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HAL Id: jpa-00209313

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Submitted on 1 Jan 1981

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The motion of atoms and molecules in a resonance light field

A.P. Kazantsev, G.I. Surdutovich, V.P. Yakovlev

To cite this version:

A.P. Kazantsev, G.I. Surdutovich, V.P. Yakovlev. The motion of atoms and molecules in a resonance light field. Journal de Physique, 1981, 42 (9), pp.1231-1237. �10.1051/jphys:019810042090123100�.

�jpa-00209313�

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The motion of atoms and molecules in a resonance light field

A. P. Kazantsev

L. D. Landau Institute of Theoretical Physics, Moscow, U.S.S.R.

G. I. Surdutovich

Institute of Physics of Semiconductors, Novosibirsk, 90, U.S.S.R.

and V. P. Yakovlev

Institute of Engineering Physics, Moscow, U.S.S.R.

(Reçu le 10 décembre 1980, accepté le 30 avril 1981)

Résumé.

2014

L’influence de la pression de radiation de la lumière sur le mouvement d’atomes et des molécules est étudiée. Une équation cinétique est obtenue qui décrit l’action sur les particules des forces moyennes de la pression

de radiation ainsi que de leurs fluctuations. On considère les forces dues aux effets des transitions spontanées et

induites ainsi que celles qui sont produites par une combinaison de ces deux effets. L’équation cinétique obtenue

est appliquée à la description de la diffusion d’atomes dans des conditions proches des expériences. De même,

on calcule la température d’atomes refroidis par une onde stationnaire à l’aide de la force de pression de radiation du type mixte.

Abstract.

2014

The influence of the resonance light pressure on the motion of atoms and molecules is studied. A kinetic equation has been obtained describing the action on particles of both the mean force of light pressure and its fluctuations. The forces that are considered are due to spontaneous and stimulated transitions and also to a certain combination thereof. The kinetic equation is used to describe the scattering of atoms in conditions close to the

experimental ones [16]. The temperature of atoms cooled in a standing wave with the aid of the force of light pressure of the combined type has also been found.

Classification Physics Abstracts

32.00 - 33.00

1. Introduction.

-

Quite considerable forces of

light pressure are exerted on resonance particles in the

field of laser radiation [1-3]. These forces depend on the intensity, the closeness of the frequency to resonance, and on the spatial structure of the external field. The

magnitude of the light pressure on separate particles,

in the long run, is determined by the rate of scattering

of the external field photons.

In a running light wave, only spontaneous transi- tions contribute to the pressure. In inhomogeneous

fields (for example, in a standing wave), the force of the stimulated light pressure (the gradient force)

also appears. Since the processes of absorption and

emission of quanta of the external field are correlated in stimulated transitions, the gradient force is a regular quantity. Spontaneous emission destroys the cohe-

rence of interaction with the field and affects the motion of an atom in two ways. First, a certain

average force appears which, depending on the

detuning, may be an accelerating or a decelerating

one [4]. Second, diffusion of the atoms in velocity

space appears. Depending on the field parameters and the duration of interaction in real conditions,

the motion of particles may be of a drift or diffusion

nature. A kinetic equation must be used to describe

such motions. The present paper is devoted to a

derivation of a kinetic equation for atoms and mole-

cules in a quasi-classical limit.

The motion of slow quasi-classical atoms satisfies

an equation of the Fokker-Planck type (Sec. 3). The

average force and the diffusion coefficient, taking into

account the spontaneous and stimulated transitions,

have been obtained. The scattering of an atomic beam (Sec. 4) in conditions close to those of the experiment

of Oka et al. [16] has been considered as an example of

an application of this equation. The average scattering angle is calculated as a function of the detuning and

the magnitude of the field.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019810042090123100

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1232

In section 5, the correct distribution function of the atoms cooled by a standing light wave is found. If it is

approximated by means of the Boltzmann distribution,

it is possible to determine the temperature of the particles in terms of the parameters of the field.

2. Initial equations.

-

Let us consider a model consisting of two-level atoms with a frequency w0 and a transition dipole moment d. We call the lower and upper levels, 1 and 2, respectively : the upper level has width y. The resonance field

has a small detuning A.

The Hamiltonian of an atom of mass M in the

resonance approximation can be written in the

following form

(1) where (J 3,:t are the spin matrices.

To take into account the recoil in spontaneous and stimulated transitions, we must proceed from the

quantum equation of the density matrix of the atoms

p(r1 r2 t) [5, 6].

(2)

The relaxation operator ÿ has the form

Here W(n) is the probability of spontaneous emission of a quantum in the direction n, k = wolc.

For typical conditions, the momenta of the atoms considerably exceed that of a resonance quantum

volv « 1 , vo = hkIM . (3)

In this case, equation (2) can be expanded in the Wigner representation according to the quasi-classical para- meter vo/v. Hence, for the distribution function of the atoms f(rvt) = p22(rvt) + pl l(rvt), the difference in

populations q(rvt) = p22(rvt) - p11(rvt) and the dipole

moment p(rvt) = P12(rvt) (in units of d), we have the following system of equations [7] :

Here d/dt = a/at + vV is the total time derivative, Q(r) = dE(r)/h,

Equations (4) and (5) correspond to the well-known system of Bloch ’equations for two-level atoms. The difference lies only in the second term in the right-

hand side of equation (4) that takes into consideration the recoil effect in stimulated transitions.

Equation (6) for the distribution function describes the change in the motion of the centre of inertia of an atom as a result of both stimulated transitions (the right-hand side) and spontaneous ones. Equation (6)

contains the spontaneous relaxation operator for an atomic dipole linearly polarized along the z-axis,

which equals

Introducing the solution of Bloch’s equations (4) and (5) into equation (6), we get a kinetic equation for the

distribution function. For slow atoms (Sec. 3), it takes

the form of the Fokker-Planck equation.

3. Kinetic equation.

-

3.1 RUNNING WAVE.

-

Let us first consider the simpler case of a running wave

which makes a spatially homogeneous solution of Bloch’s equations possible. Hence, in the quasi- stationary limit (dq/dt « yq and dp/dt « vp), equa- tions (4) and (5) are reduced to algebraic equations

for p and q. As a result, we get the following kinetic equation for f :

where F. = yhkw is the force of spontaneous light

pressure, Do = y/20 vô w is the reduced diffusion

coefficient, and

is the probability of population of the upper level.

The difference between the transverse diffusion coefficients along the y- and z-axes is associated with the orientation of the emitting dipole along the z-axis.

Since the process of absorption of external field quanta due to spontaneous emission is of a random nature, the resulting fluctuations of the momentum of an atom leads to an increase in the longitudinal diffusion

coefficient. The diffusion coefficient in equation (7)

(4)

differs in its numerical factors from the corresponding

value obtained in [9, 10] for a running wave (1).

Note that the average force contains a correction that is small relative to the parameter of equation (3).

This correction is due to the dependence of the

diffusion coefficient on the velocity.

3.2 STANDING WAVE. - Let us now consider the motion of atoms in a standing wave

In this case, the spatial inhomogeneity of the field

plays an important role and prevents an analytic

solution of the equations. In many important appli- cations, however, the motion of the atoms is slow,

so that

which makes it possible to expand the pair of equa- tions (4) and (5) with respect to small gradients. In addition, the recoil effect in the Bloch equations can

be taken into account by perturbations theory. Let

us assume that p = po + pl and q = qo + ql, where pl « po and q, « qo.

In the zero-order approximation, the atoms are

considered to be stationary, and recoil is not taken into account. Hence

where x(x) = 2 Q2(x)/1 V12 is the parameter of satu- ration in a standing wave.

For. the first-order approximation, we have the equations

The zero-order approximation (9) gives the mean

induced dipole moment of an atom depending on the

local value of the field. Thus, the gradient force acting

on a particle is determined. The first-order approxi-

mation determines the non-local part of the dipole

moment that is proportional to ôqfflx. This results in the appearance of a non-gradient force and in the

diffusion of the atoms by velocities. Introducing

(1) It also differs from the Piqué diffusion coefficient [22] which

was obtained as a result of the linear momentum transferred from the photons to the atoms through a succession of photon-scattering

processes.

Re (po + Pi) into equation (6), we arrive at the following kinetic equation

The logarithmic potential (12) for slow atoms was

obtained earlier by one of the authors [4].

The terms proportional to G, A, and D found accord-

ing to perturbation theory are small in comparison

with the gradient force ôU/ôx. They must be taken

into account, however, because upon being averaged

over the period of the field, these terms do not vanish,

whereas the mean gradient force does vanish. Conse-

quently, the influence of these terms is of a systematic

nature increasing with time.

The force Gvx is a force of friction. Depending on the detuning, the friction may be positive or negative.

Like any dissipative force, it cannot be represented in

the form of a potential gradient.

From the physical viewpoint, the nongradient force

results from mixing of spontaneous and stimulated transitions. It is natural to call it a force of the com-

bined type [3, 4]. Figure 1 shows how a force of the combined type averaged over space ( G(x) > vx (the angular brackets signify averaging over the period

of the field) depends on the detuning.

In a weak field when d 0, the force is a decele-

rating one, and when A > 0

-

an accelerating one (curve 1). In a strong held, when Q > y, G vanishes not

only at zero detuning but also at the finite detunings

y/2J3 and d fd, where A rd = J2 Qg/y.

Fig. 1.

-

Dependence of the averaged force of combined type on detuning. Curve (1) corresponds to a weak field, and curve (2) to

a strong field.

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1234

The diffusion coefficient D contains a contribution from the spontaneous transitions Do and one from

the stimulated transitions proportional to the square of the field gradient.

In weak fields, the contributions from the spon- taneous and stimulated transitions differ only in the

numerical factor.

In a strong field, the stimulated diffusion coefficient

considerably exceeds the spontaneous one :

Thus, the diffusion coefficient due to spontaneous emission becomes rapidly saturated when the field

.

grows, while the diffusion coefficient due to the

gradient force is proportional to the field intensity (see Fig. 2). The term with the combined derivative

proportional to A somewhat alters the total diffusion

coefficient, as will be shown in section 5.

We shall use the kinetic equation to describe two

concrete physical examples

-

scattering of an atomic

beam and cooling of atoms by a standing wave.

Fig. 2.

-

Dependence of the averaged diffusion coefficient in a

standing wave on the held intensity for zero detuning. The bottom

curve corresponds to the spontaneous diffusion coefficient.

4. Scattering of an atomic beam by a standing light

wave.

-

Let us consider the scattering of a mono- energetic beam of atoms travelling with the velocity vy along the y-axis by the field of a standing light wave arranged along the x-axis. Transverse velocities vx

are absent in the oncoming beam. Our task consists in

finding the transverse velocity distribution of the

particles after passing through the field region.

Earlier, the classical method [12, 13] and the quan- tum method [14, 15] were used to study the case of

small durations i of interaction with a field (yi 1)

with no account taken of spontaneous transitions. The

scattering of atoms by the non-monochromatic field

of a standing wave at large 1:’S was considered in [8].

The investigation of the case of large durations of

flight through the field when the spontaneous emission becomes significant is of interest. Recently, the scattering of a beam of sodium atoms by the field of a

monochromatic standing wave was observed in [16].

The conditions of the experiment were such that the parameter yr ~ 10, and the final transverse velocities of the atoms were low, kvx « y. For these conditions, kinetic equation (11) can be used to find the distribu-

tion function of the scattered atoms. In scattering through small angles, the longitudinal velocity vy

can be considered constant. A particle getting into the region of a periodic potential begins to vibrate with

the characteristic frequency k.J’U/M along the x-axis.

A diffusion motion is superimposed upon these vibrations.

If we limit ourselves to the case (2)

then the problem can be appreciably simplified. The

increment of the transverse momentum of an atom in

scattering is proportional to T, while the change in the

coordinate along the x-axis is of the order of ’t2.

Condition (18) allows us to disregard the change in the

transverse coordinate. In this approximation, the problem is to find the transverse momenta of the scattered particles moving in the light field along

rectilinear traj ectories. This means that in equation (ll)

we may omit vx.ôf/ôx, A.02f/ox oVx, and also Gvx because, in this case, there is no self-averaging of the gradient force (a particule does not have time to become noticeably displaced from the initial coor-

dinate along the x-axis), and ô U/ôx is greater than the force of the combined type for all detunings. The

diffusion term must be retained because for small

detunings the gradient force is small and scattering

is of a diffusive nature.

Thus, after the simplifications we have made, the stationary kinetic equation acquires the form

Assume that a light ray occupies the region 0 y 1

along the y-axis. At the entrance to the light ray, the distribution function has the form

The aperture of the beam is usually large in comparison

with the wavelength of the light ; therefore the distri- bution of the particles along the x-axis in the incident beam may be considered uniform. By solving equa-

(2) In conditions of the experiment [16], this criterion can be

fulhlled for weak belds and small detunings.

(6)

tion (19), we find the distribution of the scattered

particles

The angular brackets signify averaging over the space

period of the field because the detector registers a scattering pattem averaged over the coordinate. Let

us calculate the mean square of the velocity of the

scattered particles v;. In a weak field Qo « y, we have

The first term in the brackets corresponds to the

contribution from the gradient force, the second

-

from diffuse scattering. In the very weak field

Q0 « vy-/T, diffusion plays the most important role.

When Qo » -,/Y/Tl scattering at the expense of the

gradient force is of greater significance except for small detunings 1 A 1 A1 = v/ y/-2 T. ylqo. In figure 3

which shows the dependence of the average scattering angle 6 = I-vTIv. on the detuning, curves 1 and 2

correspond to these cases.

Fig. 3.

-

Dependence of the average scattering angle on the detun- ing. Curve 1 corresponds to a weak field Q(O) « (Y/1:)1/2 y, curve 2

to a field of intermediate strength (Y/1:)1/2 Q(O) « y, and curve 3 to a strong held Q(O) > y.

In strong fields Qo » y, we have

The contribution from the gradient force ( ~ -r2) is considerably greater than that from diffusion (~ i)

in all detuning except in the small region 1 d 1 L1 o.

When A ~ Jo both contributions become of the same order of magnitude. Thus the frequency Ao depends

primarily on the field. In the intermediate field regime

y « Q(0) « y2 i we have

In very strohg fields when Q(O) » y2 i, from (23) we

find

Thus, in a stronfl field, the graph showing how the scattering angle O(A) depends on detunings has two

characteristic frequencies, Sl(0) and 40. For large detunings Id 1 k Q(O) the scattering pattern coin- cides with the results of the theory of purely potential scattering (without taking into account spontaneous emission). In particular, the characteristic scattering angle associated with a detuning 4 - Q(O) is 8 ~ 7:k dEo/Mvy. If we substitute 7: using equation (18)

in which we replace the inequality by an equality sign,

then we have

This result is in agreement with the conclusions of

[12-15] and with the result of the experiment [16].

When Li Q(0), spontaneous emission appreciably changes the picture of the scattering. If the sponta-

neous emission is neglected we obtain the peak in the scattering at the resonance frequency, but when yi » 1 a dip appears in that frequency region. Unfor- tunately, no measurement of the frequency dependence

of the scattering angle is given in [16] (3).

5. Cooling of atoms by light.

-

The rate of cooling

or heating of atoms (depending on the sign of the detuning) in a standing light wave was calculated previously for strong saturation in [4] and for weak

saturation in [17].

Here we shall consider the final stage of the cooling

of atoms in a standing wave, find a stationary solution

of the kinetic equation, and calculate the temperature of the cooled gas. We shall consider one-dimensional motion along the x-axis, considering that the distri- bution function does not depend on vy and Vz. Actually,

this means that the energy of transverse motion is

considerably higher than that of longitudinal motion.

In the absence of collisions, such a strong anisotropy

of the temperature is quite possible.

Thus, the one-dimensional stationary equation (11)

for the distribution function has the form (vx = v)

(3) In the recent work [23] an expression for the diffuse coefficient of atoms in a standing light wave analogous to our formula (15)

was obtained.

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1236

In the following, we shall take advantage of the small- size of the terms with G, A, and D in comparison with

thé gradient force. For this purpose, we shall transform from the variables x and v to the variables x and

e = Mv2/2 + U(x), where e is the total energy of a

particle. Equation (27) thus becomes

We shall seek the solution of this equation by per- turbation theory, representing the distribution func- tion in the form f(e, x) = fo(8) + fi(8, x) ; here fi

is much smaller than fo owing to the smallness of the parameters A, G and D. From equation (28), we have

Since in stationary conditions, the function f1(e, x)

is a periodic function of the coordinate, after averaging

over the period of the field we get

The expression in braces must be equated to zero

because otherwise the function fo will not be nor-

malized. Hence, we find

The exact distribution function (31) can be simplified

if we approximate the average of the products with

the product of the average values. Then the distri- bution function acquires its standard form fo - e-£/T, ,

where

It is not difficult to see that the numerator in equa- tion (32) is a négative quantity. Consequently, a stationary distribution exists provided that

Let us consider cases that are extreme as regards

the field intensity. In the weak field Qo « y, we have

The minimum temperature (7/20) hy z ~ (1/3) hy is

reached whena = y/2. This agrees in order of magni-

tude with the results obtained in [18, 19]. The depth

of modulation of the potential U(x) in this case is

small in comparison with the température ; therefore the fraction of particles captured into potential wells

is small.

When Qo » y and Xo - 1, we find from equation (32)

Finally, in the strong field vo » 1

The condition for positive temperatures leads to the following restriction being imposed on detuning :

y/,/1-2 Li 0 or y/,/1-2 Li « Qo. It is understood

that detuning is not too close to the boundaries of the indicated regions because otherwise the condition of slowness of the atoms (8) may be violated. We see

that in a strong field the temperature of the atoms is of the order of the Rabi frequency and is comparable

with the depth of the potential wells. Therefore, the

fraction of bound atoms is of the order of unity. It

must be noted that recently the cooling of ions cap- tured in SHF-traps by means of resonance light

pressure was observed [20].

6. Conclusion.

-

To describe the drift motions of

an atom in a light field, it is sufficient to take into

account only the average force of the light pressure.

When the average acting force is small, however, it is

necessary to take into consideration the fluctuations of the force of light pressure that lead to diffusion of the atomic velocities. This occurs, as a rule, with small

detunings from resonance and for long durations of

interaction with the field, yi » 1. A kinetic equation

must be used in this case to describe the motion of the

atoms in the field. A kinetic equation for purely sti-

mulated transitions in a non-monochromatic field

of a standing wave was previously considered in [7, 8].

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The diffusion of atoms at the expense of spontaneous transitions in the field of a running wave was calculated

in [9, 10, 11, 21].

In the present paper, we have calculated the diffu- sion coefficient in a monochromatic standing wave

due to a certain combination of spontaneous and stimulated transitions. We have also determined the spontaneous diffusion coefficient in a running wave

more precisely.

If one considers the various types of diffusion from the viewpoint of the mechanism of scattering of the

external field photons, then as in the case of average

forces, there are three types of diffusion : spontaneous, stimulated, and the combined type [3]. The most appreciable difference between these diffusion coef- ficients occurs in a strong saturating field. The spon-

taneous diffusion coefficient does not depend on the field, while the diffusion coefficient of the combined type, like the coefficient of induced diffusion [7, 8],

grows in proportion to the field intensity.

Further the kinetic equation has been used to find

the distribution function of atoms scattered in a

standing wave.

It has been shown that the dependence of the average

scattering angle on the detuning in a strong field is non-monotonic. The temperature of the atoms cooled

by a standing light wave and its dependence on the

field parameters was investigated and the results are

given in the paper.

Acknowledgments.

-

We acknowledge helpful dis-

cussions with A. M. Dykhne.

References

[1] ASKARIAN, G. A., Zh. Eksp. Teor. Fiz. 42 (1962) 1567.

[2] ASHKIN, A., Phys. Rev. Lett. 24 156 ; 25 (1970) 1321.

[3] KAZANTSEV, A. P., Usp. Fiz. Nauk 124 (1978) 113.

[4] KAZANTSEV, A. P., Zh. Eksp. Teor. Fiz. 66 (1974) 1599.

[5] KOLCHENKO, A. P., RAUTIAN, S. G., SOKOLOVSKY, R. I., Zh. Eksp. Teor. Fiz. 55 (1968) 1864.

[6] VOROBYEV, F. A., RAUTIAN, S. G., SOKOLOVSKY, R. I., Opt.

Spektrosk. 27 (1969) 728.

[7] KAZANTSEV, A. P., Zh. Eksp. Teor. Fiz. 67 (1974) 1660.

[8] BOTIN, A. P., KAZANTSEV, A. P., Zh. Eksp. Teor. Fiz. 68 (1975)

2075.

[9] PUSEP, A. Yu., Zh. Eksp. Teor. Fiz. 70 (1976) 851.

[10] BAKLANOV, E. V., DUBETSKY, B. Ya., Opt. Spektrosk. 41 (1976) 3.

[11] KRASNOV, I. V., SHAPAREV, N. E., Zh. Eksp. Teor. Fiz. 77 (1979)

899.

[12] KAZANTSEV, A. P., SURDUTOVICH, G. I., Pis’ma Zh. Eksp. Teor.

Fiz. 21 (1975) 346 ;

ZHELNOV, B. L. KAZANTSEV, A. P. and SURDUTOVICH, G. I., Kvantovaya Elektronika 4 (1977) 893.

[13] DELONE, G. A., GRINCHUK, V. A., KAZANTSEV, A. P., SURDU- TOVICH, G. I., Opt. Commun. 25 (1978) 399.

[14] COOK, R. J., BERNHARDT, A. F., Phys. Rev. A 18 (1978) 2533.

2533.

[15] KAZANTSEV, A. P., SURDUTOVICH, G. I., YAKOVLEV, V. P., Pis’ma Zh. Eksp. Teor. Fiz. 31 (1980) 542.

[16] ARIMONDO, A., LEW, H. and OKA, T., Phys. Rev. Lett. 43 (1979)

753.

[17] HÄNCH, T. W., SCHAWLOW, A. L., Opt. Commun. 13 (1975) 68.

[18] LETOKHOV, V. S., MINOGIN, V. G., PAVLIK, B. D., Zh. Eksp.

Teor. Fiz. 72 (1977) 1328.

[19] KLIMONTOVICH, Yu. L., LUZGIN, S. P., Zh. Tekhn. Fiz. 48 (1978)

2217.

[20] WIENELAND, D. J., DRULLINGER, R. E., WALLS, F. L., Phys.

Rev. Lett. 40 (1978) 1639.

[21] MANDEL, L., J. Opt. 10 (1979) 51.

[22] PICQUÉ, J. L., Phys. Rev. A 19 (1979) 1622.

[23] COOK, R. J., Phys. Rev. A 22 (1980) 1078.

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