PHYSICAL REVIEW A VOLUME 49, NUMBER 5 MAY 1994 Complex-potential model of collisions of laser-cooled atoms Paul S. Julienne Molecular Physics Division, National Institute of Standards and Technology, Gaithersburg, Maryland M899 Kalle-Antti Suominen Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OXI 3PU, United Kingdom Yehuda Band Departments of Chemistry and Physics, Ben Gurion University, Beer Sheva 8)108, Israel (Received 16 December 1993) 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 oK-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. PACS number(s): 32.80. Pj, 42. 50.Vk, 03.65. w— I. INTRODUCTION Collisional processes which produce hot atoms and cause loss of laser-cooled atoms from a magneto-optical trap (MOT) have been observed for a number of trapped alkali-metal species [1 — 7), and semiclassical theories that neglect hyper6ne structure have been advanced by Gal- lagher and Pritchard (GP) [8], Julienne and Vigue (JV) [9], and Band and Julienne (BJ) [10] to account for the measured trap-loss rate coeKcients, with typical agree- ment within a factor of 2 or 3 for Gs [1], Na [2, 5], and Li [6,7] MOTs, and larger discrepancies for Rb [3,4]. These models all take into account the profound efkct on the collision of excited-state spontaneous radiative decay during the very long interaction time. A recent quantum- mechanical calculation based on a time-independent cora- plex potential method indicated that these semiclassical models might be seriously wrong at low T due to a quan- tum suppression effect [ll]. All of these models assume a factorization of the probability P(E, l, 4, I;p) for the overall trap-loss event for collisions with energy E, rel- ative angular momentum l, laser power I, detuning A from atomic resonance, and excited state decay rate p: P(E l 4 I &) = Px(E l) J(E l 4 I; 7). The rate coefficient for trap-loss events has the form K(E, E, I) = ) (2l+ l)P( E/, b, , I;p), 1,=0 where E = 5 k2/2p, is the relative collision energy for re- duced mass p and v = hk/p is the relative velocity. The colliding ground-state atoms in their molecular ground state g are excited to a quasimolecular state a with an attractive potential V (A) at very large internuclear sep- aration B and reach the small B region on state a with a probability J(E, l, 6, I; p) [12], where a process occurs with probability Px(E, t) that produces the hot atoms that escape the trap. This process, the probability of which is independent of 4, I, and p, may involve ei- ther a change of fine-structure state (X=FS), releasing the amount of the P3~z — Pqy2 Gne-structure splitting, or a radiative escape (X=RE) process by which the excited atoms decay to produce fast ground-state atoms [8,9]. The unique features of ultracold collision physics are con- tained in the excitation-survival factor J(E, l, A, I;p), which very strongly depends on the magnitude of p due to excited-state decay during the collision. We have set up a simple quantum close-coupling calcu- lation with a complex potential to test the factorization assumption (1) and to test the various semiclassical mod- els for J. Our scattering model also includes the e8'ect of bound-state resonances in the excited-state potential. A quantum treatment of such resonances was not given in pre~ious trap-loss models [8 — ll]; Ref. [11] calculated Aux in one direction only, neglecting reBection &om inner turning points. We will show here that the factorization assumption is an excellent approximation for T ( 1 mK and that the simple Landau-Zener model given in Ref. [10] provides a much better approximation at MOT tem- 1050-2947/94/49(5)/3890(7)/$06. 00 49 3890 1994 The American Physical Society