Dipolar Bose-Einstein condensates in weak anisotropic disorder

Abstract

Here we study properties of a homogeneous dipolar Bose-Einstein condensate in a weak anisotropic random potential with Lorentzian correlation at zero temperature. To this end we solve perturbatively the Gross-Pitaevskii equation to second order in the random potential strength and obtain analytic results for the disorder ensemble averages of both the condensate and the superfluid depletion, the equation of state, and the sound velocity. For a pure contact interaction and a vanishing correlation length, we reproduce the seminal results of Huang and Meng, which were originally derived within a Bogoliubov theory around a disorder-averaged background field. For dipolar interaction and isotropic Lorentzian-correlated disorder, we obtain results which are qualitatively similar to the case of an isotropic Gaussian-correlated disorder. In the case of an anisotropic disorder, the physical observables show characteristic anisotropies which arise from the formation of fragmented dipolar condensates in the local minima of the disorder potential.

Bogoliubov theory of dipolar Bose gas in a weak random potential

“…As the dipolar interaction potential (3) has two repulsive and only one attractive direction in real space, it yields a net expulsion of bosons from the ground state which is described by Q 3 2 (ǫ dd ) in (28). In contrast to that the dipolar interaction supports the localization of bosons in the random environment [37,39] …”

“…Note that (5) is not continuous at q = 0, as the limit q → 0 is direction dependent. This is the origin for various anisotropic properties, which are characteristic for dipolar Bose gases [37,39,42,43]. The operatorsâ k and a † k are the annihilation and creation operators in Fourier space, respectively, which turn out to satisfy the bosonic commutation relations…”

“…In the first case we have C dd = µ 0 d 2 m , with the magnetic dipole moment d m and the magnetic permeability of vacuum µ 0 , and in the latter case we have C dd = d 2 e /ǫ 0 , with the electric dipole moment d e and the vacuum dielectric constant ǫ 0 . Note that, due to the anisotropic character of the dipolar interaction, many static and dynamic properties of a dipolar Bose gas become tunable [4,5,40,41], consequences for a random environment have only recently be explored [37,39].…”

“…With this we go beyond the zero-temperature mean-field approach where the Gross-Pitaevskii equation is solved perturbativly with respect to the random potential [37,39]. This extension to the finite-temperature regime allows us to study in detail how the anisotropy of superfluidity can be tuned, a phenomenon which also occurs at zero-temperature [37,39], but turns out to be enhanced by the thermal fluctuations. We observe in addition that contact and dipolar interaction have different effects upon quantum, thermal, and disorder depletion of condensate and superfluid.…”

“…For a delta-correlated disorder it was found that both a condensate and a superfluid depletion occurs due to the localization of bosons in the respective minima of the external random potential which is present even at zero temperature. A generalization to the corresponding situation, where the disorder correlation function falls off with a characteristic correlation length as, for instance, a Gauss function [36,37], laser speckles [38] or a Lorentzian [39] is straightforward.…”