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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�
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.
2014L’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.
2014The 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
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)
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.
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
.