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.
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.
Bose-Einstein-condensed gases in external spatially random potentials are considered in the frame of a stochastic self-consistent mean-field approach. This method permits the treatment of the system properties for the whole range of the interaction strength, from zero to infinity, as well as for arbitrarily strong disorder. Besides a condensate and superfluid density, a glassy number density due to a spatially inhomogeneous component of the condensate occurs. For very weak interactions and sufficiently strong disorder, the superfluid fraction can become smaller than the condensate fraction, while at relatively strong interactions, the superfluid fraction is larger than the condensate fraction for any strength of disorder. The condensate and superfluid fractions, and the glassy fraction always coexist, being together either nonzero or zero. In the presence of disorder, the condensate fraction becomes a nonmonotonic function of the interaction strength, displaying an antidepletion effect caused by the competition between the stabilizing role of the atomic interaction and the destabilizing role of the disorder. With increasing disorder, the condensate and superfluid fractions jump to zero at a critical value of the disorder parameter by a first-order phase transition.
Properties of Bose–Einstein condensates (BEC) are studied in the presence of a Gaussian random potential of arbitrary strength of disorder and with arbitrary correlation length. Using the stochastic mean field approximation, a system having arbitrary strength of interactions is treated at finite temperature. A special case, in which we assume that the second moment of the random potential is Lorentzian, is treated in more detail at zero temperature. The total density of particles consists of condensate density, glassy density and normal density components. Our results are in agreement with those of Huang and Meng (HM). The system is stable below the critical value of the disorder parameter and undergoes a first-order phase transition at the critical value. We find that the critical value of the disorder parameter changes with changing the correlation length of the random potential.
Bose systems, subject to the action of external random potentials, are considered. For describing the system properties, under the action of spatially random potentials of arbitrary strength, the stochastic mean-field approximation is employed. When the strength of disorder increases, the extended Bose-Einstein condensate fragments into spatially disconnected regions, forming a granular condensate. Increasing the strength of disorder even more transforms the granular condensate into the normal glass. The influence of time-dependent external potentials is also discussed. Fastly varying temporal potentials, to some extent, imitate the action of spatially random potentials. In particular, strong time-alternating potential can induce the appearance of a nonequilibrium granular condensate.
Thermodynamical properties of an interacting boson system at finite temperatures and zero chemical potential are studied within the framework of the Skyrme-like mean-field toy model. It is assumed that the mean field contains both attractive and repulsive terms. Self-consistency relations between the mean field and thermodynamic functions are derived. It is shown that, for sufficiently strong attractive interactions, this system develops a first-order phase transition via the formation of a Bose condensate. An interesting prediction of the model is that the condensed phase is characterized by a constant total density of particles. It is shown that the energy density exhibits a jump at the critical temperature.
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.
A system with Bose-Einstein condensate is considered in the frame of the self-consistent mean-field approximation, which is conserving, gapless, and applicable for arbitrary interaction strengths and temperatures. The main attention is paid to the thorough analysis of the condensate and superfluid fractions in the whole range of the interaction strength, between zero and infinity, and for all temperatures between zero and the critical point Tc. The normal and the anomalous averages are shown to be of the same order for almost all interactions and temperatures, except the close vicinity of Tc. But even in the vicinity of the critical temperature, the anomalous average cannot be neglected, since only in the presence of the latter the phase transition at Tc becomes of second order, as it should be. Increasing temperature influences the condensate and superfluid fractions in a similar way, by diminishing them; but their behavior with respect to the interaction strength is very different. For all temperatures, the superfluid fraction is larger than the condensate fraction. These coincide only at Tc or under zero interactions. For asymptotically strong interactions, the condensate is almost completely depleted, even at low temperatures, while the superfluid fraction can be close to one.