Bogoliubov Theory for Ultra Dilute Bose Gases

Christian Brennecke

Xuguang Yue, Shujuan Liu, Biao Wu +1 more

Ultra-cold atomic gases provide a new chance to study the universal critical behavior of phase transition. We study theoretically the matter wave interference for ultra-cold Bose gases in the critical regime. We demonstrate that the interference in the momentum distribution can be used to extract the correlation in the Bose gas. A simple relation between the interference visibility and the correlation length is found and used to interpret the pioneering experiment about the critical behavior of dilute Bose gases [Science 315 1556 (2007)]. Our theory paves the way to experimentally study various types of ultra-cold atomic gases with the means of matter wave interference.

Serena Cenatiempo

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.

Valentin Anatol'evich Zagrebnov, J.-B. Bru

Thorsten Köhler, Keith R. Burnett

We derive quantum evolution equations for the dynamics of dilute condensed Bose gases. The approach contains, at different orders of approximation, for cases close to equilibrium, the Gross-Pitaevskii equation and the first-order Hartree-Fock-Bogoliubov theory. The proposed approach is also suited for the description of the dynamics of condensed gases that are far away from equilibrium. As an example the scattering of two Bose condensates is discussed.