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arXiv:0809.1448

Beyond mean field effects in the high momentum energy spectrum of a Bose gas: revisiting Beliaev's theory

Shai Ronen

Published Sep 8, 2008• 0 citations
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Abstract

The well-known Bogoliubov expression for the spectrum of a weakly interacting dilute Bose gas becomes inadequate when the density or interactions strength are increased. The corrections to the spectrum due to stronger interactions were first considered by Beliaev (S. T. Beliaev, Sov. Phys.-JETP, 7:289, (1958)). We revisit Beliaev's theory and consider its application to a dilute gas with van der Waals interactions, where the scattering length may be tuned via a Fano-Feshbach resonance. We numerically evaluate Beliaev's expression for the excitation spectrum in the intermediate momentum regime, and we also examine the consequences of the momentum dependence of the two-body scattering amplitude.These results are relevant to the interpretation of a recent Bragg spectroscopy experiment of a strongly interacting Bose gas.

DOI
10.48550/arxiv.0809.1448

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The well-known Bogoliubov expression for the spectrum of a weakly interacting dilute Bose gas becomes inadequate when the density or interactions strength are increased. The corrections to the spectrum due to stronger interactions were first considered by Beliaev (S. T. Beliaev, Sov. Phys.-JETP, 7:289, (1958)). We revisit Beliaev's theory and consider its application to a dilute gas with van der Waals interactions, where the scattering length may be tuned via a Fano-Feshbach resonance. We numerically evaluate Beliaev's expression for the excitation spectrum in the intermediate momentum regime, and we also examine the consequences of the momentum dependence of the two-body scattering amplitude.These results are relevant to the interpretation of a recent Bragg spectroscopy experiment of a strongly interacting Bose gas.
The well-known Bogoliubov expression for the spectrum of a weakly interacting dilute Bose gas becomes inadequate when the density or interactions strength is increased. The corrections to the spectrum due to stronger interactions were first considered by Beliaev (1958 Sov. Phys.—JETP 7 289). We revisit Beliaev's theory and consider its application to a dilute gas with van der Waals interactions, where the scattering length may be tuned via a Fano–Feshbach resonance. We numerically evaluate Beliaev's expression for the excitation spectrum in the intermediate-momentum regime, and we also examine the consequences of the momentum dependence of the two-body scattering amplitude. These results are relevant to the interpretation of a recent Bragg spectroscopy experiment of a strongly interacting Bose gas.
In 1947, Bogoliubov suggested a heuristic theory to compute the excitation spectrum of weakly interacting Bose gases. Such a theory predicts a linear excitation spectrum and provides expressions for the thermodynamic functions which are believed to be correct in the dilute limit. Thus far, there are only a few cases where the predictions of Bogoliubov can be obtained by means of rigorous mathematical analysis. A major challenge is to control the corrections beyond Bogoliubov theory, namely, to test the validity of Bogoliubov’s predictions in regimes where the approximations made by Bogoliubov are not valid. In these notes, we discuss how this challenge can be addressed in the case of a system of N interacting bosons trapped in a box with volume one in the Gross-Pitaevskii limit, where the scattering length of the potential is of the order 1/N and N tends to infinity. This is a recent result obtained in Boccato et al. [Commun. Math. Phys. (to be published); preprint arXiv:1812.03086 and Acta Math. 222, 219–335 (2019); e-print arXiv:1801.01389], which extends a previous result obtained in Boccato et al. [Commun. Math. Phys. 359, 975 (2018)], removing the assumption of a small interaction potential.
We present semianalytical many-body results for energies and excitation frequencies for an inhomogeneous Bose-Einstein condensate over a wide range of atom numbers NNN for both small sss-wave scattering lengths, typical of most laboratory experiments, and large scattering lengths, achieved by tuning through a Feshbach resonance. Our dimensional perturbation treatment includes two-body correlations at all orders and yields analytical results through first order by taking advantage of the high degree of symmetry of the condensate at the zeroth-order limit. Because NNN remains a parameter in our analytical results, the challenge of calculating energies and excitation frequencies does not rise with the number of condensate atoms. In this proof-of-concept paper the atoms are confined in a spherical trap and are treated as hard spheres. Our many-body calculations compare well to Gross-Pitaevskii results in the weakly interacting regime and depart from the mean-field approximation as the density approaches the strongly interacting regime. The excitation frequencies provide a particularly sensitive test of beyond-mean-field corrections. For example, for N=2000N=2000N=2000 atoms and an experimentally realized large scattering length of $a=0.433{a}_{ho}\phantom{\rule{0.3em}{0ex}}({a}_{ho}=\sqrt{\ensuremath{\hbar}∕m{\ensuremath{\omega}}_{ho}})$ we predict a 7575%75 shift from the mean-field breathing mode frequency.
Abstract This chapter develops the mean-field Bogoliubov theory of the weakly interacting Bose–Einstein condensed gas. Results for the ground state energy, momentum distribution, and excitation spectrum are discussed. Emphasis is given to the behaviour of the sound velocity and its relationship with the s-wave scattering length. Special focus is given to the infrared divergent behaviour of the momentum distribution as well as to its behaviour at high momenta. The concept of healing length is introduced. The quantum and thermal depletion of the condensate fraction are explicitly derived. Beyond-mean effects in the equation of state, accounted for by the Lee–Yang–Huang expansion, are also discussed.