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Journal of the Optical Society of America B

Analytical study of long-range collisions between two atoms undergoing laser cooling

Andrew M. Smith II, Keith R. Burnett

Published Aug 1, 1992• 12 citations
Publisher

Abstract

An analytical description of the collisions between atoms in a laser cooling field is developed. The results confirm predictions made in our recent numerical simulations for the same model. The effect of the collision can be separated analytically, to first order in a perturbation expansion in collision strength, into two parts. The first of these parts comes from the interruption of the laser cooling cycle in each atom, and the second from the direct force resulting from a photon exchange between the atoms.

DOI
10.1364/josab.9.001256

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An analytical description of the collisions between atoms in a laser cooling field is developed. The results confirm predictions made in our recent numerical simulations for the same model. The effect of the collision can be separated analytically, to first order in a perturbation expansion in collision strength, into two parts. The first of these parts comes from the interruption of the laser cooling cycle in each atom, and the second from the direct force resulting from a photon exchange between the atoms.
An analytical description of long-range collisions between atoms in a laser cooling field is developed. We begin by considering an N-atom master equation. In the regime of low atomic densities (i.e., where the mean distance between two atoms is much larger than the laser wavelength) it is possible to treat the atom-atom interactions in perturbation theory. Furthermore we assume temperatures which allow a semiclassical treatment of the cooling process. The effect of the presence of other atoms can be separated analytically into two parts; an attenuation force due to the absorption of the laser beams in the atomic cloud similar to the results of Dalibard [Opt. Commun. 68, 203 (1988)], which tends to compress the atomic cloud, and a two-atom force due to photon emission and absorption cycles between different atoms. This force proves to be repulsive for the configurations studied and prevents the cloud from collapsing. The result for the first-order perturbation expansion in collision strength generalizes the model proposed by Walker, Sesko, and Wieman [J. Opt. Soc. B 8, 946 (1991)] by including additional terms, such as those associated with Raman couplings.
A theory to describe collisions between pairs of atoms that are being cooled by a laser field is presented. The derivation uses the Heisenberg picture and results in a set of coupled optical Bloch equations and associated equations for the translational motion. By defining a correlation variable between the two atoms, we obtain a system of linear equations that may be solved numerically. If the interaction is ignored, the usual cooling curve is obtained: In the full simulation it is found that long-range collisions (0.1ƛ → 10ƛ) can significantly alter the atomic velocities. Depending on how the atoms approach each other, there are various effects: jumps in velocity at the center of the collision, interruptions of the cooling cycle, and signs of collective effects for long-range collisions. If the atom density is sufficiently large and the atoms are moving slowly, these collisional mechanisms may prevent any further cooling.
An extension to a previous model [ J. Opt. Soc. Am. B8, 1594 ( 1991)], which described the collision between two atoms in a cooling laser field, is made, which allows diffusive effects to be analyzed. A method similar to that used in Monte Carlo studies is developed, except that the diffusion is now determined exactly at all times. This approach relies directly on making a local classical path approximation for the atoms’ trajectories. For the simple case of a single two-level atom in the cooling field, the energy of the atom reaches equilibrium at the Doppler limit, as one would expect. In the full collisional situation we make the assumption that each atom undergoes a series of binary collisions. For sufficiently high atomic densities these collisions provide a limit on the lowest temperature that may be achieved by the cooling process.
A calculation of the collisional diffusion coefficient is made which estimates the effect of collisions in an optical trap. The theory is based on the method of Gordon and Ashkin [Phys. Rev. A 21, 1606 (1980)] and incorporates the equations derived by Smith and Burnett [J. Opt. Soc. Am. B 8, 1592 (1991)] to describe the interaction of two two-level atoms in a laser field. Results are obtained by using a nearest-neighbor model for the collisions and suggest a definite relationship between atomic density and collisional diffusion.
The bichromatic force allows laser cooling using stimulated processes only. We provide details of our simulations of the bichromatic force when the effects of spontaneous emission are suppressed by restricting the atom-light interaction time to be short enough that there cannot be a significant number of spontaneous emission events. This short interaction time requires that the simulation include dynamics of the atomic motion through the light field that is coupled to the internal state dynamics that determine the force on the atom. The simulation is first tested with several standard optical field configurations to confirm its accuracy. Then the results, using conditions that match our experiment, are presented and compared with our measurements. The simulation shows that the resolution of the experiment greatly obscures the degree of cooling. Our simulation predicts an observed velocity distribution reduced by up to a factor of 4 in width over a time comparable to the excited state lifetime. This technique can allow the direct laser cooling of atoms and molecules without closed cycling transitions.