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arXiv:1302.0028 · Physical Review A

Kinetics of the disordered Bose gas with collisions

Georg Schwiete, Alexander M. Finkel’stein

Published Nov 11, 2013• 8 citations
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Abstract

We discuss the kinetics of the disordered interacting Bose gas using the Boltzmann transport equation. The theory may serve as a unifying framework for studying questions of dynamics of the expanding Bose gas at different stages of the expansion. We show that the transport theory allows us to straightforwardly reproduce and generalize a number of results previously obtained from microscopic models in different formalisms. Based on estimates for the interparticle scattering rates, we discuss the relevance of interaction effects for the localization problem in the interacting disordered Bose gas. We argue that, if the number of particles is large enough, the size of the expanding cloud may exceed the localization length. We describe the spreading of the wave packet in this regime as collision-induced diffusion and compare the obtained rate of expansion to known results on subdiffusive spreading in nonlinear disordered lattices.

DOI
10.1103/physreva.88.053611

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We discuss the kinetics of the disordered interacting Bose gas using the Boltzmann transport equation. The theory may serve as a unifying framework for studying questions of dynamics of the expanding Bose gas at different stages of the expansion. We show that the transport theory allows us to straightforwardly reproduce and generalize a number of results previously obtained from microscopic models in different formalisms. Based on estimates for the interparticle scattering rates, we discuss the relevance of interaction effects for the localization problem in the interacting disordered Bose gas. We argue that, if the number of particles is large enough, the size of the expanding cloud may exceed the localization length. We describe the spreading of the wave packet in this regime as collision-induced diffusion and compare the obtained rate of expansion to known results on subdiffusive spreading in nonlinear disordered lattices.
We use ultracold atoms in a quasiperiodic lattice to study two outstanding problems in the physics of disordered systems: a) the anomalous diffusion of a wavepacket in the presence of disorder, interactions and noise; b) the transport of a disordered superfluid.a) Our results show that the subdiffusion, observed when interaction alone is present, can be modelled with a nonlinear diffusion equation and the peculiar shape of the expanding density profiles can be connected to the microscopic nonlinear diffusion coefficients.Also when noise alone is present we can describe the observed normal diffusion dynamics by existing microscopic models.In the unexplored regime in which noise and interaction are combined, instead, we observe an anomalous diffusion, that we model with a generalized diffusion equation, where noise-and interaction-induced contributions add each other.b) We find that an instability appearing at relatively large momenta can be employed to locate the fluid-insulator crossover driven by disorder.By investigating the momentum-dependent transport, we observe a sharp crossover from a weakly dissipative regime to a strongly unstable one at a disorder-dependent critical momentum.The set of critical disorder and interaction strengths for which such critical momentum vanishes, can be identified with the separation between a fluid regime and an insulating one and can be related to the predicted zero-temperature superfluid-Bose glass transition.
In this thesis we address relevant problems of the physics of quantum disordered systems from a numerical and theoretical point of view, with specific attention to the connection of our findings with ultracold atomic gases experiments. We concentrate on two main issues: the interplay between localization and interaction in disordered systems and the problem of localization in correlated random potentials. The first problem is investigated considering the expansion of a weakly interacting Bose gas in a bichromtic optical lattice. We observe that interaction has a destructive effect on the disorder-induced localization and leads to a subdiffusive expansion of the atomic gas. By comparing three characteristic energy scales of the system one can identify three different spreading regimes: weak chaos, strong chaos and self-trapping. The spreading behaviour in these regimes is predicted theoretically and verified numerically. We also interpreted existing experimental data on the basis of our findings and showed that there is a qualitative agreement between our numerical simulations and experiments. The second problem is investigated proposing a new model of correlated disorder that can be implemented experimentally using ultracold dipolar gases. We show that this model is characterized by the presence of both short and long range correlations. We study the localization properties of the model and highlight the role played by short and long range correlations in the determination of those properties. In particular we show that when short-range correlations are dominant, extended states can appear in the spectrum. The effect of long-range correlations is instead to restore localization over the whole spectrum and lead to counterintuitive behaviours of the localization length. More precisely, depending on the localization regime they can enhance or reduce the localization length at the centre of the band.
This thesis manuscript is devoted to different aspects of the non-equilibrium dynamics of a moving two-dimensional ultra-cold Bose gas, under the simultaneous action of a spatially disordered potential and interactions between particles. In the absence of interactions, the momentum distribution of the gas is the sum of two well-known components: a classical, diffusive background made of particles multiply scattered by the disorder, and a coherent backscattering interference peak (CBS peak), consequence of the weak localization phenomenon. In the presence of weak interactions, mean field simulations show that these two contributions evolve in time. This evolution is described by two coupled kinetic equations, which we derive using a microscopic quantum transport theory. These predict the destruction of the CBS peak by an effective dephasing mechanism, and the thermalization of the diffusive background. As the interaction strength increases, the effects of disorder become negligible, and the dynamics of the gas resembles that of a superfluid. In this regime, the gas quickly enters a so-called "pre-thermal" state: the spatial correlations g_1 decrease algebraically and propagate along a light cone, in accordance with theoretical predictions. Finally, at long times, our simulations describe the global evolution of the system towards a thermal equilibrium state, characterized by a Kosterlitz-Thouless transition separating a normal fluid state from a superfluid state. Some properties of this phase transition, resulting from the non-equilibrium evolution in the presence of disorder, are discussed.
We consider the phase coherent transport of a quasi-one-dimensional beam of Bose-Einstein condensed particles through a disordered potential of length LLL. Among the possible different types of flow we identified [T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, Phys. Rev. Lett. 98, 210602 (2007)], we focus here on the supersonic stationary regime where Anderson localization exists. We generalize the diffusion formalism of Dorokhov-Mello-Pereyra-Kumar to include interaction effects. It is shown that interactions modify the localization length and also introduce a length scale ${L}^{\ensuremath{\ast}}$ for the disordered region, above which most of the realizations of the random potential lead to time-dependent flows. A Fokker-Planck equation for the probability density of the transmission coefficient that takes this effect into account is introduced and solved. The theoretical predictions are verified numerically for different types of disordered potentials. Experimental scenarios for observing our predictions are discussed.
Abstract. We develop a diagrammatic scattering theory for interacting bosons in a three-dimensional, weakly disordered potential. Based on a microscopic N-body scattering theory, we identify the relevant diagrams including elastic and inelastic collision processes that are sufficient to describe diffusive quantum transport. By taking advantage of the statistical properties of the weak disorder potential, we demonstrate how the N-body dynamics can be reduced to a nonlinear integral equation of Boltzmann type for the single-particle diffusive flux. Our theory reduces to the Gross-Pitaevskii mean field description in the limit where only elastic collisions are taken into account. However, even at weak interaction strength, inelastic collisions lead to energy redistribution between the bosons – initially prepared all at the same single-particle energy – and thereby induce thermalization of the single-particle current. In addition, we include also weak localization effects and determine the coherent corrections to the incoherent transport in terms of the coherent backscattering signal. We find that inelastic collisions lead to an enhancement of the backscattered cone in a narrow spectral window for increasing interaction strength. ar