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Physical Review A

Simple quantum-mechanical picture of cold optical collisions

H. M. J. M. Boesten, BOUDEWIJN J. VERHAAR

Published May 1, 1994• 25 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.49.4240

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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}=-\ensuremathΓat{\mathrm{\ensuremath{\Gamma}}}_{\mathrm{at}}\ensuremathΓat​ to a factor 2 at \ensuremath{\Delta}=-10\ensuremathΓat{\mathrm{\ensuremath{\Gamma}}}_{\mathrm{at}}\ensuremathΓat​. 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 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 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 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.
Measurements of the temperature dependence of trap loss collisions of laser-cooled rubidium atoms are presented. A significant suppression of the rate of collisions between ground- and excited-state atoms is observed for temperatures below $\\sim$70 mu\\mumuK. In such inelastic collisions, the atomic excitation takes place at long range and must survive against spontaneous decay to short range. Our results are consistent with recent predictions for the low temperature suppression of these collisions due to reduced survival of atomic excitation at low collision velocities.^ We report on measurements of trap loss collisions between ultracold Rb atoms which are confined in a magneto-optical trap. Both isotopes sp85\\sp{85}sp85Rb and sp87\\sp{87}sp87Rb have been studied over wide range of trap laser intensities and detunings. The trap loss collision rates exhibit variations over 3 orders of magnitude. We observe significant differences between the isotopes. At large detunings the ability of the trap to recapture products of the hyperfine-changing collisions is significantly diminished. This finding is supported by our numerical simulations of the recapture process. The trap loss rates due to collisions with room temperature background gas have also been measured.^ Using laser light tuned to a repulsive molecular potential, we have been able to suppress inelastic ground-state hyperfine changing collisions between ultracold sp87\\sp{87}sp87Rb atoms. Adiabatic excitation to the repulsive curve alters the atomic trajectories and prevents the atoms from approaching close enough for the hyperfine change to occur. Experimental results show suppressions up to $\\sim$50% and are in reasonable agreement with a simple Landau-Zener model.^ Finally, we have observed a significant cooperative effect in ultracold trap loss collisions induced by two separate lasers. One laser, tuned close to the atomic resonance, excites the atom pair to an attractive potential at long range. The resulting acceleration and deflection give rise to an enhancement in the collisional flux of ground-state atoms reaching short range, as probed by a second laser. Enhancements up to a factor of $\\sim$3 have been observed. ^
We present a coupled-channel study of optical collisions, restricted to a single atom-laser manifold. Outside of a resonance region around the Condon point, we find a simple representation of the solution in terms of propagating complex dressed states. The probability flux for the 2^{2}2P3/2{\mathit{P}}_{3/2}P3/2​+S state at small interatomic distances can be interpreted in terms of two such interfering dressed states. The coupled-channel solution displays some features that have previously been obtained with semiclassical optical-Bloch-equation methods. An important quantum effect, however, is a strong reduction of atom loss rates at low collision energies that is roughly proportional to ${\mathit{E}}^{\mathrm{\ensuremath{-}}2}$.
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.