2108 J. Opt. Soc. Am. B/Vol. 6, No. 11/November 1989

Laser cooling of atoms below the single-photon classical limit

V. G. Minogin, M. A. Olshany, and S. U. Shulga

Institute of Spectroscopy, USSR Academy of Sciences, 142092 Troitsk, Moscow Region, USSR

Received March 13, 1989; accepted August 1, 1989

We describe a new mechanism for the laser cooling of atoms. The mechanism is based on the damping of the motionof three-level atoms by laser light pressure under the condition of coherent atomic population trapping. Because ofthe action of this mechanism, three-level atoms can be cooled below the temperature of the single-photon classicallimit, T, = hy/kB, to a temperature corresponding to the recoil energy, TR = h2k 2/2MkB. It is probable that themechanism described has a part in the recently observed laser cooling of sodium atoms below the temperature hy/kB-

INTRODUCTION

In recent years two basic mechanisms for laser cooling ofatoms have been proposed and experimentally realized.The mechanism of single-photon cooling 1' 2 based on deceler-ating two-level atoms by a light-pressure force can be used tocool atoms from thermal energies to an energy determinedby the single-photon classical limit,3 kBT = hy, where y ishalf of the natural linewidth of the atomic transition. Themechanism of stochastic two-photon cooling 4 based on co-herent trapping of three-level atoms in states having mo-menta of the order of the photon momentum hk can be usedfor effective cooling of atoms from energies somewhat ex-ceeding the recoil energy R = h 2k2 /2M, theoretically to zeroenergy. Practically, the limiting energy (temperature) thatcan be achieved by two-photon cooling is determined by thetime of flight of atoms through the laser field and may bemany orders of magnitude lower than the recoil energy.4

However, the temperature of laser-cooled atoms obtainedin recent experiments, 5 T = 40 ,4K, is six times lower than thesingle-photon classical limit, T = y/kB = 240 ,uK, but 200times higher than the temperature determined by recoilenergy, TR = h2k2 /2Mk = 1.2 ,K. These experiments areexplained well in Ref. 6.

Here we suggest an alternative mechanism for coolingatoms to temperatures within the range TR < T < T. Alongwith the mechanism described in Ref. 6, this mechanismprobably contributes to the effect described in Ref. 5. It isbased on damping the motion of three-level atoms under alight-pressure force whose velocity dependence has a narrowresonance caused by coherent population trapping.7

QUALITATIVE CONSIDERATIONS

The third cooling mechanism can be described as follows.Let a three-level atom having the A scheme of levels (21 = 0,where 21 is half of the natural inewidth of the atomictransition 12) - 11)) be excited by two counterpropagatinglight waves (Fig. 1), and let the atomic velocity along thewave-propagation axis be near that satisfying the conditionof the coherent population trapping:

2kv = 2 - W + 2 1 .

Under these conditions the velocity dependence of the

light-pressure force contains a narrow resonance dip (Fig. 2)of width 6v - yG/k, where G is the saturation parameter andy is a quantity of the order of 'Yi and 'Y2. This resonance dipis so sharp that, when the average velocity of the atomicensemble (v) is within the dip (I(v) I < v), the light-pressureforce produces considerable damping of atomic motion,which in turn leads to a considerable reduction in the tem-perature of the relative atomic motion near the averagevelocity.

Assume, for example, that the average velocity is negativeand lies near the middle of the narrow resonance dip in thelight-pressure force (Fig. 2). Since the light-pressure forceis of the order of F- hkyG, the atoms in the region of the dipare subjected to a friction force Ffr = -M(v - (v)), where A

F/Mt~v hk2 /M is a friction coefficient. The joint actionof the friction force and the diffusion determined by themomentum-diffusion coefficient D h2k2-yG causes coolingof the atoms to the energy8

kBT -D/M: - hyG,

which for small G may be much lower than that of the single-photon classical limit.

Consider as another example the case when the averageatomic velocity is near zero velocity: I(v) <<yG12/k. Forlow velocities the light-pressure force and the momentum-diffusion coefficient can be estimated as F- hkyG(vlv) andD - h2k 2'yG(vlbv) 2, respectively, where G is again the satura-tion parameter, y Y -y2. Let the average velocity benegative, as above (Fig. 2). In this case the light-pressureforce produces a friction force Ff = -MO(v - (v)) with afriction coefficient 3 #(2hk 2/M)(I (v) I/tv). Accordingly, theenergy of the relative atomic motion near average velocity isestimated in this case as

kBTDIM - /2hy (k (v) /,).

This energy may already be much lower than hy for arbi-trary values of the saturation parameter G.

The two examples above show that three-level atoms canbe cooled below the energy of the single-photon classicallimit. The recoil energy R = h2k2/2M naturally remains inthe lower-energy limit since our classical consideration as-sumes that the atomic momentum exceeds the photon mo-mentum, p >> hk.

0740-3224/89/112108-04$02.00 © 1989 Optical Society of America

Minogin et al.

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due to the velocity dependence of the nonadiabatic parame-ter d in Eqs. (3).

The left-hand side of the dip of the light-pressure forcehas a negative slope with respect to the velocity axis. Ac-cordingly this part of the dip is responsible for the narrowingof the atomic velocity distribution (Fig. 4) to the steady-state one described by the temperature of the relative atomicmotion:

kBT= D,/M0. (4)

0.6

0.5 ... G=0.3

2 ~-' 1r -/ 0.4L

1 _ 0.3

Fig. 1. Interaction scheme for a three-level A atom and two coun- , /' -'a.

terpropagating light waves. 0.2 \

BASIC RELATIONS 0.1 L X , . X

The qualitative picture just outlined directly corresponds to , l

the classical dynamics of A atoms in the field of two counter- - . . -

propagating plane waves: 0 - .

E = eE1 cos(kz + CO1t) + eE 2 cos(kz - co2 t). (1) -15 -10 -5 0 5 10 15

Consider, for simplicity, the case of equal Rabi frequencies Velocity [m/sec]forthetransitions 13) -I) and 13) -12): g = d3leE,/2h = (a)92 = d32 eE2/2h = g, and assume that waves 1 and 2 are inexactresonancewiththetransitions 13) -|11) and 13) - 12): 0.6 G=3. 0|W1 = C31, W2 = C32 (Fig. 1). The spontaneous decay probabil- G=3.0

ities 2,yl and 27y2 are assumed to be related as 2 > 71- - .

Under these conditions the standard procedures of the 0.5quasi-classical solution of the Wigner density matrix equa-tions for times t >> 71-, Y2l gives, for the light-pressureforce and the momentum-diffusion coefficient, the expres- 0.4sions 7

F = 2hk'Ynp 3 3 , (2) e 0.3

Da. = 2h 2k 'p3 3[/ 3 + (1 + d)bza], .. . - -----

d /2Lq21(3G + 8) 2 + 2GL[(G + 2 2 )(G - 22)2 - 604]), 0.2-

(3) -\where G = 2g2/1y2 is the saturation parameter, p33 = 2g2/-y 2 is 0.1 .\ //.

the population of the upper level, and \ .1

'Y = (y1 + 2)/2, n = ( 2 -)/y, = kv/ly, 0

L = [G2 + 2(G + 8) 2 + 444] - -1.5 -1.0 -0.5 0 0.5 1.0 1.5

The above equations show that the light-pressure force Velocity [m/sec]and the momentum-diffusion coefficient contain narrow (b)dips located near zero velocity for the chosen frequencies Fig. 2. Velocity dependence of the light-pressure force determined(Figs. 2 and 3). Note the slight difference between the by Eq. (2) for atomic parameters corresponding to the sodium-atomcurves for the force and for the diffusion coefficient, which is transition 3S1/2 (F = 1, 2)-3P,/ 2 (Ay = 1 MHz, 72 = 5 MHz).

Minogin et al.

2110 J. Opt. Soc. Am. B/Vol. 6, No. 11/November 1989

°.1 .. ......... . . ... 1... ..0~~~~~~~~~~~~

-1.5 -1.0 -0.5 0 0.5 1.0 1.5Velocity [m/sec]

(b)

Fig. 3. Velocity dependence of the momentum-diffusion coeffi-cient D, determined by Eqs. (3) for the same parameters as in Fig. 2.

of the left-hand side of the dip, Eq. (4) for (v) = -,yG/k and(Q) =-G is

In this case the temperature can be lower than the tempera-ture kBT, = h'y only for small-saturation parameters.

1. ~ -- -- 14 Asec

--- 28 usec

sn_ -

15

10

5

0

-0.30 -0.15

Velocity [/sec]Fig. 4. Time evolution of the atomic velocity distribution on thefield [Eq. (1)] according to the computer simulation. The atomicparameters are the same as in Fig. 2. The initial velocity distribu-tion was taken as constant, n = 1, in the region -2 m/sec < v < 0.

2.0

1.5

r___1

L_"J

XE4

1.0

0.5

Here the diffusion coefficient is determined by Eqs. (3) for v= (v), and the friction coefficient for v = (v) is

MO3 =-(aF/v)v=(v)

= 8k 2nGL(Q)[2L(4(Q)2 + G + 8) ()2 - 1] (5)

where (Q) = k(v)/y.When the average velocity is chosen to be near the middle

-1.5 -1.0 -0.5 0 0.5 1.0 1.5

Velocity [m/sec]Fig. 5. Dependence of the temperature given by Eq. (7) on theaverage atomic velocity (v) for the atomic parameters as in Fig. 2.

24

0.7

0.6

0.5

0.4

1 0.3

N

N 0.2

0.1

0

kBT = (32 69 2\hGn-.(3 289 (6)

-15 -10 -5 0 5 10 15

Velocity [m/sec]

(a)

0.7

0.6

0.5

I

Ix

IIIx

II

. . .

c'2

24

0.4

0.3

0

Minogin et al.

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Vol. 6, No. 11/November 1989/J. Opt. Soc. Am. B 2111

For small velocities Eq. (4) gives the following expressionfor the temperature:

kBT = - I hzy-q-%2(4 + 312) + [4(32 -62)G-2

+ (16 + 9n2)G-11J21, (7)

as shown in Fig. 5. According to this expression, for anyvalue of the saturation parameter there is a range near zerovelocity where the temperature is lower than the single-photon classical limit hy/kB. In a weak field (G << 1) thisvelocity range broadens with increasing G; in an intense field(G ' 1) it remains fixed as G increases.

CONCLUSIONS

(1) The laser-cooling mechanism considered above issemiclassical, unlike that described in Ref. 4. As withDoppler cooling of two-level atoms, the light-pressure forceacting on the A atom is conditioned by a combination ofabsorption and spontaneous emission of photons. Theatomic ensemble velocity distribution evolution is describedby the classical Fokker-Planck equation.

(2) The expression for temperature given by Eq. (4) isapproximate and corresponds to the stationary solution ofthe one-dimensional Fokker-Planck equation with the ki-netic coefficients D(V) and u(v), where

D(v) = D,,Qv)),

F(V) = -((v))v.

The substitutions F(v) - v(v) and D,,(v) - D(v) are legiti-mate if

(a) The atomic ensemble velocity distribution width Avis less than the characteristic interval v of change of ,B(v)and DZZ(v) on the velocity scale and

(b) The characteristic time of average velocity variationr is less than the relaxation time of atomic ensemble #--(hk2 /M)-l.

For the problem under consideration,

AV'- -TIM'- (RIhy)G1bv << v,

with

G >> hy/R

or

kBT hyG >> R = h 2k 2 /2M,

which complies with the limit of applicability of the classicaltheory; and

T, -M(v) IF((v)) -((v) lbv)-l-,p

The results of numerical simulation demonstrate agreementbetween the estimation of Eq. (4) and the real temperature.

(3) The only case of exact resonance of light waves withatomic transitions was considered. In the near-resonanceconditions the light-pressure force and the momentum-dif-fusion coefficient are asymmetrical about the velocity atwhich coherent population trapping occurs.

(4) It is possible that the mechanism proposed, alongwith the mechanism of Ref. 6, can make a contribution to thecooling of atoms below the energy hy observed earlier.

ACKNOWLEDGMENTS

We express our appreciation to Yu. V. Rozhdestvensky andN. N. Yakobson for useful discussions at early stages of thisresearch.

REFERENCES

1. T. W. Hiinsch and A. L. Schawlow, "Cooling of gases by laserradiation," Opt. Commun. 13, 68-69 (1975).

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4. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C.Cohen-Tannoudji, "Laser cooling below the one-photon recoilenergy by velocity-selective coherent population trapping,"Phys. Rev. Lett. 61, 826-829 (1988).

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6. S. Chu, D. S. Weiss, Y. Shevy, and T. J. Ungar, "Laser cooling dueto atomic dipole orientation," in Atomic Physics 11, S. Haroche,J. C. Gay, and G. Grynberg, eds. (World Scientific, Singapore,1989), p. 636; J. Dalibard, C. Salomon, A. Aspect, E. Arimondo,R. Kaiser, N. Vanasteenkiste, and C. Cohen-Tannoudji, "Newschemes in laser cooling," ibid., p. 199.

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8. V. G. Minogin, M. A. Olshany, Yu. V. Rozhdestvensky, and N. N.Yakobson, "Laser cooling of atoms below the Doppler limit,"Opt. Spektrosk. (to be published).

Minogin et al.