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arXiv:physics/9708011 · Physical Review A

Quantum and semiclassical calculations of cold-atom collisions in light fields

Kalle‐Antti Suominen, Y. B. Band, Inbal Tuvi, Keith R. Burnett, Paul S. Julienne

Published May 1, 1998• 30 citations
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Abstract

We derive and apply an optical Bloch equation (OBE) model for describing collisions of ground and excited laser cooled alkali atoms in the presence of near-resonant light. Typically these collisions lead to loss of atoms from traps. We compare the results obtained with a quantum mechanical complex potential treatment, semiclassical Landau-Zener models with decay, and a quantum time-dependent Monte Carlo wave packet (MCWP) calculation. We formulate the OBE method in both adiabatic and diabatic representations. We calculate the laser intensity dependence of collision probabilities and find that the adiabatic OBE results agree quantitatively with those of the MCWP calculation, and qualitatively with the semiclassical Landau-Zener model with delayed decay, but that the complex potential method or the traditional Landau-Zener model fail in the saturation limit.

DOI
10.1103/physreva.57.3724

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We derive and apply an optical Bloch equation (OBE) model for describing collisions of ground and excited laser cooled alkali atoms in the presence of near-resonant light. Typically these collisions lead to loss of atoms from traps. We compare the results obtained with a quantum mechanical complex potential treatment, semiclassical Landau-Zener models with decay, and a quantum time-dependent Monte Carlo wave packet (MCWP) calculation. We formulate the OBE method in both adiabatic and diabatic representations. We calculate the laser intensity dependence of collision probabilities and find that the adiabatic OBE results agree quantitatively with those of the MCWP calculation, and qualitatively with the semiclassical Landau-Zener model with delayed decay, but that the complex potential method or the traditional Landau-Zener model fail in the saturation limit.
We apply a fully-quantum-mechanical complex-potential model to calculate the S matrix to test the validity of semiclassical methods for describing collisions of ground and excited laser-cooled Cs, Na, or Li atoms in a magneto-optical trap (MOT). The model includes the role of bound-state resonances in closed channels when the collision energy is smaller than the detuning from the atomic cooling transition. The model also illustrates the factorization of the S matrix into inner and outer parts, as used in simpler semiclassical theories. Our fully quantum results agree with other calculations that demonstrate much smaller rate coefficients for Cs collisions at low temperature than predicted by a semiclassical optical-Bloch-equation treatment. We show that an even simpler semiclassical Landau-Zener model accurately describes the S matrix at MOT temperatures and below for a weak laser intensity. There is no evidence for any significant contribution from the off-resonant excitation that is prominent in local-equilibrium models. The effect of excited-state spontaneous decay during the collision is much less for the light species Li than for the heavy species Cs. Semiclassical models still work well for Li.
We show that in the cold collisions between laser-cooled and trapped atoms the probability to survive in the excited state during the collision can be obtained, in the weak-field limit, from a simple Landau-Zener approach. This is shown by comparing the method with a fully quantum and time-dependent wave-packet treatment. We also show that the optical Bloch equation method is not the proper semiclassical tool to use at low energies, at least in its present form. The unexpectedly low quantum survival probability reported by Boesten et al. [Phys. Rev. A 48, 1428 (1993)] is in fact predicted by the semiclassical Landau-Zener approach. This in turn shows that off-resonant excitation is not important in the low-energy regime.
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.
We present a quantum-mechanical study of the dynamics of collisions of cold atoms in a (magneto-) optical trap valid in the linear intensity regime. We find further evidence for a discrepancy with results of the semiclassical optical-Bloch-equation method at lower collision energies and detunings, referred to as ``quantum suppression'' in previous work. We find a strong dependence on the detuning: for the lowest energies considered the suppression factor ranges from more than three orders of magnitude for \ensuremath{\Delta}=-${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{at}}$ to a factor 2 at \ensuremath{\Delta}=-10${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{at}}$. The fraction of atoms surviving at small interatomic distances for a single l value can in good approximation be described by a simple formula in the form of a product of a Landau-Zener excitation function and a WKB survival factor. Summed over partial waves this formula allows a relatively easy calculation of loss rates in a two-channel linear-intensity description. The role of such an approach in a future more complete treatment including hyperfine channels is pointed out.
We develop an optical-Bloch-equation approach for modeling ultracold-atom collisions in optical traps. The method incorporates a molecular picture of the atomic collision, laser-field dressing of the molecular states participating in the dynamics, and decay of the population and polarization due to spontaneous emission of excited states. The last is important to incorporate because the duration of the cold collisions is longer than the excited-state lifetimes. The relative motion of the atoms during the course of the collision is treated semiclassically with corrections for the time-dependent relative motion of the atoms in the various channels. An application of the method to Cs trap loss due to fine-structure-changing collisions is presented. Good agreement with experiment is obtained.
Inelastic collision rates between two very cold ground-state atoms can be strongly decreased by using an optical shielding laser to repel the colliding atoms at very large internuclear separation so that they never get close to each other. This can be achieved by using a sufficiently intense laser to excite a repulsive state of the quasimolecule formed by the two colliding atoms. We study a simple two-state model and compare the predictions of fully quantum-mechanical close coupling and time-dependent wave packet methods with semiclassical approximations based on the concept of a Landau-Zener curve crossing of field-dressed states. The effect of spontaneous emission in the excited state is treated by the Monte Carlo wave-function method. Calculations are given for a model sodium system at 250 \ensuremath{\mu}K. Strong optical shielding, without significant heating due to the quasimolecular excitation, can be achieved at relatively modest laser power. We find that Landau-Zener models give near quantitative agreement with the fully quantum calculations in spite of the extremely low temperature.