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Journal of Physics B Atomic and Molecular Physics

Atomic motion in laser light: connection between semiclassical and quantum descriptions

Jean Dalibard, Claude N. Cohen-Tannoudji

Published Apr 28, 1985• 151 citations
Publisher

Abstract

The quantum kinetic equation describing slow atomic motion in laser light is derived by an operational method which provides mathematical expressions with a transparent physical structure. The authors prove in a general way that the coefficients appearing in this equation, which is of a Fokker-Planck type, are simply related to the mean value and to the correlation functions of the Heisenberg radiative force of the semiclassical approach, where the atomic position is treated classically. They derive in particular a new theoretical expression for the damping force responsible for radiative cooling and they interpret it in terms of linear response theory. The authors also obtain a new crossed r-p derivative term, which does not appear in semiclassical treatments, but which they find to be very small in most situations. Finally, all the theoretical expressions derived are valid for any J g to J e transitions and are not restricted to two-level atoms.

DOI
10.1088/0022-3700/18/8/019

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The quantum kinetic equation describing slow atomic motion in laser light is derived by an operational method which provides mathematical expressions with a transparent physical structure. The authors prove in a general way that the coefficients appearing in this equation, which is of a Fokker-Planck type, are simply related to the mean value and to the correlation functions of the Heisenberg radiative force of the semiclassical approach, where the atomic position is treated classically. They derive in particular a new theoretical expression for the damping force responsible for radiative cooling and they interpret it in terms of linear response theory. The authors also obtain a new crossed r-p derivative term, which does not appear in semiclassical treatments, but which they find to be very small in most situations. Finally, all the theoretical expressions derived are valid for any J g to J e transitions and are not restricted to two-level atoms.
Chapter 11 presents the simplest and most intuitive approach to atom-optical interactions: the atom is modelled as a damped harmonic oscillator with spontaneous emission as the damping agent. The semiclassical viewpoint, in which the atom is treated as a quantum object but the fields are classical, is adopted throughout the chapter. Next, the two-level atom is introduced, which sets the stage for establishing light forces at the atomic level: the dipole-gradient force and the radiation pressure force. The Einstein B coefficient is revisited as stimulated dipole emission. Finally, the optical Bloch equations are used to reformulate more generally the appearance of optical forces and to describe the existence of atom Doppler cooling.
This paper reviews the basic theory of the mechanical action of light in resonant interaction with atoms. At present the main application is laser cooling, but the approach is applicable to a broader range of phenomena. It is based on an adiabatic elimination philosophy, which turns out to give the lowest-order quantum corrections to the behavior found when the photon momentum goes to zero. Hence it is called a semiclassical theory. In this manner a subjective but consistent approach can be presented; other treatments are incorporated or mentioned at the appropriate places. Both the classical and the quantum-mechanical approach are discussed. Those readers who wish to obtain only a heuristic overview of the phenomena can concentrate on Sec. III, which treats both the photon momentum effects and their connection with photon counting statistics. The detailed theoretical treatment utilizes Wigner functions and Fokker-Planck techniques. The ensuing theory is applied both to the cooling of free particles and trapped ones. The paper ends with an extensive bibliography, where the author lists most papers of interest for research into the mechanical manifestations of light. For completeness, many papers are included that are not explicitly mentioned in the text.
The motion of slow atoms with degenerate energy levels in a resonant, nonuniformly polarized laser field is described by the Fokker-Planck equation for the atomic distribution function in phase space in terms of the semiclassical approach. Field gradient expansions are used for the spatially nonuniform coefficients of the equation. For closed atomic transitions J g =J→J e =J+1 (J g and J e are the total angular momenta of the ground and excited states, respectively), new analytical results are presented for the light pressure force and the friction and diffusion coefficients in momentum space. These results allow the kinetic effects (laser cooling, localization in optical potential wells, etc.) in a field of arbitrary one-, two-, or three-dimensional configuration to be investigated. In several cases, the new contributions to the friction coefficient are interpreted qualitatively.
We generalize the method of deriving a kinetic equation for a gas laser developed in the first paper of this series to include radiation-matter correlations. As a consequence, we are able to show that the first Born approximation with asymptotic conditions which contain radiation-matter correlations is sufficient to explain the nonthermal photon distribution observed in photon-counting experiments. Our derivation includes the spatial and velocity effects of the motion of the gas atoms. After adiabatically eliminating the matter variables, we obtain a master equation for the radiation density matrix RRR with nonlinear coefficients. Our radiation master equation for stationary atoms without spatial effects is the same as the generalized Fokker-Planck equation of the Langevin-noise-equation treatment of lasers. We show that the generalized Fokker-Planck equation is the result of the first-Born-approximation treatment of the j\ifmmode\cdot\else\textperiodcentered\fi{}A interaction and is not dependent upon the microscopic structure of the reservoirs. In the first Born approximation, the diagonal and off-diagonal matrix elements of RRR do not interact with each other. Consequently, an initially diagonal radiation density matrix remains diagonal. However, we show that even though the stationary state is diagonal it is necessary to know the propagator of the off-diagonal part of RRR to answer time-dependent questions about quantities, such as the line shape, that depend on phase.
The mechanical action of light on atoms is nowadays a tool used ubiquitously in cold atom physics. In the semiclassical regime where the atomic motion is treated classically, the computation of the mean force acting on a two-level atom requires in the most general case numerical approaches. Here we show that this problem can be tackled in a pure analytical way. We provide an analytical yet simple expression of the mean force that holds in the most general case where the atom is simultaneously exposed to an arbitrary number of lasers with arbitrary intensities, wave vectors, and phases. This yields a novel tool for engineering the mechanical action of light on single atoms.