Vol. 3, No. 5/May 1986/J. Opt. Soc. Am. B 719

Radiation cooling with ir pulses

A. Jay Palmer and Juan F. Lam

Hughes Research Laboratories, Malibu, California 90265

Received April 8, 1985; accepted December 2, 1985

The concept of radiation cooling with r pulses is presented and quantified with a theoretical analysis. Numericalanalysis of 7r-pulse cooling on the hydrogen Lyman-a transition is presented as an example. The cooling rate andlimiting temperature are discussed and compared with steady-state radiation cooling.

1. INTRODUCTION

The possibility of using light to modify the velocity distribu-tion of an ensemble of atoms appears to have originated withthe pioneering work of Kolchenko et al.,' who analyzed theeffect of atomic recoil in the spectral line shape of an atominteracting with a resonance radiation field. At almost thesame time, Ashkin2 proposed the concept of resonantly en-hanced radiation pressure in a gaseous system. These worksstimulated the development of two important contributionsin the area of laser cooling. First was the concept of laser-induced cooling of gases by Hansch and Schawlow,3 which isthe foundation of much of today's research in the area.Second was the concept of atomic-beam birefringence underthe action of a strong radiation field by Kazantsev,4 whichwas the cornerstone of the so-called opLc.cl Stern-Gerlacheffect. The latter was recently demonstrated by Moskowitzet al.,5 and the theoretical analysis of Kazantsev was con-firmed by Cook and Bernhardt.6

Experimental studies of laser modification of atomic ve-locities that use the phenomenon of fluorescence have beenquite extensive. Evidence of the slowing of an atomic-beamby resonance radiation pressure was demonstrated by Bjork-holm et al.,7 Balykin et al.,8 Phillips and Metcalf,9 and Pro-dan et al.10 Recently, experiments by Prodan et al.11 andErtmer et al.12 demonstrated that an effusive atomic beamwas stopped by laser light. These works were followed bythe recent experimental demonstration of radiation trap-ping of atoms by Chu et al.13

In the Hansch-Schawlow radiation cooling procedure, theatoms are illuminated with counterpropagating laser beamsthat are tuned into the upper half of the Doppler contour of aresonance line, the laser frequency either covering the fullupper half of the contour or scanned toward line center ascooling progresses. Cooling results from the net momentumtransfer to the atoms from the laser photons during reso-nance fluorescence events. The cooling rate is thus limitedby the natural decay rate of the atom due to spontaneousradiation, the maximum cooling rate occurring at laser pow-er levels that saturate the transition. An increase of laserpower above saturation does not increase the cooling ratesince absorption followed by stimulated emission from thesame beams does not transfer any net momentum to theatom.

The above limitation on the cooling rate could be lifted ifit were possible to alternate the direction and detuning of

the separate upward-stimulating and downward-stimulat-ing photons. Then, a net momentum transfer to the atomcould be made to occur at transition rates greater than thespontaneous decay rate. As will be illustrated below, thiscan be accomplished with the use of a train of alternatelyoppositely directed and oppositely detuned r pulses. Theconcept of modifying the velocity of atoms using r pulseswas advanced by Kazantsev1 4 in 1974. A similar scheme wasproposed by Nebenzahl and Szoke15 in their studies ofatomic-beam deflection using stimulated-emission process-es. A related application of using r pulses to deflect anatomic beam for the purpose of isotope separation was con-sidered by Friedman and Wilson.16

The objective of this paper is to identify the critical issuesassociated with r-pulse cooling of a distribution of atomicvelocities, to estimate the cooling rate and limiting tempera-ture, and to compare these performance figures with thosefor steady-state laser cooling.17 Below, Section 2 describesthe physical picture of r-pulse cooling and the restrictionson the incoming train of r pulses due to Doppler dephasing.A rigorous theoretical foundation for r-pulse cooling thatutilizes the Wigner distribution function is presented inSection 3. The results of a numerical integration of thequantum-mechanical transport equations applied to r-pulse cooling on the hydrogen Lyman-a transition appear inSection 4. Section 5 compares the cooling rates and re-quired power levels for steady-state and 7r-pulse cooling, andSection 6 summarizes our results.

2. PHYSICAL PICTURE

The basic physics of r-pulse cooling is as follows: Considerthe interaction of a two-level atom with a pair of r pulsesincident from opposite directions. First, a photon from the7r pulse from the right is absorbed, exciting the atom to theupper state. An increment of momentum -(hk) has beentransferred to the atom. (Here k is the photon wave numberand h is Planck's constant divided by 2r.) Next, the 7r pulsefrom the left stimulates the atom to emit a photon to the left,resulting iii a recoil momentum transfer to the atom of-(hk). The net momentum transferred to the atom due tothe interaction with both 7r pulses is-(2hk). Clearly, a trainof N alternately counterpropagating r pulses sequentiallyincident upon the atom will result in a momentum transferof N(hk) to the atom in the direction of propagation of theupward-stimulating r pulses, provided that the duration of

0740-3224/86/050719-05$02.00 © 1986 Optical Society of Ametica

A. J. Palmer and J. F. Lam

720 J. Opt. Soc. Am. B/Vol. 3, No. 5/May 1986

the pulse train remains short in comparison with the radia-tive lifetime of the transition. To retard the atom's motion,the upward-stimulating r pulses should thus be directedopposite to the atom's motion and tuned above resonance,while the downward-stimulating pulses should be oppositelydirected and tuned below resonance.

To cool a one-dimensional velocity distribution of atoms,a train of 7r pulses with a detuning sequence chosen to retardthe motion of atoms moving in one direction must be fol-lowed by a train with the start of the detuning sequenceshifted to the oppositely directed beam in order to retard themotion of the atoms moving in the opposite direction. Toensure cooling of the distribution, each r-pulse train must,of course, effect a larger momentum transfer to atoms in thehalf of the distribution for which the motion is being retard-ed than to the other half. This will occur if the frequency ofthe 7r-pulse train is chosen to be resonant with atoms with aDoppler shift of, say, half the Doppler width of the distribu-tion, since the excitation state of the atoms in that side of thedistribution will remain in coherence with the 7r-pulse trainfor a larger number of 7r pulses than will atoms in the otherside of the distribution. The two 7r-pulse trains must beseparated by a few radiative lifetimes in order to reset theexci- ition state of all the atoms to the ground state beforeswitching the detuning sequence.

It is interesting to inquire where the entropy of the atommotion goes in the case of cooling with a train of 7r pulses. Inthe case of steady-state laser cooling, the entropy is carriedaway by the spontaneously emitted photons. During cool-ing with a train of 7r pulses, spontaneous emission is assumednot to occur. However, the entropy generation still appearsin the radiated photons. Radiation from the atoms duringillumination by a 7r-pulse train is accounted for by the in-duced transition dipole moment, which is proportional tothe off-diagonal terms in the density matrix, F(t) (see be-low). The generated entropy comes from nonresonantatoms' emitting photons into a larger number of temporalmodes than do the resonant atoms. As discussed above, themore randomized phase of F(t) for the nonresonantatoms also accounts for the loss of phase sequencing with the7r-pulse train necessary for adjacent 7 pulses to produceequally directed forces on the atom. Since F(t) is cyclingmuch faster than the spontaneous emission rate, the rate ofentropy generation can be larger than in the case of steady-state laser cooling, thus permitting a faster cooling rate for7r-pulse cooling.

The accumulated rate of phase error between the excita-tion state and the 7r-pulse train that occurs for nonresonantatoms will determine the maximum number of 7r pulses thatcan be used in a given 7r-pulse train. Also, the greater thedisparity of the accumulated phase error on the differentsides of the distributio l, the faster the cooling rate will be fora given 7r pulse. It turns out that the accumulated phaseerror can be minimized and the disparity of phase errormaximized through the use of a specific zero-field periodbetween the 7r pulses. This technique is best appreciatedthrough reference to the density-matrix vector model for theevolution of the coherent excitation state of a two-levelatom.18,19

Figure 1 shows the density-matrix vector for one atomicvelocity before the application of the first 7r pulse, after thezero-field period, and after the application of the second 7r

AC 3

3 + 6

= r +6

D (Au16

A

I= RABIFREQUENCY

A@= DETUNING

6= ZERO-FIELD DWELLPERIOD

A

Fig. 1. Vector representation of the state of a two-level atom forone atomic velocity for time zero, after the first 7r pulse, t; after thezero-field period, t 2 ; and after the second 7r pulse, t3.

pulse. The density-matrix vector rotates about the vector B= (Qel + Awe3 ), where 0 = RE/h is the Rabi frequency, isthe transition dipole moment, E is the laser field, and Aw isthe detuning from resonance. It rotates about B at an angu-lar rate equal to (22 + Ac2)1/2. When the vector points in thenegative e3 direction the atom is in the ground state; when itpoints in the positive e3 direction the atom is in the upperstate. All other directions represent the atom in a superpo-sition of these two states.

It can be seen from this picture that if there is no field-offperiod between the r pulses, then the phase-error spreadaccumulates to Or within -(/QAw) pulses. However, dur-ing a zero-field interval between 7r pulses, the vector rotatesabout the e3 axis at a rate equal to the detuning. Thus, if wechoose the duration of the off-field interval to be -7r/(ACD),

where A&.D is the Doppler shift of the atom, then after eachoff-field period, the sign of the phase error with respect tothe e3 axis will be reversed, and the subsequent 7r pulse willbring the phase error back to zero. The phase-error correc-tion cannot be made exact for all atoms, of course, because ofthe spread in the precession rate about e3 caused by thefinite velocity spread. The optimum zero-field period forminimizing the average phase error for that half of the distri-bution which is being cooled is roughly 7r/((AD)/2), where(ACLD) is the Doppler width of the transition.

This phenomenon is similar to the phase-spread reductionthat occurs in photon echoes. It is clear that for a givennumber of 7r pulses, the net accumulated phase error and thedifference in the accumulated phase error between one sideof the distribution and the other will be, respectively, lessthan and greater than their values for the case of no zero-field period. Thus the use of the zero-field period between 7rpulses will permit cooling over a broader velocity distribu-tion.

We have carried out a theoretical analysis of the r-pulsecooling technique discussed above by utilizing the quantumtransport equations. This analysis is described in the fol-lowing two sections.

3. THEORETICAL FRAMEWORK: THEQUANTUM TRANSPORT EQUATIONS

For the purpose of completeness, we derive in this sectionthe fundamental equations describing the quantum evolu-

A. J. Palmer and J. F. Lam

t =

Vol. 3, No. 5/May 1986/J. Opt. Soc. Am. B 721

tion of an ensemble of two-level atoms interacting with reso-nant radiation fields. The equations were first derived byKolchenko et al. and have been rediscovered by severalgroups.20 The equations include the effects of Dopplershift, grating washout, and atomic recoil. If R denotes thecenter-of-mass displacement vector and r the internal coor-dinates, then the space-time evolution of an atom is gov-erned by the Schrodinger equation:

iha/OtI(R, r, t) = [-h 2 V2/2M + Ho + V(R, r, t)J4'(R, r, t),

(1)

where the first term on the right-hand side accounts for thetranslational motion of the atom, Ho is the electronic Hamil-tonian, and V(R, r, t) is the interaction potential. We shallconsider an interaction potential for the case of the inducedelectric dipole moment of an atom interacting with the radi-ation field.

We begin by expanding I(R, r, t) as a linear superpositionof eigenstates, {4)mi of Ho:

'(R, r, t) = E Cm(R, t)4lzm(r)exp(-iwmt), (2)m

where the probability amplitude, C(R, t), accounts for thetranslational motion of the atom in a specific eigenstate,4'm(r). Using expression (2) in the Schr6dinger equation(1), one finds that the evolution of C(R, t) is described by

ihd/dtCn =-h 2V 2/2MC + I (n V m)exp(iwmnt)Cm,m

(3)

where

(nI VI m) J | dr4*n(r)V(R, r, t)(Dm(r) (4)

and Wmn Cm Cm n-We are interested in obtaining the evolution of bilinear

combinations of probability amplitudes. Let

Pnm(Rl, R2, t) = Cn(Rj, t)Cm*(R 2, t). (5)

Then one can show that nm(Ri, R 2, t) evolves according to

ihO/Otpnm(Ri, R 2, t) = h2(VR12 -V R22)/2Mpnm(Rj, R2, t)

+ n [nl VIa)lpam(R, Rm, t)exp(iont)

where we identify R = R + R/2 and R2 = R - R'/2.Applying the same procedure, i.e., using expression (8) inEq. (6), one finds that fn,(R, p, t) evolves according to

ih(/dt + p* V/M)fnm(R, p, t)

= Z f dk[(njVIa)exp(-ik R)f,m(R, p + hk/2, t)

X exp(iwnat)

fnt(R, p, hk/2, t)(al VI m)exp(-ik R)exp(iwamt)],

(9)

where the convective derivative p VIM accounts for themotion of the atom and gives rise to Doppler shift andmotionwashout. Thequantity (nlVla) isthespatialFouri-er transform of (nIVIa). Equation (9) is the quantum-mechanical transport equation describing the interaction ofa two-level atom with near-resonant radiation fields. It isthe difference aspect of these equations that gives rise toatomic recoil.

In essence, Eq. (9) is the quantum generalization of theclassical Boltzmann equations. In the case of a two-levelatom, they govern the evolution of three distribution func-tions: M(p, t) = 22 + f the number of atoms; N(p, t) = f22- f1, the population difference between the two states; andf12(P, t), the optical coherence. For an assumed homoge-neous distribution of atoms in one dimension interactingwith a plane electromagnetic wave, the equations for thesequantities reduce to

(10)

dM(p, t)/dt = 2(gE/h)[F(p + hk, t) - F(p, t)], (11)

(d/at + iAw - ip k/m)F(p, t)

i(yE/h)N(p, t) + 1/2(AE/h)[M(p + hk, t) - M(p, t)],(12)

where the interaction potential with a plane wave is taken asV = -E exp(ikx - wt), , is the transition dipole moment,and E is the electric field of the incident plane wave. Wehave made the substitution: F = 2 exp(ikx - iAwt), whereAC = - 12 is the detuning from line center. F indicatesthe imaginary part of F.

4. NUMERICAL SOLUTIONS

-Pna(Rn, Ra, t)(al Vim) 2 exp(icowat)], (6)

where

(nI VI a)j -f dr4t)n*(r)V(Ri, r O)<Da(r). (7)

Equation (6), though exact, does not show explicitly theeffect of Doppler broadening and recoil shift. A useful tech-nique that makes these effects explicit involves the use of theWigner distribution function, which is defined as

fnm(R, p, t) = (1/27rt) 3 J dR'

X exp(-ip R'/h)pnm(R + R'/2, R - R'/2, t),

(8)

We have carried out a numerical integration of Eqs.(10)-(12) using as the applied field a train of 7r pulses, withalternate pulses oppositely directed and tuned on oppositesides of line center as required for cooling. As discussed inSection 2, optimal rephasing of the 7r pulses with respect tothe distribution of Doppler shifts will occur when a period ofzero field is used between 7r pulses whose duration is7r((AwD)/2). The results of our numerical analysis confirmthat a velocity-selective rephasing occurs as expected whenthis zero-field period is implemented and that a net coolingof the distribution results when two separate 7r-pulse trainsare applied in the manner described in Section 2.

Results of the numerical analysis for cooling on the hydro-gen Lyman-a transition, idealized as a two-level system, arepresented in Fig. 2. Shown is an initially Maxwellian mo-

A. J. Palmer and J. F. Lam

,)N(p, t)1,9t = 4(uE1h)Fj(p, t),

722 J. Opt. Soc. Am. B/Vol. 3, No. 5/May 1986

0.80 | I I I I

INITIAL M(P) M(P) AFTER0.72 N APPLICATION OF

/ in 167T-PU LSES

0.64

0.56F(P) AFTER

APPLICATION OFa 0.48 F(P) AFTER FIRST uT-PULSE

APPLICATION OF I TRAIN TUNED TOSECOND 7r-PULSE , COOL LEFT SIDE

X 0.40- TRAIN TUNED TO OF DISTRIBUTION-COOL RIGHT SIDE

0 24 OF DISTRIBUTION

0.32-

0.24-

0.16-

0.08

0.00-240 -180 -120 -60 0 60 120 180 240

P, hk

Fig. 2. Results of numerical integration for the momentum distri-bution, of the density of atoms, M(P), and for the optical coherence,F(P) I after 7r-pulse cooling.

mentum distribution and the distribution after the applica-tion of 16 ur pulses tuned in the manner discussed above withthe appropriate zero-field periods employed. Also shownare the plots of I F(P) I after the application of each of the ur-

pulse trains tuned to cool opposite sides of the distribution.One sees from these plots that coherence is maintained bet-ter on the side of the distribution that is being cooled than itis on the opposite side, as expected. As required, the phasesof all atoms were reset to zero before the application of the

second 7r-pulse train. An essentially symmetric distribu-tion, M(P), is seen to be achieved, which is cooled except at

the wings of the distribution where some apparent heating ofthe distribution has occurred.

The onset of numerical instabilities in the integrationroutine prevented the analysis of cooling with longer 7r-pulsetrains. However, it was possible to extrapolate the results toobtain a characteristic (-lie) cooling time for that portion ofthe distribution that experienced cooling. In the above ex-ample for 7r-pulse cooling on the hydrogen Lyman-a line this

extrapolated cooling time was estimated to be

,, - 1.6 X 10-8 sec. (13)

This cooling time can be compared with the cooling time forsteady-state cooling for the case when the natural transitionlinewidth is larger than the Doppler linewidth. (In 7r-pulsecooling the effective power-broadened linewidth is larger

than the Doppler linewidth.) This latter cooling time isgiven by21

Tr, = 2m/(hk 2 ) = 1.2 X 10-8 sec, (14)

where k is the photon wave number. The above two cooling

times are comparable, indicating that the net radiationforces in the two processes have similar velocity selectivity

for cooling a distribution of velocities.

Another interesting quantity associated with radiationcooling is the limiting temperature that can be reached.These limits have been discussed extensively for the steady-state case.2 1 In the case of cooling with 7r pulses the coolinglimit occurs when the Doppler width has been reduced to thenatural linewidth since under this condition the zero-fieldperiods have become comparable with the spontaneous de-cay time for the transition and coherence will be destroyedby the spontaneous decay process. Therefore the limitingtemperature is given roughly by

Tlimit _ON 2Mlk2kB, (15)

where YN is the natural linewidth of the transition, k is theradiation wave number, and kB is Boltzmann's constant.

5. SUMMARY

In conclusion, we have examined radiation cooling of a dis-

tribution of two-level atoms in the coherent-pulse regime.A train of oppositely directed and oppositely detuned 7r

pulses with specific zero-field periods between the 7r pulses isrequired for optimal cooling. The cooling rate was foundthrough numerical analysis of the quantum transport equa-tions to be comparable with the rate for steady-state radia-tion cooling for the case when the Doppler linewidth is below

the natural linewidth of the transition. The cooling tem-perature limit is defined by the Doppler width's becomingcomparable with the natural width of the transition.

To provide some perspective on the parameters for r-pulse cooling compared with steady-state laser cooling, weillustrate qualitatively in Fig. 3 the temperature as a func-tion of time for one translational degree of freedom for hy-drogen, idealized as a two-level system, under the influenceof laser cooling on the Lyman-a transition for both steady-state and 7r-pulse cooling. The Doppler width of this transi-tion becomes comparable with the natural width at a tem-perature of -10- 2 K. For steady-state cooling (solid curves)

this temperature represents the demarcation between twocooling-rate regimes. Above this temperature, the laser fre-quency (assumed narrow compared with the natural width)must be swept toward line center as cooling progresses, andthe average atom momentum decreases roughly linearlywith time at a rate that is maximum at the saturation powerof the transition. Below this temperature, the frequency

2Z

wdHr

H-

103

102

10

10-1

1 -2

10-3

-CW COOLING: POWER DENSITY REQUIRED =-50W/cm2 (CHIRPED)

\ \ -d r-PULSE COOLING: AVERAGEPOWER) \' \ \DENSITY REQUIREl .-\ \ -5 MW/cm2' v -t-0.5 MW/cm

2

1 _ x ~~~~~-5 kW/cm2 \

BT-72 M/k2 (1 Dopp YN)

3- - kBT= N/2kBT = hk)2 /2M

0 10 20304 506 70 80 9000 I00I3L140I0I6UI7II0 10 20 30 40 50 60 70 80 9010011012013014015016017018U

TIME x 10-8, sec

Fig. 3. Qualitative comparison of temperature as a function of time

and required power levels for steady-state radiation cooling (solidcurves) and 7r-pulse radiation cooling (dashed curves) of hydrogenatoms using the Lyman-a transition.

A. J. Palmer and J. F. Lam

Vol. 3, No. 5/May 1986/J. Opt. Soc. Am. B 723

need not be swept, and the cooling becomes exponential withtime with the (1/e) cooling time given by Eq. (14). Thetemperature limit most commonly assumed for the steady-state laser cooling is the so-called stochastic limit (kBT =,yN/2) .21

The dashed curves in Fig. 3 show the estimated exponen-tial cooling for 7r pulses using the characteristic cooling timegiven by expression (13). The net cooling time is seen to bedramatically shorter for the larger initial temperatures butat the expense of considerable higher average power density.In this connection one must note that the Lyman-a photonsare capable of photoionizing the upper state of the transi-tion. The rate of energy released to electrons due to ioniza-tion is small compared with the cooling rates for the steady-state cooling power levels but is comparable with the coolingrates for the r-pulse average power levels and must bewatched closely in further analysis of the practical applica-tion of 7r-pulse cooling. Photoionization of the upper levelalso gives rise to phase shifts not accounted for in the two-level treatment of r-pulse cooling given above, but theseshifts can be expected to be small owing to the much smallerdipole-matrix element between the upper state and the con-tinuum compared with that of the Lyman-a transition.

ACKNOWLEDGMENTS

The authors wish to thank C. R. Giuliano for suggesting theuse of r pulses for cooling and R. A. McFarlane, D. G. Steel,and C. R. Giuliano for technical discussions. This work wassupported by the U.S. Air Force Office of Scientific Researchunder contract no. F49620-82-C-0004.

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11. J. V. Prodan, A Migdall, W. D. Phillips, I. So, H. Metcalf, and J.Dalibard, "Stopping atoms with laser light," Phys. Rev. Lett.54, 992 (1985).

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16. H. Friedman and A. D. Wilson, "Isotope separation by radiationpressure of coherent r pulses," Appl. Phys. Lett. 28, 270 (1976).

17. A. J. Palmer and J. F. Lam, "Radiation cooling using r pulses,"J. Opt. Soc. Am. A 1, 1251 (A) (1984).

18. R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, "Geometri-cal representation of the Schrodinger equation for solving maserproblems," J. Appl. Phys. 28, 49 (1957).

19. M. Sargent, M. 0. Scully, and W. E. Lamb, Laser Physics(Addison-Wesley, Reading, Mass., 1974), Sec. 7-5.

20. E. V. Baklanov and B. Ya. Dubetskii, "Fokker-Planck equationfor gas atoms resonantly interacting with a light wave," Opt.Spectrosc. (USSR) 41, 1 (1976); V. G. Minogin, "Kinetic equa-tion for atoms interacting with laser radiation," Sov. Phys.JETP 52, 1032 (1980); J. Javansian and S. Stenholm, "Broad-band resonant light pressure," Appl. Phys. 21, 283 (1980); R. J.Cook, "Theory of resonance-radiation pressure," Phys. Rev. A22, 1078 (1980); C. Tanguy, S. Reynaud, and C. Cohen-Tan-noudji, "Deflection of an atomic beam by a laser wave: transi-tion between diffractive and diffusive regimes," J. Phys. B 17,4623 (1984).

21. See, for example, D. J. Wineland and W. M. Itano, "Lasercooling of atoms," Phys. Rev. A 20, 1521 (1979).

A. J. Palmer and J. F. Lam