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arXiv:1502.07528 · Physical Review A

Variance as a sensitive probe of correlations

Shachar Klaiman, Ofir E. Alon

Published Jun 12, 2015• 53 citations
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Abstract

Bose-Einstein condensates made of ultracold trapped bosonic atoms have become a central venue in which interacting many-body quantum systems are studied. The ground state of a trapped Bose-Einstein condensate has been proven to be 100% condensed in the limit of infinite particle number and constant interaction parameter [Lieb and Seiringer, Phys. Rev. Lett. 88, 170409 (2002)]. The meaning of this result is that properties of the condensate, noticeably its energy and density, converge to those obtained by minimizing the Gross-Pitaevskii energy functional. This naturally raises the question whether correlations are of any importance in this limit. Here, we demonstrate both analytically and numerically that even in the infinite particle limit many-body correlations can lead to a modification of the variance of any operator compared to that expected from the Gross-Pitaevskii result. The deviation of the variance stems from its explicit dependence on terms of the reduced two-body density matrix which otherwise do not contribute to the energy and density in this limit. This makes the variance a sensitive probe of many-body correlations even when the energy and density of the system have already converged to the Gross-Pitaevskii result. We use the many-particle position and momentum operators to exemplify this persistence of correlations. Implications of this many-body effect are discussed.

DOI
10.1103/physreva.91.063613

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Bose-Einstein condensates made of ultracold trapped bosonic atoms have become a central venue in which interacting many-body quantum systems are studied. The ground state of a trapped Bose-Einstein condensate has been proven to be 100% condensed in the limit of infinite particle number and constant interaction parameter [Lieb and Seiringer, Phys. Rev. Lett. 88, 170409 (2002)]. The meaning of this result is that properties of the condensate, noticeably its energy and density, converge to those obtained by minimizing the Gross-Pitaevskii energy functional. This naturally raises the question whether correlations are of any importance in this limit. Here, we demonstrate both analytically and numerically that even in the infinite particle limit many-body correlations can lead to a modification of the variance of any operator compared to that expected from the Gross-Pitaevskii result. The deviation of the variance stems from its explicit dependence on terms of the reduced two-body density matrix which otherwise do not contribute to the energy and density in this limit. This makes the variance a sensitive probe of many-body correlations even when the energy and density of the system have already converged to the Gross-Pitaevskii result. We use the many-particle position and momentum operators to exemplify this persistence of correlations. Implications of this many-body effect are discussed.
The variance of a many-particle operator is a very sensitive probe of many-body correlations. In contrast to the expectation value of such operators which depends on the condensation fraction of the Bose-Einstein condensate (BEC), their variance depends on the absolute number of depleted particles. Consequentially, the many-body correlations incorporated within the variance can be observed even in the limit of an infinite number of particles (keeping the interaction parameter, i.e., the product of the number of particles times the scattering length, constant) when the system becomes 100% condensed. In the present work we show both analytically and numerically the effect of many-body correlations on the variance and uncertainty product for an out-of-equilibrium BEC. Our work demonstrates the need to go beyond a mean-field description of the condensate in order to correctly describe these quantities even when only a single particle is depleted and the system is by definition 100% condensed.
We introduce a family of correlated trial wave functions for the NNN-particle ground state of an interacting Bose gas in a harmonic trap. For large NNN, the correlations lead to a relative energy decrease of a fraction 3/5N3/5N3/5N, compared to mean field Gross-Pitaevskii theory. The kinetic energy in the weakly confining direction turns out to be most sensitive to our correlations and, remarkably, is higher by as much as a few per cent for condensates with atom numbers of a few thousand. Thus, the predicted deviations from Gross-Pitaevskii theory originating from ground state correlations might be observed in momentum distribution measurements of small condensates.
We study the effects of many-body correlations in trapped ultracold atomic Bose gases. We calculate the ground state of the gas using a ground-state auxiliary-field quantum Monte Carlo (QMC) method [Phys. Rev. E 70, 056702 (2004)]. We examine the properties of the gas, such as the energetics, condensate fraction, real-space density, and momentum distribution, as a function of the number of particles and the scattering length. We find that the mean-field Gross-Pitaevskii (GP) approach gives qualitatively incorrect result of the kinetic energy as a function of the scattering length. We present detailed QMC data for the various quantities, and discuss the behavior of GP, modified GP, and the Bogoliubov method under a local density approximation (LDA).
The variance of the position operator is associated with how wide or narrow a wave-packet is, the momentum variance is similarly correlated with the size of a wave-packet in momentum space, and the angular-momentum variance quantifies to what extent a wave-packet is non-spherically symmetric. We examine an interacting three-dimensional trapped Bose–Einstein condensate at the limit of an infinite number of particles, and investigate its position, momentum, and angular-momentum anisotropies. Computing the variances of the three Cartesian components of the position, momentum, and angular-momentum operators we present simple scenarios where the anisotropy of a Bose–Einstein condensate is different at the many-body and mean-field levels of theory, despite having the same many-body and mean-field densities per particle. This suggests a way to classify correlations via the morphology of 100% condensed bosons in a three-dimensional trap at the limit of an infinite number of particles. Implications are briefly discussed.
A solvable model of a generic trapped bosonic mixture, N 1 bosons of mass m 1 and N 2 bosons of mass m 2 trapped in an harmonic potential of frequency ω and interacting by harmonic inter-particle interactions of strengths λ 1 , λ 2 , and λ 12 , is discussed. It has recently been shown for the ground state [J. Phys. A 50 , 295002 (2017)] that in the infinite-particle limit, when the interaction parameters λ 1 ( N 1 − 1), λ 2 ( N 2 − 1), λ 12 N 1 , λ 12 N 2 are held fixed, each of the species is 100% condensed and its density per particle as well as the total energy per particle are given by the solution of the coupled Gross-Pitaevskii equations of the mixture. In the present work we investigate properties of the trapped generic mixture at the infinite-particle limit, and find differences between the many-body and mean-field descriptions of the mixture, despite each species being 100%. We compute analytically and analyze, both for the mixture and for each species, the center-of-mass position and momentum variances, their uncertainty product, the angular-momentum variance, as well as the overlap of the exact and Gross-Pitaevskii wavefunctions of the mixture. The results obtained in this work can be considered as a step forward in characterizing how important are many-body effects in a fully condensed trapped bosonic mixture at the infinite-particle limit.