Optical molasses

2084 J. Opt. Soc. Am. B/Vol. 6, No. 11/November 1989

Optical molasses

P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook

Center for Atomic, Molecular and Optical Physics, National Institute of Standards and Technology, Gaithersburg,Maryland 20899

Received April 4, 1989; accepted July 27, 1989

We present a summary of the results of a simple two-level theory of Doppler cooling in optical molasses and contrastit with the recent theories of multilevel, polarization-gradient cooling. The effects of single-photon recoil and oftrapping in microscopic optical potential wells are also considered. Experiments are described in which thetemperature of sodium atoms released from optical molasses is measured and found to be well below the Doppler-cooling limit. Measurements of the temperature dependence on many experimental parameters are found to be ingood qualitative agreement with the new theories of polarization-gradient cooling.

1. INTRODUCTION

A. Laser CoolingIn 1975 two groups, Hansch and Schawlow at Stanford Uni-versityl and Wineland and Dehmelt at the University ofWashington,2 independently introduced the idea of lasercooling. While the language and points of view of the twogroups were different, they were describing the same basicphenomenon: the reduction in kinetic energy of movingatoms by the mechanical action of laser radiation at a fre-quency near an atomic resonance. In the Hinsch-Schaw-low view, this cooling results because the Doppler effectmakes the radiation pressure on an atom velocity depen-dent. In the Wineland-Dehmelt view, the reduction in en-ergy results from a Raman process in which transitions fromhigher to lower kinetic energy states are more likely than thereverse. The Hainsch-Schawlow view is more natural whenone is considering the cooling of free atoms, while the Wine-land-Dehmelt view is more appropriate when the atoms arestrongly bound in a potential well. The equivalence of thetwo viewpoints has been detailed by Wineland and Itano.3,4

This paper is concerned mainly with cooling free atoms, sowe work from the Hansch-Schawlow viewpoint.

Hansch and Schawlow envisaged a gas of atoms irradiatedfrom all sides by six laser beams along each of the six Carte-sian coordinate directions. With the laser frequency tunedbelow (to the red of) the atomic resonance, the Doppler shiftmakes it more likely that atoms absorb light from the laserbeams that are propagating most nearly opposite their ownmotion. This slows all the atoms, reducing the temperatureof the gas. For an atom held in a trap, such as an ion trap, itis not necessary to have six beams. With an appropriatelychosen laser-propagation direction and a sufficiently asym-metric trap potential, even a single laser beam is sufficient tocool all the translational degrees of freedom. This is be-cause during some portion of the trapped ion's orbital mo-tion, its velocity has a component opposite the laser beam.

This kind of laser cooling is often called Doppler coolingbecause of the crucial role played by the Doppler effect.Several authors5 -7 calculated the lowest temperature (theDoppler-cooling limit) attainable through this process in thelow-intensity limit and found that the thermal energy was

approximately equal to the energy width of the resonanttransition used for cooling. Later treatments showed thathigher intensities do not yield any lower temperatures.8 9

Doppler cooling was first demonstrated for trapped ions in1978.7 l1 Since that time, laser cooling has also been demon-strated for atomic beams, in both longitudinal and trans-verse directions, and for gases of free, neutral atoms. Lasercooling has permitted a variety of interesting developments,such as high-resolution spectroscopy, quantum jumps, Cou-lomb crystallization, trapping of neutral atoms, and studiesof collisions at ultralow energy. The reader is referred to themany review papers and collections of papers on the subject,including this special journal issue.11-2 0

B. Optical MolassesIn 1985 an additional insight was obtained about the processof laser cooling. While Hdnsch and Schawlow realized thatthe strong damping of the atomic motion by the red-tunedlaser beams resulted in rapid cooling, they did not discussthe effect on the atomic motion once equilibrium is reached.In fact, such laser-cooled atoms can exhibit diffusive mo-tion.21 22 The atomic velocity is damped so quickly, and theatom moves such a short distance in a damping time, thatthe atomic motion is similar to Brownian motion. Thisresults in a quasi-confinement of atoms within the three-dimensional, six-beam geometry envisaged by Hdnsch andSchawlow. Because of the viscous nature of the confine-ment, this six-beam arrangement has been called opticalmolasses. By extension, strong cooling accompanied by dif-fusive motion in one or two dimensions is also called opticalmolasses.

Optical molasses was first observed by a group at BellLaboratories.2 ' The reported confinement time and tem-perature of the atoms were consistent with a theory based onDoppler cooling. The temperature reported was at thattime the lowest ever achieved by laser cooling, and it wasseen to be the ultimate realization of the Hdnsch-Schawlowcooling idea. Shortly thereafter, additional experimentsbrought this view into serious question. Measurements inour laboratory 2 3 showed that several features of optical mo-lasses were in strong disagreement with the theoretical pre-dictions, and the Bell Labs group24 discovered an anoma-

0740-3224/89/112084-24$02.00 © 1989 Optical Society of America

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Vol. 6, No. 11/November 1989/J. Opt. Soc. Am. B 2085

lously long-lived supermolasses that had no place in thetraditional theory.

Finally, our group discovered that the temperature of at-oms cooled by optical molasses could actually be much lowerthan the Doppler-cooling limit, the lowest temperature thatwas thought to be achievable by laser cooling.25 This dis-covery, and its confirmation,26 27 prompted a reexamination,focusing on the multilevel or multistate nature of the realatoms involved in laser cooling. Eventually two groups pro-posed similar theories based on the idea of nonadiabaticmotion of a multilevel atom through the optical polarizationgradient in molasses.2 6-29 More-recent experimental resultsfrom our laboratory have given qualitative support to thenew theories.

C. Scope of This PaperThis paper concentrates on the research done in our labora-tory on optical molasses, with brief discussions of some ofthe work of other groups. In Section 2 we present the impli-cations of a view of optical molasses based on the Hdnsch-Schawlow picture of laser cooling and the idea of viscousconfinement. This classical molasses theory is a simplegeneralization of the well-known two-level, one-dimension-al, low-intensity laser-cooling theory. We also discuss thetwo-level, arbitrary intensity theories worked out by severalauthors 89 3 0'3' and how they differ from the low-intensitytwo-level theory. In Section 3 we give the results of the newtheory of polarization-gradient laser cooling (presented indetail by two other groups in papers in this special journalissue28 29) and discuss their implications. We also considerthe influence of discrete recoil, and we discuss possible ef-fects due to the presence of dipole-force potential wells. InSection 4 we briefly summarize the early experimental re-sults on optical molasses, both in our own laboratory andelsewhere, emphasizing the ways in which these results arein disagreement with the classical theory of molasses. Wediscuss in some detail more-recent measurements in ourlaboratory designed to test explicitly the new theories ofpolarization-gradient laser cooling and compare the resultswith the predictions of the theory. In Section 5 we discussaspects of optical molasses that are still not well understoodand speculate about the future of this field. We do notprovide a complete review of the work of other groups study-ing optical molasses. Rather, we discuss such work as itrelates to our own and provide references.

2. CLASSICAL MOLASSES

By optical molasses we mean not simply laser cooling incounterpropagating laser beams but also the associated phe-nomenon of diffusive atomic motion that occurs when thelaser cooling produces a sufficiently strong damping force.For the theory of laser cooling, we adopt the viewpoint pro-posed by Hansch and Schawlowl and developed in moredetail by other authors (see, for example, Refs. 3, 6, 8, and 9)that laser cooling results from the Doppler-effect-inducedvelocity dependence of the radiation pressure force. Thistheory, which we call classical molasses, assumes a two-levelatom interacting with multiple laser beams, that the laserbeams do not interfere with one another, and that the atomdoes not experience coherent interaction involving succes-sions of absorptions and stimulated emissions. For irradia-

tion by a weak laser field in one dimension, this theory givesresults identical with the low-intensity limit of less restrict-ed theories of two-level, one-dimensional laser cooling.8'9 "'1We generalize the simple theory in a straightforward (butapproximate) way to include moderate-intensity light fieldsand two or three dimensions, discussing the utility and va-lidity of the generalizations. Within this classical molassesframework we discuss results such as the Doppler-coolinglimit and the diffusive motion of atoms in optical molasses.

A. One-Dimensional Classical Molasses

1. Damping ForceConsider a two-level atom with a frequency interval (ignor-ing recoil energy) between ground and excited states of co,irradiated by a laser beam that we assume to be a plane waveof angular frequency and wavelength X. We take thefrequency width of the laser to be small compared with allother frequencies in the problem. The detuning of the laserfrequency from resonance is given by A = - o. An atommoving with velocity v in the direction of propagation of thelaser sees the laser frequency Doppler shifted down by 2rv/X= kv, for a total detuning of A - kv. The excited-statepopulation decays radiatively to the ground state at a rate F;the strength of the laser-induced coupling between theground and excited states is characterized by a saturationintensity Io such that when the laser intensity I = Io thetransition is power broadened by a factor of 7_. The relationbetween the intensity and the Rabi frequency co, is I/o =2coL2/J 2 . For light with a wavelength X, the momentumcarried by one photon is h/X = hk. The average force on anatom moving in the positive direction is this momentumtimes the average rate of absorbing photons:

P I/I.F = ±hkh - -u

2 1 + I/10 + [2(A F kv)/] 2 (1)

Here the upper (lower) sign refers to the force from a travel-ing plane wave propagating in the positive (negative) direc-tion. This force, often called the radiation-pressure force,scattering force, or spontaneous force, is in the direction ofpropagation of the light. Note that the maximum value ofthis force is hkr/2 for I/b» >> 1. For sodium atoms irradiatedby light at the D2 line, X = 589 nm, a transition for which r/27r = 10 MHz, the acceleration corresponding to this maxi-mum force is 106 m/sec2. Io for the strongest transition (F =2, mF = 2 - F' = 3, mF' = 3) is 6 mW/cm

2.

There is no contribution to this average force from themomentum transfer on spontaneous emission because thatemission is symmetric. The momentum transfer does, how-ever, cause fluctuations of the force, as we shall see in Sub-section 2.A.2.

Now let us consider the case of two counterpropagatinglaser beams. Assuming that the two waves act independent-ly on the atoms, the average force on the atom is given by F++ F_. This assumption is true for a two-level system only inthe limit of low intensity: I/Ib << 1. In this case we have

hkr I kv 16A/r (2)2 Io r 1 + 8 (A 2 + k2

V2 ) + 16 (A2 -k2v2)2

p,2 r4

We write the damping force in this way to emphasize that it

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1

U,

a0

0U-

0.5

0

-0.5

- 1-4 -2 0 2 4

2 k v /rFig. 1. Force versus velocity for A = -r/2 and low intensity. Thedashed curves are the individual forces due to the two counterpro-pagating beams, and the solid curve is the net force. Note the linearregion around v = 0.

is the product of the maximum radiation-pressure force, anormalized intensity, the ratio of the Doppler shift to thelinewidth, and a factor depending on detuning and velocity.Figure 1 shows F+, F_, and their sum for 2A/r = -1 and lowintensity.

In the approximation that kvl << r and Ikvl << l we have

F = 4k I kv(2A/r)IO [1 + (2A/r) 2 ] 2

For A < 0 this is a friction force, linear in and opposing v.The atoms see the laser beam opposing their motion Dopplershifted closer to resonance, so they absorb photons directedopposite their motion more often than photons directedalong their motion. The force is damping for all velocities ifA < 0, and it is linear if Ikvl << AI or IkvI << r. If A > 0, theforce accelerates the atoms. We assume that A < 0 in allthat follows. In order for Eq. (3) to be meaningful as adamping force, we must further assume that the recoil froma single-photon absorption does not change the atomic ve-locity so much that the atom is Doppler shifted by a signifi-cant fraction of the transition linewidth. That is, kvR << F,where VR = hk/M is the velocity change experienced by anatom on the absorption or emission of a single photon. Forsodium this recoil is 3 cm/sec and kvR/27r = 50 kHz, so kvR <<r is satisfied.

Writing this damping force as F = -v, we have thedamping coefficient

ag = -4hk2II-[1 + (2A/)2]2 (4)

The maximum damping (with I/o << 1 and kvI << IAI, r) isobtained at 2A/r = -1i/V, although as we see below this isnot the detuning for lowest temperature.

2. Doppler-Cooling LimitThe theory of the cooling limit given here is similar in spiritto those given in the usual treatments of laser cooling (see,for example, Refs. 3, 8, 9, and 11). The damping force F =

-av leads to a rate of losing kinetic energy of

(dE/dt)c001 = Fv =-CYV 2. (5)

This cooling rate is proportional to the kinetic energy, lead-ing to an energy-loss time constant

E = M/2acol= (dE/dt)0 oo1

and a velocity-damping time constant

'damp ( /dtv M/a.

(6)

(7)

Now let us consider the rate of increase in energy that isdue to heating. The damping force reduces the averagevelocity of the atoms to zero, but, while the mean velocitybecomes zero, the mean squared velocity does not. At thesame time that the kinetic energy is being removed by thedamping force, it is being supplied by heating from therandom nature of the absorption and emission of photons.The damping force that we have derived is only the averageforce; the fluctuations of the force produce heating. Anatom with zero mean velocity is equally likely to absorb aphoton from the positive or negative traveling waves. As aresult, each absorption represents a step of size hk in arandom walk of the momentum of the atom, with equalprobability of positive and negative steps. In the same way,each spontaneous emission represents a random-walk step.In a truly one-dimensional problem the spontaneous emis-sions are along either the positive or negative direction, sothat each cycle of absorption followed by spontaneous emis-sion represents two random-walk steps. In a more realisticsituation (such as one-dimensional transverse cooling of anatomic beam) in which the spontaneous emissions are intoall directions, the average number of steps in the direction ofinterest (along the laser-propagation axis) is reduced. Anisotropic radiation pattern would produce an average of 4/3steps per absorption-emission, while a dipole pattern wouldyield 7/5 steps, compared with 2 steps for the truly one-dimensional problem. After a given number of steps, themean square momentum of the atom grows by the number ofsteps times the square of the photon momentum, h2k2 . If wedenote the total photon scattering rate by R, then for thetruly one-dimensional case we have

d(p2 )/dt = 2h2k2R = 20p, (8)

where fp is the momentum diffusion constant. Thus thekinetic energy p2 /2M increases at a rate

(dE/dt)heat = h2 k2R/M = LJl /M. (9)

The total scattering rate, in our approximation, is the sum ofthe scattering rates from the positive and negative travelingwaves, which is just the sum of the magnitudes of the indi-vidual average forces from these beams divided by the pho-ton momentum hk. With the usual approximation that kvl<< AI and KV << r (and that I/10 << 1), we have

(dE/dt)heat = M r I/Io (10)

Note that the cooling rate, Eq. (5), depends on the square ofthe velocity, whereas the heating rate, Eq. (10), depends onlyon the intensity and the detuning.

At equilibrium, the heating and cooling rates are equal.We set (dE/dt)heat + (dE/dt)c 001 = 0 and use Eqs. (4), (5), and(10) to obtain a condition on v2

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v2 = h 1 + (2A/r)2 (11)4M 21AI/r'

Since the expressions for the heating and cooling rates aretime or ensemble average rates, we interpret Eq. (11) asgiving the mean square velocity of a group of atoms undergo-ing laser cooling, or the time average of the squared velocityof a single atom. Taking the thermal energy to be kBT/2 perdegree of freedom, kBT/2 = MUrms2/2, we have in this one-dimensional example

kBT = =_ h 1 + (2A/r) 2 (12)

ax 4 21A1/r'

This temperature has a minimum value when 2A/r = -1,giving

kBTmin = hF/2. (13)

This is the so called Doppler-cooling limit. For sodium it isequal to 240 ytK, which corresponds to a rms velocity of 30cm/sec in one dimension. This result justifies the assump-tion that kvl << r, which allowed us to write a damping forcelinear in velocity. Note also that the recoil velocity VR <<

vrms, which further justifies the treatment of the velocity as acontinuous variable for the case of sodium.

This Doppler-cooling limit is equivalent to the statementthat the minimum kinetic energy is essentially the same asthe energy width of the cooling transition. This is not theonly natural energy unit that one might guess as being thecooling limit. One such natural energy, suggested as thecooling limit in Ref. 1, is the kinetic energy where the atomicvelocity gives a Doppler shift kv equal to the half linewidthof the cooling transition r: kBTDop width = M(/2k) 2 . An-other, suggested in Ref. 32, is the recoil energy imparted toan atom, initially at rest, when it absorbs the momentum hkof a single photon: kBTR = h2k 2/M. The actual Doppler-cooling limit, Eq. (13), is the geometric mean of kBTR andkBTDop width-

One may ask whether a laser-cooled atom or collection ofatoms can be said to have a temperature. Normally oneconsiders a system with a temperature to be in equilibriumwith a reservoir, and in this case the identity of that reservoiris not clear. Even if there are many atoms, they do notinteract with one another but only with the radiation field.The radiation field, including the vacuum, may in somesense be a reservoir, but the sense in which this may be truehas not been made rigorous. Nevertheless, it can be shown3 3

that the solution of the Fokker-Planck equation for a systemacted on by a friction force proportional to velocity, as in Eq.(3), and with a random noise input independent of velocity,such as is provided by the random nature of photon absorp-tion and emissions in the small recoil limit, is a Maxwell-Boltzmann distribution. Furthermore, our own Monte Car-lo simulations of the laser-cooling process for a single atom,in which every emission and absorption is treated as a ran-dom event, leads to a Maxwell-Boltzmann distribution,when averaged over time (see Subsection 3.B below). Forthis reason, we believe that we are justified in saying thateven a single Doppler-cooled atom, under the conditions ofsmall recoil velocity and small thermal velocity, has a tem-perature.

B. Simple Generalizations of Classical MolassesThe expression given in Eq. (1) for the force of a single planewave acting on a two-level atom is valid even when I/ > 1.When we include two waves, however, we must make the I/Io<< 1 approximation to obtain Eq. (2). Now a rather obviousway to include a nonnegligible I/Io in Eq. (2) would be toinclude a saturation term in the denominator correspondingto the total intensity from both waves.23 (An alternativemethod is described in a footnote in Ref. 34.) If, in addition,we wish to include the effect of other pairs of beams directedalong the orthogonal coordinate axes in a two- or three-dimensional situation, we might simply multiply the satura-tion term by N, the number of dimensions along which wechoose to cool (similar to a treatment in Ref. 6). Underthese assumptions the damping coefficient is

ax = -4hk 2 I 2A/rIo [1 + 2NI/Io + (2A/r) 2 ] 2 (14)

If N = 2 or 3, and all axes are equivalent, the velocity andforce are vectors with F = -av.

Now we might as easily have chosen the saturation term tobe simply I/IO, which would be equivalent to assuming thateach wave independently saturates the atom, with no effectfrom the other 2N - 1 waves. Of course, neither choice iscorrect. Consider just the one-dimensional case for a two-level atom: the counterpropagating waves will presumablyhave the same polarization (the one needed to drive the two-level transition), and there will be a standing wave. As weshall see below, this standing wave will have profound effectson the cooling and heating that are not contained in Eq. (14).If we wish to avoid a standing wave by having orthogonalpolarizations in the two waves, the two polarizations willdrive different transitions, and we require more than twolevels.3 5 Nevertheless, we shall find that the simple ap-proach of Eq. (14) leads to results that are approximatelycorrect for moderate intensity.

Differentiating Eq. (14) with respect to A or I, we find theconditions for maximum damping:

With fixed detuning: I (2A/2 + 1

With fixed intensity:2= (1 + 2NI/o)1/2

r \

(15a)

* (15b)

Note that the optimum detuning of Eq. (15b) is identical tothat derived from Eq. (4) if we replace the linewidth r withthe (assumed) power-broadened linewidth r(1 + 2NI/Io)1/2.Simultaneously optimizing the damping with respect toboth detuning and intensity, we find that

hk2

cx 4 for 2A/P=-1, 110 = 1N. (16)

For sodium, the minimum velocity damping time M/cx corre-sponding to this maximum value for ax is 13 sec in onedimension and 40 ,sec in three dimensions.

The total photon-scattering rate from all 2N waves is

r 2NI/Io -1,2IA- /2 1 + 2NI/I + (2A/r)2

k I )

leading to a diffusion constant Dp = h2k2R. With the condi-tions of Eq. (16) this gives a heating rate p/M =

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(N/2)kBdT/dt, which for sodium is equivalent to 75 ,4K/lsecin one dimension or 25 pK/psec in three dimensions.

Equating the heating and cooling rates at arbitrary I andA, we find from kBT = O)p/Nx that

kBT = hr 1 + 2NIIO + (2A/r)2 (18)

Note that, in the limit of I/Io << 1, Eq. (18) is identical to Eq.(12): The low-intensity cooling limit in two or three dimen-sions is the same as in one dimension. If we substitute themaximum damping conditions of Eq. (16) into Eq. (18) wefind that kBT = hr. This is twice the minimum tempera-ture of hr/2 obtained for I/Io << 1. At the intensity formaximum damping, the heating has increased more than thedamping, resulting in the higher temperature.

While the total kinetic energy is larger in higher dimen-sion, the energy per degree of freedom is the same in alldimensions. This is because we have assumed that in one ortwo dimensions the emitted photons are all emitted in theline or plane under consideration. In an experimentallyrealizable situation, the one- or two-dimensional tempera-ture will be lower and will depend on the details of theradiation pattern. In three dimensions we assume that thephotons are emitted isotropically. If the emission is notisotropic we may have anisotropic temperatures, that is,different values of (vi 2). The anisotropy could depend, forexample, on the relative polarization of the various lightbeams used in the cooling.

As noted above, the generalization to nonnegligible inten-sity and multiple dimensions is not strictly valid. There is,however, a procedure suggested by Dalibard 3 6 that makesthis generalization exact: If one alternates each of the 2Ntraveling waves so that each is on for 1/2N of the time andthey are never on simultaneously, there is no standing wave.The atoms will still see the same average force given by Eq.(14) as long as the alternation time is short compared withthe damping time Tdp but long compared with the radia-tive decay time r-1. In that case, we obtain Eq. (14), with I/lo being the average intensity in each beam and 2NI/1 0 theinstantaneous intensity. The conditions for optimumdamping are then the same as for our simple generaliza-tion.3 6

C. Cooling with a Single Traveling WaveAlthough we are concerned primarily with cooling by pairs ofcounterpropagating waves, it is interesting to note that cool-ing by a single traveling wave gives the same equilibriumtemperature as the case discussed above. The single-travel-ing-wave configuration is used to slow and cool an atomicbeam and has been treated extensively elsewhere.22 3738

Here we will deal only with the question of the cooling limit.A central concern in cooling an atomic beam is to prevent

the atoms from detuning far out of resonance with the laseras they decelerate. Two main methods are used in practice:chirp cooling, in which the laser frequency is changed tocompensate for the changing atomic Doppler shift, and Zee-man-tuned cooling, in which a spatially varying magneticfield compensates for the Doppler shift. Here we shall as-sume chirp cooling and note that the situation for Zeemantuning is essentially the same.

We assume that the atomic beam has a velocity distribu-tion centered about a positive velocity V' with a small spread

of velocities. From Eq. (1), the force on an atom with veloci-ty V' + v is

F = -hk r /Io

2 1 + I/IO + 2[A' + k(V' + v)]/lr 2

where A' is the detuning. The group of atoms with a velocityV' will experience a deceleration a = F'/M. We shall trans-form into a frame decelerating with these atoms. Further-more, we shall assume that the detuning of the laser ischanged at such a rate that the total detuning, including theDoppler shift associated with the moving frame, remainsconstant. That is, A'(t) = A'(0) - kV'(t), where V'(t) =V'(O) + at. This ensures that the force and hence thedeceleration remain constant. Writing the force in the de-celerating frame, taking into account the transformed veloc-ities and the Doppler shift of the laser frequency, we have

1r I/I.F=-Maa-hhk (2 (20)2 1 + I/10 + [2(A + kv)/] 2

where Ma is the (fictitious) inertial force in the deceleratingframe and is just Eq. (19) evaluated at v = 0. Substitutingfor Ma, using the Doppler-shift transformation A' = A -kV', and taking kvJ << 1A1 and |kv| << r, we find the force inthe decelerating frame:

F = 2hkr I kv(2A/r) (21)Io [1 + I/Io + (2A/r) 2]2

This is exactly the force found from Eq. (14) for cooling bytwo counterpropagating waves (N = 1), except that 2I/Io hasbeen replaced with I/Io since there is now only one wave.Using Eq. (21) to get cx and the single-beam scattering rate toget 5Jp, we find that the cooling limit in the deceleratingframe is equivalent to the one derived for counterpropagat-ing beams,3 9 40 Eq. (18). When we make no approximationin assuming nonnegligible intensity, the temperature in thedecelerating frame is

hF 1 f I/1 0 + (2A/F) 2

kBT=-4 21A1/r (22)

which is identical to Eq. (18) if we interpret I/Io as the totalintensity seen by the atoms. We have implicitly assumedthat the deceleration is allowed to continue long enough forequilibrium to be established. Note also that in the deceler-ating frame we must have A < 0 in order to cool. Thisproblem is equivalent to opposing a constant external forcewith radiation pressure from a single traveling wave. Whenthe radiation pressure force and the external force balance,the cooling limit is given by Eq. (22).

D. Validity of the GeneralizationsThe extension of the classical molasses theory to multipledimensions is valid, in a two-level system, for low intensity.However, as we have noted above, the extension of the the-ory to nonnegligible intensity is not valid, except in the casewhen the molasses beams are alternated so that only onebeam is on at a time. When multiple beams are presentsimultaneously, we cannot consider the beams as acting in-dependently once I/o 1. Inclusion of saturation, as inSubsection 2.B, does not properly account for the effects ofhaving two strong waves simultaneously. One of the key

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processes left out of that treatment is the possibility that anatom may absorb a photon from one of the waves and bestimulated to emit into the other one. At high intensity thiscan be an important process leading to dipole (gradient)forces and to fluctuations of these forces, which contributeadditional heating. Dipole forces have been treated in somedetail by other authors (see, for example, Refs. 8, 9, 11, and30).

Here we briefly summarize some features of atomic mo-tion in counterpropagating waves with nonnegligible inten-sity. We consider two cases: a two-level system in whichthe counterpropagating beams form a standing wave and amultilevel system in which the the counterpropagatingbeams are orthogonally polarized and do not form a standingwave. We give only the results and indicate the degree towhich these more exact treatments differ from the classicaltreatment given above.

A number of authors have treated the case of atomic mo-tion in an intense standing wave. Minogin and Serimaa3'used a continued fraction approach, which yields exact solu-tions for arbitrary atomic velocity. Gordon and Ashkin8and Cook9 adopted a perturbative approach valid for smallvelocities. Dalibard 36 gives a convenient perturbative ex-pression for the force on an atom moving in a standing wave,where the force has been averaged over a wavelength of light.With F = -av, he finds that

hk 2 4s(2A/F) (1+2s)(1 + 4s)3/2 1 + (2A/r)2

_ 2A [1 + 6s + 6s2 - (1 + 4s)3/2]} (23)

for Ikvi << r. Here, s is the detuning-dependent saturationparameter for one of the two traveling waves that composethe standing wave:

I/IO

1 + (2A/r) 2 (24)

The features of this standing-wave molasses that are sodifferent from those of classical molasses are due mainly tothe ability of the atom to interact successively with both ofthe counterpropagating traveling waves. Dalibard et a. 41

have considered a multilevel system interacting with orthog-onally polarized counterpropagating waves that eliminatesthe possibility of such successive interaction. Specifically,they considered a J = 0 - J = 1 transition irradiated byoppositely directed + and waves. Such a configurationdoes not form a standing wave, and, because of the selectionrules on transitions between the m levels, absorption fromone wave cannot be followed by stimulated emission fromthe other. For Ikv << r they find a damping coefficient

a = -hk 2 2s 2(2A/r) , (25)1 + 2s (2A/r) 2 + J1 + 1/ 2s[(2A/r) 2 + 1]12

where s is given by Eq. (24) [this differs slightly from Eq.(3.22) of Ref. 41 because of a different definition of s].

Figure 2 compares the damping coefficient for classicalmolasses, Eq. (14), standing-wave molasses, Eq. (23), ando'+-o molasses, Eq. (25). For s << 1 or I/Io << 1, these allreduce to Eq. (4). For moderate powers, I/Io < 1, the ex-pressions are not much different. For I/I > 1 the standing-

wave damping becomes significantly different from the clas-sical and o-+-o-r damping. Only at high power does the -+-a- damping differ markedly from the classical damping.

One of the most remarkable features of standing-wavemolasses is that for moderate to large I/Io and detuningslarger than needed to maximize cx, the sign of cx reverses,producing acceleration rather than damping. When thedetuning is positive rather than negative, the sign reversalleads to strong damping. This phenomenon has been usedto produce blue (stimulated) molasses and is discussed ex-tensively in Refs. 30 and 42.

The standing-wave case also differs from classical molas-ses in the rate at which heating occurs. Successive interac-tions can lead to high heating rates at high intensity. At lowintensity, however, the heating is approximately the same asfor classical molasses. For plane waves in the +-o-- case,the heating is essentially the same as for classical molasses.Explicit expressions for the momentum diffusion constantsare given in Refs. 8, 9, and 36 for the standing-wave case andRef. 41 for the r+-o-- case. The temperature for both ofthese cases is never lower than the Doppler-cooling limit.

In conclusion, for high intensity the exact solutions foratomic motion in counterpropagating waves can differ sub-stantially from the results of classical molasses. For moder-ate powers (I/Io < 1), the results of classical molasses are inapproximate agreement with the more exact treatments.Both the damping and the heating are somewhat higher inthe standing-wave case than in classical molasses, but thegeneral behavior is quite similar. Thus, at least for moder-ate intensity, classical molasses gives an approximate de-scription of a two-level-system in counterpropagating waves.For the a+-o--, J = 0 - J = 1 configuration of Ref. 41, forwhich successive stimulated processes are not possible, clas-sical molasses gives a good approximation even at moderateintensity.

E. Diffusive Motion in Classical MolassesWe now consider the motion of atoms in optical molasses inmore detail. In Subsection 2.B we found that the minimumvelocity-damping time for sodium atoms in three-dimen-sional classical molasses is 40 Asec. The three-dimensionalrms velocity at the Doppler-cooling limit is 50 cm/sec, so anatom with that average velocity will move only 20 Am duringa damping time. This short mean free path indicates thatthe motion will be diffusive if the extent of the molasses ismuch larger than 20 ,im.

Following the viewpoint adopted by Chu et al.,21 we treatthe motion of atoms in optical molasses as the Brownianmotion of particles in a viscous fluid.4 3 The position diffu-sion constant D., is given by

(26)kBT ,

cx Ncx2

where we have used kBT = p/Na. After a time tD the meansquare diffusion distance in N dimensions is N times themean square distance along a single axis:

(r2 ) = N(x 2 ) = 2NtD0x = D2pCx2

(27)

Substituting the results of Subsection 2.B into Eq. (27), weobtain

Lett et al.


I/1 0 0.3 a

. . I'i ' ' I , ' I '0.5 1 1.5

Detuning2(A/r)

2.5 3

0.6

C

CO-C04

0.4

0.2

0

-0.20 0.5 1 1.5 2 2.5 3

Detuning (A/r)

0 0.5 1 1.5Detuning

2 2.5 3

(A/r)

0.6

0.4

-C0:

0.2

0

-0.2

I/I o= 10 dI \

I \

I %

1 -I iiI %I %

I -- - - _ _ - - - -,

0 0.5 1 1.5 2Detuning (A/r)

2.5 3

Fig. 2. Damping coefficient a versus detuning red of resonance for classical molasses (solid), standing wave (dashed), and +-a- (dotted), fordifferent intensities. a, I/Io = 0.3; b, I/Io = 1.0; c, I/IO = 3.0; d, I/IO = 10.0

8k2(r2 ) I (2A/r) 2

tD Nr Io [1 + 2NI/I0 + (2A/r)2]3(28)

Maximizing the diffusion time with respect to both intensityand detuning, we obtain

tD = 4k2 2 ) for2A/r = -1, I/Is = 1/2N. (29)

Note that the condition for maximum diffusion time isslightly different from the condition for maximum dampingcoefficient given in Eq. (16).

We have assumed that the molasses is of infinite extent.For sodium the time required to diffuse 0.5 cm in an infinitemolasses is 750 msec for the conditions of Eq. (29). In a realsituation the molasses region will be formed at the intersec-tion of laser beams of finite extent. Let us assume2 ' thatthis intersection region is approximated by a sphere of radi-us r and that any atom reaching the boundary of the sphereis lost. For a variety of initial distributions within thesphere, it can be shown that the number of atoms remainingwithin the sphere after a time t is a sum of exponentiallydecaying terms.44 The dominant term is

n(t) = no exp(-t/1M), TM = r 2/'r 2 Ovx. (30)

Using Eqs. (27) and (30) with N = 3, we also have TM = 6tD/

-w2. For sodium the maximum exponential time constant fordecay of a 0.5-cm-radius classical molasses is 450 msec.

Now consider the effect of an external force applied to anatom in optical molasses. The atom will acquire a driftvelocity such that the damping force cancels the externalforce:

Vdrift = Fext/cx (31)

If the external force is gravity, and we take optimum damp-ing conditions in three-dimensional sodium molasses, wefind that Vdrift = 12Mg/hk2

= 0.37 mm/sec. This is muchsmaller than the thermal velocity of 50 cm/sec, and the drifttime through a 0.5 cm radius molasses is I14 sec, which islong compared with the molasses decay time. Therefore weexpect little effect from drift due to gravity.

Another, more severe, source of drift is imbalance betweenthe intensities of counterpropagating molasses beams. Weassume that in one pair of molasses beams the intensity inone beam is (1 + ) times the intensity in the other beam.

0.6

C

-C.

0.4

0.2

0

-0.20

0.6

cq 0. 4

-

CO 0.2

0

-0.2

Lett et al.


Generalizing Eq. (1), we have an unbalanced force along thedirection of the stronger beam of

hkre I/IO2 1 + 2NI/Io + (2A/r) 2 (32)

Using Eq. (14) for cx, we have

E 1 + 2NI/Io + (2A/r)2

Vdrift= 8k 21A1/F (33)

For the optimum damping conditions of Eq. (16) in threedimensions we have Vdrift = PE/2k. For sodium r/2k = 3 m/sec. A 1% imbalance leads to a 3 cm/sec drift velocity, or adrift time for 0.5 cm of 167 msec, considerably shorter thanthe maximum molasses decay time. Therefore we expectclassical molasses to be highly sensitive to imbalance be-tween the opposing beams.

3. NEW THEORIES OF LASER COOLING

None of the theories discussed in Section 2 predicts tem-peratures any lower than the Doppler-cooling limit. In fact,no theoretical treatment of free-atom laser cooling before1988 of which we are aware predicted a steady-state tem-perature below that limit. Our laboratory's observation oftemperatures well below the Doppler-cooling limit initiateda search for another mechanism for laser cooling.

By mid-1988 two groups, Dalibard and Cohen-Tannoudjiat the Ecole Normale Sup6rieure in Paris and Chu andcolleagues at Stanford University, had proposed a newmechanism for laser cooling, one that relies in a fundamentalway on the degeneracy of the ground state, optical pumping,and the spatial gradient of the polarization of the coolinglight. The Ecole Normale group has called this mechanismpolarization-gradient laser cooling.

The polarization-gradient damping force is in some waysradically different from the damping force in classical mo-lasses or other earlier theories of laser cooling. The moststriking and counterintuitive feature of the new force is thatthe damping coefficient is independent of laser intensity atlow intensity. Since the heating rate is proportional to theintensity, as in classical molasses, this leads to a temperaturethat decreases linearly as the intensity. At the same time,the range of coolable velocities also decreases with intensity.Recall that for classical molasses the temperature is inde-pendent of intensity at low intensity. Furthermore, themagnitude of the new damping coefficient can be consider-ably higher than the maximum coefficient achievable withclassical molasses. As we shall see, this larger dampingcoefficient results in temperatures well below the classicalcooling limit under certain conditions of intensity and de-tuning.

In Subsection 3.A we briefly summarize the qualitativefeatures of the new theory of multilevel, polarization-gradi-ent laser cooling advanced by the Ecole Normale and Stan-ford groups. We also quote the major quantitative resultsfor comparison with experiments discussed in Section 4.For more detail concerning these theories, we refer the read-er to the early expositions (Refs. 26 and 27) and especially tothe more comprehensive treatments given by these groups inthis special journal issue.2 8 29

Besides the polarization-gradient laser-cooling mecha-nism, other mechanisms have been discussed that are similarin spirit but do not explicitly involve polarization gradi-ents.45 These mechanisms involve spatially dependent op-tical pumping among either degenerate or nondegenerateground states but will not be discussed further here. Allthese mechanisms may play some role in a system as compli-cated as sodium in three-dimensional optical molasses.

In Subsection 3.B we consider the influence of the dis-creteness of the momentum transfer in laser cooling, an issueof particular significance since the new theories predictminimum kinetic energies comparable with the energy im-parted by a single photon recoil. In Subsection 3.C wediscuss the possible effect of wavelength-sized dipole-forcepotential wells on the motion of atoms in optical molasses.

A. Polarization Gradient CoolingThe Ecole Normale and Stanford groups have consideredtwo separate one-dimensional cases in which the polariza-tion gradient of the light field is responsible for the damping.The first case is counterpropagating waves with orthogonalcircular polarization. This a+---r, or corkscrew, configura-tion produces a local polarization that is linear everywherewith a direction of polarization that rotates a full turn everywavelength. The corkscrew polarization corresponds to apure rotation of polarization and gives rise to polarization-rotation cooling.

The second case is counterpropagating waves with orthog-onal linear polarizations. This linear-perpendicular-linear,or 7rx7ry, configuration has a local polarization that changesfrom a+ through elliptical to linear, to elliptical, to a-, andback to v+ in a distance of X/2, but with a period of X whenphase is included. This changing character of polarizationgives rise to ellipticity gradient cooling.

Polarization-rotation and ellipticity-gradient cooling pro-duce quite different damping coefficients. (See, for exam-ple, the plots by Ungar et al.28 comparing these dampingforces as obtained from numerical solutions of the opticalBloch equations describing the cooling of sodium.) Dali-bard and Cohen-Tannoudji29 obtain analytic results for thesimplest model systems exhibiting the effect and identifydistinct physical mechanisms for each case.

In the corkscrew polarization case, for a J = 1 -J' = 2optical transition an atom at rest is optically pumped into adistribution of ground m states having alignment (differentpopulations for states with different ml) along the polariza-tion axis but no orientation (asymmetry of positive andnegative m-level population). When the atom moves paral-lel to the laser-propagation axis, an orientation developsalong this axis. The population asymmetry is such that themost populated state is the one that absorbs light moststrongly from the circularly polarized wave opposing its mo-tion. This asymmetry in the absorption accounts for thedamping force in polarization-rotation cooling.29

In the linear-perpendicular-linear case for a J = 1/2 - J'= 3/2 optical transition an atom at rest is optically pumpedinto a distribution of m states that depends on the local,spatially varying, polarization character. The light-shiftedenergies of these states also vary in space so that a movingatom climbs and descends the hills and valleys of the modu-lated optical potential. At low velocity the optical pumping

Lett et al.


process always redistributes the m-level population suchthat as much population climbs a hill as descends. Forfinite velocity, the time lag of the optical pumping processresults in more population climbing the hills than descend-ing them, so the velocity is damped. This mechanism is

similar to the Sisyphus cooling described in Ref. 42 and isresponsible for the ellipticity-gradient cooling.29 In bothmechanisms, the existence of multiple ground-state sublev-els and a long optical pumping time among the sublevels iscrucial for obtaining large damping forces and low tempera-tures.

Ungar et al.28 have treated the same one-dimensional po-larizations but for the J = 2 - J' = 3 case corresponding tothe 3S112 (F = 2) - 3P3/2 (F' = 3) transition used in lasercooling of sodium. Their numerical solutions of the opticalBloch equations for this more complicated case are in quali-tative agreement with the analytic results of Dalibard andCohen-Tannoudji for the simpler transitions. In particular,for small intensity, and kv << r, the theories give a dampingforce that is independent of laser intensity, linear in thevelocity for small velocity, and reaching a maximum valuefor some critical velocity. Ungar et al. find that the depen-dence of the damping on detuning is also in agreement withthe analytic results.

For the ar+-a J = 1 - J' = 2 system Dalibard and Cohen-Tannoudji find [Eq. (5.14) of Ref. 29] a linear dampingcoefficient

c 60 hk 2 2A/r (34)17 5 + (2A/r) 2

This has a maximum value of -0.8 hk2 at A = -V57 r. Sucha damping coefficient would give a velocity-damping time of-4 sec in sodium. By comparison, for the a+-a- J = 2 - J'= 3 system Ungar et al. find numerically a similar detuningdependence, with a maximum cx near A = -r/2, of approxi-mately 4.5 hk2. It is of particular interest to compare Eq.(34) with Eq. (4) or (14) for classical molasses. At largedetuning the classical damping decreases as 1/A3 , while thepolarization-rotation damping decreases only as 1/A. Re-call also that the maximum classical damping coefficient is0.25 hk2 in one dimension.

In the ellipticity-gradient case for J = 1/2 - J' = 3/2Dalibard and Cohen-Tannoudji find [Eq. (4.26) of Ref. 29]that

cx = 3 h2 22 r (35)

This dependence on detuning is in especially sharp contrastto the case of classical molasses, with the damping actuallyincreasing with detuning. Ungar et al. do not evaluate thedetuning dependence for this case, but at a detuning of A =-2.73 r they find an a of roughly 15 hk2, whereas for thatdetuning Eq. (35) would predict a = 8 hk2 for the lowerangular momentum transition. The latter damping coeffi-cient would imply a velocity damping time of -0.5 ,usec insodium.

Dalibard and Cohen-Tannoudji have derived analytic ex-pressions for the momentum diffusion coefficients in the twocases, under conditions of low velocity and intensity. Thisleads to the temperature as in Section 2 above. For polar-ization rotation they find [Eqs. (5.18) and (4.37) of Ref. 29]that

hr. / [ 29 254 1 1kT=(21A1/r) L300 75 1 + (2A/r)2J

whereas for the ellipticity gradient, for A >> r,

kBT = ( I11/°

(36)

(37)

For both cases the temperature is proportional to the inten-sity and, at large detuning, inversely proportional to detun-ing. This would seem to imply that arbitrarily low tempera-tures are possible at sufficiently low intensity or large detun-ing; however, another feature of the polarization-gradientcooling limits the temperature.

An important result of both the Ecole Normale and Stan-ford treatments of polarization-gradient cooling is the exis-tence of a critical velocity v, around which the damping forceis no longer linear in velocity and above which the dampingcoefficient decreases. Velocities higher than v, do not expe-rience the strong polarization-gradient damping force im-plied by Eqs. (34) and (35), so v, represents a capture rangefor the polarization-gradient force. Velocities much higherthan v, will be only slowly damped, mainly by the classicalmolasses force, whereas those comparable with or lower thanv, will be quickly damped to still lower velocities. Typically,this capture range is significantly smaller than that for clas-sical molasses, where v, P F/k. For polarization rotation,the condition guaranteeing linearity of the damping force iskv << 5, where 6 is the difference in light shifts between thesublevels, whereas for ellipticity-gradient cooling the condi-tion is kv << 1/Tp where Tp is the optical pumping time.29

The decrease of vc as the intensity decreases (or as thedetuning increases, since light shifts and optical pumpingrates depend on detuning as well) implies that there will be alower limit to the temperature. Considering first the polar-ization-rotation case, in the limit of small intensity and largedetuning we have

(38)6 (I/I 0 )r 2 /A.

In this limit Eq. (36) implies that

MVrms2 h(I/ 0o)(r 2/A).

Combining the inequality kv << with relations Eqs. (38)and (39) yields

MVrms >> hk. (40)

Thus for polarization-rotation cooling the rms atomic veloci-ty must be much larger than the recoil velocity to have alinear damping force. If the rms velocity approaches therecoil velocity, the nonlinear damping will lead to a non-Maxwell-Boltzmann velocity distribution.

For ellipticity-gradient cooling the condition on v for lin-earity of damping is Ikvl << r(I/Io)(r/2A) 2 , since the opticalpumping rate is proportional to the optical excitation rate.A similar argument leads to

Mvrms >> hk(21A1/r). (41)

For linearity of the damping force both inequalities (40)and (41) require that the velocity be much larger than therecoil velocity. At the same time, the expression for thetemperature, Eq. (36), allows the thermal velocity to gobelow the recoil velocity for some intensity and detuning.

Lett et al.

(39)


As such a condition is reached, the polarization-gradientforce will no longer be linear in v and will be smaller thanthat given by Eq. (34) or (35). Dalibard and Cohen-Tan-noudji have given an explicit form for the nonlinearity of theforce with velocity for the case of ellipticity-gradient cooling,finding [Eq. (4.28) of Ref. 29] that

F = -/v (42)1+ 2/vc2'

where the critical velocity v is given by k, = 1/2rp. Usingtheir expression for the optical pumping time rp, and assum-ing that these results are valid for sodium, for I/b = 1.25 andA = -2.73 r we find for vc _ 3 cm/sec, the recoil velocity vR.For these conditions, the numerical results of the Stanfordgroup give a v, of roughly 1 cm/sec. When vc is less than vR, asingle photon emission or absorption can take the atom'svelocity outside the critical velocity. Under such condi-tions, the effectiveness of the polarization-gradient coolingwill be seriously reduced. The polarization-gradient coolinglimit for these conditions leads to a rms velocity that isactually higher than v, so it is clear that the velocity distri-bution will be nonthermal and have an average energy high-er than that predicted by the linear damping.

Of course, the intensity at which v goes below either VR orthe cooling-limit velocity will depend on the specific atomicsystem and the nature of the polarization-gradient coolingbeing employed. For example, the Stanford group findsthat v = 1 cm/sec for ellipticity-gradient cooling, for thesame conditions that give v = 10 cm/sec for polarization-rotation cooling. Nevertheless, for any system there will bea combination of intensity and detuning at which v is lessthan VR or the cooling limit. At that point, the polarization-gradient force becomes ineffective, and the classical molas-ses force begins to govern the cooling process. The questionof the critical velocity and the laser intensity conditionsneeded to avoid problems are also discussed in Ref. 29.

The damping coefficients discussed above were calculatedin Refs. 28 and 29 by determining the force on an atomconstrained to move at constant velocity through the opticalfield. In fact, an atom does not move in this way but isconstantly subject to fluctuations in its velocity owing to theabsorption and emission of photons. Ungar et al. havepointed out that the steady-state damping force is reachedonly after a time comparable with the optical pumping time,during which time the fluctuations in the velocity can beconsiderable. They have performed Monte Carlo calcula-tions of the atomic motion and obtained force-versus-veloci-ty curves that take into account the hysteresis and averagingthat occur. The damping coefficient so obtained is smallerand the critical velocity is higher than given by the steady-state calculation. The change is particularly dramatic forthe ellipticity-gradient case, in which the critical velocity iscomparable with the recoil velocity. The inclusion of theseeffects makes the damping for the ellipticity-gradient andthe polarization-rotation cases quite similar. For I/I = 1.25and A = -2.73 r, a _ 0.7 hk2 with a critical velocity of 10cm/sec. This implies a velocity-damping time of 5 ttsec.

B. Recoil LimitThe previous discussions of new laser-cooling mechanismsshow that the minimum achievable temperatures may beclose to those associated with the recoil energy from a single

photon, ER = h2k2/2M. We expect this energy to set a lowerlimit on the temperature. Wineland and Itano first exam-ined this problem, assuming Maxwell-Boltzmann velocitydistributions.3 We reexamine the problem assuming two-level Doppler cooling, without putting any restrictions onthe form of the velocity distributions, and find that thediscrete recoil does indeed set a limit corresponding to onerecoil energy in each degree of freedom. The theoreticaldiscussions of Sections 1-3.A essentially ignore the discretenature of laser cooling. When the minimum temperature iscomparable with the recoil energy, as in the case of polariza-tion-gradient cooling, a continuous treatment may not bejustified. In addition, there are some cases for which lasercooling has been proposed, such as hydrogen and positroni-um on the Lyman-a transition, for which the recoil energy isso large (h2 k2 /2M > hy) that the effects of a finite recoilcannot be neglected even in the context of Doppler cooling.It is important then to examine the effect of a large recoil onthe laser-cooling process, including its effect on detuning,equilibrium temperatures, and velocity distributions.

The approximation that ensures a linear damping force,1k! << r, can be recast in terms of the recoil velocity. UsingEq. (13) and the equipartition theorem, we can replace r by2 Mvmin2 /h, where vmin represents the rms velocity at theDoppler-cooling limit. This yields the condition

kVmin/F = hk/2Mvmin = VR/2vmin << 1, (43)

where hk/M = VR is the recoil velocity. It is apparent thatthe condition for a linear damping force is equivalent tostating that the recoil velocity is small compared with thesmallest rms velocity of the atoms in the molasses. Notethat in the limit of large detunings, where, from Eq. (12),MV

2 = hAl, the condition that kvl << Al can still be rewrittenas VR << vrms. For sodium VR = 3 cm/sec, while the rmsvelocity for one-dimensional molasses at the minimum tem-perature is 30 cm/sec.

As the condition on the velocities is weakened, the firstdeparture from the simple analysis applied to Eq. (2) is theappearance of terms of higher order in velocity in the damp-ing force and the appearance of a velocity-dependent term inthe heating component. This perturbative approach is dis-cussed by Castin et al.46 Since we are also interested in theextreme condition in which kR >> r, we shall consider asolution to the problem that can be applied for arbitrarylinewidths and recoil velocities.

The discrete nature of laser cooling, especially when therecoil velocity is large, is ideally suited for Monte Carlosimulations. We performed these simulations in three di-mensions in the low-power limit, with no standing-waveeffects included. The emitted radiation pattern was as-sumed to be isotropic. The linewidth of the laser was alwaysassumed to be negligible compared with r. Velocity andspeed distributions were obtained by sampling the velocityof a single particle in molasses, in the spirit of the ergodictheorem. Temperatures were extracted from fits to thevelocity distributions. It is important when doing simula-tions to sample the velocity of the atom at fixed times, notper photon, since when kR>> r an atom can spend vastlydifferent amounts of time between absorptions, dependingon its velocity. This will turn out to have a critical effect onrecoil-dominated situations.

A test of the Monte Carlo simulation is the case of sodium,

Lett et al.


1 0000

1 0 0

C._

Cu

D

0.

1

0.01

0.0001 I I l I I _

0 0.5 1 1.5 2 2.5V2 (m/sec 2 )

Fig. 3. Velocity distribution from a Monte Carlo simulation ofthree-dimensional sodium molasses with A = -r/2 and I/Io << 1.The straight line represents an exponential fit to the data, giving atemperature of 239 ,uK.

for which classical molasses theory predicts a minimum tem-perature of 238 AK at A = -r/2. To obtain a Maxwell-Boltzmann velocity distribution one needs not only a forcelinear over the range of velocities relevant for cooled atomsbut a velocity-independent momentum diffusion constant,as well. This is nearly the case for sodium. Figure 3 is a plotof the velocity distribution from a Monte Carlo simulation ofsodium molasses. It is convenient to plot the logarithm ofthe distribution versus v2. A Maxwell-Boltzmann distribu-tion is then a straight line whose slope is inversely propor-tional to the temperature. A fit to the distribution is dis-played, resulting in a temperature of 239,uK, in good agree-ment with classical molasses theory. Note also that thedistribution of velocities is well described by a Maxwell-Boltzmann distribution. As shown by Ref. 46, the effect of afinite recoil raises the temperature by ER/2kB = 1.2 ,uK forsodium at this detuning of -r/2. This is not resolvablebecause of the -3% statistical uncertainty. Simulationswere run for cases with somewhat larger recoils, for whichDoppler cooling is still expected to be a good approximationbut recoil effects are more prominent. For large detunings,classical molasses theory predicts a temperature that ap-proaches -hA/2kB. We find that, at large detunings, thefinite recoil actually lowers this temperature by ER/2kB, inagreement with the perturbative treatment of Ref. 46.

Numerical simulations were run for cases in which therecoil velocity is not small compared with the rms velocity ofthe cooled atoms. We characterize such systems with arecoil parameter, n = kvR/(r/2) = 4ER/hr. The simulationsreveal two main features of laser cooling in this recoil-domi-nated (q > 1) regime. Wineland and Itano 3 showed withsimple energy balance arguments that, when recoil is includ-ed, cooling is possible only for a laser detuning of A < -kvR,

where we define A = 0 to be the center of the transition for aground-state atom at zero velocity.47 This is to be contrast-ed with the condition A < 0 for Doppler cooling, where recoilis neglected. Our Monte Carlo results show that there is adetuning condition that is even more severe, namely, thatthe detuning must be red of resonance by approximately A <

-2kVR = 1r or more. A similar result, obtained by using adifferent method, is found in Ref. 46. It is difficult to speci-fy the critical detuning condition exactly from the MonteCarlo simulations, because for detunings near this conditionthe fluctuations in the energy are large, requiring a greatdeal of computer time to determine whether equilibrium hasbeen reached. Figure 4 shows the rms velocity of an atomwith Xj = 104 at a detuning of -2.OkvR as a function of thenumber of scattered photons. The large fluctuations in thevelocity are characteristic of this region near the criticaldetuning condition. For detunings comfortably satisfyingthe condition, the atom remains stably cooled without anylarge velocity excursions.

The reason for the detuning condition, A < -2kvR, and thediscrepancy with the condition A < -kVR from Ref. 3 areconnected with the velocity distributions. Figure 5 shows aseries of velocity distributions for differing values of -q. As inthe case of sodium, when the recoil is negligible the distribu-tions look much like a Maxwell-Boltzmann distribution.When the recoil parameter is large, a hole in the velocitydistribution is burned at the place where the laser is tuned.This is not too surprising: the recoil due to the absorptionand emission of a photon is sufficient to take the atom out ofresonance-a velocity-space optical pumping process. At-oms that random walk into the correct velocity range to be

4

In

E

3

2

1

0

-. .0

0

- .0 0

_ ; O*

0

00 .0

0 0 0 0

_ * 0 * * *S 00 40 80 120 160 20 0

Photon number (x1000)

Fig. 4. Vrms versus number of scattered photons for i = 10, A =-?,r, and VR = 3 cm/sec.

1.0

1 0 '

7 1 0

a 10

1 00 0.0125 0.025

v2 (m/sec 2 )0.0375 0.05

Fig. 5. Velocity distributions for = 0.1, 1, 5, 100, showing theappearance of a hole for large tn. In all cases yR = 3 cm/sec. A =-1.1lr, except for n = 0.1, where A = -r/2.

. _

Lett et al.


100 -

1 0

1 - 0 .0

0.1 -0.01 0.1 1 10 100 1000

Recoil parameter Fig. 6. Minimum temperature versus recoil parameter. Note thatthe minimum temperature approaches TR for ii> 1, and hr/2 (solidline) for i < 1. For >> 1, the temperature was derived by usingonly the low-velocity atoms of the distribution.

resonant with the laser rapidly recoil out of resonance. Thiswill tend to create a hole where the excitation is high, leadingto nonthermal velocity distributions. The hole is much wid-er than the linewidth of the transition. Laser cooling ofatoms with a large recoil parameter demonstrates the impor-tance of considering the velocity dependence not only of thecooling force but also of the scattering rate that produces theheating. The existence of the hole in the velocity distribu-tion is a direct consequence of the strong dependence of theexcitation rate on velocity.

The hole appears to be a rather minor perturbation on thevelocity distribution. One needs only a small number ofatoms to fill the hole. The hole is difficult to observe in thespeed distribution, since the velocities in the three dimen-sions are essentially independent. It is most apparent whenone is measuring the velocity distribution along a laser beamaxis. As we shall see, its importance is significant, althoughits existence is easy to overlook.

We may assign an effective temperature to the distribu-tion by fitting the low-velocity atoms to a Maxwell-Boltz-mann distribution. Alternatively, we can assign an effectivetemperature from the mean square velocity. This yieldssimilar results because the hole is in a velocity region wherethere are few atoms for a Maxwell-Boltzmann distribution.As is shown in Fig. 6, the minimum temperature achievablefor arbitrarily narrow lines (large i7) is kBT/2 = h2 k2 /2M(within 5%), corresponding to one recoil energy in each de-gree of freedom. We refer to this temperature as TR.

Our result that the minimum temperature is TR should becompared with the result obtained by Wineland and Itano.3They examined recoil-dominated laser cooling analyticallyand found temperatures well below TR for very narrow-linewidth transitions (7 > 400). This discrepancy is due to a.different set of assumptions. Wineland and Itano assumethat the velocity distribution is always Maxwell-Boltzmann:the atoms effectively rethermalize between every absorp-tion-emission cycle, and a hole never appears. This as-sumption would be nearly impossible to satisfy for atoms inoptical molasses but is more reasonable for species stored ina trap. The rethermalization always supplies some atoms atthe velocities that interact strongly with the laser. It ap-

pears that the lack of the hole increases the effectiveness oflaser cooling, permitting a detuning closer to resonance anda temperature well below TR. When the hole is formed, as inour model, the cooling is less effective. The extent of theLorentzian wings of the excitation line over the Gaussianshape of the velocity distribution determines the outcome.The large breadth of the hole (many linewidths) indicatesthat the wings of the excitation are quite important. Whenthe detuning A > -2kvR apparently the wings do more heat-ing than cooling. They extend over many more low- thanhigh-velocity atoms because of the shape of the distribution.The heating of the low-velocity atoms dominates over thecooling of the high-velocity ones, leading to an unboundedincrease in the temperature. When A < -2kvR our resultsindicate that the excitation of high velocities is favoredenough for cooling to occur. Although we have seen thatvelocity-space optical pumping limits the temperature forrecoil-dominated cooling, it is possible to use such opticalpumping to populate coherent dark states to achieve a tem-perature below the recoil limit in one dimension. 4 8

A number of groups including our own have proposed andare working on cooling atomic hydrogen by using the 122-nmLyman-a transition.4 95 0 Although the linewidth for thistransition is very broad (100 MHz), hydrogen is so light andthe photon is so energetic that = 0.5. As can be seen fromFig. 5, the effect of recoil is beginning to appear in thevelocity distribution and may affect the cooling process. Inaddition, the added benefits from polarization-gradientcooling mechanisms will be limited, since the recoil tempera-ture is so close to the Doppler-cooling limit. Another atomictransition that would show the effect of recoil is the s2s 3Sr

l ls3p3P, transition at 389 nm in helium where n = 0.9.The laser cooling of positronium has been suggested.51

This is a completely recoil-dominated situation owing to theextremely light mass of the atom. For positronium, therecoil parameter = 240. The lowest obtainable tempera-ture would essentially be the recoil energy of 300 mK. Al-though this temperature is much larger than h/2, it wouldstill be of great utility for positronium experiments. Unfor-tunately, the detuning condition A < -2kvR allows detuningonly to within 240 linewidths, where the transition probabil-ity is so low that extremely high laser powers would berequired (recall that the lifetime of ortho-positronium isonly 140 nsec). It might be possible to use a broadbandlaser, which effectively increases the transition width, butthis would also require much more power to drive the transi-tion. It is still possible to decelerate a beam of positronium,but the velocity spread will be increased substantially be-cause of heating if the laser is detuned less than 240linewidths, which would almost certainly be necessary toobtain reasonable absorption rates.

C. PinballAtoms in three-dimensional optical molasses move in a com-plicated standing-wave field having both polarization andintensity gradients. We have considered the damping andheating effects of this field but not the possibility that atomsmight be trapped or channeled or undergo scattering fromthe static dipole-force potentials created by the standingwaves. Atoms have been confined and channeled by dipoleforces.42 52 57 Cold atoms, with energies of the same order ofmagnitude as the size of the modulations in the dipole poten-

Lett et al.


tials, can scatter, possibly quite strongly, off these modula-tions. Since the intensity modulations in a standing-wavelight field occur on a scale of X/2 (1/4 m), the motion of

atoms in molasses, which would otherwise have an -20-,4mmean free path in the case discussed in Subsection 2.E, maybe dramatically altered.

We use a two- or three-dimensional generalization of thetwo-level theory of Gordon and Ashkin8 to study the effectsof such potential scattering. This approach, applicable onlyto a two-level system, takes proper account of all locallyvarying quantities such as the dipole potential forces, thediffusion coefficient (position-dependent heating), and thevelocity-dependent damping forces. We have investigatedthis system with a detailed Monte Carlo simulation and givehere a qualitative description of our results. The depth ofthe potential wells for A = -r and I = Io is -50 pK forsodium. When the atoms' mean kinetic energy is severaltimes the depth of these wells, the wells have little effect onthe atomic motion. Even the random fluctuations in theatoms' velocity seldom reduce the energy of the atoms to alevel at which they are scattered significantly by the poten-tial energy modulations of the light field.

On the other hand, when the atoms have an average kinet-ic energy equal to or less than the peak-to-valley differencein the local potential energy modulations, their motion willbe dominated by scattering effects. These atoms will fre-quently be trapped inside local potential wells until theirenergy fluctuations allow them to escape (trapping in three-dimensional optical potential wells was considered as earlyas 1977 by Letokhov et al.6 and recently by Kazantsev et

al.58). This temporary localization of the atoms in the lat-tice of a standing-wave potential severely retards their mac-roscopic diffusive motion. Even when the atoms are notactually trapped inside local potential minima, they canscatter off the lattice of potential wells much like a ballrolling across a board with hills and depressions in it. Theball will scatter off bumps or channel along troughs much asin a game of pinball, seldom moving in a direct path for long.In this situation it is impossible for the ball to build up alarge velocity before being scattered. Similarly, an atom inthe lattice of standing-wave potential wells in molasses willscatter if its energy is low enough.

The inclusion of potential well scattering in these MonteCarlo simulations of otherwise classical molasses has notice-able, but not large, effects on the temperature and lifetimeof, and drift velocities produced in, molasses. Artificiallylowering the heating (diffusion) term in the simulations by afactor of 2, however, greatly increases the scattering effects.The polarization-gradient damping mechanisms discussedabove would be expected further to enhance the role playedby such scattering processes. Nonetheless, the pinball ef-fect does not seem to be required to explain any of theobserved features of molasses, with the possible exception ofthe lifetime.

4. EXPERIMENTS WITH OPTICAL MOLASSES

A. Early ExperimentsThe earliest experiments that used symmetric or counter-propagating laser beams to cool atoms were performed byBalykin et al.59 In those experiments a two-dimensionalconfiguration of laser beams was used to cool the transverse

500 - , , - ,|

.-. 400 _

m. 300-

0O 10 0-_

. . ,, . . . . . . . . .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time (sec)

Fig. 7. Fluorescence intensity of the optical molasses versus timein the R&R method (see text). The molasses beams are turned offfor a period toff lasting, in this case, 20 msec. The data are fromR&R measurements taken in our laboratory.

motion of atoms in a sodium atomic beam. By bouncing acollimated light beam off the inside of a reflecting cone, theyformed an axisymmetric light field. An atomic beam thattraveled along the axis of this cone was thus irradiated uni-formly from all directions transverse to its propagation di-rection. This configuration damped the transverse motionof the atoms, increasing the phase space density and colli-mating the beam. Their first experiments demonstratedtransverse cooling from 40 to 3.5 mK in some tens of micro-seconds spent in the light field. Later efforts providedtransverse cooling from 190 to 17 mK.59

The concept of optical molasses as a three-dimensionalviscous confinement mechanism (as opposed to simplythree-dimensional cooling') was proposed and demonstrat-ed by Chu et al. in 198521 and independently proposed inRef. 22 near the same time. In the Bell Labs experiments apulsed source of sodium atoms was used from which some ofthe atoms were laser cooled by the frequency-chirp method3 8

to velocities (-30 m/sec) capturable by the molasses. Themolasses confinement beams were formed by combining twolasers with frequencies separated by approximately the 1.7-GHz sodium ground-state hyperfine splitting to prevent op-tical pumping (see Subsection 4.B). After the molasses wasloaded, it decayed by diffusion of the atoms out of the con-finement region. The molasses lifetime was measured bymonitoring the decaying fluorescence. Decay times of -0.1sec from a 0.8-cm-diameter region were observed.

These experiments also introduced the first ballistic tem-perature measurement technique: The loaded molasses isfirst allowed to decay for a brief period of time. The molas-ses laser beams are then blocked for a time toff, of the order ofmilliseconds. During toff the atoms move ballistically, andsome fraction of them leaves the confinement volume. Thelaser beams are then unblocked, causing the remaining at-oms again to fluoresce. (See Fig. 7.) The ratio of the fluo-rescence intensity just after the beams are turned back on tothe intensity just before they are blocked is a measure of thefraction of atoms that remains in the region during theperiod of ballistic motion. The plot of this ratio versus toff iscompared with model calculations. If one assumes a Max-well-Boltzmann initial velocity distribution and an initial

Lett et al.


spatial distribution, a temperature can be extracted fromthe data. We call this technique release and recapture(R&R).

Using this R&R technique, the Bell Labs group found atemperature of 240o200 jiK for a detuning that was not wellknown. This was in good agreement with the minimumtemperature predicted by Doppler cooling (see Subsection2.A.2). Later, the temperature of cesium optical molasseswas also measured by the same technique to be 100+ii0 jK,consistent with the Doppler-cooling limit of 120 jK for cesi-um.60 These experiments suggested that the classical the-ory of optical molasses was adequate to describe the experi-ments and supported the belief that the Doppler-coolinglimit was the lowest temperature achievable in such experi-ments.

Soon, however, experiments showed serious discrepancieswith the classical theory of molasses under conditions whenone had good reason to believe that the classical theory wasvalid. Early experiments23 in our laboratory measured themolasses brightness and diffusive lifetime as a function ofthe laser intensity and detuning. A typical experiment in-volved scanning the molasses laser and simultaneously re-cording a saturated absorption spectrum as a frequency ref-erence. During the scan, brightness was recorded with aphotomultiplier tube. A decay rate experiment was per-formed as follows: As the laser frequency was slowlyscanned the source of cold atoms was chopped on and off,allowing the molasses first to load and then to decay. Atypical loading-decay curve is shown in Fig. 8. The figureshows the brightness increasing as the molasses accumulatesslow atoms. Once the stopping laser and the atomic beamare chopped off, the molasses decays slowly as the atomsdiffuse out of the molasses region.

These measurements revealed startling discrepancieswith the classical theory. The first surprise was that theoptical molasses was still effective at slowing and accumulat-ing atoms at A -3r. Indeed, the molasses was brightest atdetunings near -2r to -2.5r. The simple classical molas-ses analysis given in Section 2 indicates that the best molas-ses (lowest temperatures, largest damping, longest lifetimes,and therefore probably the brightest fluorescence) would be

U,

C

Cu

a)

600

500

400

300

200

1 00

00 2 4 6

TIME (sec)8 1 0

Fig. 8. Molasses fluorescence versus time as the source of slowatoms is chopped on and off. The molasses is allowed to load for -4sec, whereupon the atomic beam and its cooling laser are choppedoff, allowing the molasses to decay.

0.6

0 .5

Ua)

a)

a)

0.4

0.3

0.2

0. 1

0 1 2 3 4

Detuning (linewidths)

Fig. 9. Molasses lifetime versus laser detuning to the red of reso-nance. The open boxes are early experimental measurements madein our laboratory, while the solid curve is the prediction of Eq. (30)with I/Io = 0.5.

expected at detunings near -0.5r. The theoretical classicalmolasses lifetime derived in Subsection 2.E is shown in Fig. 9and compared with our early experimental lifetime measure-ments. The lifetime TM of the molasses peaks at A -3r.A further surprise was that the lifetime of the molasses wasseriously degraded by the application of magnetic fields assmall as 50 jT. This was unexpected since the Zeemanshifts (0.014 MHz/jiT) for such a field are small comparedwith both A and r.

An even clearer discrepancy from the classical theory in-volves the effect on molasses decay rate of intensity imbal-ance within a beam pair. The classical theory [see Eq. (33)]predicts that an imbalance of 1%, for an average one-beamintensity of 7 mW/cm2 , leads to a drift velocity of -4 cm/secand an added decay rate of 10 sec- 1 for a 4-mm-radiusmolasses. In fact, we found experimentally that the imbal-ance had to be >30% to produce such a large added decayrate 1/TM. Figure 10 shows experimental results for theincrease in molasses decay rate due to intensity imbalanceversus the percentage imbalance. The data were obtainedby attenuating one of the retroreflected beams. A theoreti-cal curve,, showing the classical molasses prediction derivedfrom the calculations of Subsection 2.E is also presented.The drift-induced decay rate is a factor of 10 to 20 timessmaller than that predicted. A similar insensitivity to im-balance was reported by Chu et al. for supermolasses.2 4

These early experiments demonstrate that the depen-.dence of molasses lifetime on detuning, magnetic field, andimbalance contradicts the classical theory. As we shall seebelow, all these features can now be understood, at leastqualitatively, in terms of polarization-gradient laser cooling.For example, the two dashed curves shown in Fig. 10 arederived from calculations in Ref. 29 of the polarization-gradient damping coefficients (albeit for different angularmomenta) given in Eqs. (34) and (35) and fit the experimen-tal data much better than does the classical theory.

While these experiments indicated difficulties with theclassical theory, it was the measurement of the temperatureof optical molasses that finally provided convincing evidenceof the complete inadequacy of the classical theory to de-scribe optical molasses.

I~~~~~~S in*

I I

O 1.~~~~0

I IS

Lett et al.


4 0

0a)U)

a3)(a

C.)

0

a)

a)

A0

3 0

2 0

1 0

00 1 0 20 30

Percent ImbalanceFig. 10. Added decay rate of the molasses due to imbalancepair of beams versus the percent intensity imbalance. Thecircles and open boxes indicate two independent experimentsets. The solid curve indicates the classical molasses predictSubsection 2.E with A = -2r and I/IO = 0.5. The dashed curfrom similar calculations made using the damping coefficient!polarization-gradient forces of Section 3. The upper dashecis from the polarization-rotation damping of Eq. (34), and thidashed curve is from the ellipticity-gradient damping of Eq.

In experiments first reported in Ref. 25 we measurtemperature of the atoms released from the molasseploying a technique different from the R&R meth(scribed above. We added a probe laser beam beneamolasses, and separated by one to two molasses dianand observed the time of flight (TOF) of the releasedfrom the molasses region to the probe.6 ' This techniless sensitive to certain systematic errors associated wiR&R method. For example, determination of the tenture from R&R measurements requires knowledgespatial distribution of the atoms before they are relSuch geometric considerations become insignificantTOF method as the separation between the molasses aprobe is increased. If the viewing volume for recalatoms is not sufficiently restricted, the measuremencome confused by atoms that fluoresce in a pair of ]outside the molasses. Although the TOF method these problems, it has uncertainties of its own, whiidiscussed in some detail in Subsection 4.B.3 below.

We measured the temperature by recording the Iinduced fluorescence as a function of the time aftiatoms were released from the molasses. The tempe:was determined by comparing the peak of the arrivadistribution with that of the expected distribution caled by assuming an initial temperature for the atoms. I11 shows a recent example of a TOF measurement wiresults of such a calculation for sodium atoms at 25 aijiK. The measured temperatures here and in Ref. obviously inconsistent with the Doppler-cooling lim:are consistent with a temperature near 25 jK. We esied a systematic error of 20 jiK for the TOF method.

To check this surprising result we performed threetional kinds of temperature measurement (includi:R&R measurement in which careful account was talthe problems listed above), each sensitive to a differentatomic velocity components. These have been descrilRef. 25. They supported the TOF measurements and

- out the possibility that the molasses was as hot as the classi-cal Doppler-cooling limit. Later, temperatures below theDoppler limit were also observed at Stanford,27 using sodi-um, and at the Ecole Normale,26 using cesium and, in onedimension, metastable helium.

Since the initial measurements of sub-Doppler-limit tem-peratures, we have made improvements in our experimentaltechnique that have allowed us to refine the conclusionsgiven at that time. We stated in Ref. 25 that, based on ourR&R measurements, the temperature did not change within

. - experimental error when an acousto-optic modulator wasused to turn off the molasses beams in less than 1 isecinstead of the usual 30-jsec turnoff time of our mechanical

4 0 shutter. We also observed no dependence of the tempera-ture on the intensity over a limit range. Subsection 4.B

in one below discusses more precise measurements, revealing thata filled both intensity and turnoff time affect the data and that theal data slow turnoff suppresses the intensity dependence of the tem-ions of perature.ves are We also reported 2 5 that the temperature exhibited aocfutrhve strong dependence on the magnetic field. At that time the

e lower magnetic field had been nulled by optimizing the brightness(35). of the molasses with respect to field. We later found that

the temperature is a more sensitive measure of the field.After adjusting the magnetic field to give the lowest tem-

ed the perature, we found that the field that had been used in Ref.s, em- 25 was -50 jiT different from that which gives the lowest)d de- temperature. Measurements with a flux gate magnetome-th the ter confirmed that the optimum average field is in fact zeroieters,' to within 3 jT. The gradient of any field component is lessatoms than 30 T/cm in any direction, and increasing this gradientIque is by at least a factor of 2 has no measurable effect on theIth the temperature. We had stated, on the basis of Hall probeapera- measurements, that the field was zero to within 20 jT, so weof the were apparently in error by more than twice our estimatedeased. uncertainty.in thend the B. Recent Experiments)tured In this section we shall first give a brief description of ourits be- apparatus as it was used in our most recent experiments.beams We shall discuss general characteristics of the experimentalavoids system and details of the laser configuration used in ourch are

)robe-.er therature1-timelculat-Figureth theid 25025 areit and

timat-

addi-.ng anken of

set ofbed inruled

1 000

a

0Ca)

0a)0cJ

800

600

400

200

00 20 40 60

Time (msec)80 1 00

Fig. 11. TOF data and results of numerical calculations at 25 and250 K, all normalized to the same height. A = -2.5r. The datapoints show fluorescence intensity versus time, and the solid curvesindicate the expected signal.

II-

a e0 - - - - -0 ~ ~ ~ ~ ~ ----------

Lett et al.


to Frequency to SatruratedStabilization AbsorptionSpectrometer Cell

EC vl Shutter

Linear 1/4 Wave 4 mm AperShutter Polarizer Plate

Slopping n ALaser U

BeamExpandingTelescope

Fig. 12. Block diagram of the optical system. Thwas not present for most of the data. An electro-optidesignated by EOM.

MolassesBeams

7-9 mm Aperture

Probe Beam

ure

stopping Beam

e spatial filterc modulator is

laboratory to produce sodium optical molasses. (See Fig.12.) Typical values for some experimental parameters aregiven and, unless otherwise noted, are used in the experi-ments described in the following sections. We discuss pro-gress that we have made in improving our experimentaltechniques since the earlier measurements and presentmore-recent investigations of the properties of optical mo-lasses. In particular, we have studied the dependence of thetemperature on a variety of experimental conditions forcomparison with the predictions of the polarization-gradienttheories summarized in Section 3. We shall see that themajority of the recent data agree, at least qualitatively, withthe predictions of the polarization-gradient theories. Wealso discuss in detail the experimental errors associated withour temperature measurements.

1. Experimental ConfigurationOur optical molasses is loaded with slow atoms from a sodi-um atomic beam that propagates along the axis of a Zeeman-tuning tapered solenoid and is continuously cooled by acounterpropagating laser beam.2 2 Atoms that are not quitestopped on reaching the end of the solenoid leak past theedges of the cooling laser and travel into the experimentalregion. Measurements of the velocity distribution of theseescaping atoms in the molasses region show that the ratherbroad distribution peaks near 30 m/sec. The molasses re-gion, which is located -20 cm downstream from the end of,and -2.5 cm above, the magnet axis (see Fig. 13), interceptssome small percentage of these escaping slow atoms andviscously captures and accumulates them.

The molasses is formed at the intersection of three retro-reflected beams propagating along orthogonal axes. Thelight in each pair of beams is linearly polarized and orthogo-nal to the polarization of the other two pairs. The beams areroughly Gaussian (1/e2 radius = 4 mm) and are apertured toa 9-mm diameter. In the most recent experiments a spatialfilter ensures Gaussian beams. A total power less than orequal to 45 mW is split into three equal beams, each of whichis retroreflected to form one of the beam pairs. This maxi-mum value gives a single-beam on-axis intensity of approxi-mately 60 mW/cm2. The average single-beam intensitywithin the 1/e2 radius of the Gaussian is then -26 mW/cm2 .Averaging over the sublevels in the F = 2 - F' = 3 transitiongives a saturation intensity Io = 13.5 mW/cm 2. Therefore,for the maximum available power, the average single beam I/

Io 1.9, and we typically use approximately a factor of 4 less.We adopt the convention of quoting the measured laserpower before the beam is split.

The laser is tuned near the 3S1/2, F = 2 - 3P3 /2, F' = 3transition, and an electro-optic modulator is used to impress1732-MHz sidebands upon the carrier. Each sideband con-tains -10% of the total power, and the upper sideband isused to prevent optical pumping to the F = 1 ground state(see Fig. 14). The chosen frequency provides a resonantrepumper on the F = 1 - F' = 2 transition when the carrieris tuned 20 MHz below the F = 2 - F' = 3 resonance.Adjusting the sideband frequency by small amounts (5MHz) has little effect on the molasses.

The beams are collimated to better than 1 mrad, passthrough antireflection-coated vacuum windows, and are ret-roreflected external to the vacuum chamber. Window andmirror losses are <1%, and the absorption losses in the mo-lasses itself, even with the densest molasses obtained, arenegligible.

The laser used to produce these beams is optically isolatedfrom the retroreflected beams by an 80 MHz acousto-opticmodulator (AOM), which can also serve as a fast shutter. Amechanical shutter at the focus of a beam-expanding tele-scope operates more slowly but with complete extinction.The laser is locked to an external cavity that is in turn lockedto a polarization-stabilized tunable He-Ne laser. The satu-

opticalmolasses

coolinglaser

solenoid stoppedatoms

Fig. 13. Sketch of the apparatus, showing the relative position ofthe Zeeman tuning magnet and the molasses region (not to scale).

F=3

60 MHz

F=2

F=1

3p 3/2

cyclingtransition

, 589 nm)

F=2

1772 MHz 3S 1/2

F=1

Fig. 14. Sodium energy-level diagram (not to scale) indicating thecycling transition used for laser cooling.

Lett et al.


rated absorption spectrum from an auxiliary vapor cell iscontinuously monitored and provides an absolute frequencyreference near the atomic resonance frequency. Magneticfields present in the molasses region are nulled, within -3yT, by using three pairs of orthogonal coils. The measuredbase vacuum pressure in the experimental region is less than10-6 Pa (10-8 Torr).

The probe laser is derived from the same laser as themolasses beams and includes the repumping sidebands. Itis split off before the mechanical shutter so that it may be onwhile the molasses beams are off.

2. Experimental ResultsThe new polarization-gradient theories summarized in Sub-section 3.A make a number of predictions about the behaviorof molasses when experimental parameters such as laserdetuning, intensity, and magnetic field are varied. Withrefinements in our experimental technique we have beenable to investigate these predictions in some detail. All thedata in this subsection were acquired with the TOF methodin a magnetic field nulled by minimizing the temperature.In Subsection 4.B.3 we discuss in detail other systematicerrors of the order of 20 gK remaining in our absolute-temperature measurements. Day-to-day changes in the ge-ometry often resulted in 10-20-ptK shifts. However, duringa day's run the geometry remains relatively fixed, and we aresensitive to changes in the temperature of a few microkel-vins. Therefore we are confident that the dependences ofthe temperature on the varied parameters are significant.

We begin with a discussion of experiments investigatingthe effect of the shut-off time of the molasses beams, be-cause it affects the interpretation of our other measure-ments. We measured the temperature, using an AOM toturn off the beams with a variable ramped turn-off time.This type of experiment was first reported by the Stanfordgroup.6 2 Our data are shown in Fig. 15. The initial molas-ses power was 13 mW in the sum of all three beams, and thedetuning was approximately -20 MHz. The light intensitywas ramped down linearly in a time tr, which could be varied.The extinction ratio of the AOM was -700:1 within a fewhundred nanoseconds after a fast turnoff. We used a me-chanical shutter to extinguish the light completely 20-80

zL

I-L

0)

0.a)

C-

3

Q2

90

80

70

60

5 0

40

3 00.1 1 1 0

t r (sec)1 00 1 000

Fig. 15. Temperature measured by TOF as a function of turn-offtime tr. The quantity tr is the time that it takes to ramp down themolasses laser beam intensity with an AOM.

lisec after the AOM had switched. The data for a 30-ttsecAOM ramp agree well with those obtained with a mechanicalshutter. The data are similar to the Stanford results, whichwere obtained by using much higher laser powers, exceptthat our temperatures are lower.

The Stanford group suggested that the turn-off time ef-fect could be due to cooling by adiabatic lowering of dipolepotential wells.62 They pointed out that, at these low tem-peratures, the dipole force due to the standing waves in themolasses region could trap some of the atoms. As the lasersare shut off, the receding walls of the dipole wells could coolthe atoms by adiabatic expansion if the atomic oscillationperiod in the wells were small compared with the shut-offtime.

We propose a different explanation for the temperaturedecrease with increasing tr in Fig. 15. If the light intensity isturned down slowly compared with the molasses equilibra-tion time Tdap defined by Eq. (7), then the temperature willreflect that of the molasses during the final Tdamp of the shut-off. Since the new theory predicts that the temperaturevaries as the intensity, we expect lower temperatures forlonger tr. From the figure we deduce that Tdap is of theorder of 3 Asec. This is in rough agreement with the analyticand numerical predictions quoted in Subsection 3.A and ingood agreement with the Monte Carlo results of Ref. 28 for avelocity range comparable with that implied by our mea-sured temperature. This time is significantly shorter thanthe equilibration time predicted by the classical theory ofSubsection 2.B.

We have confirmed this interpretation in two ways. First,we have observed directly the dependence of the tempera-ture on the light intensity predicted by the polarization-gradient theories. The data are shown in Fig. 16. Thesquares represent the data obtained with a fast AOM shut-off; circles show the temperature measured by using a me-chanical shutter. As predicted by Eqs. (36) and (37), thedependence is linear. The smaller slope of the mechanicalshutter data is easily explained if one supposes that for themechanical shutter data the atoms simply equilibrate with amolasses whose average power is that of the last few micro-seconds of the shut-off. The slope of the fast shut-off data is-80 jiK per I/IO in a single beam. We may compare this, forexample, with Eq. (36), which has a slope of 35 ,K per I/Ib.We note, however, that a quantitative comparison is difficultbecause of the one-dimensional nature of the theory.

A second confirmation was an experiment with a slightlymodified shut-off procedure from that used to obtain thedata in Fig. 15. In Fig. 17a we show schematically themolasses beam intensity as a function of time for this experi-ment. We used an AOM to turn down the light from thenormal power of 13 mW to Plow in 200 nsec, where it re-mained for 50 ,sec before being extinguished completely bythe AOM. A mechanical shutter followed 10 Msec after theAOM shut-off. The temperature was measured as a func-tion of P1ow. Figure 17b shows the data. Just as in the AOMdata of Fig. 16, the lower temperature of low-power molassesis clearly evident. Because the shut-off time in Fig. 17a isfast, adiabatic cooling cannot account for the low tempera-tures, nor is it necessary to invoke adiabatic cooling duringthe shut-off to explain the results of Fig. 15. All this doesnot rule out the possibility that adiabatic cooling occurs, butit does imply that the equilibration to the temperature asso-

SS 0 l

. . S -

-f al | alS*S

SI , ''.1 I ' l' S

Lett et al.


80

70

0)a-a)a-

0.

Ea)1--

6 0

5 0

40

30

2 00 1 0 20 30 40

Total power (mW)Fig. 16. Temperature measured by TOF versus molasses laserpower. The power was measured before the beginning of the shut-off. The AOM shutoff took 200 nsec (squares); the mechanicalshutoff took 20-30 jusec (filled circles). The power indicated is thepower in the molasses laser beam before it was split into thirds andretroreflected. A =-2r.

7 0 -

0)

a-

Ea)

60

50

40

an

b

S

.

.

.

..

0 2 4 6 8 10 12 14

Plow (mW)Fig. 17. a, Molasses laser power as a function of time for the data inb. The mechanical shutter shuts off approximately 10 gsec afterthe AOM. b, Temperature versus PLOW showing that low-powermolasses beams cool the atoms despite a fast shut-off. A = -2r.

performed an experiment in which the molasses power wasturned off slowly (in -20 sec), but not completely, with anAOM. The slow turn-off ramps the molasses to a low tem-perature. After the ramp, a power of 0.14 mW, 1% of theinitial power, remained incident upon the molasses regionfor a variable heating time th. Figure 18a shows the laserintensity versus time for this experiment: Figure 18b showsthe temperature as a function of th. The data clearly show alinear heating. Using Eq. (10) and assuming that I/O =0.01, we find that the heating rate is of the same order ofmagnitude as that shown in the data. This experiment alsodemonstrates that one must be careful when using an AOMas a shutter to make sure that the light is sufficiently extin-guished. Following the AOM turn-off with a mechanicalshutter, a procedure also reported in Ref. 62, avoids theseheating problems when the extinction of the AOM is insuffi-cient.

The data of Fig. 15 reveal that the true temperature of theatoms while the laser beams are at their steady-state poweris best measured by shutting the molasses lasers off in a timemuch less than 3 ,usec; otherwise the temperature will reflectthe laser power during the last several microseconds of theshut-off. In the experiments discussed later in this sectionthe data were taken by using a slow mechanical shutter.Therefore all the temperature measurements correspond tomolasses at a much lower intensity than before the start ofthe shut-off. The dependences that we report are still valid.

The polarization-gradient theory suggests that there is asensitive dependence of the temperature on magnetic field.(Recall that the classical molasses theory does not predict alarge sensitivity to magnetic field). The new theory, howev-er, depends on optical pumping and ac Stark shifts betweendifferent states in the F = 2 ground-state manifold. Thepresence of a magnetic field can cause Larmor precessionand Zeeman shifts of magnetic sublevels that are compara-ble with the optical pumping rates and ac Stark shifts. Onecan estimate the field at which the Larmor precession ofmagnetic sublevels should begin to interfere with opticalpumping by observing that the optical pumping time shouldbe of the order of 1-10 gisec. The ground-state Zeeman shiftand therefore the precession frequency among different hy-perfine states is of the order of 10 kHz/tT. Thus, at a field

ciated with lower intensity is the dominant factor in produc-ing lower temperatures for slow turn-off.

Figure 17b also shows that at powers below 3 mW (total I/IO of -1) the temperature ceases to fall. This supports theprediction of Subsection 3.A [see the discussion followingEq. (42)] that the polarization-gradient force loses effective-ness at low powers as the velocity range over which it actsbecomes small. The sharp rise in temperature shown in Fig.15 at times longer than 100 ,sec can also be explained by theloss of polarization-gradient cooling. The polarization-gra-dient cooling disappears well before the end of a 100-gsecramp, so the atoms spend a significant amount of time inlow-power laser beams. The classical molasses force shouldstill operate, but its equilibrium temperature is very high,-500 MK at A = -2r, where these data were taken, so weexpect the atoms to be heated.

We have additional evidence for this heating. We have

1 4 0 -_ _\ _ 01 W _ - .

120

y 100 -

) 80

X 60

E 40 - *_

0 1 2 3t h (msec)

Fig. 18. a, Molasses laser power versus time for the data in Fig. b.b, Temperature versus th, showing rapid heating of the atoms invery-low-power (0.14-mW) molasses.

13 mW a

Plow 50 Asec Shutter

< 0.02 mW _ 0

I . . . . . . . .

Lett et al.


500 -I II C * 011.5 mW

400 -5.4 mW

<|) 300- 0 o US0)~~~~~~a- 250 2*C20 0 0) 5

E 1 0 .

10-

-200 -100 0 100 200Magnetic field (T)

Fig. 19. Temperature versus magnetic field. The data were takenwith a mechanical shutter. The two powers displayed were mea-sured before the shutoff. A = -21.

of 10-100 MT, the Larmor precession frequency is of thesame order of magnitude as the optical pumping rate. TheLarmor precession will compete with the optical pumpingand interfere with the polarization-gradient damping. Thepolarization-gradient theory of laser cooling also suggeststhat the magnetic field dependence of the temperature willvary with laser intensity. As the intensity is increased, theoptical pumping time will decrease. This means that athigher intensity it will require a higher magnetic field todegrade the polarization-gradient force by the same amount.Figure 19, which shows molasses temperature versus mag-netic field for two different laser intensities, qualitativelyverifies both of these predictions. The effective variation inpower was significantly smaller than what is shown in thefigure because mechanical shutters were used.

The new theories of laser cooling all depend on the exis-tence of a polarization gradient in the radiation field.Hence if this gradient is reduced we expect the temperatureto rise. We have examined the dependence of the tempera-ture on the polarization of the molasses laser beams. Undernormal conditions, the (linear) polarization is such that thepolarization of each pair of beams is orthogonal to the othertwo pairs. When the polarization of one of the beam pairs isrotated parallel to the polarization of one of the other pairs,the degree of polarization gradient is smaller. We haverotated the polarization of each pair of molasses beams. Wefind that when the polarization is rotated from 0° to 900, thetemperature increases monotonically from 25 to -50 4K.Behavior similar to this has also been observed by the au-thors of Ref. 34.

We have measured the dependence of the temperature onthe laser detuning A. Figure 20 shows data taken with theTOF method. Shown also is the prediction of the classicalmolasses theory, Eq. (12). The shape of the curve is radical-ly different from that of classical molasses. The most strik-ing feature illustrated by Fig. 20 is that the temperaturecontinues to decrease when the detuning is increased farbeyond the classical minimum of A = -F/2. In the classicaltheory of molasses the increase in temperature at large de-tuning comes about because the cooling rate, proportional toa, is decreasing as A3 , while the heating rate, proportionalto Op, is decreasing only as A-2. In the polarization-gradi-

ent theories of laser cooling the situation is quite different.Both the ellipticity-gradient and the polarization-rotationcases produce temperatures that decrease with increasingdetuning, as do the data, although the detailed dependencesare different [see Eqs. (36) and (37)]. Our three-dimension-al molasses with three orthogonally polarized standingwaves will in general have a mixture of ellipticity gradientand polarization rotation, with an additional intensity mod-ulation from the standing waves, preventing a direct quanti-tative comparison with theory. The increase in tempera-ture at detunings larger than 35 MHz shown in Fig. 20 isprobably due to the fact that the F' = 2 level is only 60 MHzto the red of the F' = 3 level. At these large detunings thelaser is more nearly resonant with, and to the blue of, thelower transition.

Finally, we report measurements of the molasses decayrate as a function of both magnetic field and laser power.Recall from Subsection 2.E that, if one assumes that themotion of the atoms is diffusive, then the molasses decayrate 1/TM is proportional to DOp/a2. The decay rate versusmagnetic field is shown in Fig. 21. The behavior of 1/TM issimilar to that of the temperature, shown in Fig. 19.

Figure 22 shows the dependence of the decay rate on laserpower. In polarization-gradient cooling, Op is proportionalto the laser intensity, while a is independent of it. Thus, ifthe decay rate is governed purely by polarization-gradientcooling, one expects the decay rate to increase as the intensi-ty increases. The data show quite the opposite. It is inter-esting to note that the decay rate decreases with power in amanner similar to the prediction of the classical molassestheory. Because of the low temperature and high damping,note that the diffusion lifetime of a molasses governed bypolarization-gradient cooling is many seconds. The longestTM that we have observed is only 625 msec. We have experi-mentally verified that this lifetime was not limited by ourvacuum. Perhaps the decay of the molasses is determinedby the probability that an atom's velocity is outside therange over which polarization-gradient cooling acts. If anatom's velocity is outside that range, its motion will be gov-erned by classical molasses. The classical molasses will tendto damp its velocity to a point where the polarization-gradi-ent force is effective again. However, during this time theatom also has an increased probability of escaping the mo-

600

500

0)a-Cu

0)

ECD

400

300

200

1 0 0

00 1 0 20 30 40

Detuning (MHz)

Fig. 20. Measured temperature versus laser detuning A, to the redof resonance. Also shown is the prediction of classical, low-power,one-dimensional molasses (solid curve), Eq. (12).

.

SS** 0 0 "0 5 s *mSS0

Lett et al.


30

Q 20u,

A)

a)

a

0-200 -1 00 0 1 00

Magnetic Field (T)Fig. 21. Molasses decay rate versus magnetic field.power, 10 mW.

8

I

0

0)a)co

0)a

6

4

2

00 5 10 15

Power (mW)20

Fig. 22. Molasses decay rate versus laser power. A = -2.5r.

lasses, since the damping force of classical molasses is muchsmaller. Both the likelihood of an atom's velocity beingoutside of the polarization gradient range and the escapeprobability from the classical molasses are higher at lowerintensity. This could explain the behavior observed in Fig.22. The Stanford group has reported observations of bi-modal velocity distributions,3 4 which may be due to similareffects. We note that, if we had a bimodal distribution, withwell-separated temperatures such as those quoted in Ref. 34,the presence of the hot peak would have essentially no effecton the determination of our low temperature.

3. Data AnalysisThe uncertainties in the temperatures quoted above arelargely systematic in nature; the short-term repeatability ofour data is a small fraction of the total uncertainty. TOFdata taken on the same day under the same conditions resultin a peak arrival time scatter of -0.5 msec, implying a 1arstatistical uncertainty of -3 MK.

The systematic uncertainties, on the other hand, are muchlarger and are due mainly to uncertainties in the distancetraveled by the atoms as they fall to and through the probe.To get a feel for the difficulty in making an absolute-tem-perature measurement, consider that the gravitational po-tential energy difference (mgh) between atoms at the top

and the bottom of a 0.9-cm-diameter molasses correspondsto an energy difference (kBT/2) equivalent to -500 tK.This is much larger than the tens-of-microkelvins kineticenergy that the atoms have when they are first released fromthe molasses. In this section we shall describe the way inwhich a temperature is extracted from the experimentaldata and then discuss individual contributions to the totaluncertainty.

To determine the temperature, the data are comparedwith the results of a numerical calculation of the expectedsignal. Input includes the relative positions of the molassesand the probe, the molasses and the probe diameters, thelength of the probe being imaged by the detection optics,and an experimentally determined estimate of the spatialdistribution of atoms in the molasses. For each volumeelement in a spherically bounded molasses, the programassumes a Maxwell-Boltzmann velocity distribution corre-sponding to a given temperature and calculates the numberof atoms falling, under the influence of gravity, to the probeas a function of time. Integrating over the molasses volumegives an arrival-time distribution corresponding to a TOFsignal. Examples of the calculations are shown in Fig. 11 fortemperatures of 25 and 250 sK.

To analyze the TOF data, we compare the experimentallydetermined peak arrival time with peak arrival times calcu-lated for different temperatures. Other quantities, such asthe centroid and the FWHM, also vary with temperatureand may be used to interpret the data, although they aregenerally less sensitive. It is possible, under suitable limitsof temperature and/or molasses-probe separation, to deriveanalytic expressions for the TOF distribution if we assume apoint molasses and a point probe. Numerical calculationsusing these initial conditions are in excellent agreement withthe analytic results.

We now discuss the individual contributions to the tem-perature uncertainty. In general, these are found by vary-ing the parameter of interest in the numerical calculationand observing its effect on the arrival-time distribution.

The main source of geometric error is the measurement ofmolasses-probe separation. The vertical distance betweenthe centers of the probe beam and the horizontal molassesbeam can be measured to 0.5 mm. For temperatures below-100 AK, this leads to an uncertainty of -6 K for a typicalcenter-to-center separation of 1.25 cm. Error due to hori-zontal displacement of the geometric centers is negligible, asare uncertainties in the size of the molasses. Uncertaintiesdue to the probe will be handled separately.

The uncertainty associated with the distribution of atomsin the molasses is closely connected to uncertainties in deter-mining the molasses-probe separation. Obviously, if all theatoms were at the top of the molasses, the peak arrival timewould increase dramatically, and we would assign much toolow a temperature. From video images of the molasses wecan obtain the distribution of excited-state atoms projectedonto a plane. Given the single view that we have of themolasses and the non-Gaussian nature of our molasses laserbeams, we can only estimate the position-dependent atomicdensity. We place the centroid of the distribution at thegeometric center of the molasses and assign an uncertaintyof 0.5 mm to this estimate. This translates into a tempera-ture uncertainty of 6 sK.

The detection process in the probe is the source of most of

I-II

S

S

SS

0~~~~~~

S

I I I~~~~

Lett et a.

2104 J. Opt. Soc. Am. B/Vol. 6, No. l/November 1989

the systematic error in these experiments. Recall that theprobe is configured to produce one-dimensional optical mo-lasses. As such, it quickly damps any atomic velocity alongthe direction of the laser beams. Unlike in the strictly one-dimensional problem of Subsection 2.A, atoms in the probeemit photons into all three dimensions. Photons emittedperpendicular to the laser beam produce the same heating asin Eq. (9) but without a counteracting damping force. Thiscauses the velocity distribution perpendicular to the probeto spread as the atoms fall through the probe. Under typicalconditions of probe intensity, detuning, and molasses-probeseparation, the velocity change is large enough to expel theatoms before they fall completely through the probe. Thisaffects our assignment of a temperature since it changes theeffective separation between the molasses and probe. Inaddition, it becomes more important as the velocity at theprobe decreases.

In addition, there is a second, geometric source of probeerror. During the course of the data taking, there were day-to-day changes in the probe's profile. While the aperturedefining the probe remained a constant 4 mm, the intensityprofile across that aperture varied from being fairly uniformto having a high-intensity hot spot a few millimeters indiameter. Uncontrolled changes in the probe's intensityprofile across the aperture could shift the effective molas-ses-probe separation by 1 mm. We assign 12 AK to thissystematic uncertainty.

We have performed a detailed Monte Carlo simulation toaddress the probe detection process.63 Here we discuss theresults of that simulation for typical probe parameters:power 3 mW, beam diameter 4 mm, laser detuning -2.5 F,and a Gaussian profile with 1/e2 radius = 0.15 cm centeredon the aperture. Under these conditions, an atom fallingfrom the center of a 40-AK molasses to a probe centered 1.25cm away spends roughly 4 msec inside the probe and scatters-3000 photons before being expelled. In addition, most ofthe photons are scattered within the top 1.5 mm of the probe,and the atom rarely falls even halfway through the probe.This has been confirmed experimentally: blocking the bot-tom half of the probe at this power did not change the size orthe shape of the TOF signal.

If the power in the probe is reduced to 0.7 mW, the behav-ior is radically different. An atom is now no longer scatteredout of the probe. Instead, it spends -6 msec in the probe(compared with an 8-msec ballistic transit time) and scatters-1000 photons, most of those within -1 mm of the probecenter. This has also been verified experimentally: reduc-ing the probe power from 0.7 to 0.2 mW changed the peakarrival time by only 0.5 msec, a number comparable with thestatistical scatter.

In the experiments discussed in Subsection 4.C.3 the pow-er and profile of the probe were not well controlled. Weassign an error of 12 AK to these uncertainties in addition tothe uncertainty involving the placement of the probe withinthe aperture. Combining all uncertainties in quadrature,we arrive at a total uncertainty of 20 AK for the TOF datapresented in this paper. We note that this error is roughlyindependent of temperature and therefore gives quite largerelative uncertainties at the lowest temperatures.

The numerical calculations have also allowed us to placelimits on other possible sources of error. We find that mag-netic forces due to field gradients twice those measured at

the molasses have no effect on the shape of the TOF curve.In addition, the TOF method is rather insensitive to largeradial changes in the molasses density profile.

Finally, we have run our numerical calculations for a vari-ety of different assumptions about the velocity distributionof atoms in the molasses. We looked at the behavior ofvertical temperature gradients, radial temperature gradi-ents, and the effect of having different temperatures in thevertical direction and in the horizontal plane. We also ex-amined the possibility of molasses having a homogeneous,isotropic, nonthermal velocity distribution, for example, amixture of temperatures or a flat velocity distribution hav-ing a sharp cutoff. In each situation, the results are consis-tent with the following simple analysis: As the averagevertical velocity of the molasses increases, the peak of thedistribution shifts to shorter times. As the average horizon-tal velocity increases, the atoms spread as they fall to theprobe, so fewer atoms are detected. Atoms with a long falltime, such as those with a small vertical velocity or thosetraveling upward, suffer more horizontal displacement andmiss the probe. The result is that the TOF method is mostsensitive to atoms near the bottom of the molasses and tothose atoms with the lowest horizontal velocities.

We now consider the consequences of this differentialsensitivity to velocities when the molasses does not have aMaxwell-Boltzmann velocity distribution. Figure 23 showsthe time integral of the TOF distribution as a function oftemperature, assuming a point molasses, a point probe, anda molasses-probe separation of 1.25 cm. As the tempera-ture of the molasses rises, the total number of atoms beingdetected by the probe falls rapidly. If the molasses werecomposed of equal amounts of atoms at 10,20,30, and 40 uK,the temperature seen by the TOF method would look muchlike a pure temperature of 10 gK. Similarly, a distributionwith equal amounts of 20- and 100-,uK atoms would againclosely resemble a pure temperature of 20 ALK. In general,for isotropic, homogeneous mixtures of temperatures, theTOF method sees mainly the lowest temperature, while, fortwo widely separated temperatures, it produces a two-peaked distribution. When considering bimodal TOF dis-tributions, such as presented in Ref. 34, it is necessary to beaware of this differential sensitivity.

1.5

_3 1

-

a) 0.5-

._0E

00 125 250

Temperature375

(gK)500

Fig. 23. Time integral of the fluorescence intensity versus molassestemperature showing the differential sensitivity to velocity of theTOF method.

Lett et al.


C. Improved GeometryThe systematic error limiting the temperature determina-tion in Subsection 4.B is the shape of the probe and uncer-tainties about exactly where in the probe the atoms werebeing detected. This error could be largely eliminated if theprobe had a well-defined intensity profile and moderatelylow power. Atoms falling through such a probe would fluo-

resce mainly near the center and would be perturbed onlyslightly. Under these conditions, the detection process canbe modeled and included in an improved temperature deter-mination. Furthermore, the molasses beams used in theabove experiments were decidedly non-Gaussian. There-fore, to improve the uniformity and symmetry of the molas-ses and probe beams, the distribution of atoms in the molas-ses, and the modeling of the detection process, we added aspatial filter. The resulting molasses beams, althoughroughly 30% lower in power, now have a smooth Gaussianprofile with a l/e2 radius of 5 mm. Similarly, the probebeam is also a well-defined Gaussian with a l/e 2 radius of 2.5mm.

The lifetime of the molasses when the spatially filteredbeams are used is now much longer. Lifetimes, as measuredat the l/e point, before spatial filtering were -300 msec inthe best of conditions. After filtering, the lifetime is now625 msec, and there is a corresponding increase in molassesbrightness. These results are quoted for a molasses detuned2.5r to the red of resonance and a total power in the mo-lasses beams of 10 mW. Also, we are not vacuum limited ata measured pressure of 7 X 10-7 Pa (5 X 10-9 Torr). Wenote that other groups have used spatially filtered beamsfor producing optical molasses and have obtained im-proved performance with them compared with unfilteredbeams.2 6 '2 7 '5 2 '6 4

Using this improved molasses and the well-defined probe,we have made new TOF measurements. An example isshown in Fig. 11 for a total molasses laser beam power of 10

mW, a molasses-probe separation of 1.25 cm, a laser detun-ing 2.5r to the red of resonance, and a probe power of 0.6mW. Also shown are the results of the numerical calcula-tion for the best fit of 25,gK. The results of the Monte Carloprobe simulations have been included in this calculation.After completing a full analysis, we expect our error to begreatly reduced from 20 ptK.

In addition, we have begun exploring the velocity distribu-tion of the atoms in the molasses. Our previous tempera-ture measurements have assumed a well defined Maxwell-Boltzmann velocity distribution characterized by a singletemperature, and most of the data are in reasonable agree-ment with that assumption. However, given the differentialsensitivity of the TOF method, high-velocity atoms might bepresent in the molasses but not efficiently detected. Also,the polarization-gradient forces that currently best explainmolasses work only over a limited velocity range. Presum-ably, outside this range, classical molasses is still effective,and the resulting velocity distribution is expected to be non-Maxwell-Boltzmann. The exact nature of this distributionsupplies important information about the polarization-gra-dient forces. Therefore determining the velocity distribu-tion of atoms in the molasses is an interesting test of the newtheories.

The shower method2 5 is well suited to this task. Recallthat the shower method measures the time integral of the

TOF distribution versus horizontal displacement of theprobe and is sensitive to the atoms' horizontal velocity.Since there are no significant horizontal forces in these ex-periments, the horizontal velocity does not change as theatoms fall to the probe, and the shower method permits amore straightforward determination of the initial velocityprofile. In addition, if the molasses is far away, the effect ofthe atomic spatial distribution in the molasses is minimized,and the technique is less sensitive to all the geometric errorsdiscussed in Subsection 4.B.3. Finally, the shower methodhas less differential sensitivity to velocity than the TOFmethod.

We have made recent improvements in the shower meth-od originally described in Ref. 25. The probe is now located-6 cm below the molasses. At this position, the verticalvelocity of the atoms at the probe no longer depends muchon the atoms' initial vertical velocity or position in the mo-lasses. In addition, a better optical system has been devisedto eliminate the uncontrolled variations present in the earli-er experiment.

We have obtained preliminary results with this new ar-rangement, and, indeed, the. quality of the data is muchimproved. Temperatures as derived from measurementstaken with the TOF method at the standard 1.25-cm molas-ses-probe separation and at the new 6-cm separation agreewith each other. In addition, both temperatures agree withthat obtained from the shower method taken with small tomoderate horizontal displacements. Shower data takenwith large horizontal displacements show evidence that thevelocity distribution of atoms in the molasses is not a pureMaxwell-Boltzmann. However, much work needs to bedone in interpreting these results, and a full discussion ofthese new experiments will be forthcoming.

5. CONCLUSIONS

The past year has witnessed a sudden and dramatic increasein our knowledge of the processes involved in laser cooling.The theories of Cohen-Tannoudji and Dalibard and of Chuet al. describe cooling mechanisms entirely unanticipatedbefore the discovery of sub-Doppler temperatures. Thesemechanisms are much more effective in producing viscousconfinement and low mean energies than is Doppler coolingalone. The data of Section 4 show that there is a consider-able body of evidence supporting these new theories of lasercooling. Our measurements of temperature versus intensi-ty, detuning, magnetic field, and polarization are all in quali-tative agreement with the predictions of polarization-gradi-ent theories, as are measurements of decay rate versus beamimbalance. We emphasize that one must be cautious inmaking quantitative comparisons because the theories areone dimensional and the analytic expressions apply to tran-sitions different from the one used in sodium. In addition,there are differences in the definition of Io among the Stan-ford group, the Ecole Normale group, and our group, and it isunclear how to include the effects of the other beams in threedimensions.

A number of problems remain unresolved. Several theo-retical predictions, as well as some experimental evidence,indicate that there may be non-Maxwell-Boltzmann veloci-ty distributions in optical molasses. Detailed quantitativepredictions of these velocity distributions have yet to be

Lett et al.


made, and more research needs to be done on measurementsof the true velocity distribution. Perhaps a better under-standing of the velocity distribution together with the inclu-sion of the pinball effects will allow us to explain the behav-ior of the power dependence of the decay rate as well as thepeculiar properties of supermolasses.

A detailed theoretical explanation of how the magneticfield affects the molasses temperature is also needed. Beingable to describe the role played by Larmor precession andZeeman shifts may allow us to use the magnetic field mea-surements as a sensitive probe of the polarization-gradientdynamics. It may also help us to distinguish experimentallyamong various laser-cooling mechanisms.

A full treatment of the polarization-gradient force prob-lem is needed for the J = 2 - J' = 3 system in threedimensions. This would permit quantitative comparisonswith the data in Section 4. Other groups are at work on orhave proposed cooling other atomic species. They offer newsystems to test polarization gradient theories. It is possibleto choose an atomic system in which recoil effects are eithermuch smaller or much larger than those in sodium. Thispermits experimental investigation of polarization-gradientcooling either unobscured by the effects of recoil or stressingthe role of recoil in the cooling process. Atoms with a differ-ent hyperfine structure will also permit the study of polar-ization-gradient cooling at detunings A < -3r without thecomplications presented by the F = 2 - F' = 2 transition insodium. Finally, experiments that involve cooling in fewerthan three dimensions will greatly simplify the theoreticalinterpretation of the data since the complicated nature of athree-dimensional standing wave can be avoided. Such ex-periments are being performed at the Ecole Normale, Stan-ford, and Stony Brook.264 5

We believe that three-dimensional optical molasses willbecome an important tool for future research in atomicphysics. As an example, consider the high-resolution spec-troscopic technique of the atomic fountain in which atomsinteract twice, once while traveling up and a second timewhile falling down, with an oscillatory field (optical or rf) in avariation of the Ramsey technique.6 5 We propose using themolasses to launch the atoms collectively so that they havean upward velocity of -1 m/sec. This will be done by mov-ing the lower molasses mirror upward at this velocity andthen rapidly shutting off the molasses laser beams. Owingto the rapid equilibration time in molasses, all the atoms willfollow the motion of the mirror, remaining at rest with re-spect to the translating standing-wave pattern, and this willproduce a group of ultracold atoms with a uniform upwarddrift velocity. This launch avoids the severe heating prob-lems that would be associated with an upward force pro-duced with a single laser beam. Figure 24 shows a numericalsimulation of the number of atoms passing through a plane 2cm above the molasses as a function of time after a 1-m/secdrift velocity has been imparted to a 30-,4K molasses. Thelater peak, which is from the atoms as they fall back down, isbroadened owing to the thermal velocity distribution withinthe molasses. Note that this well-resolved second peak willbe visible for atoms only with the low temperatures nowachievable in optical molasses. The extremely long interac-tion time (200 msec) should be of great benefit for very-high-resolution spectroscopy.

As a final comment, we note that most theoretical treat-

2000

1 500

amC

C.)U)

1 000

500

0

0 50 1 0 0 1 50 200 250Time (msec)

Fig. 24. Fluorescence intensity from a probe versus time after acollective launch of 1 m/sec has been given to a 30-AK sample ofmolasses 2 cm below the probe.

ments of laser cooling to date are based on the assumptionthat the position and momentum coordinates of the atomcan be treated classically (the de Broglie wavelength XB <<X). This is clearly not the case when the temperature of theatoms approaches the recoil limit where B = X. Further-more, in the case of sodium molasses, typical dipole poten-tial wells are -50 jiK deep, while TR = 24 ,K. Semiclassi-cally, the atom would clearly be trapped most of the time if T_ TR. The length scale of the standing-wave potential wellsis /2, smaller than the de Broglie wavelength of the atom.Quantum mechanically, such a trapped atom would have amomentum spread larger than its thermal momentum.This problem with the semiclassical treatment is alreadyrelevant since, at the present 20-tK temperatures obtainedin sodium optical molasses, XB X/3. A fully quantum-mechanical method for treating some cases of laser cooling inthis limit is outlined by Dalibard and Cohen-Tannoudji2 9

and also discussed by Castin et al.46

ACKNOWLEDGMENTS

This research was partially supported by the U.S. Office ofNaval Research. In addition, C. E. Tanner and R. N. Wattsacknowledge the support of National Research Council-Na-tional Institute of Standards and Technology postdoctoralfellowships. We acknowledge the help of our collaborators,Hal Metcalf and Phil Gould. We thank Ed Williams, SteveChu, Dave Wineland, Wayne Itano, Alain Aspect, and AlanMigdall for helpful discussions. We are especially gratefulto Jean Dalibard and Claude Cohen-Tannoudji for commu-nicating their results before publication. In addition, wethank Tom Lucatorto for critically reading this manuscript.Finally, we thank Barry Taylor for his enthusiastic supportof our laser-cooling efforts.

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*

0

0 S

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- I~~ I I


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1751 (1985).23. P. Gould, P. Lett, and W. Phillips, in Laser Spectroscopy VIII,

W. Persson and S. Svanberg, eds. (Springer-Verlag, Berlin,1987), p. 64.

24. S. Chu, M. Prentiss, A. Cable, and J. Bjorkholm, in Laser Spec-troscopy VIII, W. Persson and S. Svanberg, eds. (Springer-Verlag, Berlin, 1987), p. 58.

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26. J. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N.Vansteenkiste, and C. Cohen-Tannoudji, in Atomic Physics 11,S. Haroche, J. C. Gay, and G. Grynberg, eds. (World Scientific,Singapore, 1989), p. 199, and personal communications.

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1118 (1989).35. One could imagine a two-level atom achieved with a J = 0 - J =

1 transition with a magnetic field applied to split the J = 1 levelsso much that only one transition, say the 0 - 0 transition, was inresonance with the laser. Two counterpropagating waves, po-larized at +45O to the magnetic field, would both drive the two-level transition, but they would create no standing wave in theusual sense of an intensity modulation. There would be, how-ever, a periodic spatial variation in the coupling between theatom and the field since the local polarization from the superpo-sition of the two waves would vary in space even though theintensity of the light would not. This variation would have thesame kind of effect as would a true standing wave.

36. J. Dalibard, Thase de doctorat d'6tatas Sciences Physique (Uni-versite de Paris, Paris, 1986).

37. J. Javanainen, M. Kaivola, U. Nielsen, 0. Poulsen, and E. Riis,J. Opt. Soc. Am. B 2,1768 (1985);

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44. See, for example, E. Buchwald, Ann. Phys. 66, 1 (1921).45. J. Dalibard, Laboratoire de Spectroscopie Hertzienne de l'Ecole

Normale Sup6rieure 24 rue Lhomond, F-75231 Paris Cedex 05,France (personal communication); D. Weiss, E. Riis, Y. Shevy,P. Ungar, and S. Chu, J. Opt. Soc. Am. B 6, 2072 (1989); H.Metcalf, Department of Physics, State University of New Yorkat Stony Brook, Stony Brook, New York 11794 (personal com-munication).

46. Y. Castin, H. Wallis, and J. Dalibard, J. Opt. Soc. Am. 6, 2046(1989).

47. We set hw0 equal to the internal energy plus one recoil energy.Wineland and Itano3 use a different convention: our wo isequivalent to their co'.

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49. P. Lett, P. Gould, and W. Phillips, Hyperfine Interact. 44, 335(1988).

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57, 314 (1986).53. C. Salomon, J. Dalibard, A. Aspect, H. Metcalf, and C. Cohen-

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Weiner, Phys. Rev. Lett. 60, 788 (1988).55. C. Tanner, B. Masterson, and C. Wieman, Opt. Lett. 13, 357

(1988).56. V. Balykin, V. Letokhov, Yu. Ovchinnikov, A. Sidorov, and S.

Shul'ga, Opt. Lett. 13, 958 (1988).57. A. Anderson, M. Boshier, S. Harocke, E. Hinds, W. Jhe, and D.

Meschede, in Atomic Physics 11, S. Haroche, J. C. Gay, and G.Grynberg, eds. (World Scientific, Singapore, 1989), p. 626.

58. A. P. Kazantsev, G. Surdutovich, and V. Yakovlev, Opt. Com-mun. 68, 103 (1988).

59. V. Balykin, V. Letokhov, and A. Sidorov, Pis'ma Zh. Eksp. Teor.Fiz. 40, 251 (1984) [JETP Lett. 40, 1026 (1984)]; V. Balykin, V.Letokhov, V. Minogin, Yu. Rozhdestvenskii, and A. Sidorov, Zh.Eksp. Teor. Fiz. 90,871 (1986) [Sov. Phys. JETP 63,508 (1986)].

60. D. Sesko, C. Fan, and C. Wieman, J. Opt. Soc. Am. B 5, 1225(1988).

61. This technique was suggested to us by H. Metcalf.62. Y. Shevy, D. Weiss and S. Chu, in Spin Polarized Quantum

Systems, S. Stringari, ed. (World Scientific, Singapore, 1989), p.287.

63. The analysis presented in the next two paragraphs was not usedin assigning TOF temperatures in Ref. 25. Instead, we assignedan uncertainty based on our then limited knowledge of theprobe effect.

64. S. Chu, Department of Physics, Stanford University, Stanford,California 94305 (personal communication); J. Dalibard, Labor-atoire de Spectroscopie Hertzienne de l'Ecole Normale Sup6r-ieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France (per-sonal communication).

65. A. DeMarchi, Metrologia 18, 103 (1982); R. Beausoleil and T.Hansch, Phys. Rev. A 33,1661 (1986). Such an experiment wasrecently performed by the Stanford group [M. Kasevich, E. Riis,S. Chu, and R. DeVoe, Phys. Rev. Lett. 63, 612 (1989)], usingatoms from a Zeeman-assisted radiation-pressure trap butwithout the launch mechanism that we suggest here.

Lett et al.

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