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arXiv:0806.2322 · Physical Review Letters

Excitations of One-Dimensional Bose-Einstein Condensates in a Random Potential

Victor Gurarie, Gil Refael, J. T. Chalker

Published Oct 24, 2008• 33 citations
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Abstract

We examine bosons hopping on a one-dimensional lattice in the presence of a random potential at zero temperature. Bogoliubov excitations of the Bose-Einstein condensate formed under such conditions are localized, with the localization length diverging at low frequency as l(omega) approximately 1/omega(alpha). We show that the well-known result alpha=2 applies only for sufficiently weak random potential. As the random potential is increased beyond a certain strength, alpha starts decreasing. At a critical strength of the potential, when the system of bosons is at the transition from a superfluid to an insulator, alpha=1. This result is relevant for understanding the behavior of the atomic Bose-Einstein condensates in the presence of random potential, and of the disordered Josephson junction arrays.

DOI
10.1103/physrevlett.101.170407

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We study the onset of localization from excited states of trapped Bose- Einstein Condensates expanding in presence of Gaussian uncorrelated random disorder. In 1D systems, we observe that for a fixed ratio between the disorder strength and the initial energy, excited states localize exponentially with a localization length that decreases as the energy of the initial state increases. Moreover, the localized state keeps the shape of the initial state wave function with an exponential tail. In 2D, we analyze the interplay between vorticiy and localization by examining the dispersion of a state containing a vortex on it in a disordered media. Despite localization can be associated to islands of constant phase, the presence of a vortex in the initial state leads to dislocations and phase jumps in the localized state. The study of dispersion of a bosonic condensate with vorticity bears similarities to the stability of topological excitations in 2D p-wave fermionic superfluids.
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 study the one-dimensional Bose gas in spatially correlated disorder at zero temperature, using an extended density-phase Bogoliubov method. We analyze, in particular, the decay of the one-body density matrix and the behavior of the Bogoliubov excitations across the phase boundary. We observe that the transition to the Bose-glass phase is marked by a power-law divergence of the density of states at low energy. A measure of the localization length displays a power-law energy dependence in both regions, with the exponent equal to -1 at the boundary. We draw the phase diagram of the superfluid-insulator transition in the limit of small interaction strength.
A spin-wave approach of the zero temperature superfluid-insulator transition for two-dimensional hard-core bosons in a random potential μ=±W is developed. While at the classical level there is no intervening phase between the Bose-condensed superfluid (SF) and the gapped disordered insulator, the introduction of quantum fluctuations leads to a much richer physics. Upon increasing the disorder strength W, the Bose-condensed fraction disappears first, before the SF. Then a gapless Bose-glass phase emerges over a finite region until the insulator appears. Furthermore, in the strongly disordered SF regime, a mobility edge in the spin-wave excitation spectrum is found at a finite frequency Ω(c) decreasing with W, and presumably vanishing in the Bose-glass phase.
We study the thermodynamics of a one-dimensional system of noninteracting bosons in the presence of random fixed finite-range repulsive potentials. It is shown that a Bose---Einstein condensation into the lowest state occurs. The critical temperature increases with the number of impurities. The specific heat exhibits a discontinuity and is infinitely differentiable from both sides of the critical point. For a suitably defined pressure the isothermals exhibit a flat portion below the "transition length."