A Hartree–Fock mean-field theory of a weakly interacting Bose-gas in a quenched white noise disorder potential is presented. A direct continuous transition from the normal gas to a localized Bose-glass phase is found which has localized short-lived excitations with a gapless density of states and vanishing superfluid density. The critical temperature of this transition is as for an ideal gas undergoing Bose–Einstein condensation. Increasing the particle-number density a first-order transition from the localized state to a superfluid phase perturbed by disorder is found. At intermediate number densities both phases can coexist.
A Hartree–Fock mean-field theory of a weakly interacting Bose-gas in a quenched white noise disorder potential is presented. A direct continuous transition from the normal gas to a localized Bose-glass phase is found which has localized short-lived excitations with a gapless density of states and vanishing superfluid density. The critical temperature of this transition is as for an ideal gas undergoing Bose–Einstein condensation. Increasing the particle-number density a first-order transition from the localized state to a superfluid phase perturbed by disorder is found. At intermediate number densities both phases can coexist.
The properties of systems with Bose-Einstein condensate in external time-independent random potentials are investigated in the frame of a self-consistent stochastic mean-field approximation. General considerations are presented, which are valid for finite temperatures, arbitrary strengths of the interaction potential, and for arbitrarily strong disorder potentials. The special case of a spatially uncorrelated random field is then treated in more detail. It is shown that the system consists of three components, condensed particles, uncondensed particles, and a glassy density fraction, but that the pure Bose glass phase with only a glassy density does not appear. The theory predicts a first-order phase transition for increasing disorder parameter, where the condensate fraction and the superfluid fraction simultaneously jump to zero. The influence of disorder on the ground-state energy, the stability conditions, the compressibility, the structure factor, and the sound velocity are analyzed. The uniform ideal condensed gas is shown to be always stochastically unstable, in the sense that an infinitesimally weak disorder destroys the Bose-Einstein condensate, returning the system to the normal state; but the uniform Bose-condensed system with finite repulsive interactions becomes stochastically stable and exists in a finite interval of the disorder parameter.
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 investigate the thermodynamic properties of a dilute Bose gas in a correlated random potential using exact path integral Monte Carlo methods. The study is carried out in continuous space and disorder is produced in the simulations by a three-dimensional (3D) speckle pattern with tunable intensity and correlation length. We calculate the shift of the superfluid transition temperature due to disorder and we highlight the role of quantum localization by comparing the critical chemical potential with the classical percolation threshold. The equation of state of the gas is determined in the regime of strong disorder, where superfluidity is suppressed and the normal phase exists down to very low temperatures. We find a T 2 dependence of the energy in agreement with the expected behavior in the Bose glass phase. We also discuss the major role played by the disorder correlation length and we make contact with a Hartree–Fock mean-field approach that holds if the correlation length is very large. The density profiles are analyzed as a function of temperature and interaction strength. Effects of localization and the depletion of the order parameter are emphasized in the comparison between local condensate and total density. At very low temperature, we find that the energy and the particle distribution of the gas are very well described by the T =0 Gross–Pitaevskii theory, even in the regime of very strong disorder.
Abstract We study a three-dimensional Bose–Einstein condensate in an isotropic harmonic trapping potential with an additional delta-correlated disorder potential and investigate the emergence of a Bose-glass phase for increasing disorder strength. At zero temperature a first-order quantum phase transition from the superfluid phase to the Bose-glass phase is detected at a critical disorder strength, which agrees with the findings in the literature. Afterwards, we study the interplay between temperature and disorder fluctuations on the respective components of the particle density. In particular, we find for smaller disorder strengths that a superfluid region, a Bose-glass region, and a thermal region coexist. Furthermore, depending on the respective system parameters, three phase transitions are detected, namely, one from the superfluid to the Bose-glass phase, another one from the Bose-glass to the thermal phase, and finally one from the superfluid to the thermal phase. All these results are obtained by extending a quite recent Hartree–Fock mean-field theory for the dirty boson problem, which is based on the replica method, from the homogeneous case to a harmonic confinement.
The localization-disorder paradigm is analyzed for a specific system of weakly repulsive Bose gas at zero temperature placed into a quenched random potential. We show that at low average density or weak enough interaction the particles fill deep potential wells of the random potential whose radius and depth depend on the characteristics of the random potential and the interacting gas. The localized state is the random singlet with no long-range phase correlation. At a critical density the quantum phase transition to the coherent superfluid state proceeds. We calculate the critical density in terms of the geometrical characteristics of the noise and the gas. In a finite system the ground state becomes nonergodic at very low density. For atoms in traps four different regimes are found; only one of it is superfluid. The theory is extended to lower (one and two) dimensions. Its quantitative predictions can be checked in experiments with ultracold atomic gases and other Bose systems.
The localization‐disorder paradigm is analyzed for a specific system of weakly repulsive Bose gas at zero temperature placed into a quenched random potential. We show that, at low average density or weak enough interaction the particles fill deep potential wells of the random potential whose radius and depth depend on the characteristics of the random potential and the interacting gas. The localized state is the random singlet with no long‐range phase correlation. At a critical density the quantum phase transition to the coherent superfluid state proceeds. We calculate the critical density in terms of the geometrical characteristics of the noise and the gas. In a finite system the ground state becomes non‐ergodic at very low density. For atoms in traps four different regimes are found, only one of it is superfluid. The theory is extended to lower (1 and 2) dimensions. Its quantitative predictions can be checked in experiments with ultracold atomic gases and other Bose‐systems.This article is an extended version of a presentation delivered by one of the authors (V.L.P.) at a Conference on Modern Trends in Theoretical Physics dedicated to the Lev Landau Centenary (Chernogolovka, Russia, June 2008).