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Annual Meeting Optical Society of America

Laser cooling theory of multistate atoms

Juha Javanainen

Published Jan 1, 1990• 0 citations
Publisher

Abstract

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.

DOI
10.1364/oam.1990.ws6

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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 describe a numerical implementation of our semiclassical laser-cooling theory [J. Javanainen, Phys. Rev. A 44, 5857 (1991)] for an arbitrary multilevel atom and light field. There is satisfactory agreement between experiments and temperatures obtained from Langevin-equation simulations of the motion of an atom subject to light-pressure force and the accompanying diffusion in three-dimensional (3D) optical molasses in which there are three orthogonal pairs of counterpropagating waves with a common linear polarization for the two beams in a pair, but orthogonal polarizations between the pairs. However, the velocity distribution of the atoms is anisotropic and may deviate strongly from a Gaussian, heterodyne spectra may have a sideband, and the atomic density may be a maximum at the maximum of the potential of the dipole forces of the molasses beams. Subsequent velocity-linearized analysis of atomic damping and diffusion shows that the damping tensor may have a negative eigenvalue corresponding to heating in large regions of space, even though the light is red tuned for cooling. It is also found that the position-averaged damping and diffusion tensors make a poor predictor of the temperature obtained in Langevin-equation simulations and real experiments. Based on additional simulations of a simple 1D model, we formulate a hypothesis that connects our findings. The key idea is that an atom trapped in a minimum of the potential of the dipole forces is subject to a weaker-than-average damping, and therefore tends to boil out of the trap.
We derive a Fokker-Planck equation (FPE) for the center-of-mass Wigner function of an atom with an arbitrary level structure, moving in a given three-dimensional light field. This FPE is obtained by eliminating adiabatically the internal degrees of freedom from the expansion of the equation of motion of the atomic density operator into a power series of photon momentum. The force and the diffusion tensor in the FPE may be computed by solving a set of differential equations closely related to the time-evolution equations of the density matrix for the internal state of the atom, as encountered in the conventional recoilless spectroscopy. By studying spontaneous, induced, and free-evolution terms in the equations of motion of the density operator for various level configurations, we devise algorithms that enable a computer to build all the required equations automatically for an arbitrary atomic level scheme.
We describe a theoretical approach to laser cooling with broadband real Gaussian laser fields that is based on adiabatic elimination methods and the Fokker-Planck equation for the Wigner function. This approach is used to model the recent broadband cooling experiment of Zhu, Oates, and Hall [Phys. Rev. Lett. 67, 46 (1991)] and gives correct order-of-magnitude results. The theory is then applied to J=0\ensuremath{\rightarrow}J=1 and J=1\ensuremath{\rightarrow}J=2 transitions in a ${\mathrm{\ensuremath{\sigma}}}_{+\mathrm{\ensuremath{-}}}$${\mathrm{\ensuremath{\sigma}}}_{\mathrm{\ensuremath{-}}}$ optical molasses configuration with correlated broadband laser fields. An instability in the cooling scheme at zero velocity is found for red laser detunings, and sub-Doppler-width velocity distributions are predicted for J=1\ensuremath{\rightarrow}J=2 transitions with both red and blue laser detunings.
We adapt an earlier semiclassical theory of laser cooling of an arbitrary multistate atom [J. Javanainen, Phys. Rev. A 44, 5857 (1991)] to the limit of low light intensity. The formal theory is implemented analytically on a computer using mathematica. Expressions of light-pressure force and diffusion are provided for j1{\mathit{j}}_{1}j1​=1/2\ensuremath{\rightarrow}j2{\mathit{j}}_{2}j2​=3/2 and j1{\mathit{j}}_{1}j1​=1\ensuremath{\rightarrow}j2{\mathit{j}}_{2}j2​=2 atoms in a one-dimensional optical confinement area in which the pair of counterpropagating waves has orthogonal polarizations. In our models the cooling temperature decreases as the level degeneracy increases.
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 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.