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Physical Review A

Complex-potential model of collisions of laser-cooled atoms

Paul S. Julienne, Kalle‐Antti Suominen, Y. B. Band

Published May 1, 1994• 35 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.49.3890

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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.
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 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.
This paper examines two collisional mechanisms by which cold alkali-metal atoms can escape from a neutral-atom trap, due to collison of excited- and ground-state atoms. One is fine-structure-changing collisions, by which the atoms are heated by the amount of the fine-structure splitting. The other produces hot ground-state atoms following emission of a red-shifted photon during the course of a collision. A rate expression is obtained that applies to both normal and ultracold temperatures (1 mK). This expression assumes a canonical distribution of initial states, low-intensity excitation, and a semiclassical treatment of the survival probability relative to excited-state decay during the long time of the ultracold collision, and uses fine-structure-changing probabilities found by quantum scattering calculations. The known properties of the attractive molecular states of the alkaline-metal dimers are used to identify and calculate the probabilities for the specific mechanisms. We conclude that the earlier semiclassical analysis by Dashevskaya [Opt. Spectrosk. 46, 423 (1979)] of the fine-structure-changing mechanisms for the various alkali-metal species is qualitatively correct. We present quantum-mechanical calculations of the rate coefficient for fine-structure-changing collisions between ground- and excited-state Cs atoms from 1000 K to 100 \ensuremath{\mu}K. Collision-rate coefficients for the trap-loss processes are calculated for pairs of Li, Na, K, Rb, and Cs atoms at low temperature. There is a wide variation of predicted loss rate among alkali-metal species from the fastest for K and the slowest for Li. Retardation corrections to molecular lifetimes must be taken into account to predict the correct rate coefficients for Na, K, and Rb. Good agreement is obtained with the observed trap loss rate in a Cs trap.
We report new calculations of two-body associative ionization collisions between sodium atoms confined in a magneto-optic trap at ultracold temperatures. These collisions represent a new kind of {open_quotes}open{close_quotes} or dissipative collision for which the energy of the atom + applied light field need not be conserved due to spontaneous emission coupling to the vacuum modes of the radiation field and require density matrix methods rather than wavefunction methods to describe the collision. In these collisions the Na atoms must absorb a photon when they are in close enough proximity that a significant fraction of the excited population survives relaxation back to the ground state. The Na atoms accelerate towards each other on the excited molecular state and absorbs a second photon to a doubly excited state which then proceeds to associatively ionize at close range. Our calculations incorporate a molecular picture of the atomic collision, laser optical-field dressing of the molecular states participating in the dynamics, and population and coherence decay due to spontaneous emission. The effect of excited state hyperfine structure, which plays a critical role in the dynamics, is simulated by simplified models with a reduced number of states based on the full set of molecular-hyperfine adiabatic potentialmore » curves. Comparison is made with recent measurements of photo-associative ionization. rates as a function of optical field intensity from about 20 to 130 mW cm{sup -2}.« less
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.