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arXiv:cond-mat/0507681 · Physical Review A

Thermalization in mixtures of ultracold gases

Marco Anderlini, David Guéry-Odelin

Published Mar 3, 2006• 28 citations
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Abstract

Starting from a set of coupled Boltzmann equations, we investigate the thermalization of a two-species cold atomic gas confined either in a box or in an isotropic harmonic trap. We show that the thermalization times, by contrast to the collision rate, depend on the interferences between scattering partial waves. The dynamics of thermalization in a harmonic trap is also strongly dependent upon the ratio between the collision rate and the trap frequencies.

DOI
10.1103/physreva.73.032706

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Starting from a set of coupled Boltzmann equations, we investigate the thermalization of a two-species cold atomic gas confined either in a box or in an isotropic harmonic trap. We show that the thermalization times, by contrast to the collision rate, depend on the interferences between scattering partial waves. The dynamics of thermalization in a harmonic trap is also strongly dependent upon the ratio between the collision rate and the trap frequencies.
Abstract We report on the dynamics of thermalization by extending a generalization of the Caldeira–Leggett model, developed in the context of cold atomic gases confined in a harmonic trap, to higher dimensions. Universal characteristics en route to thermalization which appear to be independent of dimensionality are highlighted, which additionally suggest a scaling analogous to turbulent mixing in fluid dynamics. We then focus on features dependent on dimensionality, with particular regard to the role of angular momentum of the two atomic clouds, having in mind the goal of efficient thermalization between the two species. Finally, by considering asymmetry in species numbers, we find that nonlinear interspecies interactions provide a mode locking mechanism between the majority and minority species, relevant to recent experiments involving Fermi–Bose mixtures in the normal phase.
We present a theoretical study of the dynamical behavior of a gas made of ultracold fermionic atoms, which during their motions can collide with a much smaller number of thermal bosonic impurities. The atoms are confined inside harmonic traps and the interactions between the two species are treated as due to sss-wave scattering with a negative scattering length modeling the $^{40}\mathrm{K}\penalty1000-\hskip0pt^{87}\mathrm{Rb}$ fermion-boson system. We set the fermions into motion by giving a small shift to their trap center and examine two alternative types of initial conditions, referring to (i) a close-to-equilibrium situation in which the two species are at the same temperature (well below the Fermi temperature and well above the Bose-Einstein condensation temperature); and (ii) a far-from-equilibrium case in which the impurities are given a Boltzmann distribution of momenta while the fermions are at very low temperatures. The dynamics of the gas is evaluated by the numerical solution of the Vlasov-Landau equations for the one-body distribution functions, supported by some analytical results on the collisional properties of a fermion gas. We find that the trapped gaseous mixture is close to the collisionless regime for values of the parameters corresponding to current experiments, but can be driven towards a collisional regime even without increasing the strength of the interactions, either by going over to heavier impurity masses or by matching the width of the momentum distribution of the impurities to the Fermi momentum of the fermion gas.
Two-body inelastic collisions arising from chemical reactions are prevalent in ultracold fermionic and bosonic molecular gases. Although recent advancements have achieved quantum degeneracy in these systems, loss dynamics are typically modeled phenomenologically using rate equations that often assume thermalization during chemical reactions. In this study, we employ the inelastic quantum Boltzmann equation to analyze particle loss, temperature evolution, and momentum distributions in single-component Fermi gases from first principles. Our results demonstrate that the conventional particle-number rate equation accurately describes the dynamics in trapped systems but fails to capture the behavior in homogeneous systems. Notably, under pure p-wave inelastic collisions and zero elastic collisions, we find that systems prepared near or above quantum degeneracy remain in a thermal state, whereas systems initialized deep within degeneracy exhibit nonequilibrium dynamics. Our theoretical predictions align well with recent experimental observations in trapped systems, and our claim can be further verified in atomic systems with induced two-body loss in box potentials.
This article is devoted to the study of two-dimensional Bose gases harmonically confined. We first summarize their equilibrium properties. For such a gas above the critical temperature, we also derive the frequencies and the damping of the collective oscillations and we investigate its expansion after releasing of the trap. The method is well suited to study the collisional effects taking place in the system and in particular to discuss the crossover between the hydrodynamic and the collisionless regimes. We establish the link between the relaxation times relevant for the damping of the collective oscillations and for the time-of-flight expansion. We also evaluate the collision rate and its relationship with the relaxation time.
The collision cross section that controls thermalization of gas mixtures is the transport cross section ${\ensuremath{\sigma}}_{\ensuremath{\eta}}^{(1)}$ and not the elastic cross section ${\ensuremath{\sigma}}_{\mathrm{el}}$. The two are the same for pure sss-wave scattering but not when higher partial waves contribute. We investigate the differences between them for prototype atomic mixtures and show that the distinction is important at energies above $100\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{K}$ for LiYb and $3\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{K}$ for RbYb and RbCs. For simple systems, both ${\ensuremath{\sigma}}_{\ensuremath{\eta}}^{(1)}$ and ${\ensuremath{\sigma}}_{\mathrm{el}}$ follow universal energy dependence that depends only on the sss-wave scattering length when expressed in reduced length and energy units.
The linearized time-dependent coupled Gross-Pitaevskii equations describing the long-wavelength excitations of Bose-Einstein condensed binary mixtures are solved in harmonic traps for the case where only the binary phase is present. The dispersion law contains two branches whose dependence on the quantum numbers of the modes is the same as for a one-component condensate, but with different prefactors, depending on the ratios of the three sss-wave scattering lengths of the two atomic species. In the general case where the binary phase in the trap coexists with one or both one-component phases, the mode spectrum depends on the geometry of the interphase boundaries due to the boundary conditions there. Measurements of the oscillation frequencies as in recent experiments with modulated traps would yield very detailed information on this system.