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Journal of the Optical Society of America B

Wigner-function approach to laser cooling in the recoil limit

Sung Mi Yoo, Juha Javanainen

Published Jun 1, 1991• 13 citations
Publisher

Abstract

We derive an equation of motion for the center-of-mass Wigner function (WF) of a two-level atom interacting with a linearly polarized, monochromatic standing plane wave of light. The derivation is based on a truncation of the spatial Fourier expansion of the WF and is consequently valid only at low intensities of the driving light, but no assumption regarding the size of the recoil velocity need be made. The stationary state of laser cooling is studied analytically and numerically. The asymptotic velocity distribution at large velocities is found to obey a power law, with a power independent of the linewidth of the optical transition.

DOI
10.1364/josab.8.001341

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We derive an equation of motion for the center-of-mass Wigner function (WF) of a two-level atom interacting with a linearly polarized, monochromatic standing plane wave of light. The derivation is based on a truncation of the spatial Fourier expansion of the WF and is consequently valid only at low intensities of the driving light, but no assumption regarding the size of the recoil velocity need be made. The stationary state of laser cooling is studied analytically and numerically. The asymptotic velocity distribution at large velocities is found to obey a power law, with a power independent of the linewidth of the optical transition.
We derive a Fokker-Planck equation for the center-of-mass Wigner function of an atom moving in a Zeeman neutral-atom trap. The Zeeman structure of atomic sublevels is taken into account. Our analysis is limited to optical transitions between two sets of sublevels corresponding to angular momenta jb{\mathit{j}}_{\mathit{b}}jb​=0 and ja{\mathit{j}}_{\mathit{a}}ja​=1. Expressions for the light pressure, friction, and diffusion forces in the case of low intensity of the laser light are presented. An analytical solution for the stationary center-of-mass Wigner function as well as the size and geometry of the trap are found and discussed.
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.
We give a general theoretical description of radiative forces on atoms in a monochromatic radiation field. The driven transition has a lower level consisting of several substates, and we consider the limit of low velocities and weak intensities. This situation comprises the schemes where cooling to temperatures below the Doppler limit is possible. After expressing the atomic dipole in terms of the polarizability tensor, we obtain an expression of the force as the sum of the radiation pressure, the dipole force, and a contribution from the gradient of the polarization. This latter contribution contains a part resulting from redistribution of photons between the plane waves that compose the field, and a part resulting from fluorescence. We express the average force and the momentum-diffusion tensor in terms of a closed evolution equation for the lower-state density matrix. We show by a scaling argument that in any case of cooling, the final temperature goes down linearly with the intensity, until the recoil limit is approached. The description is valid for a weak and monochromatic, but otherwise arbitrary field, and it allows for the presence of an external magnetic field, and of hyperfine splitting. It is applicable to cooling in one, two, or three dimensions. We give numerical results for the average force in a number of specific cases of two counterrunning plane waves.
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.
The setting time of the stationary distribution over translational degrees of freedom of two-level atoms in the field of a one-dimensional standing light wave is studied. The dependences of this time on the problem parameters such as the light wave intensity, frequency detuning, and atom mass are obtained. Calculations are performed on the basis of the quantum-mechanical equation for the atomic density matrix taking completely into account the recoil and spatial localization effects in an arbitrarily intense light field.