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arXiv:1202.3489 · New Journal of Physics

Condensate deformation and quantum depletion of Bose–Einstein condensates in external potentials

Cord Axel Müller, Christopher Gaul

Published Jul 27, 2012• 41 citations

Abstract

The one-body density matrix of weakly interacting condensed bosons in external potentials is calculated using inhomogeneous Bogoliubov theory. We determine the condensate deformation caused by weak external potentials at the mean-field level. The momentum distribution of quantum fluctuations around the deformed ground state is obtained analytically and finally, the resulting quantum depletion is calculated. The depletion due to the external potential, or potential depletion for short, is a small correction to the homogeneous depletion, validating our inhomogeneous Bogoliubov theory. Analytical results are derived for weak lattices and spatially correlated random potentials, with simple, universal results in the Thomas–Fermi limit of very smooth potentials.

DOI: 10.1088/1367-2630/14/7/075025

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The one-body density matrix of weakly interacting condensed bosons in external potentials is calculated using inhomogeneous Bogoliubov theory. We determine the condensate deformation caused by weak external potentials at the mean-field level. The momentum distribution of quantum fluctuations around the deformed ground state is obtained analytically and finally, the resulting quantum depletion is calculated. The depletion due to the external potential, or potential depletion for short, is a small correction to the homogeneous depletion, validating our inhomogeneous Bogoliubov theory. Analytical results are derived for weak lattices and spatially correlated random potentials, with simple, universal results in the Thomas–Fermi limit of very smooth potentials.

We describe repulsively interacting Bose-Einstein condensates in spatially correlated disorder potentials of arbitrary dimension. The first effect of disorder is to deform the mean-field condensate. The second is that the quantum excitation spectrum and condensate population are affected. By a saddle-point expansion of the many-body Hamiltonian around the deformed mean-field ground state, we derive the fundamental quadratic Hamiltonian of quantum fluctuations. Importantly, a basis is used such that excitations are orthogonal to the deformed condensate. Via Bogoliubov-Nambu perturbation theory, we compute the effective excitation dispersion, including mean free paths and localization lengths. Corrections to the speed of sound and average density of states are calculated, due to correlated disorder in arbitrary dimensions, extending to the case of weak lattice potentials.

Abstract At zero temperature, homogeneous interacting Bose-condensed fluids are entirely superfluid, with remarkable transport properties. A non-superfluid, normal component is induced by finite temperatures and spatial inhomogeneity, the combined effects of which are rather intriguing, and difficult to describe quantitatively. By inhomogeneous Bogoliubov theory, applicable to weakly interacting condensed Bose gases in static external potentials with arbitrary spatial correlations, we calculate the normal fluid density via the transverse current–current correlation. We obtain finite-temperature disorder corrections to the normal fraction known since Laudau’s seminal two-fluid theory, using diagrammatic perturbation theory for systems of any dimensionality, with closed analytical expressions to leading, quadratic order in disorder strength.

We analyze the mean-field equilibrium states of a homogeneous Bose-Einstein condensate of boson particles in different internal states, possibly in the presence of an internal Josephson interaction, and we describe the quantum fluctuations by constructing an explicit form of the Bogoliubov transformation that characterizes the quasiparticle excitations of the system. The derivation of the Bogoliubov transformation is based on a simple geometrical picture which leads to a convenient parametrization of the transformation. Our calculations of physical properties, such as depletions and phonon velocities, as well as our discussion of stability, indicate the sense and the extent to which the internal Josephson interaction controls the many-body physics of multicomponent phase coherent matter.

We study the effects of repulsive interaction and disorder on bosons in a two-site Bose–Hubbard system, which provides a simple model of the dirty boson problem. By comparison with exact numerical results, we demonstrate how a straightforward application of the Bogoliubov approximation fails even to deliver a qualitatively correct picture. It wrongly predicts an increase of the condensate depletion due to disorder. We show that, in the presence of disorder, the noncommutative character of the condensate operator has to be retained for a correct description of the system. While these corrections to a simple meanfield Bogoliubov theory are usually small in extended systems, we show that care has to be taken when modeling small inhomogeneous Bose systems.