Laser cooling and trapping of atoms - Universiteit Utrechtstrat102/preprint/visby.pdf · Laser cooling and trapping of atoms H. J. Metcalf Department of Physics, State University

H. J. Metcalf and P. van der Straten Vol. 20, No. 5 /May 2003/J. Opt. Soc. Am. B 887

Laser cooling and trapping of atoms

H. J. Metcalf

Department of Physics, State University of New York, Stony Brook, New York 11794

P. van der Straten

Debye Institute, Department of Atomic- and Interface Physics, Utrecht University, P.O. Box 80.000, 3508 TA Utrecht,The Netherlands

Received August 27, 2002; revised manuscript received November 25, 2002

A review is presented of some of the principal techniques of laser cooling and trapping that have been devel-oped during the past 20 years. Its approach is primarily experimental, but its quantitative descriptions areconsistent in notation with most of the theoretical literature. It begins with a simplified introduction to op-tical forces on atoms, including both cooling and trapping. Then its three main sections discuss its three se-lected features, (1) quantization of atomic motion, (2) effects of the multilevel structure of atoms, and (3) theeffects of polychromatic light. Each of these features is an expansion in a different direction from the simplestmodel of a classical, two-level atom moving in a monochromatic laser field. © 2003 Optical Society of America

OCIS codes: 020.7010, 300.6210.

1. INTRODUCTIONThe combination of laser cooling and atom trapping hasproduced astounding new tools for atomic physicists.1

These experiments require the exchange of momentumbetween atoms and an optical field, usually at a nearlyresonant frequency. The energy of the light \v changesthe internal energy of the atom, and the angular momen-tum \ changes the orbital angular momentum , of theatom, as described by the well-known selection rule D,5 61. By contrast, the linear momentum of the lightp 5 \v/c (p 5 \k) cannot change the internal atomicdegrees of freedom and therefore must change the mo-mentum of the atoms in the laboratory frame. The forceresulting from this momentum exchange between thelight field and the atoms can be used in many ways to con-trol atomic motion and is the subject of this paper.

If the light is absorbed, the atom makes a transition tothe excited state, and the return to the ground state canbe either by spontaneous or by stimulated emission. Thenature of the optical force that arises from these two dif-ferent processes is quite different and will be describedseparately. The spontaneous emission case is differentfrom the familiar quantum mechanical calculations thatuse state vectors to describe the state of the system, be-cause spontaneous emission causes the state of the sys-tem to evolve from a pure state into a mixed state. Sincespontaneous emission is an essential ingredient for thedissipative nature of the optical forces, the density matrixis needed to describe it.

A. Radiative Optical ForcesIn the simplest case, the absorption of well-directed lightfrom a laser beam, the momentum exchange between thelight field and the atoms results in a force

F 5 dp/dt 5 \kgp , (1)

0740-3224/2003/050887-22$15.00 ©

where gp is the excitation rate of the atoms. The absorp-tion leaves the atoms in their excited state, and, if thelight intensity is low enough that they are much morelikely to return to the ground state by spontaneous emis-sion than by stimulated emission, the resulting fluores-cent light carries off momentum \k in a random direction.The momentum exchange from the fluorescence averageszero, so the net total force is given by Eq. (1).

The scattering rate gp depends on the laser detuningfrom atomic resonance d [ v, 2 va , where v, is the la-ser frequency and va is the atomic resonance frequency.This detuning is measured in the atomic reference frame,and it is necessary that the Doppler-shifted laser fre-quency in the moving atoms’ reference frame be used tocalculate the absorption and scattering rate. The excita-tion rate gp for a two-level atom is given by the Lorentz-ian

gp 5s0g/2

1 1 s0 1 @2~d 1 vD!/g#2 , (2)

where g [ 1/t is an angular frequency corresponding tothe decay rate of the excited state. Here s0 5 I/Is is theratio of the light intensity I to the saturation intensityIs [ phc/3l3t, which is a few mW/cm2 for typical atomictransitions. The Doppler shift seen by the moving atomsis vD 5 2k • v (note that k opposite to v produces a posi-tive Doppler shift). The force is thus velocity dependent,and the experimenter’s task is to exploit this dependenceto reach the desired goal, for example, optical friction forlaser cooling.

The maximum attainable deceleration is obtained forvery high light intensities. High-intensity light can pro-duce faster absorption, but it also causes equally faststimulated emission; the combination produces neitherdeceleration nor cooling. The momentum transfer to theatom in stimulated emission is in the direction opposite towhat it was in absorption, resulting in a net transfer of

2003 Optical Society of America

888 J. Opt. Soc. Am. B/Vol. 20, No. 5 /May 2003 H. J. Metcalf and P. van der Straten

Table 1. Overview of Paper Contents

Section Levelsa Frequencies Motion Dimensions Topics

1 Forces and temperature2 TLA Single Classical 1, 3 Beam slowing

MolassesOptical traps

3 TLA Single Quantum 1, 3 Dark states and VSCPTOptical lattices

4 Multiple Single Classical 1 Sub-Doppler coolingMagneto-optical traps

5 TLA Multiple Classical 1 Bichromatic forceOther polychromatic forces

a TLA, two-level atom.

zero. Since high intensity causes the atom to divide itstime equally between ground and excited states, the forceis limited to F 5 \kgp ; so the deceleration saturates at avalue amax 5 \kg/2M [see Eq. (2)].

B. Dipole Optical ForcesWhen the detuning d @ g, spontaneous emission may bemuch less frequent than stimulated emission. The dissi-pative radiative force found from Eqs. (1) and (2) is nec-essary for laser cooling, but the conservative dipole forcearises from stimulated emission and derives from the gra-dient of the light shift.

To find the light shift, the dressed-atom picture pro-vides the easiest description, and, for a single laser beamtraveling in the x direction with Rabi frequency V5 gAs0/2, the light shift is given by

v ls 5 ~AV2 1 d 2 2 d!/2. (3)

For sufficiently large detuning d @ V, approximation ofEq. (3) leads to v ls ' V2/4d 5 g2s0/8d.

In a standing wave in one dimension with d @ V, thelight shift v ls(x) varies sinusoidally from node to anti-node, and also spontaneous emission is suppressed, sothat \v ls(x) may be treated as a potential U(x). The re-sulting dipole force is

F~x ! 5 2“U~x ! 5 2\g2

8dIs“I~x !, (4)

where I(x) is the total intensity distribution of thestanding-wave light field of period l/2. For such a stand-ing wave, the optical electric field (and the Rabi fre-quency) at the antinodes is double that of each travelingwave that composes it, and so the total intensity Imax atthe antinodes is four times that of the single travelingwave.

C. TemperatureThe idea of ‘‘temperature’’ in laser cooling requires somecareful discussion and disclaimers. In thermodynamics,temperature is carefully defined as a parameter of thestate of a closed system in thermal equilibrium with itssurroundings. This, of course, requires that there bethermal contact, i.e., heat exchange, with the environ-ment. In laser cooling this is clearly not the case, be-cause a sample of atoms is always absorbing and scatter-

ing light. Furthermore, there is essentially no heatexchange (the light cannot be considered as heat eventhough it is indeed a form of energy). Thus the systemmay very well be in a steady-state situation but certainlynot in thermal equilibrium, so that the assignment of athermodynamic ‘‘temperature’’ is completely inappropri-ate.

Nevertheless, it is convenient to use the label of ‘‘tem-perature’’ to describe an atomic sample whose average ki-netic energy Êk& in one dimension has been reduced bythe laser light, and this is written simply as kBT/25 Êk&, where kB is Boltzmann’s constant. It must be

remembered that this temperature assignment is abso-lutely inadequate for atomic samples that do not have aMaxwell–Boltzmann velocity distribution, whether or notthey are in thermal contact with the environment: Thereare infinitely many velocity distributions that have thesame value of Êk& but are so different from one anotherthat characterizing them by the same temperature is a se-vere error. In the special case where there is a truedamping force, F } 2v, and where the diffusion in mo-mentum space is a constant independent of momentum,solutions of the Fokker–Planck equation can be foundanalytically and lead to a Maxwell–Boltzmann distribu-tion that does indeed have a temperature.

It is convenient to define a single dimensionless param-eter « [ vr /g that is ubiquitous in describing laser cool-ing. It is the ratio of the recoil frequency vr [ \k2/2M tothe natural width g, and as such embodies most of the im-portant information that characterizes laser cooling on aparticular atomic transition. For atoms under consider-ation for laser cooling, « is of the order of 1023 –1022.

D. Organization of this PaperThis paper is organized as follows (see Table 1): After abroad introduction, the simplest case of optical forces ontwo-level atoms from a single-frequency light field is pre-sented in Section 2. Several aspects of the classical mo-tion of neutral atoms under these forces are introducedand discussed. These include beam slowing, optical mo-lasses, and optical traps. With this as a basis, Sections3–5 address cases where these simplifications are nolonger valid:

• In Section 3 the phenomena associated with quanti-zation of the atomic motion are considered. Atoms


are no longer regarded as point particles whose po-sition and momentum can be simultaneously knownto arbitrary precision, but instead they are de-scribed by wave functions that include their center-of-mass motion in the laboratory frame.

• In Section 4 the simplification of the two-level atomis lifted. Here the importance of light polarizationand selection rules appear, and the concept of sub-Doppler cooling is described. Laser cooling is nolonger limited by the natural decay rate g, and at-oms can be cooled all the way to the recoil limitkBT/2 5 \vr determined by vr ! g. The influenceof magnetic fields becomes much more importantthan for the two-level case, and it not only allowsthe existence of the magneto-optical trap (MOT) butalso leads to sub-Doppler temperatures.

• In Section 5 the extraordinarily large forces associ-ated with polychromatic light are described. Theradiative force is limited by the natural lifetime to\kgp , and the velocity-dependent part of the Sisy-phus force is similarly limited by g. But multifre-quency light, even on a two-level atom, can exertmuch stronger cooling forces, enabling smaller andmore efficient beam slowing for trapping or otherpurposes.

Much of the material here was taken from our recenttextbook,1 and the reader is encouraged to consult thatsource for the origin of many of the formulas presented inthe present paper, as well as for further reading and moredetailed references to the literature.

2. OPTICAL FORCES ON TWO-LEVELATOMS

A. Slowing Atomic BeamsAmong the earliest laser-cooling experiments was decel-eration of atoms in a beam.2 Phillips and Metcalf ex-ploited the Doppler shift to make the momentum ex-change (hence the force) velocity dependent. Theapproach worked by directing a laser beam opposite to anatomic beam so that the atoms could absorb light, and

hence momentum \k, very many times along their pathsthrough the apparatus as shown in Fig. 1.2,3 Of course,excited-state atoms cannot absorb light efficiently fromthe laser that excited them, so between absorptions theymust return to the ground state by spontaneous decay, ac-companied by emission of fluorescent light. The spatialsymmetry of the emitted fluorescence results in an aver-age of zero net momentum transfer from many such fluo-rescence events. Thus the net force on the atoms is inthe direction of the laser beam, and the maximum decel-eration is limited by the spontaneous emission rate g.

Since the maximum deceleration amax 5 \kg/2M isfixed by atomic parameters, it is straightforward to calcu-late the minimum stopping length Lmin and time tmin forthe rms velocity of atoms v 5 2AkBT/M at the sourcetemperature. The result is Lmin 5 v2/2amax and tmin5 v/amax . In Table 2 are some of the parameters forslowing a few atomic species of interest from the peak ofthe thermal velocity distribution.

Maximizing the scattering rate gp requires d 5 2vD inEq. (2). If d is chosen for a particular atomic velocity inthe beam, then, as the atoms slow down, their changingDoppler shift will take them out of resonance. They willeventually cease deceleration after their Doppler shift hasbeen decreased by a few times the power-broadened widthg8 5 gA1 1 s0, corresponding to Dv of a few timesvc 5 g8/k. Although this Dv of a few m/s is considerablylarger than the typical atomic recoil velocity vr 5 \k/Mof a few cm/s, it is still only a small fraction of the atoms’average thermal velocity v, so that significant furthercooling or deceleration cannot be accomplished.

To achieve deceleration that changes the atomic speedsby hundreds of m/s, it is necessary to maintain (d 1 vD)! g by compensating such large changes of the Doppler

shift. This can be done by changing vD, or d through ei-ther v, or va . The two most common methods for main-taining this resonance are sweeping the laser frequencyv, along with the changing vD of the deceleratingatoms,4–6 or spatially varying the atomic resonance fre-quency with an inhomogeneous dc magnetic field to keepthe decelerating atoms in resonance with the fixed fre-quency laser.1,2,7

The use of a spatially varying magnetic field to tune theatomic levels along the beam path was the first method tosucceed in slowing atoms.2 It works as long as the Zee-

Fig. 1. Schematic diagram of apparatus for beam slowing. The tapered magnetic field is produced by layers of varying length on thesolenoid.


man shifts of the ground and excited states are different,so that the resonant frequency is shifted. The field canbe tailored to provide the appropriate Doppler shift alongthe moving atom’s path. A solenoid that can producesuch a spatially varying field has layers of decreasinglengths. The technical problem of extracting the beam ofslow atoms from the end of the solenoid can be simplifiedby reversing the field gradient and choosing a transitionwhose frequency decreases with increasing field.8

For alkali atoms such as sodium, a time-of-flight (TOF)method can be used to measure the velocity distributionof atoms in the beam. It employs two additional beamslabeled pump and probe from laser 1 as shown in Fig. 1.Because these beams cross the atomic beam at 90°, vD5 2k • v 5 0, and the beams excite atoms at all veloci-ties. The pump beam is tuned to excite and empty a se-lected ground hyperfine state (hfs), and it transfers morethan 98% of the population as the atoms pass through its0.5-mm width. To measure the velocity distribution ofatoms in the selected hfs, this pump laser beam is inter-rupted for a period Dt 5 10–50 ms with an acoustic opti-cal modulator (AOM). A pulse of atoms in the selected

Fig. 2. The velocity distribution measured with the TOFmethod. The experimental width of approximately 1

6(g/k) isshown by the dashed vertical lines between the arrows. TheGaussian fit through the data yields a FWHM of 2.97 m/s (fromRef. 9).

Table 2. Parameters of Interest for SlowingVarious Atomsa

AtomToven(K)

v(m/s)

Lmin(m)

tmin(ms)

H 1000 5000 0.012 0.005He* 4 158 0.03 0.34He* 650 2013 4.4 4.4Li 1017 2051 1.15 1.12Na 712 876 0.42 0.96K 617 626 0.77 2.45Rb 568 402 0.75 3.72Cs 544 319 0.93 5.82

a The stopping length Lmin and time tmin are minimum values. Theoven temperature Toven that determines the peak velocity is chosen to givea vapor pressure of 1 Torr. Special cases are H at 1000 K and He in themetastable triplet state, for which two rows are shown: one for a 4-Ksource and another for the typical discharge temperature.

hfs passes the pump region and travels to the probe beam.The time dependence of the fluorescence induced by theprobe laser, tuned to excite the selected hfs, gives the timeof arrival, and this signal is readily converted to a velocitydistribution. Figure 2 shows the measured velocity dis-tribution of the atoms slowed by laser 2.

B. Optical Molasses

1. Doppler CoolingIt is straightforward to calculate the radiative force on at-oms moving in a standing wave (counterpropagating laserbeams) by using Eq. (2). In the low-intensity case, wherestimulated emission is not important, the forces from thetwo light beams are simply added to give FOM 5 F1

1 F2 , where F6 are found from Eqs. (1) and (2). Thenthe sum of the two forces is

FOM >8\k2ds0v

g@1 1 s0 1 ~2d/g!2#2 [ 2bv, (5)

where terms of order (kv/g)4 and higher have been ne-glected. The slowing force is proportional to velocity forsmall enough velocities, resulting in viscous damping10,11

that gives this technique the name optical molasses (OM).These forces are plotted in Fig. 3. For d , 0, the sum

of the forces opposes the velocity and therefore viscouslydamps the atomic motion. The force FOM has maximanear v ' 6gAs0 1 1/2k and decreases rapidly for largervelocities.

In laser cooling and related aspects of optical control ofatomic motion, the forces arise because of the exchange ofmomentum between the atoms and the laser field. Sincethe energy and momentum exchange are necessarily indiscrete quanta rather than continuous, the interaction ischaracterized by finite momentum kicks. This is oftendescribed in terms of steps in a fictitious space whose axesare momentum rather than position. These steps in mo-mentum space are of size \k and thus are generally smallcompared with the magnitude of the atomic momenta atthermal velocities v. This is easily seen by computing\k/Mv 5 ATr /T ! 1.

If there were no other influence on the atomic motion,all atoms would quickly decelerate to v 5 0, and thesample would reach T 5 0, a clearly unphysical result.There is also some heating caused by the light beams thatmust be considered, and it derives from the discrete sizeof the momentum steps that the atoms undergo with eachemission or absorption. Since the atomic momentumchanges by \k, their kinetic energy changes on averageby at least the recoil energy Er 5 \2k2/2M 5 \vr . Thismeans that the average frequency of each absorption isvabs 5 va 1 vr , and the average frequency of each emis-sion is vemit 5 va 2 vr . Thus the light field loses an av-erage energy of \(vabs 2 vemit) 5 2\vr for each scatter-ing. This loss occurs at a rate 2gp (two beams), and theenergy is converted to atomic kinetic energy because theatoms recoil from each event. The atomic sample isthereby heated because these recoils are in random direc-tions.

The competition between this heating with the damp-ing force of Eq. (5) results in a nonzero kinetic energy


in steady state where the rates of heating and cooling areequal. Equating the cooling rate, FOM • v, to the heatingrate, 4\vrgp , the steady-state kinetic energy is (\g/8)(2udu/g 1 g/2udu). This result is dependent on udu, and ithas a minimum at 2udu/g 5 1, whence d 5 2g/2. Thetemperature found from the kinetic energy is then TD5 \g/2kB , where kB is Boltzmann’s constant and TD iscalled the Doppler temperature or the Doppler coolinglimit. For ordinary atomic transitions, TD is typically be-low 1 mK.

Another instructive way to determine TD is to note thatthe average momentum transfer of many spontaneousemissions is zero, but the rms scatter of these about zerois finite. One can imagine these decays as causing a ran-dom walk in momentum space, similar to Brownian mo-tion in real space, with step size \k and step frequency2gp , where the factor of 2 arises because of the twobeams. The random walk results in an evolution of themomentum distribution as described by the Fokker–Planck equation, and it can be used for a more formaltreatment of the laser-cooling process. It results in dif-fusion in momentum space with diffusion coefficientD0 [ 2(Dp)2/Dt 5 4gp(\k)2. Then the steady-statetemperature is given by kBT 5 D0 /b. This turns out tobe \g/2 as above for the case s0 ! 1 when d 5 2g/2.This remarkable result predicts that the final tempera-ture of atoms in OM is independent of the optical wave-length, atomic mass, and laser intensity (as long as it isnot too large).

2. Atomic Beam Collimation—One Dimensional OpticalMolasses—Beam BrighteningWhen an atomic beam crosses a one-dimensional (1D) OMas shown in Fig. 4, the transverse motion of the atoms isquickly damped, while the longitudinal component is es-sentially unchanged. This transverse cooling of anatomic beam is an example of a method that can actuallyincrease its brightness [atoms/(sec sr cm2)] because suchactive collimation uses dissipative forces to compress thephase-space volume occupied by the atoms. By contrast,the usual focusing or collimation techniques for light

Fig. 3. Velocity dependence of the optical damping forces for 1DOM. The two dotted traces show the force from each beam, andthe solid curve is their sum. The straight line shows how thisforce mimics a pure damping force over a restricted velocityrange. These are calculated for s0 5 2 and d 5 2g, so there issome power broadening evident.

beams and most particle beams is restricted to selectionby apertures or conservative forces that preserve thephase-space density of atoms in the beam.

This velocity compression at low intensity in one di-mension can be simply estimated for two-level atoms to beapproximately vc /vD 5 Ag/vr 5 A1/«. Here vD is thevelocity associated with the Doppler limit for laser coolingdiscussed below: vD 5 A\g/2M. For Rb, vD 5 12cm/s, vc 5 g/k . 4.6 m/s, vr . 2p 3 3.8 kHz, and 1/«. 1600. Including two transverse directions along withthe longitudinal slowing and cooling discussed above, thedecrease in phase-space volume from the momentum con-tribution alone for laser cooling of a Rb atomic beam canexceed 106. Clearly optical techniques can create atomicbeams enormously more times intense than ordinarythermal beams, and also many orders of magnitudebrighter.

3. Experiments in Three-Dimensional Optical MolassesBy use of three intersecting orthogonal pairs of oppositelydirected beams, the movement of atoms in the intersec-tion region can be severely restricted in all three dimen-sions, and many atoms can thereby be collected andcooled in a small volume.

Even though atoms can be collected and cooled in theintersection region, it is important to stress that this isnot a trap. That is, atoms that wander away from thecenter experience no force directing them back. They areallowed to diffuse freely and even escape, as long as thereis enough time for their very slow diffusive movement toallow them to reach the edge of the region of the intersec-tion of the laser beams. Because the atomic velocitiesare randomized during the damping time M/b 5 2/vr ,atoms execute a random walk in position space with astep size of 2vD /vr 5 l/(pA2«) > few mm. To diffuse adistance of 1 cm requires ;107 steps or ;30 s (Refs. 13and 14).

Three-dimensional (3D) OM was first observed in1985.11 Preliminary measurements of the average ki-netic energy of the atoms were done by blinking off thelaser beams for a fixed interval. Comparison of thebrightness of the fluorescence before and after the turnoffwas used to calculate the fraction of atoms that left theregion while it was in the dark. The dependence of thisfraction on the duration of the dark interval was used to

Fig. 4. Scheme for optical brightening of an atomic beam.First the transverse velocity components of the atoms aredamped out by an OM, then the atoms are focused to a spot, andfinally the atoms are recollimated in a second OM (from Ref. 12).


estimate the velocity distribution and hence the tempera-ture. This method, which is usually referred to as re-lease and recapture, is specifically designed to measurethe temperature of the atoms, since the usual way of mea-suring temperatures cannot be applied to an atomic cloudof a few million atoms. The result was consistent withTD as calculated from the Doppler theory, as described inSubsection 2.B.1.

Later a more sensitive ballistic technique was devisedat the National Institute of Standards and Technology15

(NIST) that showed the astounding result that the tem-perature of the atoms in OM was very much lower thanTD . These experiments also found that OM was lesssensitive to perturbations and more tolerant of alignmenterrors than was predicted by Doppler theory. For ex-ample, if the intensities of the two counterpropagating la-ser beams forming an OM were unequal, then the force onatoms at rest would not vanish, but the force on atomswith some nonzero drift velocity would vanish. This driftvelocity can be easily calculated by using unequal inten-sities s01 and s02 , to derive an analog of relation (5).Thus atoms would drift out of an OM, and the calculatedrate would be much faster than observed when the beamswere deliberately unbalanced in the experiments.16

It was an enormous surprise to observe that the ballis-tically measured temperature of the Na atoms15 was asmuch as 10 times lower than TD 5 240 mK, the tempera-ture minimum calculated from the theory. This breach-ing of the Doppler limit forced the development of an en-tirely new picture of OM that accounts for the fact that inthree dimensions a two-level picture of atomic structureis inadequate. The multilevel structure of atomic states,and optical pumping among these sublevels, must be con-sidered in the description of 3D OM, as discussed in Sub-section 4.B.5.

C. Dipole Force Optical Traps

1. Single-Beam Optical Traps for Two-Level AtomsThe simplest imaginable trap consists of a single, stronglyfocused Gaussian laser beam17,18 (see Fig. 5) whose inten-sity at the focus varies transversely with r as

I~r ! 5 I0 exp~2r2/w02!, (6)

where w0 is the beam waist size. Such a trap has a well-studied and important macroscopic classical analog in aphenomenon called optical tweezers.19–21

With the laser light tuned below resonance (d , 0),the ground-state light shift is everywhere negative, butlargest at the center of the Gaussian beam waist.Ground-state atoms therefore experience a force attract-ing them toward this center given by the gradient of thelight shift, which is found from Eq. (3) and for d/g @ s0 isgiven by Eq. (4). For the Gaussian beam, this transverseforce at the waist is harmonic for small r and is given by

F .\g2

4d

I0

Is

r

w02 exp~22r2/w0

2! (7)

In the longitudinal direction there is also an attractiveforce, but it is more complicated and depends on the de-tails of the focusing. Thus this trap produces an attrac-tive force on atoms in three dimensions.

Although it may appear that the trap does not confineatoms longitudinally because of the radiation pressurealong the laser beam direction, careful choice of the laserparameters can indeed produce trapping in three dimen-sions. This can be accomplished because the radiationpressure force decreases as 1/d 2 [see Eq. (2)], but, by con-trast, the dipole force only decreases as 1/d for d @ V [seeEq. (3)]. If ud u is chosen to be sufficiently large, atomsspend very little time in the untrapped (actually repelled)excited state because its population is proportional to1/d 2. Thus a sufficiently large value of udu both produceslongitudinal confinement and maintains the atomic popu-lation primarily in the trapped ground state.

The first optical trap was demonstrated in Na withlight detuned below the D lines.18 With 220 mW of dyelaser light tuned ;650 GHz below the atomic transitionand focused to a ;10-mm waist, the trap depth was;15\g, corresponding to 7 mK. Single-beam dipole forcetraps can be made with the light detuned by a significantfraction of its frequency from the atomic transition. Sucha far-off-resonance trap (FORT) was developed for Rb at-oms by use of light detuned by nearly 10% to the red ofthe D1 transition at l 5 795 nm (Ref. 22). Between 0.5and 1 W of power was focused to a spot about 10 mm insize, resulting in a trap 6 mK deep where the light scat-tering rate was only a few hundred per second. The traplifetime was more than half a second.

There is a qualitative difference when the trappinglight is detuned by a large fraction of the optical fre-quency. In one such case, Nd:YAG light at l 5 1064nm was used to trap Na, whose nearest transition is atl 5 596 nm (Ref. 23). In a more extreme case, a trap us-ing l 5 10.6 mm light from a CO2 laser was used to trapCs, whose optical transition is at a frequency ;12 timeshigher24 (l 5 852 nm). For such large values of udu, cal-culations of the trapping force cannot exploit the rotating-wave approximation as was done for Eq. (3), and theatomic behavior is similar to that in a dc field. It is im-portant to remember that for an electrostatic trap Earn-shaw’s theorem precludes a field maximum, but that inthis case there is indeed a local 3D intensity maximum ofthe focused light because it is not static.

2. Blue-Detuned Optical TrapsOne of the principle disadvantages of the optical trapsdiscussed above is that the negative detuning attracts at-oms to the region of highest light intensity. This resultsin significant spontaneous emission unless the detuningis a large fraction of the optical frequency, such as in theNd:YAG laser trap23 or the CO2 laser trap.24 More im-portant in some cases is that the trap relies on Starkshifting of the atomic energy levels by an amount equal tothe trap depth, and this severely compromises the capa-bilities for precision spectroscopy in a trap.25

Fig. 5. A single focused laser beam produces the simplest typeof optical trap.


Attracting atoms to the region of lowest intensity wouldameliorate both of these concerns, but such a trap re-quires positive detuning (blue), and an optical configura-tion having a dark central region. One of the first experi-mental efforts at a blue-detuned trap used the repulsivedipole force to support Na atoms that were otherwise con-fined by gravity in an optical cup.26 Two rather flat, par-allel beams detuned by 25% of the atomic resonance fre-quency were directed horizontally and oriented to form aV-shaped trough. Their Gaussian beam waists formed aregion .1 mm long where the potential was deepest, andhence provided confinement along their propagation di-rection as shown in Fig. 6. The beams were the l5 514 nm and l 5 488 nm from an Ar laser, and thechoice of two frequencies was not simply to exploit the fullpower of the multiline Ar laser, but also to avoid the spa-tial interference that would result from use of a single fre-quency.

Obviously a hollow laser beam would also satisfy therequirement for a blue-detuned trap, but conventionaltextbook wisdom shows that such a beam is not an eigen-mode of a laser resonator.27 Some lasers can make hol-low beams, but these are illusions because they consist ofrapid oscillations between the TEM01 and TEM10 modesof the cavity. Nevertheless, Maxwell’s equations permitthe propagation of such beams, and in the recent pastthere have been studies of the LaGuerre–Gaussian modesthat constitute them.28–30 The several ways of generat-ing such hollow beams have been tried by many experi-mental groups and include phase and amplitude holo-grams, hollow waveguides, axicons or related cylindricalprisms, stressing fibers, and simply mixing the TEM01and TEM10 modes with appropriate cylindrical lenses.

An interesting experiment was performed by using theideas of Sisyphus cooling (see Subsection 4.B) with eva-nescent waves combined with a hollow beam formed withan axicon.31 In the previously reported experiments withatoms bouncing under gravity from an evanescent wavefield,32,33 they were usually lost to horizontal motion forseveral reasons, including slight tilting of the surface,surface roughness, horizontal motion associated withtheir residual motion, and horizontal ejection by theGaussian profile of the evanescent wave laser beam. Theauthors of Ref. 31 simply confined their atoms in the hori-zontal direction by surrounding them with a wall of blue-detuned light in the form of a vertical hollow beam.Their gravito-optical surface trap cooled Cs atoms to .3mK at a density of .3 3 1010 cm23 in a sample whose 1/eheight in the gravitational field was only 19 mm. Simpleballistics gives a frequency of 450 bounces/s, and the .6-slifetime (limited only by background gas collisions) corre-sponds to several thousand bounces. However, at suchlow energies the de Broglie wavelength of the atoms is.1/4 mm, and the atomic motion is no longer accuratelydescribed classically, but requires de Broglie wave meth-ods.

3. QUANTUM STATES OF MOTIONAs the techniques of laser cooling advanced from a labo-ratory curiosity to a tool for new problems, the emphasisshifted from attaining the lowest possible steady-state

temperatures to the study of elementary processes, espe-cially the quantum mechanical description of the atomicmotion. In the completely classical description of lasercooling, atoms were assumed to have well-defined posi-tion and momentum that could be known simultaneouslywith arbitrary precision. However, when atoms are mov-ing sufficiently slowly that their de Broglie wavelengthprecludes their localization to less than l/2p, these de-scriptions fail and a quantum mechanical description isrequired. Such exotic behavior for the motion of wholeatoms, as opposed to electrons in the atoms, was not con-sidered before the advent of laser cooling simply becauseit is too far out of the range of ordinary experiments. Aseries of experiments in the early 1990s provided dra-matic evidence for these new quantum states of motion ofneutral atoms and led to the debut of de Broglie waveatom optics.

The quantum description of atomic motion requiresthat the energy of such motion be included in the Hamil-tonian. The total Hamiltonian for atoms moving in alight field would then be given by

H 5 Hatom 1 Hrad 1 Hint 1 Hkin , (8)

where Hatom describes the motion of the atomic electronsand gives the internal atomic energy levels, Hrad is the en-ergy of the radiation field and is of no concern here be-cause the field is not quantized, Hint describes the excita-tion of atoms by the light field and the concomitant lightshifts, and Hkin is the kinetic energy operator of the mo-tion of the atoms’ center of mass. This Hamiltonian haseigenstates of not only the internal energy levels and theatom–laser interaction that connects them, but also of thekinetic energy operator Hkin [ P2/2M. These eigen-states will therefore be labeled by quantum numbers ofthe atomic states as well as the center-of-mass momen-tum p. Here and in the remainder of this section the mo-mentum is measured in units of \k. An atom in theground state, u g; p&, has an energy Eg 1 p2\vr , whichcan take on a range of values. Here the kinetic energy ofthe atom is given in terms of the recoil frequency vr[ \k2/2M.

Bose–Einstein condensation (BEC) is another exampleof atomic motion in the quantum domain. It occurs inthe absence of light, and its onset is characterized by cool-ing to the point where the atomic de Broglie wavelengths

Fig. 6. Light intensity experienced by an atom located in aplane 30 mm above the beam waists of two quasi-focused sheetsof light traveling parallel and arranged to form a V-shapedtrough. The x and y dimensions are in micrometers (from Ref.26).


are comparable with the interatomic spacing. The topicsdiscussed here concern the properties of cold atoms in anoptical field, where the atomic de Broglie wavelengths arecomparable with the optical wavelength l.

A. Dark StatesOne special area of interest is the study of dark states,atomic states that cannot be excited by the light field.Some atomic states are trivially dark; that is, they cannotbe excited because the light has the wrong frequency orpolarization. The more interesting cases are superposi-tion states created by coherent optical Raman coupling.A very special case are those superpositions whose excit-able component vanishes exactly when their external (deBroglie wave) states are characterized by a particular mo-mentum. Such velocity-selective coherent populationtrapping (VSCPT) has been a subject of considerable in-terest since its first demonstration in 1988.34 VSCPT en-ables arbitrarily narrow momentum distributions andhence arbitrarily large delocalization for atoms in thedark states.35 Adequate discussion of VSCPT requires aquantum mechanical description of the atomic motion.

1. VSCPT in Two-Level AtomsTo see how the quantization of the motion of a two-levelatom in a monochromatic field allows the existence of avelocity-selective dark state, consider the states of a two-level atom with single internal ground and excited levels,u g; p& and ue; p8&.

Two ground eigenstates u g; p& and u g; p9& are onlycoupled to one another by an optical field in certain spe-cial cases. For example, in a single pair of counterpropa-gating light beams there can be absorption-stimulatedemission cycles that connect u g; p& to itself or to u g; p6 2&, depending on whether the stimulated emission is

induced by the beam that excited the atom or by the coun-terpropagating one. In the first case the states of theatom and field are left unchanged, but the interactionshifts the internal atomic energy levels, thereby produc-ing the light shift. In the second case the initial and finalkinetic energies of the atom differ by 64( p 6 1)\vr , soenergy conservation requires p 5 71 (the energy of thelight field is unchanged by the interaction, since all thephotons in the field have energy \v,). Thus energy con-servation corresponds to Raman resonance between thedistinct states u g; 21& and u g; 11& and is therefore ve-locity selective.

The coupling of these two degenerate states by the lightfield produces off-diagonal matrix elements of the totalHamiltonian H of Eq. (8), and subsequent diagonalizationof it results in the new ground eigenstates of H given by(see Fig. 7).

u6& [ ~ u g; 21& 6 u g; 11&)/A2. (9)

The excitation rate of the eigenstates u6& given in Eq. (9)to ue; 0& is proportional to the square of the electric dipolematrix element m given by

uê; 0umu6&u2 5 uê; 0umu g; 21& 6 ê; 0umu g; 11&u2/2.(10)

This vanishes for u2& because the two terms on the right-hand side of Eq. (10) are equal, since m does not operateon the external momentum of the atom (dotted line of Fig.7).

Excitation of u6& to ue; 62& is much weaker, since it isoff resonance because its energy is higher by 4\vr , sothat the required frequency is higher than to ue; 0&. Theresultant detuning is 4vr 5 8«(g/2), and for « 5 0.22,the value for the 2 3S → 3 3P transition in He* used inour experiments,36 this is large enough to reduce the ex-citation rate, making u2& semidark. Excitation to anystate other than ue; 62& or ue; 0& is forbidden by momen-tum conservation. Atoms are therefore optically pumpedinto the dark state u2&, where they stay trapped, andsince their momentum components are fixed, the result isVSCPT.

2. Another View of VSCPTA useful view of this dark state can be obtained by con-sidering that its components u g; 61& have well-definedmomenta and are therefore completely delocalized. Thusthey can be viewed as waves traveling in opposite direc-tions but having the same frequency, and therefore theyform a standing de Broglie wave. The fixed spatial phaseof this standing wave relative to the optical standingwave formed by the counterpropagating light beams re-sults in the vanishing of the spatial integral of the dipoletransition matrix element so that the state cannot be ex-cited. This view can also help to explain the conse-quences of p not exactly equal to 61, where the de Brogliewave would be slowly drifting in space with respect to thestanding wave of light. It is common to label the averageof the momenta of the coupled states as the family mo-mentum, P, and to say that these states form a closedfamily, having family momentum P 5 0 (Refs. 34 and 37).

3. Bragg Diffraction and VSCPTA completely new view of VSCPT has emerged from morecareful consideration of the motion of such dark-state at-oms in the spatially periodic field of oppositely propagat-ing light beams.38 This is Bragg diffraction, where theradiation field forms a periodic structure from which deBroglie waves are diffracted instead of vice versa. Herethe de Broglie matter wave is Bragg reflected by the spa-tially periodic optical field: Matter and field have been

Fig. 7. Schematic diagram of the transformation of the eigen-functions from the internal atomic states u g; p& to the eigen-states u6&. The coupling between the two states u g; p& andu g; p9& by Raman transitions mixes them, and since they are de-generate, the eigenstates of H are the nondegenerate states u6&.


interchanged from the usual case of Bragg reflection of anelectromagnetic field by crystalline planes of atoms!

The usual case of x-ray Bragg reflection can be viewedas arising from multicenter scattering of radiation by at-oms at each lattice site in a crystal. It follows thatpropagation of the reflected wave can occur only in thepreferred direction. Such waves are the only ones notdiffusively scattered by atoms in the lattice. The equiva-lent view of atoms in dark states is simply that the deBroglie wave fields propagate without scattering (i.e., nospontaneous emission) in the light field only when the at-oms are indeed in dark states.

4. Case When P is Not Exactly 0Since the momenta of the constituents of the eigenstatesu6& must differ by 62, the states are not degenerate whenthe momenta are not exactly 61. Their superposition istherefore not a stationary state of the Hamiltonian in Eq.(8), so atoms escape because u2& will eventually evolveinto u1&. The rate for this evolution can be calculatedfrom the Hamiltonian of Eq. (8) for the three-level systemof one excited and two ground states. In the basisu6(P)& [ @ u g; P 2 1& 6 u g; P 1 1&]/A2 and ue; P&, thismatrix has off-diagonal elements given by

^6~P!uHkinu 7 ~P!& 5 2\vrP, (11)

where 2vrP 5 vD is the Doppler shift associated with theaverage momentum P of the ground states’ constituents.The states u g; 21& and u g; 11& are just these basisstates with P 5 0. Thus u2(P)& is coupled to u1(P)& bythe atomic motion unless P 5 0.

The spontaneous decay of ue; P& is modeled by makingits energy complex, and so the three eigenvalues of H areall complex as a result of the mixing. The imaginaryparts of the eigenvalues (decay rates) depend on P, andone of them vanishes at P 5 0, corresponding to theVSCPT state. This state is called NC for noncoupled,and near P 5 0 its decay rate is39,40

GNC 5 ~gs0/2!@1 2 A1 2 @P/P0#2#, (12)

where P0 [ s0/8« (if P . P0 , then GNC 5 gs0/2). With« [ vr /g ' 0.22, we find P0 ; s0/2.

5. VSCPT in Real AtomsReal atoms have multiple internal levels that include theeffects of the magnetic, hyperfine, and other sublevels.The most well-studied example occurs in metastable Heon the 2 3S → 2 3P transition at l 5 1.083 mm, usingthe J 5 1 → 1 component in counterpropagating beamsof opposite circular polarization.34 Decay from the ex-cited MJ 5 0 state to the ground MJ 5 0 state is forbid-den by the selection rules, so the ground MJ 5 0 state isemptied by optical pumping, and the only populatedground states are MJ 5 61. Then the DMJ 5 61 tran-sitions can populate only the excited MJ 5 0 state, thusforming a L system of levels.

The two ground states having MJ 5 61 can be coupledby a Raman transition requiring the participation of bothlight beams, and thus their momenta must be different by62 in units of \k. For the case where the s 1 ( s2) beampropagates in the positive (negative) z direction, the su-perposition states are given by

u6& 5 @ u g21 ; 21& 6 u g11 ; 11&]/A2, (13)

where the subscripted quantum number denotes MJ andthe other number denotes the atomic momentum. As forthe two-level atom case discussed above, one of the statesgiven in Eq. (13) is dark.

Unlike the usual cases of laser cooling, there is nodamping force in this most commonly studied case ofVSCPT because the Doppler and polarization gradientcooling cancel each other as a result of a numerical acci-dent for this particular J 5 1 → 1 case. Thus atoms ex-ecute a random walk in momentum space until spontane-ous emission lands them in the dark state of Eq. (13).Atoms that diffuse too far from P 5 0 are lost from theprocess. By contrast, in the two-level atom experiment36

there is strong Doppler cooling because 2 3S1 → 3 3P2 isa J 5 1 → 2 transition, and this numerical accident doesnot occur. Moreover, Doppler cooling on this transition isvery effective because the large value of « makes the Dop-pler cooling limit TD 5 \g/2kB ' 36 mK, not very differ-ent from Tr ' 32 mK.

6. Entangled StatesOne of the most interesting aspects of dark-state physicsarises from the entanglement of motional and internalstates. This leads to the opportunity for fundamentalstudies of many topics whose basis is at the heart of quan-tum mechanics, such as the well-known Einstein–Podolsky–Rosen paradox, Schrodinger’s cat, quantumcommunications, and quantum computing.41,42 The keyfeature of entangled states is embodied in the form of Eq.(13). Here u6& is written as a sum of products, and it canbe shown that is not possible to find a basis where thisstate can be described as an outer product.

Clearly dark-state entanglements with multilevel neu-tral atoms offer several advantages over related opticalexperiments. First, atoms arrive as discrete objects, un-like optical fields with the notorious difficulties of produc-ing Fock states. Perhaps more important, the number ofHilbert spaces that are available, as well as their dimen-sionality, can each be larger than two.

A quantum-controlled NOT gate can be realized directlywith the states that are entangled in VSCPT. This is be-cause two independent Hilbert spaces, the external mo-tion and the internal MJ levels, are entangled in thestate. Therefore a measurement of one determines theother. This neutral atomic beam version of a controlledNOT gate is complementary to one realizable with trappedions while retaining the relatively high isolation from en-vironmental decoherence (the momentum states are natu-rally very robust, and the internal states are composedentirely of ground levels).

B. Optical Lattices

1. IntroductionIn 1968 V. S. Letokhov43 suggested that it is possible toconfine atoms in the wavelength size regions of a standingwave by means of the dipole force that arises from thelight shift. This was first accomplished in 1987 in one di-mension with an atomic beam traversing an intensestanding wave.44 Since then the study of atoms confinedin wavelength-size potential wells has become an impor-


tant topic in optical control of atomic motion because itopens up configurations previously accessible only incondensed-matter physics using crystals.

The limits of laser cooling discussed in Subsection 4.B.5suggest that atomic momenta can be reduced to a ‘‘few’’times \k. This means that their de Broglie wavelengthsare equal to the optical wavelengths divided by a ‘‘few’’.If the depth of the optical potential wells is high enoughto contain such very slow atoms, then their motion in po-tential wells of size l/2 must be described quantum me-chanically, since they are confined to a space of size com-parable with their de Broglie wavelengths. Thus they donot oscillate in the sinusoidal wells as classical localizableparticles, but instead occupy discrete, quantum mechani-cal bound states,45 as shown in the lower part of Fig. 8.

The basic ideas of the quantum mechanical motion ofparticles in a periodic potential were laid out in the 1930swith the Kronig–Penney model and Bloch’s theorem, andoptical lattices offer important opportunities for theirstudy. For example, these lattices can be made essen-tially free of defects with only moderate care in spatiallyfiltering the laser beams to assure a single transversemode structure. Furthermore, the shape of the potentialis exactly known, and doesn’t depend on the effect of thecrystal field or the ionic energy level scheme. Finally, thelaser parameters can be varied to modify the depth of thepotential wells without changing the lattice vectors, andthe lattice vectors can be changed independently by redi-recting the laser beams. The simplest optical lattice toconsider is a 1D pair of counterpropagating beams of thesame polarization, as was used in the first experiment.44

Of course, such tiny traps are usually very shallow, soloading them requires cooling to the microkelvin regime.Even atoms whose energy exceeds the trap depth must bedescribed as quantum mechanical particles moving in aperiodic potential that display energy band structure.45

Such effects have been observed in very careful experi-ments.

Fig. 8. Energy levels of atoms moving in the periodic potentialof the light shift in a standing wave. There are discrete boundstates deep in the wells that broaden at higher energy and be-come bands separated by forbidden energies above the tops of thewells. Under conditions appropriate to laser cooling, opticalpumping among these states favors populating the lowest onesas indicated schematically by the arrows.

Atoms trapped in wavelength-sized spaces occupy vi-brational levels similar to those of molecules. The opticalspectrum can show Raman-like sidebands that resultfrom transitions among the quantized vibrationallevels46,47 as shown in Fig. 9. These quantum states ofatomic motion can also be observed by stimulatedemission46,48 and by direct rf spectroscopy.49,50

2. Properties of 3D LatticesThe name ‘‘optical lattice’’ is used rather than optical crys-tal because the filling fraction of the lattice sites is typi-cally only a few percent (as of 1999). The limit arises be-cause the loading of atoms into the lattice is typicallydone from a sample of trapped and cooled atoms, such asa MOT for atom collection, followed by an optical molas-ses for laser cooling. The atomic density in such experi-ments is limited to a few times 1011/cm3 by collisions andmultiple light scattering. Since the density of latticesites of size l/2 is a few times 1013/cm3, the filling fractionis necessarily small. With the advent of experiments

Fig. 9. (a) Fluorescence spectrum in a 1D lin ' lin opticalmolasses. Atoms are first captured and cooled in an MOT;then the MOT light beams are switched off, leaving a pairof lin ' lin beams. Then the measurements are made withd 5 24g at low intensity. The open symbols are scaled up toemphasize the sidebands by a factor of 20 compared with theoriginal data indicated by the filled symbols. The center peak isdue to spontaneous emission of the atoms to the same vibrationalstate from which they are excited, whereas the sideband on theleft (right) is due to spontaneous emission to a vibrational statewith one vibrational quantum number lower (higher) (see Fig. 8).The presence of these sidebands is a direct proof of the existenceof the band structure. (b) Same as (a) except the 1D molasses iss 1–s 2, which has no spatially dependent light shift and henceno vibrational states (from Ref. 47).


that load atoms directly into a lattice from a Bose–Einstein condensate (BEC), the filling factor can be in-creased to 100%, and in some cases it may be possible toload more than one atom per lattice site.51,52

Because of the transverse nature of light, any mixtureof beams with different k-vectors necessarily produces aspatially periodic, inhomogeneous light field. The impor-tance of the ‘‘egg-crate’’ array of potential wells arises be-cause the associated atomic light shifts can easily be com-parable to the very low average atomic kinetic energy oflaser-cooled atoms. A typical example projected againsttwo dimensions is shown in Fig. 10.

In 1993 a very clever scheme was described.53 It wasrealized that an n-dimensional lattice could be created byonly n 1 1 traveling waves rather than 2n. The realbenefit of this scheme is that in the case of phase insta-bilities in the laser beams the interference pattern isshifted only in space, but the interference pattern itself isnot changed. Instead of producing optical wells in twodimensions with four beams (two standing waves), Gryn-berg et al.53 used only three. The k vectors of the copla-nar beams were separated by 2p/3, and they were all lin-early polarized in their common plane (not parallel to oneanother) The same immunity to vibrations was estab-lished for a 3D optical lattice by use of only four beamsarranged in a quasi-tetrahedral configuration. The threelinearly polarized beams of the 2D arrangement describedabove were directed out of the plane toward a commonvertex, and a fourth circularly polarized beam was added.All four beams were polarized in the same plane. Gryn-berg et al.53 showed that this configuration produced thedesired potential wells in three dimensions.

3. Spectroscopy in 3D LatticesThe NIST group studied atoms loaded into an optical lat-tice by using Bragg diffraction of laser light from the spa-tially ordered array.54 They cut off the laser beams thatformed the lattice, and before the atoms had time to moveaway from their positions, they pulsed on a probe laserbeam at the Bragg angle appropriate for one of the sets oflattice planes. The Bragg diffraction not only enhancedthe reflection of the probe beam by a factor of 105 but, byvarying the time between the shutoff of the lattice andturnon of the probe, they could measure the temperatureof the atoms in the lattice. The reduction of the ampli-tude of the Bragg scattered beam with time providedsome measure of the diffusion of the atoms away from thelattice sites, much like the Debye–Waller factor in x-raydiffraction.

The group at NIST also developed a new method thatsuperposed a weak probe beam of light directly from thelaser upon some of the fluorescent light from the atoms ina 3D optical molasses, and they directed the light fromthese combined sources onto on a fast photodetector.55

The resulting beat signal carried information about theDoppler shifts of the atoms in the optical lattices.47

These Doppler shifts were expected to be in the submega-hertz range for atoms with the previously measured50-mK temperatures. The observed features confirmedthe quantum nature of the motion of atoms in thewavelength-size potential wells15 (see Fig. 9).

4. Quantum Transport in Optical LatticesIn the 1930s Bloch realized that applying a uniform forceto a particle in a periodic potential would not accelerate itbeyond a certain speed, but instead would result in Braggreflection when its de Broglie wavelength became equal tothe lattice period. Thus an electric field applied to a con-ductor could not accelerate electrons to a speed fasterthan that corresponding to the edge of a Brillouin zone,and at longer times the particles would execute oscilla-tory motion. Ever since then experimentalists have triedto observe these Bloch oscillations in increasingly pure ordefect-free crystals.

Atoms moving in optical lattices are ideally suited forsuch an experiment, as was beautifully demonstrated in1996.56 Ben Dahan et al. loaded a 1D lattice with atomsfrom a 3D molasses, further narrowed the velocity distri-bution, and then, instead of applying a constant force,simply changed the frequency of one of the beams of the1D lattice with respect to the other in a controlled way,thereby creating an accelerating lattice. Seen from theatoms’ reference frame, this was the equivalent of a con-stant force trying to accelerate them. After a variabletime ta the 1D lattice beams were shut off, and the mea-sured atomic velocity distribution showed beautiful Blochoscillations as a function of ta . The centroid of the verynarrow velocity distribution was seen to shift in velocityspace at a constant rate until it reached vr 5 \k/M, andthen it vanished and reappeared at 2vr as shown in Fig.11. The shape of the dispersion curve allowed measure-ment of the effective mass of the atoms bound in the lat-tice.

4. MULTILEVEL ATOMSA. IntroductionIn the late 1980s it became apparent that the simplifiedview of optical forces on moving atoms that was based ontwo-level atoms in a single-frequency light field would notsuffice for many phenomena. Although this picture wasadequate for beam slowing, optical traps, and lattices,and even provided a primitive description of optical mo-lasses, the consideration of the multilevel structure of at-oms was necessary for other phenomena.

Fig. 10. Egg-crate potential of an optical lattice shown in twodimensions. The potential wells are separated by l/2.


Perhaps the greatest stimulus for the extension to mul-tilevel atoms came from the measurement of laser-cooledtemperatures below the Doppler limit15 TD that eventu-ally led to the idea of Sisyphus cooling in polarization gra-dients. Other phenomena that depended on the internalstructure of real atoms included VSCPT, MOTs, variousspecialized kinds of optical lattices, and many schemesthat depended on optical pumping among various atomicsublevels such as magnetically induced laser cooling.Many of these topics are discussed below.

B. Cooling Below the Doppler Limit

1. IntroductionIn response to the surprising measurements of tempera-tures below TD discussed at the end of subsection 2.B.3,two groups developed a model of laser cooling that couldexplain the lower temperatures.57,58 The key feature ofthis model that distinguishes it from the earlier picturewas the inclusion of the multiplicity of sublevels thatmake up an atomic state (e.g., Zeeman and hfs). The dy-namics of optically pumping moving atoms among thesesublevels provides the new mechanism for producing theultralow temperatures.59

The dominant feature of these models is the nonadia-batic response of moving atoms to the light field. Atomsat rest in a steady state have ground-state orientationscaused by optical pumping processes that distribute thepopulations over the different ground-state sublevels. Inthe presence of polarization gradients, these orientationsreflect the local light field. In the low-light-intensity re-gime, the orientation of stationary atoms is completelydetermined by the ground-state distribution: The optical

Fig. 11. Plot of the measured velocity distribution versus timein the accelerated 1D lattice. (a) Atoms in a 1D lattice are ac-celerated for a certain time ta , and the momentum of the atomsafter the acceleration is measured. The atoms accelerate only tothe edge of the Brillouin zone, where the velocity is 1vr , andthen the velocity distribution appears at 2vr . (b) Mean velocityof the atoms as a function of the quasi-momentum, i.e., the forcetimes the acceleration time (from Ref. 56).

coherences and the exited-state population follow theground-state distribution adiabatically.

For atoms moving in a light field that varies in space,optical pumping acts to adjust the atomic orientation tothe changing conditions of the light field. In a weakpumping process, the orientation of moving atoms alwayslags behind the orientation that would exist for stationaryatoms. It is this phenomenon of nonadiabatic followingthat is the essential feature of the new cooling process.

Production of spatially dependent optical pumping pro-cesses can be achieved in several different ways. As anexample, consider two counterpropagating laser beamsthat have orthogonal polarizations, as discussed below.The superposition of the two beams results in a light fieldhaving a polarization that varies on the wavelength scalealong the direction of the laser beams. Laser cooling bysuch a light field is called polarization gradient cooling.In a 3D optical molasses the transverse wave character oflight requires that the light field always have polarizationgradients.

2. Linear ' Linear Polarization Gradient CoolingOne of the most instructive models for discussion of sub-Doppler laser cooling was introduced in Ref. 57 and verywell described in Ref. 59. If the polarizations of twocounterpropagating laser beams are identical, the twobeams interfere and produce a standing wave. When thetwo beams have orthogonal linear polarizations (same fre-quency v,) with their E vectors perpendicular (e.g., x andy), the configuration is called lin ' lin or lin-perp-lin.Then the total field is the sum of the two counterpropa-gating beams given by

E 5 E0x cos~v,t 2 kz ! 1 E0y cos~v,t 1 kz !

5 E0@~ x 1 y !cos v,t cos kz 1 ~ x 2 y !sin v,t sin kz#.

(14)

At the origin, where z 5 0, this becomes

E 5 E0~ x 1 y !cos v,t, (15)

which corresponds to linearly polarized light at an angle1p/4 to the x axis. The amplitude of this field is A2E0 .Similarly, for z 5 l/4, where kz 5 p/2, the field is alsolinearly polarized but at an angle 2p/4 to the x axis.

Between these two points, at z 5 l/8, where kz5 p/4, the total field is

E 5 E0@ x sin~v,t 1 p/4! 2 y cos~v,t 1 p/4!#. (16)

Since the x and y components have sine and cosine tem-poral dependence, they are p/2 out of phase, and so Eq.(16) represents circularly polarized light rotating aboutthe z axis in the negative sense. Similarly, at z 5 3l/8,where kz 5 3p/4, the polarization is circular but in thepositive sense. Thus in this lin ' lin scheme the polar-ization cycles from linear to circular to orthogonal linearto opposite circular in the space of only half a wavelengthof light, as shown in Fig. 12. It truly has a very strongpolarization gradient.

Since the coupling of the different states of multilevelatoms to the light field depends on its polarization, atomsmoving in a polarization gradient will be coupled differ-


ently at different positions, and this will have importantconsequences for laser cooling. For the Jg 5 1/2 → Je5 3/2 transition (one of the simplest transitions thatshow sub-Doppler cooling60), the optical pumping processin purely s 1 light drives the ground-state population tothe Mg 5 11/2 sublevel. This optical pumping occursbecause absorption always produces DM 5 11 transi-tions, whereas the subsequent spontaneous emission pro-duces DM 5 61, 0. Thus the average DM > 0 for eachscattering event. For s 2 light the population is pumpedtoward the Mg 5 21/2 sublevel. Thus atoms travelingthrough only a half-wavelength in the light field need toreadjust their population completely from Mg 5 11/2 toMg 5 21/2 and back again.

The interaction between nearly resonant light and at-oms not only drives transitions between atomic energylevels but also shifts their energies. This light shift ofthe atomic energy levels plays a crucial role in thisscheme of sub-Doppler cooling, and the changing polariza-tion has a strong influence on the light shifts. In the low-intensity limit of two laser beams, each of intensity s0Is ,the light shifts DEg of the ground magnetic substates aregiven by a slight variation of Eq. (3) that accounts for themultilevel structure of the atoms. We write

DEg 5\ds0Cge

2 /2

1 1 ~2d/g!2 , (17)

where Cge is the Clebsch–Gordan coefficient that de-scribes the coupling between the particular levels of theatom and the light field.

In the present case of orthogonal linear polarizationsand J 5 1/2 → 3/2, the light shift for the magnetic sub-state Mg 5 1/2 is three times larger than that of the Mg5 21/2 substate when the light field is completely s 1.

On the other hand, when an atom moves to a place wherethe light field is s 2, the shift of Mg 5 21/2 is three timeslarger. So in this case the optical pumping discussedabove causes there to be a larger population in the statewith the larger light shift. This is generally true for anytransition Jg to Je 5 Jg 1 1. A schematic diagramshowing the populations and light shifts for this particu-lar case of negative detuning is shown in Fig. 13.

3. Origin of the Damping ForceTo discuss the origin of the cooling process in this polar-ization gradient scheme, consider atoms with a velocity vat a position where the light is s 1 polarized, as shown atthe lower left of Fig. 13. The light optically pumps suchatoms to the strongly negative light-shifted Mg 5 11/2state. In moving through the light field, atoms must in-

Fig. 12. Polarization gradient field for the lin ' lin configura-tion.

crease their potential energy (climb a hill) because the po-larization of the light is changing and the state Mg5 1/2 becomes less strongly coupled to the light field.After traveling a distance l/4, atoms arrive at a positionwhere the light field is s 2 polarized and are opticallypumped to Mg 5 21/2, which is now lower than the Mg5 1/2 state. Again the moving atoms are at the bottomof a hill and start to climb. In climbing the hills, the ki-netic energy is converted to potential energy, and in theoptical pumping process the potential energy is radiatedaway because the spontaneous emission is at a higher fre-quency than the absorption (see Fig. 13). Thus atomsseem to be always climbing hills and losing energy in theprocess. This process brings to mind a Greek myth andis thus called Sisyphus laser cooling.

The cooling process described above is effective over alimited range of atomic velocities. The force is maximumfor atoms that undergo one optical pumping process whiletraveling a distance l/4. Slower atoms will not reach thehilltop before the pumping process occurs, and faster at-oms will already be descending the hill before beingpumped toward the other sublevel. In both cases the en-ergy loss is smaller, and therefore the cooling process lessefficient. Nevertheless, the damping constant b for thisprocess is much larger than for Doppler cooling, andtherefore the final steady-state temperature is lower.57,59

In the experiments of Ref. 61 the temperature wasmeasured in a 3D molasses under various configurationsof the polarization. Temperatures were measured by aballistic technique, where the flight time of the releasedatoms was measured as they fell through a probe a fewcentimeters below the molasses region. The lowest tem-

Fig. 13. Spatial dependence of the light shifts of the ground-state sublevels of the J 5 1/2⇔3/2 transition for the case of thelin ' lin polarization configuration. The arrows show the pathfollowed by atoms being cooled in this arrangement. Atomsstarting at z 5 0 in the Mg 5 11/2 sublevel must climb the po-tential hill as they approach the z 5 l/4 point where the lightbecomes s 2 polarized, and there they are optically pumped tothe Mg 5 21/2 sublevel. Then they must begin climbing an-other hill toward the z 5 l/2 point, where the light is s 1 polar-ized and they are optically pumped back to the Mg 5 11/2 sub-level. The process repeats until the atomic kinetic energy is toosmall to climb the next hill. Each optical pumping event resultsin absorption of light at a lower frequency than emission, thusdissipating energy to the radiation field.


perature obtained is 3 mK, which is a factor 40 below theDoppler temperature and a factor 15 above the recoil tem-perature of Cs.

4. Magnetically Induced Laser CoolingAlthough the first models that described sub-Dopplercooling relied on the polarization gradient of the lightfield as above, it was soon discovered that a light field ofconstant polarization in combination with a magneticfield could also produce sub-Doppler cooling.62 In thisprocess the atoms are cooled in a standing wave of circu-larly polarized light.

There is a simple model using the Jg 5 1/2 to Je5 3/2 transition to describe this phenomenon.63 In theabsence of a magnetic field, the s 1 light field drives thepopulation to the Mg 5 11/2 sublevel. Since the Mg5 11/2 sublevel is more strongly coupled to the lightfield than is Mg 5 21/2, the light shift of this state islarger. Thus atoms traveling through this standing wavewill descend and climb the same potential hills corre-sponding to Mg 5 1/2 and will experience no averageforce.

The situation changes if a small transverse magneticfield is applied. Optical pumping processes determinethe atomic states in the antinodes of the standing wavelight field where the light is strong. But in the nodes,where the intensity of the light field is zero, the smalltransverse magnetic field precesses the population fromMg 5 1/2 toward Mg 5 21/2. Atoms that leave thenodes with Mg 5 21/2 are returned to Mg 5 11/2 in theantinodes by optical pumping in the s 1 light.

This cooling process is depicted in Fig. 14 for negativedetuning d , 0. Potential energy is radiated away in theoptical pumping process as before, and kinetic energy isconverted to potential energy when the atoms climb thehills again into the nodes. The whole process is repeatedwhen the atoms travel through the next node of the lightfield. Again the cooling process is caused by a Sisyphuseffect, similar to the case of lin ' lin. Since this damp-ing force is absent without the magnetic field, it is calledmagnetically induced laser cooling (MILC).

Efficient cooling by MILC depends critically on the re-lation between the Zeeman precession frequency vZ andthe optical pumping rate gp in the antinodes. It isclearly necessary that gp @ vZ in the antinodes wherethe light is strong. But, as in any cooling process that de-pends on nonadiabatic processes, there is a limited veloc-ity range where the force is effective. For efficient cool-ing by MILC, the velocity cannot be too small comparedwith vZ /k or atoms will undergo many precession cyclesnear the nodes, and no effective cooling will result. Onthe other hand, if the velocity is large compared withgp /k, then atoms will pass through the antinodes in atime too short to be optically pumped to Mg 5 11/2, andno cooling will result either. Thus, in addition to the re-quirement d , 0, there are two other conditions on theexperimental parameters that can be combined to give

vZ , kv , gp . (18)

Sub-Doppler cooling has been observed for MILC in Rbatoms cooled on the l 5 780 nm transition in onedimension.63 The width of the velocity distribution near

v 5 0 is as low as 2 cm/s, much lower than the 1D Dop-pler limit vD 5 A7\g/20M ' 10 cm/s for Rb.

5. Limits of Laser CoolingThe extension of the kind of thinking of temperature inthe case of Doppler cooling to the case of the sub-Dopplerprocesses must be done with some care, because a naıveapplication of the consequences of the Fokker–Planckequation would lead to an arbitrarily low final tempera-ture. In the derivation of the Fokker–Planck equation itis explicitly assumed that each scattering event changesthe atomic momentum p by an amount that is a smallfraction of p, and this clearly fails when the velocity is re-duced to the region of vr [ \k/M.

This limitation of the minimum steady-state value ofthe average kinetic energy to a few times 2Er [ kBTr5 Mvr

2 is intuitively comforting for two reasons. First,one might expect that the last spontaneous emission in acooling process would leave atoms with a residual mo-mentum of the order of \k, since there is no control overits direction. Thus the randomness associated with thiswould put a lower limit on such cooling of vmin ; vr . Sec-ond, the polarization gradient cooling mechanism de-scribed above requires that atoms be localizable withinthe scale of ;l/2p in order to be subject to only a singlepolarization in the spatially inhomogeneous light field.The uncertainty principle then requires that these atomshave a momentum spread of at least \k.

The recoil limit discussed here has been surpassed byevaporative cooling of trapped atoms64 and two different

Fig. 14. Spatial dependence of the light shifts of the ground-state sublevels of the J 5 1/2⇔3/2 transition for the case of apurely s 1 standing wave that has no polarization gradient andis appropriate for magnetically induced laser cooling. The ar-rows show the path followed by atoms being cooled in this ar-rangement. Atoms starting at z 5 0 in the strongly light-shifted Mg 5 11/2 sublevel must climb the potential hill as theyapproach the node at z 5 l/4. There they undergo Zeeman mix-ing in the absence of any light and may emerge in theMg 5 21/2 sublevel. They will then gain less energy as they ap-proach the antinode at z 5 l/2 than they lost climbing into thenode. Then they are optically pumped back to the Mg 5 11/2sublevel in the strong light of the antinode, and the process re-peats until the atomic kinetic energy is too small to climb thenext hill. Each optical pumping event results in absorption oflight at a lower frequency than emission, thus dissipating energyto the radiation field.


optical cooling methods, neither of which can be based insimple notions. One of these uses optical pumping into avelocity-selective dark state and is described in Subsec-tion 3.A above. The other one uses carefully chosen,counterpropagating laser pulses to induce velocity-selective Raman transitions and is called Ramancooling.65

C. Magneto-Optical Traps

1. IntroductionThe most widely used trap for neutral atoms is a hybrid,employing both optical and magnetic fields, to make aMOT, first demonstrated in 1987.66 The operation of aMOT depends on both inhomogeneous magnetic fieldsand radiative selection rules to exploit both optical pump-ing and the strong radiative force.66,67 The radiative in-teraction provides cooling that helps in loading the trap,and enables very easy operation. The MOT is a very ro-bust trap that does not depend on precise balancing of thecounterpropagating laser beams or on a very high degreeof polarization. The magnetic field gradients are modestand can readily be achieved with simple, air-cooled coils.The trap is easy to construct because it can be operatedwith a room-temperature cell where alkali atoms are cap-tured from the vapor. Furthermore, low-cost diode laserscan be used to produce the light appropriate for many at-oms, so the MOT has become one of the least expensiveways to make atomic samples with temperatures below 1mK.

Trapping in a MOT works by optical pumping of slowlymoving atoms in a linearly inhomogeneous magnetic fieldB 5 B(z) [ Az, such as that formed by a magnetic quad-rupole field. Atomic transitions with the simple schemeof Jg 5 0 → Je 5 1 have three Zeeman components in amagnetic field, excited by each of three polarizations,whose frequencies tune with field (and therefore with po-sition) as shown in Fig. 15 for one dimension. Two coun-terpropagating laser beams of opposite circular polariza-tion, each detuned below the zero-field atomic resonanceby d, are incident as shown.

Because of the Zeeman shift, the excited state Me5 11 is shifted up for B . 0, whereas the state withMe 5 21 is shifted down. At position z8 in Fig. 15 themagnetic field therefore tunes the DM 5 21 transitioncloser to resonance and the DM 5 11 transition furtherout of resonance. If the polarization of the laser beam in-cident from the right is chosen to be s 2 and correspond-ingly s 1 for the other beam, then more light is scatteredfrom the s 2 beam than from the s 1 beam. Thus the at-oms are driven toward the center of the trap where themagnetic field is zero. On the other side of the center ofthe trap, the roles of the Me 5 61 states are reversed,and now more light is scattered from the s 1 beam, againdriving the atoms towards the center.

The situation is analogous to the velocity damping inan optical molasses from the Doppler effect as discussedin Subsection 2.B, but here the effect operates in positionspace, whereas for molasses it operates in velocity space.Since the laser light is detuned below the atomic reso-nance in both cases, compression and cooling of the atomsis obtained simultaneously in a MOT.

So far the discussion has been limited to the motion ofatoms in one dimension. However, the MOT scheme caneasily be extended to three dimensions by using six in-stead of two laser beams. Furthermore, even thoughvery few atomic species have transitions as simple asJg 5 0 → Je 5 1, the scheme works for any Jg → Je5 Jg 1 1 transition. Atoms that scatter mainly fromthe s 1 laser beam will be optically pumped toward theMg 5 1Jg substate, which forms a closed system withthe Me 5 1Je substate.

2. Cooling and Compressing Atoms in a MOTFor a description of the motion of the atoms in a MOT,consider the radiative force in the low-intensity limit [seeEqs. (1) and (2)]. The total force on the atoms is givenby F 5 F1 1 F2 , where F6 can be found from Eqs. (1)and (2), and the detuning d6 for each laser beam isgiven by d6 5 d 7 k • v 6 m8B/\. Here m8 [ ( geMe2 ggMg)mB is the effective magnetic moment for thetransition used. Note that the Doppler shift vD [2 k • v and the Zeeman shift vZ 5 m8B/\ both have op-posite signs for opposite beams.

When both the Doppler and Zeeman shifts are smallcompared with the detuning d, the denominator of theforce can be expanded as for relation (5) and the result be-comes

F 5 2bv 2 kr, (19)

where the damping coefficient b is defined in relation (5).The spring constant k arises from the similar dependenceof F on the Doppler and Zeeman shifts and is given byk 5 m8Ab/\k.

The force of Eq. (19) leads to damped harmonic motionof the atoms, where the damping rate is given by GMOT5 b/M and the oscillation frequency vMOT 5 Ak/M.For magnetic field gradients A ' 10 G/cm, the oscillationfrequency is typically a few kilohertz, and this is muchsmaller than the damping rate that is typically a few hun-dred kilohertz. Thus the motion is overdamped, with a

Fig. 15. Arrangement for a MOT in one dimension. The hori-zontal dashed line represents the laser frequency seen by anatom at rest in the center of the trap. Because of the Zeemanshifts of the atomic transition frequencies in the inhomogeneousmagnetic field, atoms at z 5 z8 are closer to resonance with thes 2 laser beam than with the s 1 beam and are therefore driventoward the center of the trap.


characteristic restoring time to the center of the trap of2GMOT /vMOT

2 ' several milliseconds for typical values ofthe detuning and intensity of the lasers.

3. Capturing Atoms in a MOTAlthough the approximations that lead to Eq. (19) for theforce hold for slow atoms near the origin, they do not ap-ply for the capture of fast atoms far from the origin. Inthe capture process the Doppler and Zeeman shifts are nolonger small compared with the detuning, so the effects ofthe position and velocity can no longer be disentangled.However, the full expression for the force still applies, andthe trajectories of the atoms can be calculated by numeri-cal integration of the equation of motion.68

The capture velocity of a MOT is serendipitously en-hanced because atoms traveling across it experience a de-creasing magnetic field just as in the beam decelerationdescribed in Subsection 2.A. This enables resonanceover an extended distance and velocity range because thechanging Doppler shift of decelerating atoms can be com-pensated by the changing Zeeman shift as atoms move inthe inhomogeneous magnetic field. Of course, it willwork this way only if the field gradient A does not de-mand an acceleration larger than the maximum accelera-tion amax . Thus atoms are subject to the optical forceover a distance that can be as long as the trap size, andthey can therefore be slowed considerably.

The very large velocity capture range vcap of a MOT canbe estimated by using Fmax 5 \kg/2 and choosing a maxi-mum size of a few centimeters for the beam diameters.Thus the energy change can be as large as a few kelvin,corresponding to vcap ; 100 m/s (Ref. 67). The number ofatoms in a vapor with velocities below vcap in the Boltz-mann distribution scales as vcap

4 , and there are enoughslow atoms to fall within the large MOT capture rangeeven at room temperature, because a few kelvin includes1024 of the atoms.

4. Variations on the MOT TechniqueBecause of the wide range of applications of this most ver-satile kind of atom trap, a number of careful studies of itsproperties have been made,67,69–75 and several variationshave been developed. One of these is designed to over-come the density limits achievable in a MOT. In the sim-plest picture, loading additional atoms into a MOT pro-duces a higher atomic density because the size of thetrapped sample is fixed. However, the density cannot in-crease without limit as more atoms are added. Theatomic density is limited to ;1011 cm23 because the fluo-rescent light emitted by some trapped atoms is absorbedby others.

One way to overcome this limit is to have much lesslight in the center of the MOT than at the sides. Simplylowering the laser power is not effective in reducing thefluorescence because it will also reduce the capture rateand trap depth. But those advantageous properties canbe preserved while reducing fluorescence from atoms atthe center if the light intensity is low only in the center.

The repumping process for the alkali atoms provides anideal way of implementing this idea.76 If the repumpinglight is tailored to have zero intensity at the center, thenatoms trapped near the center of the MOT are optically

pumped into the wrong hfs state and stop fluorescing.They drift freely in the dark at low speed through the cen-ter of the MOT until they emerge on the other side intothe region where light of both frequencies is present andthey begin absorbing again. Then they feel the trappingforce and are driven back into the dark center of the trap.Such a MOT has been operated at MIT76 with densitiesclose to 1012 cm23, and the limitations are now from col-lisions in the ground state rather than from multiple lightscattering and excited-state collisions.

5. POLYCHROMATIC LIGHTA. IntroductionIn the introductory Sections 1 and 2 of this paper, theideas of laser cooling are discussed in terms of two-levelatoms moving in a monochromatic laser beam. Section 4shows that this simple view is inadequate and that themultiple level structure of atoms is necessary to explainsome experiments. The extension from two-level to mul-tilevel atoms gives an unexpected richness to the topic ofatomic motion in optical fields. It seems natural to ex-pect that a comparable multitude of new phenomena is tobe found for the motion of two level atoms in multifre-quency fields, but this subject has not received as muchattention.

In the Doppler cooling process of two-level atoms de-scribed in Subsection 2.B, it is the radiative force thatproduces both the slowing force and the dissipation nec-essary for cooling. The force arises from the incoherentsequence of absorption followed by spontaneous emission.Thus it is limited to F < \kg/2.

By contrast, in most cases of laser cooling in multilevelatoms, such as Sisyphus cooling in a polarization gradientas discussed in Subsection 4.B, it is the dipole force thatworks on the atoms. This force is usually present in mul-tiple beams of monochromatic light such as standingwaves. Since this force results from the rapid coherentsequence of absorption followed by stimulated emission, itdoes not suffer from such limits.

Although this force can be very strong, its sign alter-nates in space on the wavelength scale, and thus its spa-tial average vanishes. Moreover, this force is completelyconservative so its cooling aspect, or velocity dependence,must arise from the relatively infrequent spontaneousemission events, and so such sub-Doppler cooling forcesare similarly limited to a maximum of Frad 5 \kg/2.Thus not only is the cooling aspect of the dipole force lim-ited in strength, but also the spatial average of the strongconservative component vanishes.

In 1997 there was a dramatic demonstration of the useof a bichromatic field with two beams of equal intensitiesand detunings that provided both a strong force and widevelocity range.77 Using only modest laser power, Sodinget al. demonstrated that this bichromatic force could de-celerate a thermal beam of Cs to ;20 m/s in just a fewcentimeters with no Doppler compensation.

Unlike the limited velocity range of the rectified dipoleforce,78,79 the bichromatic force has both a very large ve-locity range and magnitude. Since the force covers amuch larger range of velocities, Doppler compensation,such as using a multikilowatt Zeeman tuning magnet, is


rendered unnecessary for slowing a thermal beam. Thusit is a valuable method for fast, short-distance decelera-tion of thermal atoms that minimizes atom loss, therebymaking it a most useful and important tool in the produc-tion of cold, dense atomic samples for traps, lithography,and other purposes.

Another way to make polychromatic light is to fre-quency modulate a single beam. In this case the param-eters available are the rate of frequency modulation vmand the strength of the modulation, characterized by a di-mensionless parameter b. For weak modulation thelight can be considered a carrier with sidebands, but thattopic is not pursued in this paper.80 For stronger modu-lation the dynamics of the atomic response changes, andthe adiabatic rapid passage (ARP) picture is appropriate(see Subsection 5.C below).

B. Bichromatic Force

1. Basic IdeasThe bichromatic force has a velocity range very muchlarger than the usual velocity range g/k for slowing by ra-diative forces, and it also has a strong velocity depen-dence at its range boundaries so that it can cool.77 Natu-rally this (dissipative) velocity dependence originatesfrom the occasional spontaneous emission events at a ratedetermined by g, but the magnitude of the force is notlimited by this rate. Using two frequencies provides ex-tra degrees of freedom for the light field, and these havebeen exploited to make forces substantially larger thanthe radiative force and cover a wider velocity range thanthe rectified force or Sisyphus cooling.80–84

Suppose an atom is exposed to resonant light of Rabifrequency V for a time tp that satisfies Vtp 5 p. Thenthe atom is driven from the ground to excited state (orvice versa). Such a ‘‘p pulse’’ provides an intuitive modelfor the bichromatic force. Consider atomic motion alongthe axis of counterpropagating bichromatic light beams.Each beam contains the same two frequencies, and theyare detuned from atomic resonance by 6d (differencefrequency 5 2d). Each beam can be described as anamplitude-modulated single carrier frequency at theatomic resonance frequency va having modulation periodp/d, and d is chosen such that p/d ! t or d @ g. Theequal intensities of each beam are chosen so that the en-velope of one pulse of the beats satisfies the condition of ap pulse for the atoms: Ground-state atoms are coher-ently driven to the excited state and vice versa. This con-dition is V 5 pd/4, where V [ gAI/2Is is the Rabi fre-quency associated with each frequency component of eachbeam.

Atoms in this light field are subject to these p pulsesalternately from one beam direction and then from theother, so the force on them can become very large. Thisis because the first p pulse causes excitation along withmomentum transfer in one direction, and the secondcounterpropagating p pulse causes stimulated emission,producing a momentum transfer along the same directionas the first pulse.

Thus atoms are coherently driven between the groundand the excited states, and momentum is monotonically

exchanged with the light field. The magnitude of the mo-mentum transfer in each cycle is 2\k, and the repetitionrate of these controlled processes is d/p, so that the opti-mum total force is of the order of 2\kd/p. This is verymuch larger than the usual maximum radiative force\kg/2, principally because it is a coherently controlledrapid momentum exchange whose rate is limited only bylaser power through the p-pulse condition. The bichro-matic force is not subject to saturation.

The mechanism described above requires two addi-tional features to be applicable to atomic beam slowing:(1) There must be some directional asymmetry becausethe excitation pulse could come from either direction, andthus the force could point either way, and (2) it must bevelocity dependent so that it can compress the phase-space volume of the atomic sample.

The first condition is satisfied by a careful choice of therelative intervals between the counterpropagating pulsesof light (i.e., the relative phase of the modulation) and ex-ploitation of the random nature of spontaneous emission.If the pulses are evenly spaced, spontaneous emissionspoils the cycle described above, since an excited atom canspontaneously decay to the ground state so that the nextpulse causes absorption rather than stimulated emission.Thus each spontaneous emission produces a reversal inthe direction of the force, resulting in zero average force.

However, if the pulses are timed unevenly so that theycome in closely spaced pairs, one from each direction,their asymmetry indeed produces a preferred direction.When the first pulse of each closely spaced pair producesexcitation, the atom spends less time in the excited statebefore stimulated emission by the second pulse, resultingin a lower probability for spontaneous emission thatwould reverse the force. Just the opposite situation pre-vails for an atom excited by the second pulse of a pair, be-cause spontaneous emission is more likely to occur in thelonger interval than in the shorter one when t @ p/d.While it is still possible for an atom to get into a cycle thatwill produce a force in either direction, atoms that get outof phase with the preferred order by suffering spontane-ous emission during the short interval are more likely toundergo correction than to remain out of phase, so therewill be more events that produce a force in the direction oftravel of the leading pulse. This model was experimen-tally realized in Ref. 85 on Cs by use of picosecond ppulses from a pulsed laser.

The optimum phase difference between the beats in thecounterpropagating beams has been found by numericalcalculations to be >p/4, so that atoms typically experi-ence a force in the wrong direction for 25% of their time.Then the same 25% is needed to negate this undesiredforce, leaving 50% of the time for the desired force. Thusthe average force is \kd/p, half the optimum estimatedabove, and this is also borne out by the numerical calcu-lations of Refs. 77, 86, and 87.

The second condition is that the force be velocity depen-dent in order to cool. Understanding how this arises re-quires simultaneous consideration of both the light shiftand the Doppler shift, and the simplified picture of ppulses given above is insufficient. Instead, a dressed-state description of atoms moving in a bichromatic, time-dependent field is required. This has not yet been pro-


vided, and so numerical calculations of the bichromaticforce are still required.

2. Experimental ResultsNumerical calculations of the bichromatic force using theoptical Bloch equations (OBE) have been done for com-parison with experiments. The program written by theauthors of Ref. 77 works by integrating the OBE for ashort time to get past the transients associated with ini-tial conditions, and then calculating the optical force1

from F 5 2tr(r¹H). The velocity dependence is imple-mented by Doppler shifting the two frequencies of thebeam traveling in one direction up by kv and those fromthe other direction down by kv.

Figure 16 shows a typical example of the output. Ford 5 20g the magnitude of the force as given by the modelabove is expected to be F 5 \kd/p 5 (2d/pg)Frad' 13Frad , and here the average value is about 10 Frad ,not very different. Also, the p-pulse condition for thevarying envelope of the beat pattern from the two fre-quencies gives V 5 pd/4, a value considerably smallerthan the V 5 22g shown here. Thus these simple mod-els begin to fail when quantitative evaluations are made.

Nevertheless, the velocity range of the force in Fig. 16is 'd/k, and its magnitude corresponds fairly well withthe model. The sharp spikes that are ubiquitous in thegraph are not artifacts or noise, but correspond to multi-photon effects closely related to Dopplerons.88 They oc-cur at velocities given by md/nk, where m and m aresmall integers.

There are small features at v 5 6d/k 5 620g/k inFig. 16. These occur at the velocity where the upwardDoppler shift of va 2 d and the downward Doppler shiftof va 1 d bring both of these frequencies to atomic reso-nance, and they correspond to the ordinary Doppler force.Note that their magnitude is confined to less than Fradand their velocity range to a small fraction of g/k.

The bichromatic force has been used to slow a He*beam (see Ref. 89). The He* source produced a beamwith velocity '1000 m/s ' 550 g/k as shown by thedashed curve in Fig. 17. Thus a significant beam decel-eration requires a bichromatic force having a range of theorder of a few hundred times g/k.

The measurements used d 5 184g 5 300 MHz, so thatthe range of the bichromatic force is 'd/k 5 184g/k' 350 m/s, and the slowing measurements shown in Fig.17 demonstrate this velocity range. Moreover, the inter-action length was limited by the geometry of the beamoverlaps to ,5 cm, which is a significant reduction inslowing length compared with slowing with Frad .

The '100 m/s width of the peak of slowed atoms in thevelocity distributions indicates significant cooling in addi-tion to the bichromatic slowing. The factor of three re-duction in the velocity spread of the slowed atoms fromthe initial velocity spread of 300 m/s corresponds to an or-der of magnitude in temperature reduction.

While short-distance deceleration is applicable for He*trapping experiments, acceleration of the He* beam isalso desirable. The direction of the bichromatic force iseasily reversed by changing the phase from f 5 p/2 to2p/2, and velocity distributions showing acceleration to1150 m/s have also been observed. The ability to switch

the sign of the bichromatic force by controlling the rela-tive phase of the counterpropagating bichromatic fields isa signature effect of the bichromatic force.

C. Adiabatic Rapid Passage ForceThe use of p pulses is not the only way to reverse coher-ently the ground and excited state populations of two-level atoms and thereby mediate controlled momentumexchange. Another way uses the well-known frequency-sweep technique called adiabatic rapid passage (ARP) toinvert the population with nearly 100% efficiency.90–92

ARP is a far more robust method than p pulses because itis not sensitive to variations of the interaction time or theintensity as is the p-pulse method. Changes in the startand end point of the sweep are easily tolerated with littleconsequence as long as certain conditions on the param-eters are satisfied. Thus it seems very attractive toimplement this method as a tool for coherently controlledexchange of momentum between atoms and a light field.

A simple model calculation of the magnitude of theARP-mediated force begins by considering that the mo-mentum transfer in one half-cycle p/vm of the frequency-modulated light is 2\k. First, a frequency-swept laserbeam from one direction excites the atoms and transfers

Fig. 16. Bichromatic force on moving atoms in two standingwaves of frequencies va 6 d, where d 5 20g, plotted in units ofFrad 5 \kg/2. The Rabi frequency for each component of eachstanding wave was V 5 22g, and the spatial phase shift betweenthe two standing waves was f 5 p/2.

Fig. 17. He* atoms were produced in a LN2-cooled, dc dischargesource with a longitudinal velocity distribution shown by adashed curve. For these data the detuning d 5 184 g, V5 225 g, and x 5 p/2, appropriate for slowing. The laser andatomic beams intersect at a few tens of milliradians, and theirdiameters determine the interaction length to be a few centime-ters.


\k, and then another beam from the opposite directionwhose sweep is delayed drives them back to the groundstate and also transfers \k. Since the time for this isp/vm , the force is FARP ; 2\k/(p/vm) 5 2\kvm /p.

ARP is particularly well described by use of an artificialBloch vector R whose components are determined by thecomplex coefficients of the superposition of atomic groundand excited states caused by interaction between a two-level atom and the laser light.93,94 The Schrodingerequation is equivalent to a form where the time depen-dence of these coefficients satisfies dR/dt 5 V8 3 R,where V8 is a torque vector whose two horizontal compo-nents are the real and the imaginary parts of the atomicRabi frequency V in the laser field, and the vertical com-ponent is the detuning from atomic resonance d. ThusuV8u 5 AuVu2 1 d 2. (Usually V can be chosen to be real.)

This geometric view of the Schrodinger equation allowsa particularly graphic interpretation of ARP (see Fig. 18).At the beginning of the frequency sweep, where the initialdetuning d0 is much larger than the Rabi frequency V, Rexecutes small, rapid orbits near the south pole (atom inthe ground state), and the axis of these orbits slowlydrifts up toward the equator as d approaches zero. Thesweep continues toward the opposite detuning, so thatnear the end of the sweep, where again d @ V, R ex-ecutes small rapid, orbits near the north pole and is fi-nally left at the north pole (the atom is in the excitedstate). The direction of the frequency sweep, namely thesign of d, is of no consequence as long as V8 is essentiallypolar at the ends of the sweep.

The electric field Ei(z, t) of the frequency-modulatedplane waves traveling in the 6 directions can be writtenas

E6~z, t ! 5 E0« cos@vct 7 kz 1 b sin~vmt 6 x/2!#, (20)

where « is the unit polarization vector and E0 is the time-independent amplitude of the waves. The waves haveequal modulation amplitude b 5 d0 /vm , where d0 is therange of the frequency sweep. The only difference inmodulation properties are phase differences x of eachwave. The familiar steady-state spectrum of frequency-modulated light is usually viewed as a carrier with side-bands, but it doesn’t seem appropriate for ARP. How-ever, the instantaneous frequency is the time derivative ofthe phase of E in Eq. (20), namely, vc 1 d0 cos(vmt),which is just the desired frequency sweep of 2d0 in timep/vm .

There are certain requirements for ARP to occur effi-ciently. First, V must be large enough so that R pre-cesses about V8 much faster than V8 rotates during thesweep, which means that V @ d/V 5 bvm

2 /V [see Eq.(20)]. This adiabatic following of V8 by R produces the‘‘A’’ in ARP. For a uniform sweep rate, d 5 2d0vm /p,where 6d0 is the sweep range. So adiabaticity requiresvm ! pV2/2d0 .

Second, the entire sweep must occur in a time that isshort compared with the atomic excited-state lifetime tominimize the effects of spontaneous emission during thesweep and thereby preserve coherence between the atomand the radiation field. This requires vm @ g, or d@ 2d0g/p, and constitutes the rapid condition in the

name ARP. These are two conditions on the sweep timeand the Rabi frequency V that must be met indepen-dently, in addition to their combination V @ g.

Finally, it is required that d0 @ V, so that V8 is nearlypolar at the extremes of the sweep. Thus d0 is the high-est frequency in the system. All these conditions can bewritten together as

gAb ! vmAb ! V ! d0 . (21)

The ARP force on He* has been measured by recordingthe atomic spatial distribution after deflection by appro-priate transverse laser beams (see Refs. 84 and 89). Theaverage measured velocity change was 3.8 g/k, corre-sponding to a force of 3.8 Frad , about 3/5 of the predictionof the model. The velocity range of this measured forcewas ;63.8g/k, corresponding to ;7.6vc , considerablylarger than that for Frad . Even though the ARP condi-tions of Eq. (21) are not very well fulfilled in the measure-ments of Ref. 84, the observed force was @Frad and had amagnitude comparable with the simple ARP model.

D. SummaryThe ARP and bichromatic forces share many similaritiesin spite of their very different model descriptions. Botharise from controlled momentum transfer between modu-lated counterpropagating laser beams; the ARP forcemakes use of frequency-modulated light, while the bichro-matic force relies on amplitude modulation. Althoughthe physical models presented in the previous sectionsgive predictions for the optimum parameters for the coun-terpropagating laser beams, in both experiment and cal-culation, for both the ARP force and the bichromatic forcethe strongest force is observed under conditions that donot satisfy their respective models.

6. CONCLUSIONSIn this paper we have reviewed some of the fundamentalsof optical control of atomic motion. In addition, we havechosen a few broad areas to discuss in more detail. Thereader is cautioned that this is by no means an exhaus-

Fig. 18. At the beginning of the frequency sweep with the atomsin the ground state (south pole), the initial detuning d0 is muchlarger than the Rabi frequency V. Thus R executes small, rapidorbits near the south pole because the torque vector V8 is nearlypolar as shown in part (a). As shown in part (b), d 5 0, R ex-ecutes orbits in a vertical plane because the only remaining com-ponent of the torque vector V8 is V, and thus V8 is in the equa-torial plane. Near the end of the sweep d is again very large, Rorbits near the north pole as shown in part (c), and is finally leftat the north pole with the atom in the excited state. Atoms thatstart at the north pole (excited state) are similarly driven in thiscoherent way to the south pole.


tive review of the field and that many important and cur-rent topics have been omitted.

In place of a comprehensive (book-length) paper, wehave simply selected a few special defining characteristicsas a focus for discussion of several specific topics and thenspecialized to a few topics in domains where these char-acteristics do not apply (see Table 1). As we wrote inSubsection 1.C, we considered the characteristics of quan-tum states of atomic motion instead of classical trajecto-ries for atoms, multilevel instead of two-level atoms, andpolychromatic instead of monochromatic light. Each ofthese are subfields, along with many others, of this bur-geoning field of controlling the motion of free, neutral at-oms.

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