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Journal of the Optical Society of America B

Effect of diffusion on laser cooling of atoms with long-range collisions

Andrew M. Smith II, Keith R. Burnett

Published Aug 1, 1992• 5 citations
Publisher

Abstract

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.

DOI
10.1364/josab.9.001240

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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.
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.
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.
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.
The purpose of our master thesis is to present an original synthesis of various works dealing with the Monte-Carlo wavefunction (MCWF) method in quantum optics and its applications to laser cooling of atoms with a particular focus on Doppler and Sisyphus cooling mechanisms. We introduce the MCWF method which is used to solve master equations describing the dynamics of open quantum systems in quantum optics. This method turns out to be much more efficient than a method based on the full integration of the density matrix. We perform laser cooling simulations in 1D both with a classical and a quantum treatment of the atomic centre-of-mass motion and investigate Doppler and Sisyphus cooling. Our numerical results turn out to be in very good agreement with those predicted by semiclassical theories, except for large detuning and/or low intensity laser where semiclassical theories fail. For such cases, our method provides a good alternative and predicts temperatures much larger than the recoil temperature. Finally, we also consider a classical treatment of the centre-of-mass motion and we compare the subsequent results with those obtained with a full quantum approach.
We give a full quantum treatment of the radiative heating mechanism which arises in atomic clouds at low temperatures due to the collisions in the presence of a light field. The time scale of these collisions is comparable to the time scale of spontaneous decay, hence one needs to combine excitation and decay processes with the quantum motion of the colliding atoms into a single model, which we have done. Our study produces a statistical approximate to the full density matrix by combining wave-packet dynamics with the Monte Carlo state-vector method. We also present a quantitative semiclassical treatment based on the Landau-Zener model, and show that it agrees with the quantal results in certain parameter regimes. However, for low temperatures and strong laser fields the full quantum treatment is needed. We describe our numerical procedures and the statistical aspects of the Monte Carlo state-vector method in detail. Our main result is that the low-energy collisions can produce appreciable heating of the cloud.