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International Conference on Path Integrals

CRITICAL TEMPERATURE OF DIRTY BOSONS

Ben Klünder, Axel Pelster, R. Graham

Published Nov 1, 2008• 0 citations
Publisher

Abstract

We consider a dilute Bose Gas moving in a harmonic trap with a superimposed frozen random potential which arises in experiments either naturally in wire traps or artificially and controllably with laser speckles. The critical temperature, which characterises the onset of Bose-Einstein condensation, depends on the disorder realization within the ensemble. Therefore, we introduce an effective grand-canonical potential from which we determine perturbatively the disorder averages of both the first and the second moment of the critical temperature in leading order. We discuss our results for a finite number of particles by assuming a Gaussian spatial correlation for the quenched disorder potential. Keywords: Disorder, Bose-Einstein condensation, critical temperature 1.

DOI
10.1142/9789812837271_0061

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We consider a dilute Bose Gas moving in a harmonic trap with a superimposed frozen random potential which arises in experiments either naturally in wire traps or artificially and controllably with laser speckles. The critical temperature, which characterises the onset of Bose-Einstein condensation, depends on the disorder realization within the ensemble. Therefore, we introduce an effective grand-canonical potential from which we determine perturbatively the disorder averages of both the first and the second moment of the critical temperature in leading order. We discuss our results for a finite number of particles by assuming a Gaussian spatial correlation for the quenched disorder potential. Keywords: Disorder, Bose-Einstein condensation, critical temperature 1.
We determine the leading shift of the Bose-Einstein condensation temperature for an ultracold dilute atomic gas in a harmonic trap due to weak disorder by treating both a Gaussian and a Lorentzian spatial correlation for the quenched disorder potential. Increasing the correlation length from values much smaller than the geometric mean of the trap scale and the mean particle distance to much larger values leads first to an increase of the positive shift to a maximum at this critical length scale and then to a decrease.
The Bose–Einstein condensation in the presence of quenched disorder due to random boundary conditions is considered. We have shown that for bosons confined in a box with volume Ld and for random L the Bose condensation temperature is lowered by the presence of disorder and the onset of condensation may even be completely prohibited. Similar results hold for bosons confined in a harmonic oscillator potential well, the geometry used in recent experiments on Bose–Einstein condensation in atomic traps. Comments on possible consequences of quenched disorder in boundary conditions for many fermion systems are also given.
We consider the thermodynamics of a homogeneous superfluid dilute Bose gas in the presence of weak quenched disorder. Following the zero-temperature approach of Huang and Meng, we diagonalize the Hamiltonian of a dilute Bose gas in an external random $\ensuremath{\delta}$-correlated potential by means of a Bogoliubov transformation. We extend this approach to finite temperature by combining the Popov and the many-body TTT-matrix approximations. This approach permits us to include the quasiparticle interactions within this temperature range. We derive the disorder-induced shifts of the Bose-Einstein critical temperature and of the temperature for the onset of superfluidity by approaching the transition points from below, i.e., from the superfluid phase. Our results lead to a phase diagram consistent with that of the finite-temperature theory of Lopatin and Vinokur which was based on the replica method, and in which the transition points were approached from above.
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.
We consider a finite number of noninteracting bosons trapped in an isotropic three-dimensional harmonic oscillator. Using the grand canonical ensemble, we calculate the heat capacity of the system as a function of temperature, for several values of the number of particles. We find that a new definition of critical temperature is necessary for a finite number of trapped particles, and present a definition that is experimentally convenient and in good agreement with a recent definition given by W. Ketterle and N. J. van Druten [Phys. Rev. A 54, 656 (1996)].
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.