In this regime, oneshould then correct expression (4.24) for the force, whichwas obtained by assuming a constant velocity. A MonteCarlo simulation, similar to the one in Ref. 28, is probablythe best way o derive precise results concerning the station-

ary state in this configuration.Finally, let us look for the lowest temperatures achievablein this configuration. Expression (4.37) suggests that anarbitrarily low temperature could be reached, for instance,by decreasing the laser power. Actually, this is not true.Indeed one must check that the rms velocity deduced fromexpression (4.37) is well below the critical velocity v, [Eq.(4.29)], so that the cooling force is indeed linear for all veloci-ty classes contributing to the stationary state. We thereforegetVrms hr 3 (4.38)

which puts a lower bound on the laser power required for ourtreatment to be valid. This in turn gives a lower bound onthe achievable rms velocity:hh 161

Vrms>> r (4.39)Consequently, the lowest achievable rms velocity in thismodel remains larger than the recoil velocity. Let us recallthat the recoil velocity was, in any event, a limit for thevalidity of our semiclassical treatment.We are currently working on a full quantum treatment ofcooling in the lin I lin configuration, analogous to the onepresented at the end of Subsection 5.D for the a+-of config-uration. Note that such a treatment in the present configu-ration is more complicated than for the o+-ar one, owing to

the possibility that photons are coherently redistributedbetween the two waves.5. THEORY OF LASER COOLING IN THE +-o- CONFIGURATIONNow we come to this final section of this paper, which isdevoted to the quantitative study of laser cooling n a a+-of-configuration for a Jg = 1 Je = 2 atomic transition.As shown in Section 2, the polarization gradient is thenquite different from the one studied in Section 4: Thepolarization is linear in any place and rotates along thepropagation axis Oz on the length scale X.Cooling in this configuration originates from two quitedistinct processes. The first is the usual Doppler coolingthat results from differential absorption of the C+and crwaves when the Doppler shift kv is a nonnegligible part ofthe natural width r. The second mechanism, qualitativelystudied in Subsection 3.B, originates from the enhancementof radiation pressure imbalance that is due to a sensitivemotion-induced atomic orientation. We study here the caseof a Jg = 1 Je = 2 transition (Fig. 5), which, as explained inSubsection 3.B, is the simplest atomic transition exhibitingthis sensitive velocity-induced orientation. For simplifica-tion, we also use at the end of this section a fictitious Watom, which can be formally obtained from the real 1-2transition by removing the V system formed by Igo), le-,),le+,).A. Various Velocity DomainsAs in Section 4, we restrict our treatment to the low-intensi-ty domain (Q > ror equivalently to r


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Vol. 6, No. 11/November 1989/J. Opt. Soc. Am. B 2037reference frame. Using expressions (2.2)(2.3) for the laser electric field, we get

V = FIgl)e2l + if Igo)(ell + A Ig-l) (eol2 [ JX exp[i(w-t - kz)] + h.c.

* hg1) e-2 + I 1O)(e-lI+-1g)e12 [ aW2X exp[i(w+t - hz)] + h.c., (5.1)

= -2d~o/h [identical to Eq. (4.3)],W = L + kV. (5.2)

first (third) line of Eq. (5.1) describes the coupling witha+(cr) laser wave propagating toward z > 0 (z < 0). AsSection 4, the semiclassical force is obtained from thethe gradient of the coupling V. Assuming= 0 in its reference frame, we

= ihkf{p(e2 , g1) + I p(el, go) + I (e gl)] + c.c.- ihkQ (e- 2 , gl) + A (e-., go)+ 1(eo gl) +C-C-

(5.3)p are defined by

P(eisl gi)= (ei, 1Ipigi)exp(iwt). (5.4)moving rotating frame defined in Subsec-

We now need to evaluate the steady-state value of thep(ej, g3). This is done using optical Blochp.2 6 Let us, for example, write down the equationj(el, go):

.(el,go)= [i(6 - kV)- P(e1 , go)

+ i[(ellplel) +(e1IpIel)exp(-2ikvt)(golplgo)].5.5)this equation, (ellplel) and (eilplel) can be neglectedwith (golplgo) ecause of the low-power hypothe-We then get in steady state

p(el go) = /2 no, (5.6)6 kv + i 22Hi = (gilpjgi). (5.7)

way, calculate all optical coherences in termsthe ground-state populations Hiand coherences amongNote that, because of the structure of the laser

excitation, the only nonzero Zeeman coherence in steadystate is the coherence between g, and g-1. We put in thefollowing:Cr = Re[(glplgl)exp(-2ikvt)],Ci = Im[(glplgl)exp(-2ikvt)J. (5.8)

Once all optical coherences have been calculated, it is possi-ble to get a new expression for the force [Eq. (5.3)]. Thedetailed calculation is rather long and tedious and is pre-sented in Appendix B. Here, we give only the main results.First we get for the forcef = hk 2 - + (s - + -++C(h+-1Ck)+(5where

S= Q2/2


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2038 J. Opt. Soc. Am. B/Vol. 6, No. 11/November 1989g2 /2S = 62+ 2/ (5.12)

We can note that these results are in agreement with theones obtained in Section 3. First, for an atom at rest, it iseasy to show that the nondiagonal density matrix obtainedhere in the Igm)z asis indeed leads to the diagonal densitymatrix obtained in Subsection 3.B in the Igm)ybasis. Sec-ond, the population difference 11 -II-, calculated fromEqs. (5.11) is equal to the one predicted in Eq. (3.20) in thelimit 161> . (In this limit, one indeed gets for the light shiftAO'of go)y the value A0 ' = f22/36.)We are now able to calculate the force in this low-velocitydomain. First we note that Eq. (5.9) can be greatly simpli-fied; by neglecting all terms in kv/r and keeping only termsin kv/r', we get

f= hkr so (1 - ,) - 2 Ci] (5.13)The first term of Eq. (5.13) has already been estimated inSubsection 3.B, whereas the second one describes the effectof limited photon redistribution.Now by inserting the result [Eqs. (5.11)] for the atomicdensity matrix into the force, we obtain

f = -av,120 -6P 2a = - 6 hh2 . (5.14)17 5p 2 + 46

The contribution of the population term 5/6(H - -1)represents 4/5 of the total result for the friction coefficient a,the remaining 1/5 part being due to 26C,/3.As expected, we get in this low-velocity domain a forcelinear with velocity, with a damping coefficient (for negativedetunings) independent of power. This damping coefficientis maximal for a detuning 6 = -r5N, where the force is ofthe order of -0.8hk2 v. We find here a result that was al-ready mentioned in Subsection 3.B: For 161>r, he frictioncoefficient in the 0+-cr configuration, which varies as r/a, ismuch smaller than the friction coefficient for the lin linconfiguration, which is proportional to 6/r [Eq. (4.26)].Furthermore, we see here that, even for the optimum detun-ing, the a+-a- damping coefficient is four times smaller thanthe lin lin one.We now come to the complete calculation of the force forany atomic velocity in the low-power approximation. Thisis done by keeping all the velocity-dependent terms in ex-pressions like (5.6) and by using Eq. (5.9) for the force in-stead of the simplified expression (5.13). The result of sucha calculation (which is detailed in Appendix B) is represent-ed in Fig. 8. In the low-velocity domain, we again find thefriction force just calculated above (see, in particular, theinset of Fig. 8). Outside this domain, the force appears to beclose to the usual Doppler force, represented by dottedcurves. This Doppler force is calculated by neglecting allcoherences among ground-state sublevels, so that ground-state populations are obtained only from rate equations:

Fig. 8. Variations with velocity of the steady-state radiative force for a Jg = 1 Je = 2 transition in the a+-a- configuration (1 = 0.25 r; a =-0.5 r). The slope of the force near v = 0 is very high (see also inset), showing that there is polarization gradient cooling. This new coolingforce acts in the velocity range kv - A'. Outside this range, the force is nearly equal to the Doppler force (shown by the dotted curve) calculatedby neglecting all coherences between ground-state sublevels Ig)z..

J. Dalibard and C Cohen-Tannoudji


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Vol. 6, No. 11/November 1989/J. Opt. Soc. Am. B 20393s,(s, + 5s,)

15(s+2 + 52) + 14s+s-8s+s-

15(s+2 + s_2) + 14s+s-

The result [Eq. (5.16)] can now be used to get the equilibri-um temperature(5.15) k = h92 F 29 + 254 r

2/4 1hBT- 161 300 75 62 + (r 2 /4)J (5.18)

therefore confirm quantitatively that the new coolingmainly acts in the kv < r' domain.

Equilibrium Temperature in the 0+-acConfigurationvaluate the equilibrium temperature in thisfirst need to evaluate the momentum dif-D. The exact calculation of D isAppendix B. It uses a method sug-by Castin and Molmer9 and leads toD = D, + D 2 , (5.16)D = 18 h2k2rso,170 h h P

2 [17 1+ (4 2/5 2) + h22so. (5.17)can be understood by simple momen-Since there are no dipolein the 0+-cr configuration, the momentum diffusionto the first two mechanisms mentioned at the

The first term of Eq. (5.16) (D1) corresponds to the fluctu-of the momentum carried away by fluorescence pho-The second term (D2) corresponds to the fluctuationsthe difference between the number of photons absorbed inwave. For large detunings (161 > ), D, and D2 are ofJg = 0 - Je = 1 transition. 2 Byor small , D2 becomes much larger than D,. Asout to us by Molmer and Castin,29 this enhancementD2 is a consequence of correlations introduced by opticalImmediately after a cycle involving theof a o-+ hoton, the atom is more likely o be in thestate and is therefore more likely to absorb another a+than a cr one. As a consequence, the steps of

be several hk instead of hk, and this can increase D 2 for asaturation parameter by a factor as large as 10. Ofsuch an argument holds only if the atom remains infor a time Tp= 1/rP, hich is the mean time between twocycles. If this is not the case, i.e., ifare redistributed among the three Zeemang in a time shorter than r, the previous memorylarge-step random walk, disap-a small value of D2 and thus to low tempera-At low velocities (v CM>>kM (5.19)

This limit is smaller than the one found in the lin I lin case[expression (4.39)] when the detuning 6 is large comparedwithD. Principle of a Full Quantum TreatmentIt is clear from the result obtained above that, for sufficient-ly low power, the cooling limit becomes of the order, of theone-photon recoil energy. As emphasized in the Introduc-tion, a semiclassical treatment then is no longer possible.We present here the principle of a treatment in whichatomic motion is treated in a full quantum way. For sim-plicity, we considered a W transition, sketched in Fig. 9(a),instead of the real J 1 J = 2 transition. We use theconcept of closed families of states introduced in Refs. 18and 19: In the 0+-cr configuration, the set of states5r(p) le 2,p - 2hk),g,p - hk),

Ieo,p), Ig,p + hk), Ie2,p+ 2hk)Iis closed with respect to absorption and stimulated-emissionprocesses, and transfers between families occur only viaspontaneous-emission processes. Then it is possible toshow that the total atomic density operator in steady statehas nonzero matrix elements only among states belonging tothe same family. This greatly simplifies the study of theevolution of this density matrix: If one wants to take, forinstance, 1000points in momentum space, one has to consid-er only a Np X 1000 vector instead of a Np X [1000]2 squarematrix; N is the size [(2Jg + 1) + (2Je + 1)]2 of the internalatomic density matrix.Using the generalized optical Bloch equations includingrecoil,'8 "9 we studied the evolution of the atomic momentumdistribution for a W atom with a linewidth hr equal to 400times the recoil energy h2k2/2M (analogous to sodium). Atypical result is presented in Fig. 9(b). The initial distribu-tion is Gaussian with a rms width of 16hk/M. (The standardDoppler limit kBT= hF/3 for an isotropic radiation patternleads to a rms velocity of 12hk/M.) One clearly sees thatduring the evolution a narrower peak appears around thezero velocity with a width of -2hk/M. This peak is super-imposed upon a much broader background. The reasonthat such a background remains present is because the con-

Io =



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:t2000r-1

t= 1000 r-9 -1 9+1 t= 300r-'

t=O-50tk 0 50MkFig. 9. (a) Fictitious W atom: This atom isthe simplest atomic-levelschemeleading to extra cooling n the a+-ar configuration. (b) Timeevolution of the atomic velocity distribution for a Wtransition (6 =-r, Q= 0.2 r, hr = 200h2k2/Mas for a sodium atom). The initial velocitydistribution is chosen to be Gaussian with a rms width of 16 recoil velocities. The evolution is calculated via the generalized optical Bloch equa-tions includingrecoil,which permit a full quantum treatment of atomic motion. During the evolution,the velocitydistribution becomesnon-Gaussian, with a very narrow peak (HWHM _ 2 recoil velocities) superimposed upon a much broader background.

dition kv


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Vol. 6, No. 11/November 1989/J. Opt. Soc. Am. B 2041paper experimentally is to investigate the velocity depen-dence of the friction force in 1-D molasses, for example, bystudying the transverse collimationof an atomic beam cross-ing two counterpropagating laser waves with controllablepolarizations. Actually, preliminary results have alreadybeen obtained on the 23S, - 23P2 transition of helium.32They clearly showthat coolingwith orthogonal linear polar-izations (i.e.,with a polarization gradient) is more efficientthan with parallel linear polarizations (no polarization gra-dient). Transverse temperatures below the Doppler limitTDwere measured in the lin lin configuration. For a+-o--,the residual kinetic energy remainsof the order of the Dopp-ler limit, but the velocity profiles are far from Gaussian. Afull quantum treatment is certainly required in order toanalyze them (see Subsection 5.D) because of the vicinity ofthe Doppler (TD = 23,uK)and recoil (TR = 4,gK) limits.To extend our results to real 3-Dmolasses,oneshould firstnote that Doppler cooling s still present in these results (see,for example, Figs. 7 and 8). Therefore one takes advantageof both coolings: atoms with velocitiesof up to P/k are firstDoppler-cooledand then reachlowertemperatures by polar-ization gradient cooling. On the other hand, it does notseem easy to single out one of the two new cooling mecha-nisms since both types of polarization gradient are usuallysimultaneously present in a real 3-D experiment.Finally, let us emphasize that the steady-state rms veloci-ties that can be achieved in these new cooling schemes arevery low, since they can approach the recoil velocity [seeexpressions (4.39) and (5.19)]. For such low velocities, a fullquantum approach of both internal and external atomicdegrees of freedom is required. The basis of such an ap-proach and preliminary results are presented at the end ofSection 5. We are currently extending this approach tomore realistic situations, but these preliminary results al-ready appear to be quite promising (see, for example, Ref.31).APPENDIX A: INTERNAL ATOMIC STATE INTHE MOVING ROTATING FRAME FOR THE9+-T- CONFIGURATIONIn this appendix, we establish the expression of the newHamiltonian that governs the atomic evolution in the mov-ing rotating frame introduced inSubsection 3.B.2. We thendetermine the new ground-state energy sublevels in such amoving rotating frame (for a Jg = 1 ground state) and theexpression of the steady-state density matrix in this energybasis.Hamiltonian in the Moving Rotating FrameIn the laboratory frame and in the a+-o- laser configurationwith Go = Go' [see Eq. (2.4)], the laser-atom coupling V isproportional to the component along Ey givenby Eq. (2.5b)]of the dipolemoment operator D. In the atomic-rest frame,we ust replace z with vt, and we get from Eq. (2.5b)

V -D-ey =D. sin kot +DY coshot.

The only term of the initial Hamiltonian H that changes insuch a transformation is V. Now, since D is a vector opera-tor, its components satisfy the well-known commutationrelations with J,[Jr, D] = ihDy, (A3)J, Dy] = -ihD.,

from which it is easy to showthatT(t) [DX in kt + Dycos kt] T+(t) = Dy. (A4)

Thus it is clear that the transform of V by T(t) describes thecoupling of D with a laser field keeping a fixed polarizationey,which proves that T(t) is the unitary operator associatedwith the transformation to the moving rotating frame.The fact that T(t) is time dependent introduces a newterm, ihtdT(t)/dt]T+(t), in addition to T(t)H(t)T+(t) in theHamiltonian H' governing the time evolution in the newrepresentation. According to Eq. (A2), we haveih[dT(t) JT+(t) = hoJ_, (A5)

which is nothing but Eqs. (3.13). To sum up, in the movingrotating frame, the atomic dynamics is due to coupling witha laser field with fixedpolarization ey nd to the inertial term[Eq. (A5)].Ground-State Energy Sublevels in the Moving RotatingFrameIf, in a first step, we neglect the inertial term [Eq. (A5)], heenergy sublevelsin the ground state are the eigenstates Igm)yof Jy (see Subsection 3.B.1). These states can be easilyexpanded on the basis flgm)z1of eigenstates of J,. One gets

1 1Ig,1)Y = + 2 1g~l) + , Igo), i IIgO)y = 1t Ig+1) + 1 Ig l),.

(A6a)

(A6b)

From Eqs. (A6), one can then calculate the matrix ele-ments of the inertial term [Eq. (A5)] between the Igm)yIstates that are necessaryfor a perturbative treatment of theeffect of the term given by Eq. (A5) [see expression (3.14b)].One finds that the only nonzero matrix elements of Vrot=kvJz arey(g+lIVrotlgo)y = y(golVrotlg+i)y = hk/V,y(g-1IVrotgO)y = y(b'oIVrotIg- 1 )y = -hkv/1.

(A7a)(A7b)

Since Vot has no diagonal elements in Igo)y and in themanifold Ig,)y, he energy diagram in the ground state (seeFig. 6) is not changed to order 1 in Vrot. On the other hand,the wavefunctions are changed to order 1 and become

(Al)

The transformation to the moving rotating frame is thenachievedby applying a unitary transformation

I Ig,)~ + I 0 ) y(goIVrotIl'i)yF = Igo)y+ Ig+ ) y(g+lrotAlo)y

ig l~y g i rotlgo)y+ Mg)~hAO'Al')

(Aa)

(A8b)T(t) = exp(-ikvtJl/h).


(A2)


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2042 J. Opt. Soc. Am. B/Vol. 6, No. 11/November 1989In Eq. (A8b), we used y(g+1IVrotlg_-)y = 0. Inserting Eqs.(A7) into (A8) gives Eqs. (3.15) and (3.16). (e2, g,) = -- III,26- P(e-2, -l) = - II-1,Steady-State Density Matrix in the Energy Basis l~g)yJThe terms describing optical pumping in the master equa-tion are diagonal in the lIgm)yJ asis. This is no longer thecase in the {Igm)ybasis. From Eqs. (3.15) and (3.16), onesees that there will be source terms of order 1 in kv/A' in theequation of motion ofy(go g)y and of order 2 in kv/A' inthe equations of motion of the populations y(gmIPgm)yndof the coherence y(g+,lpg-_l)y. The only nonzero matrixelements of p, to order 1 in kv/A', are thus y(golplgai)y Onthe other hand, since the evolution frequencies of thesecoherencesare (A' - Al'), with a damping rate of the orderof r, and since the optical-pumping source term is static (atfrequency 0), the steady-state value of y(go plgt,)y will bereduced by an extra nonsecular term of the order of r/A.Consequently, if we neglect all terms of order (kV/A'), (kv/A')(r'/A') or higher, the steady-state density matrix PSt sdiagonal in the JIgm)y)asis and has the same diagonal ele-ments as those calculated in Subsection 3.B.1, which provesEqs. (3.17).

APPENDIX B: Ig= 1 1 = 2 TRANSITION INTHE +-a- CONFIGURATIONThe purpose of this appendix is to outline the calculationsofthe radiative force and of the momentum diffusion coeffi-cient in the +-ar configuration for the case of a Jg = 1 Je- 2 transition. The calculation is done in the low-powerlimit (


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-(el,c-1)= -(r + 2ikv)(el, e-)+ - [(el, g) - (go, -)]. (B9)2Tt

The steady-state value for p(el, e1) is then, using Eqs. (B2),p(el, e-) = ( 2 + i 2)H0, (B10)

where we have put 3r v1r - 2 1kv#2 2 r' + 4k VI3r Air + 2vlkv (Bll)22 P+4h1o

We also need for the following the quantityA(e2 , eo) + A(eo, e- 2 ) = ((e2Ipleo) + (e0 Iple- 2 ))exp(-2ikt),(B12)

which is found to bep(e2, eo)+ p(eo, e- 2 ) = 2 (2 + iV2 )(H,1 + H.-1 )

+ (4 + jV 4 )(Cr + iCe)I (B13a)with

3 r+ 4a22 [r(vl + V) - 2kv(zll + $3)],=4 'fI 2 + 4h o 2

2 2 + 2V2 [2kv(vl + V3 ) + r(Al + $)]2 + 4hko2(B13b)

and$3 = [(6+ 3kV)S3- - (6 - 3kv)s3+]/6r,V3= (S3+ + S30)/12. (B13c)

Calculation of Ground-State Populations and CoherencesWe now write the equations of motion of the ground-statepopulations. For example, we have for IIIft, = rp(e2 , e2) + - p(el, el) + p(eo,eo)2 6

+ i - [P(gl, e 2) - (e2 , g 1] + i - [p(g1, eo) - p(eo, gl)].2(B14)

The first line describes how the state gl is fed by spontane-ous emission from the three excited states e2,el, e, and thesecond line describes absorption and stimulated-emissionprocesses. In steady state, we can put II, = 0 and replaceexcited populations and optical coherencesby their value interms of ground-state variables. We obtain in this way 6 s-II + 32 +H+ 6 s+IIl + (2v, S-)Cr

+ 2($ -6 +ho S-)Ci. (B15)

In a similar way, we obtain from 11-I = 0= 3E1 +5 snI -5 s+IIl + (21-S+)Cr

+ 2(,l + kv S+)CiFor the population IIO,we ust use

1 = Hr + II0 + II-,

(B16)

(B17)since excited-state populations are negligibleat this order in1. We now write down the evolution equation of theground-state coherence in the moving rotating frame:

P(gl,g-1) = (glIplg.-)exp(-2ikt) = Cr + iCi, (B18a)X(gl,g-1) = -2iko(gl, g-1) + 2 p(el, e-1)

+ rt-[p(e2, eo) + p(eo, e-2)]+ i . [A(gl, o) A(eo, .-)]+ i 2 [(gl, e-2) - (e2,g-0' (B18b)

We again replace the excited-state coherencesand the opti-cal coherences by their steady-state values. Taking the realpart and the imaginarypart of the equation p(gl,g-1) = 0, weget0 = (2 - S)II + 2 $210+ (2 - ][1-+$5Cr-VAi

(B19a)6+ho 3o0= V2+ _a )II + 2][10~

\2. -p)H+ (v2 -s+ a 4r ) + V5Cr+ AC,

where$5= ) & - 2 V1 93,Vs5 - + 3V4 + 3 A + 93P5-J2 2

(B19b)

(B19c)

(B19d)The set formed by the five equations [Eqs. (B15)-(B19)]allows one to calculate numerically the five quantities 1+1,H1, Cr, and C for any value of the atomic velocity. Insertingthe result into Eq. (5.9), one then gets the value of theradiative force for any atomic velocity in the low-powerregime (see, e.g., Fig. 8).In the very low-velocity domain (ho


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2044 J. Opt. Soc. Am. B/Vol. 6, No. l/November 1989

=- II + 1+ 6 II--2 C 6-2ci,6 2 6 3 60=IH,+H 0 ~H, C+2C

1 = H, + H0 + H-1,1=Irl3 1nnJ_5 C 6hoCi6 I, 6 II 6ho 5O= 46II146 II1 - r0Cr -- Ci. (B21)4r 41' rs 4

Then it is straightforward to check that the quantities givenin Section 5 [Eqs. (5.11)]are indeed solutions of this simpli-fied set.On the other hand, for velocities such that kv >>1P, thecoherencebetween the twoground states g, and g-1 becomesnegligible. One can then neglect Cr and C in the threeequations [Eqs. (B16)-(B17)],which givesthe simplified set-5s-Hl + 9s+IIo+ s+II- = 0,

sIl +9s_1Io 5s+Il- =0,II, + o + .- = 1. (B22)

The solution of this set is given in Eqs. (5.15). Inserting thecorresponding values into expression (5.9) for the force, werecover the usual Doppler force, which has been plotted indotted curves in Fig. 8.Calculation of the Atomic Momentum DiffusionCoefficientIn order to calculate the momentum diffusion coefficient,weuse a method introduced by Castin and Milmer.29 Thismethod is welladapted to the present o-+-a-laser configura-tion with a lowlaser power. It consists of writing down thegeneralized optical Bloch equations including recoil. Sincewe are interested here in the momentum diffusion coeffi-cient, we take the limit of infinite atomic mass. Thisamounts to considering an atom with zero velocity but stillexchanging momentum with the laser field, so that (p2 )increases linearly with time as 2Dt. In order to get d(p 2)/dt,we multiply the generalized optical Bloch equations (with v= p/M = 0) by p2, and we integrate overp.Westart with the equation of motion for the population ofIgo, ), i.e., the atom in the ground state g with momentump:(go,plplgo,P) = 2 (el, pplel, P~)+ I (eo,plpleo, )

2 3c.,p (e_1, lple-1, )i2

+ 2 [(g0,plplei, + hk)+ (go,plple-,p - hk)]exp(-iWLt) + c.c.

(B23)The first two lines describe feeding by spontaneous emis-sion,and the last two linesdescribe departure due to absorp-tion. The single or double bar over (ei, pIPlei, ) means an

average over the momentum carried away by the fluores-cence photon, either o polarized or r polarized:r,,: (el, plplel,) = dp' 8 (1 + h2k2)

X(el, p +p'lplel,p +p'),7r: (eo,plpleop) fdP 4hk h2k2

X (eo,p +plpleo, +p'). (B24)We can in the same wayrewrite all the other optical Blochequations. This is done in detail in Refs. 18 and 19 for a Jg= 0 Je = 1 transition or Jg = 1 Je = 0 transition, so thatwe ust givethe main results here. First, owingto the con-servation of angular momentum, the followingfamily:lemp + mhk); Ig,,p + nhk)j,

m=0, 1, 2, n=0,J1remains globally invariant in absorption and stimulated-emission processes. Transfers among families occur onlyvia spontaneous-emission processes. Taking now the fol-lowingnotation:

IIm(p) = (gm,p + mhklplgn, p + mhk), m = 0, +1,(Cr+ iCi)(p)= (g, p + hklpIgl, p - hk), (B25)we can get a closed set of equations for these variables byelimination of the optical coherences and the excited-statepopulations and coherences. Here we ust givethe result ofthis elimination:

2 I(p) + [o(p-hk) + (p+hk)]- Ho(p) II-(P) + g Cr(P)}

JI,(p) 111(p hk) + - IH1(p hk)- H(P)21 36 6+1 0(p) - IIl( +hk)+ - C p+hk)4 36 18

1 Cr(P) 3r Ci(P)6 3Cp)= -2s{ [II(p) II ()J

+ 6 [Ci(p-hk)+ Ci(p+hk)]-7 Ci(P)}66(B26)

If one integrates these equations overp, onejust recovers hestationary values deduced from Eqs. (5.11) for zero velocity.Now,multiplying these equations byp and integrating overp, we get(pCi) = dppCi(p) = 2r (PI1),

11 36k I (pI1); (B27)17 1 + (462 /5r 2 )=-(H-) (27

J. Dalibard and C.Cohen-Tannoudji


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for symmetry reasons one has(pH10) = (PC) = 0. (B28)

multiply Eqs. (B26) by p2 and integrate over p into get the rate of variation of (p2 ):d ( 2 ) = d ((p 2 H) + (p2Ho) + (p 2 H.-)).dt dt

(B29)to symmetry around p = 0, one has (p2 fl.,) =2 1ll). After some calculations, we obtain from Eqs. (B26)(B27)

d (p2 ) =h krso 72 1 + 8represents twice the total momentum diffusioncoeffi-The two contributions D, and D2 appearing in Eq.can easily be deduced from this calculation. For ex-D, represents the fluctuations of the momentumaway by the fluorescence photons. If we replace in(B23) the kernel describing the spontaneous-emissionby just a 6(p') function, we cancel out this cause ofThen, repeating the same calculation again, weexpression similar to Eq. (B30), where 58/85 is re-by 8/17. This represents the contribution of theof the differenceamong the numbers of photonsin each wave (D2 term). We then get D, by differ-D and D2-

authors warmly thank all their colleagues from theSuperieure laboratory and A. Aspect, E. Ari-R.Kaiser, K. Molmer,C. Salomon,and N.Vansteenkiste for many helpful discussions and remarksabout this paper. They are also grateful to W. Phillips andhis group for their comments on this work and communicat-ing their experimental results previous to publication. Fi-nally, they acknowledge stimulating discussions with S. Chu,H. Metcalf, and T. W. Hansch last summer.

REFERENCES AND NOTES1. T. W. Hansch and A. Schawlow, Opt. Commun. 13,68 (1975).2. D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637(1975).3. D. Wineland and W. Itano, Phys. Rev. A 20,1521 (1979); V. S.

Letokhov and V. G. Minogin, Phys. Rev. 73,1 (1981).4. S. Stenholm, Rev. Mod. Phys. 58, 699 (1986).5. S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin,Phys. Rev. Lett. 55, 48 (1985).6. D. Sesko, C. G. Fan, and C. E. Wieman, J. Opt. Soc. Am. B 5,1225 (1988).7. P. Lett, R. Watts, C. Westbrook, W. D. Phillips, P. Gould, andH. Metcalf, Phys. Rev. Lett. 61, 169 (1988).

(B30)

8. Y. Shevy, D. S. Weiss, and S. Chu, in Proceedings of the Confer-ence on Spin Polarized Quantum Systems, S. Stringari, ed.(World Scientific, Singapore, 1989); Y. Shevy, D. S. Weiss, P. J.Ungar, and S. Chu, Phys. Rev. Lett. 62, 1118 (1989).9. J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N.Vansteenkiste, and C. Cohen-Tannoudji, in Proceedingsof the11th Conference onAtomic Physics, S. Harsche, J. C. Gay, andG. Grynberg, eds. (World Scientific, Singapore, 1989).10. S. Chu, D. S. Weiss, Y. Shevy, and P. J. Ungar, in Proceedings ofthe 11th Conference on Atomic Physics, S. Harsche, J. C. Gay,and G. Grynberg, eds. (World Scientific, Singapore, 1989).11. We restrict ourselves here to neutral atoms. Note that fortrapped ions, mechanisms overcomingthe Doppler limit werealsoproposed. They involveRaman two-photon processes: H.Dehmelt, G. Janik, and W. Nagourney, Bull. Am. Phys. Soc. 30,612 (1985); P. E. Toschek, Ann. Phys. (Paris) 10, 761 (1985); M.Lindberg and J. Javanainen, J. Opt. Soc. Am. B 3, 1008 (1986).12. J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).13. J. Dalibard and C.Cohen-Tannoudji, J. Phys. B 18,1661(1985).14. Other consequences of long atomic pumping times are describedin W. Gawlik, J. Kowalski, F. Trager, and M. Vollmer, J. Phys. B20,997 (1987).15. J. Javanainen and S. Stenholm, Appl. Phys. 21, 35 (1980).16. J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2,1707(1985).17. A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, and C. Co-hen-Tannoudji, Phys. Rev. Lett. 57, 1688 (1986).18. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C.Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988); J. Opt. Soc.Am. B 6, 2112 (1989).19. Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. B 6, 2046(1989).20. E. Arimondo, A. Bambini, and S. Stenholm, Phys. Rev. A 24,898(1981).21. J. Dalibard, S. Reynaud, and C. Cohen-Tannoudji, J. Phys. B17,4577 (1984).22. J.-C. Lehmann and C. Cohen-Tannoudji, C. R. Acad. Sci. 258,4463 (1964).23. J. Dupont-Roc, S. Haroche, and C. Cohen-Tannoudji, Phys.Lett. 28A, 638 (1969).

24. M. Lombardi, C. R. Acad. Sci. 265, 191 (1967); J. Phys. 30, 631(1969).25. C. Cohen-Tannoudji and J. Dupont-Roc, Opt. Commun. 1, 184(1969).26. C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, R.Balian, S. Haroche, and S. Liberman, eds. (North-Holland, Am-sterdam, 1977).27. We also reincluded in the definition of the excited states thephase factors exp(+i7r/4) that appear when Eq. (2.7) is insertedinto Eq. (4.1).28. J. Dalibard, A. Heidmann, C. Salomon, A. Aspect, H. Metcalf,and C. Cohen-Tannoudji, inFundamentals of Quantum OpticsII, F. Ehlotzky, ed. (Springer-Verlag, Berlin 1987), p. 196.29. K. Molmer and Y. Castin, J. Phys. B (to be published).30. D. S. Weiss, E. Riis, Y. Shevy, P. J. Ungar, and S. Chu, J. Opt.Soc. Am. B 6, 2072 (1989).31. Y. Castin, K. Molmer, J. Dalibard, and C. Cohen-Tannoudji, inProceedingsof the Ninth International Conference on LaserSpectroscopy, M. Feld, A. Mooradian, and J. Thomas, eds.(Springer-Verlag, Berlin, 1989).32. E. Arimondo, A. Aspect, R. Kaiser, C. Salomon, and N. Van-steenkiste, Laboratoire de Spectroscopic Hertzienne, EcoleNormale Superieure, Universit6 Paris VI, 24 Rue Lhomond, F-75231 Paris Cedex 05, France (personal communication).

Dalibard and C. Cohen-Tannoudji

Laser Cooling Below the Doppler Limit by Polarization

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