We theoretically study the influence of weak interactions on the diffusive expansion of a Bose-Einstein condensate in a three-dimensional random potential. For this purpose we develop a perturbative approach and calculate analytically the first-order nonlinear correction to the ensemble-averaged atomic density as a function of position and time. We find that interactions renormalize the typical diffusion coefficient of the condensate. The magnitude of the nonlinear correction is controlled by a single dimensionless parameter that includes both interaction and disorder strengths.
We theoretically study the influence of weak interactions on the diffusive expansion of a Bose-Einstein condensate in a three-dimensional random potential. For this purpose we develop a perturbative approach and calculate analytically the first-order nonlinear correction to the ensemble-averaged atomic density as a function of position and time. We find that interactions renormalize the typical diffusion coefficient of the condensate. The magnitude of the nonlinear correction is controlled by a single dimensionless parameter that includes both interaction and disorder strengths.
Expansion of a Bose-Einstein condensate (BEC) is studied in the presence of a random potential. The expansion is controlled by a single parameter, (microtau(eff)/variant Planck's over 2pi), where micro is the chemical potential, prior to the release of the BEC from the trap, and tau(eff) is a transport relaxation time which characterizes the strength of the disorder. Repulsive interactions (nonlinearity) facilitate transport and can lead to diffusive spreading of the condensate which, in the absence of interactions, would have remained localized in the vicinity of its initial location.
Abstract We study the diffusion of an expanding Bose–Einstein condensate released from a harmonic trap in a three-dimensional speckle disorder potential. To this end, we use the first Born approximation and examine the density profiles at short and long times. Analytical results are presented in different regimes. The evolution of the density profiles in space and time is deeply examined. We find that at long times and for a fixed disorder strength, the profile of the average atomic density decreases in power law. The time evolution of the typical size of the condensate is explored numerically.
We describe repulsively interacting Bose-Einstein condensates in spatially correlated disorder potentials of arbitrary dimension. The first effect of disorder is to deform the mean-field condensate. The second is that the quantum excitation spectrum and condensate population are affected. By a saddle-point expansion of the many-body Hamiltonian around the deformed mean-field ground state, we derive the fundamental quadratic Hamiltonian of quantum fluctuations. Importantly, a basis is used such that excitations are orthogonal to the deformed condensate. Via Bogoliubov-Nambu perturbation theory, we compute the effective excitation dispersion, including mean free paths and localization lengths. Corrections to the speed of sound and average density of states are calculated, due to correlated disorder in arbitrary dimensions, extending to the case of weak lattice potentials.
Weakly interacting Bose gases in a disorder environment have long been a challenging topic in the field of solid-state physics due to the intriguing interplay between superfluidity and localization. In this thesis we investigate theoretically the equilibrium properties of one- and three-dimensional harmonically trapped dirty Bose-Einstein condensates (BECs). In particular we focus on the decomposition of the total particle density into three components, namely, the condensate density, the thermal density, and the density of fragmented local Bose-Einstein condensates within the respective minima of the random potential landscape. In order to determine the different density components we develop a Hartree-Fock mean-field theory for a weakly interacting trapped BEC in a random environment, then specialize it to contact interaction and delta-correlated disorder. This non-perturbative theory is worked out on the basis of the replica method, which represents a well- established tool to deal with disorder problems. With this we calculate the corresponding free energy at finite temperature and derive from it the underlying self-consistency equations for the respective density components. Then, as a first step, we investigate a quasi one-dimensional Bose-Einstein condensed gas in a harmonic trapping potential with an additional delta- correlated disorder potential at zero temperature. To this end we solve the self-consistency equations within the Thomas-Fermi approximation and find the emergence of a Bose-glass region, where the condensate vanishes. We corroborate this analysis by an elaborate numerical treatment, where the corresponding one-dimensional time-independent Gross-Pitaevskii equation is numerically solved and disorder-ensemble averages of the condensate wave function are performed. The variance of the condensate wave function quantifies the number of bosons which condense in the local minima of the random potential. For weak disorder these mini-condensates turn out to occur preferentially at the border of the condensate, while for intermediate disorder strength this happens in the trap center. Additionally we use a variational ansatz in order to describe analytically the numerically observed redistribution of the fragmented mini-condensates with increasing disorder strength. In close analogy to the one-dimensional case we then treat the three-dimensional dirty Bose gas in an isotropic harmonic trap and a delta- correlated disorder potential at zero temperature using the corresponding self-consistency equations obtained via the Hartree-Fock mean-field theory within the Thomas-Fermi approximation. Additionally we use a variational ansatz, whose results turn out to coincide qualitatively with those obtained from the Thomas-Fermi approximation. Inparticular, a first order quantum phase transition from the superfluid phase, where the condensate density contributes to the total density, to the Bose-glass phase, where all particles are in the mini-condensates, is detected at a critical disorder strength, which agrees with findings in the literature. Furthermore, in a generaltriaxial harmonic trap, we investigate the geometric effect of different trap aspect ratios on the respective properties of the dirty BEC system. Finally, we consider the three- dimensional dirty BEC at finite temperature. This allows us to study the impact of both temperature and disorder fluctuations on the respective components of the density as well as their Thomas-Fermi radii. In particular, we find that the superfluid region, the Bose-glass region, and the thermal region coexist. Furthermore, depending on the system parameters, three phase transitions are detected, namely, one from the superfluid to the Bose-glass phase, one from the Boseglass to the thermal phase, and finally one directly from the superfluid to the thermal phase. All these results could be particularly useful for a quantitative analysis of world-wide on-going experiments with dirty bosons in quasi one-dimensional and three-dimensional harmonic traps.
We present a detailed numerical study of the effect of a disordered potential on a confined one-dimensional Bose–Einstein condensate, in the framework of a mean-field description, using a highly efficient and fast converging numerical scheme. For repulsive interactions, we consider the Thomas–Fermi and Gaussian limits and for attractive interactions the behaviour of soliton solutions. We find that the average spatial extension of the stationary density profile decreases with an increasing disorder strength both for repulsive and attractive interactions among bosons. In the Thomas–Fermi limit, a strong localization of the bosons is obtained in momentum space around the state k = 0. The time-dependent density differs considerably in the cases we have considered. For attractive and disordered Bose–Einstein condensates, we show evidence of a bright soliton with an overall unchanged shape, but a disorder-dependent width. For weak disorder, the soliton is delocalized and for stronger disorder, it bounces back and forth between high potential barriers.