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Princeton University Press eBooks

5. Density Matrix Equations: Atomic Center-of-Mass Motion, Elementary Atom Optics, and Laser Cooling

P. R. Berman, Vladimir S. Malinovsky

Published Dec 31, 2011• 0 citations
Publisher
DOI
10.1515/9781400837045-006

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We present a calculation of one-dimensional sub-Doppler cooling of neutral atoms, in which the atomic center-of-mass motion is quantized. The cooling field consists of a pair of counterpropagating, linearly polarized laser beams, with their polarizaton vectors at an angle \ensuremath{\theta} with respect to each other. The steady-state atomic density matrix is analyzed as a function of \ensuremath{\theta}. It is shown that, depending on the internal atomic level scheme, the equilibrium population of the quantized ground state of atomic center-of-mass motion can either increase or decrease significantly when \ensuremath{\theta} is reduced from \ensuremath{\pi}/2. These effects are interpreted based on the spatial localization of atoms in the light-induced potential wells.
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.