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2 DISORDERED BOSE EINSTEIN CONDENSATES WITH INTERACTION IN ONE DIMENSION

Robert Seiringer, Jakob Yngvason, Valentin, A. Zagrebnov

Published Jan 1, 2012• 26 citations
OpenAlex

Abstract

Abstract. We study the effects of random scatterers on the ground state of the one-dimensional Lieb-Liniger model of interacting bosons on the unit interval in the Gross-Pitaevskii regime. We prove that Bose Einstein condensation survives even a strong random potential with a high density of scatterers. The character of the wave func-tion of the condensate, however, depends in an essential way on the interplay between randomness and the strength of the two-body interaction. For low density of scatterers or strong interactions the wave function extends over the whole interval. High density of scatterers and weak interaction, on the other hand, leads to localization of the wave function in a fragmented subset of the interval. 1.

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Starting with the many-body Schrdinger Hamiltonian in R~2, we prove that the ground state energy of a two-dimensional interacting Bose gas with the pairwise attractive interaction approaches to the minimum of the Gross-Pitaevskii energy functional in the meanfield regime, as the particle number N→∞ and however the scattering length κ→0. By fixing N|κ|, this leads to the mean-field approximation of Bose-Einstein condensates with attractive interactions in R~2.

Acta Mathematica Scientia•Jan 1, 2016•0 citations
Abstract. We study the effects of random scatterers on the ground state of the one-dimensional Lieb-Liniger model of interacting bosons on the unit interval in the Gross-Pitaevskii regime. We prove that Bose Einstein condensation survives even a strong random potential with a high density of scatterers. The character of the wave func-tion of the condensate, however, depends in an essential way on the interplay between randomness and the strength of the two-body interaction. For low density of scatterers or strong interactions the wave function extends over the whole interval. High density of scatterers and weak interaction, on the other hand, leads to localization of the wave function in a fragmented subset of the interval. 1.
We study the effects of random scatterers on the ground state of the one-dimensional Lieb-Liniger model of interacting bosons on the unit interval in the Gross-Pitaevskii regime. We prove that Bose Einstein condensation survives even a strong random potential with a high density of scatterers. The character of the wave function of the condensate, however, depends in an essential way on the interplay between randomness and the strength of the two-body interaction. For low density of scatterers or strong interactions the wave function extends over the whole interval. High density of scatterers and weak interaction, on the other hand, leads to localization of the wave function in a fragmented subset of the interval.
We study the effects of random scatterers on the ground state of the one-dimensional Lieb–Liniger model of interacting bosons on the unit interval in the Gross–Pitaevskii regime. We prove that Bose–Einstein condensation survives even a strong random potential with a high density of scatterers. The character of the wavefunction of the condensate, however, depends in an essential way on the interplay between randomness and the strength of the two-body interaction. For low density of scatterers and strong interactions the wavefunction extends over the whole interval. A high density of scatterers and weak interactions, on the other hand, lead to localization of the wavefunction in a fragmented subset of the interval.
It has been proven theoretically for bosons with two-body repulsive interaction potentials in the dilute limit that the Gross-Pitaevskii equation provides the exact energy and density per particle as does the basic many-particle Schr\"odinger equation [E. H. Lieb and R. Seiringer, Phys. Rev. Lett. 88, 170409 (2002)]. Here, we investigate the overlap of the Gross-Pitaevskii and exact ground-state wave functions. It is found that this overlap is always smaller than unity and may even vanish despite the fact that both wave functions provide the same energy and density per particle. Consequences are discussed.
This paper studies the localization behavior of Bose--Einstein condensates in disorder potentials, modeled by a Gross--Pitaevskii eigenvalue problem on a bounded interval. In the regime of weak particle interaction, we are able to quantify exponential localization of the ground state, depending on statistical parameters and the strength of the potential. Numerical studies further show delocalization if we leave the identified parameter range, which is in agreement with experimental data. These mathematical and numerical findings allow the prediction of physically relevant regimes where localization of ground states may be observed experimentally.
We present a detailed numerical study of the effect of a disordered potential on a confined one-dimensional Bose–Einstein condensate, in the framework of a mean-field description, using a highly efficient and fast converging numerical scheme. For repulsive interactions, we consider the Thomas–Fermi and Gaussian limits and for attractive interactions the behaviour of soliton solutions. We find that the average spatial extension of the stationary density profile decreases with an increasing disorder strength both for repulsive and attractive interactions among bosons. In the Thomas–Fermi limit, a strong localization of the bosons is obtained in momentum space around the state k = 0. The time-dependent density differs considerably in the cases we have considered. For attractive and disordered Bose–Einstein condensates, we show evidence of a bright soliton with an overall unchanged shape, but a disorder-dependent width. For weak disorder, the soliton is delocalized and for stronger disorder, it bounces back and forth between high potential barriers.