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Journal of Experimental and Theoretical Physics

Steady state of a low-density ensemble of atoms in a monochromatic field taking into account recoil effects

Прудников Олег Николаевич, R. Ya. Ilenkov, A. V. Taǐchenachev, Anatolii M Tumaikin, Valeriy I. Yudin

Published Jun 1, 2011• 32 citations
Publisher

Abstract

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.

DOI
10.1134/s1063776111050189

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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.
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.
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.
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.
The analytic solution of the steady-state density matrix is presented for a closed two-level atom with an arbitrary ground-state angular momentum Jg{\mathit{J}}_{\mathit{g}}Jg​ and an excited-state angular momentum Je{\mathit{J}}_{\mathit{e}}Je​ interacting with a linearly polarized laser field. From this solution, analytic formulas for the total rate of laser-atom scattering, the rate of coherent scattering, and radiative forces are derived. Interestingly, it is found that population inversion in true atomic states, as opposed to the population inversion in the dressed states, can occur under certain conditions. A method is proposed to take advantage of this inversion for light amplification. The mathematical formulation is built around the concept of the resolvent operator in Liouville space, with some conclusions that are generally applicable to any system with a unique steady state.
We study the laser cooling of atoms with a narrow-line optical transition in monochromatic laser light, i.e. in regimes of quantum nature of laser-light interactions resulting in a significant recoil effect. It is demonstrated that a minimum laser cooling temperature for two-level atom in standing wave reached for red detuning close to 3 recoil frequency, greatly different from the theory used for a semiclassical description of Doppler cooling. A set of dimensionless parameters uniquely characterizing the time evolution and the steady state of different atoms with narrow-line optical transitions in the laser field is introduced. The results can be used for analysis of optimal conditions for laser cooling of atoms with narrow lines such as Ca, Sr, and Mg, which are of great interest for atomic clocks.