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Journal of Physics B Atomic Molecular and Optical Physics

The optimum Fokker-Planck equation for laser cooling

Klaus Mølmer

Published May 14, 1994• 11 citations
Publisher

Abstract

The momentum distribution of laser cooled atoms may be identified approximately by a semiclassical treatment. Among different possibilities we present empirical and analytical arguments for choosing ( delta / delta t) omega (p,t)=$ -( delta / delta p)(F(p) omega (p,t))+( delta / delta p)(D(p)( delta / delta p) omega (p,t)) as the optimum Fokker-Planck equation for laser cooling, where the mean force and the momentum diffusion coefficients are determined from the solutions of the internal state atomic dynamics.

DOI
10.1088/0953-4075/27/9/024

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The momentum distribution of laser cooled atoms may be identified approximately by a semiclassical treatment. Among different possibilities we present empirical and analytical arguments for choosing ( delta / delta t) omega (p,t)=$ -( delta / delta p)(F(p) omega (p,t))+( delta / delta p)(D(p)( delta / delta p) omega (p,t)) as the optimum Fokker-Planck equation for laser cooling, where the mean force and the momentum diffusion coefficients are determined from the solutions of the internal state atomic dynamics.
Abstract We report a comparative analysis of the limits of laser cooling of atoms in light fields both on the basis of a numerical solution of the quantum kinetic equation for the atomic density matrix, which makes it possible to accurately take into account quantum recoil effects during the interaction of atoms with field photons, and within the framework of the semiclassical approach using various forms of the Fokker – Plank equation. The analysis allows one to outline the limits of applicability of the semiclassical approach, as well as to choose an unambiguous form for introducing the diffusion contribution to the Fokker – Planck equation in the semiclassical description of atomic kinetics.
In the limit of asymptotically small photon momentum, a Fokker–Planck equation for the center-of-mass Wigner function suffices to describe laser cooling of an atom. We have devised a procedure to compute the light pressure force and the diffusion tensor for an atom with an arbitrary level structure, traveling with an arbitrary velocity in any given three-dimensional light field. The scheme is based on solving a set of differential equations closely related to the density matrix equations of motion encountered in conventional recoilless laser spectroscopy. Computer programs have been written that automatically construct and solve the required equations from broad specifications of the problem, such as energies and angular momenta of the atomic levels. A Langevin equation stochastic simulation has been implemented to solve the Fokker-Planck equation of laser cooling using the computed force and diffusion. Comparisons of optical-molasses experiments and simulations carried out for a model sodium atom with a J = 2→J = 3 transition show qualitative similarities such as sub-Doppler limit temperatures, but the quantitative agreement of theory and experiment is not satisfactory.
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 semiclassical dynamics of atoms, when the atoms are confined inside a standing-wave high-finesse resonator, is theoretically studied. The atoms are cooled by scattering processes in which the photons of a transverse laser are coherently scattered into the cavity mode. We derive a Fokker-Planck equation for the atomic center-of-mass variables which allows us to determine the equations of motion in the semiclassical limit for any value of the intensity of the laser field. We extract its prediction for the dynamics when the resonator is essentially in the vacuum state and the atoms are cooled by scattering photons into the cavity mode, which then decays. Its predictions for the stationary atomic distribution are compared with the ones of the Fokker-Planck equation by Domokos et al. [J. Phys. B 34, 187 (2001)], which has been derived under different assumptions. We find full agreement in the considered parameter regime.
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.