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Quantum Electronics

Itô – Stratonovich dilemma in the problem of laser cooling of atoms: limits of applicability of the semiclassical approximation

Anastasia Kirpichnikova, Прудников Олег Николаевич, А. В. Тайченачев, Valeriy I. Yudin

Published Feb 1, 2022• 4 citations
Publisher

Abstract

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.

DOI
10.1070/qel17980

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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.
Based on the numerical solution of the quantum kinetic equation for the atomic density matrix, which makes it possible to accurately take into account the recoil effects in the interaction of atoms with field photons, we have studied the limits of laser cooling of atoms using closed optical transitions characterised by different recoil parameters (the ratio of the recoil energy to the natural linewidth). It is shown that for optical transitions with an insufficiently small recoil parameter, the polarisation effects, which lead to the possibility of sub-Doppler laser cooling, lose their efficiency and do not ensure an attainment of the temperature below the Doppler limit. The analysis performed allows one to outline the boundaries of the sub-Doppler theory of laser cooling of atoms.
The theoretical description of the kinetics of neutral atoms in the polarized light fields with all the atomic levels, the coherence, the recoil effect is both important and challenging problem. The first step toward understanding mechanisms of interaction between atoms and light was called quasi-classical approach. [1,2] It lies in the fact that the equations for the density matrix can be reduced to the Fokker-Planck equation for the Wigner function in the phase space. But the semiclassical approximation is inapplicable for investigating the cooling of atoms at clock transitions, because the quasiclassical parameter (the recoil frequency) is not small (in comparison with the natural line width). Later quantum methods were developed [3], for example, the secular approach which describes cooling and localization of atoms in the optical potential. Secular approximation fails high vibrational levels and for atoms in high vibrational states.
A method has been developed for obtaining the steady-state solution of a quantum kinetic equation for the atomic density matrix in an arbitrarily polarized monochromatic field with the complete inclusion of recoil effects and degeneracy of atomic levels in the projection of the angular momentum. This method makes it possible to obtain the most general solution beyond the previously accepted approximations (semiclassical approximation, secular approximation, etc.). In particular, it has been shown that the laser cooling temperature is a function of not only the depth of the optical potential (as was previously thought), but also the mass of an atom.
We study laser cooling of atoms with narrow-line optical transition, i.e. in regimes when quantum nature of laser-light interactions results significant effect on atom kinetics. We demonstrate the minimum of laser cooing temperature is reached for the light field detuning close to -3 recoil frequency, i.e. is much differ from standard Doppler cooling theory is used for semiclassical description of laser cooling. We define a set of dimensionless parameters that describe the steady state solution of cold atom distribution in laser light and cooling time. The results can be used for analysis an optimal conditions of laser cooling of atoms with narrow lines Ca, Sr and Mg are of interest for optical time standards.
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.
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.