The time-dependent Hartree-Fock-Bogoliubov equations for Bosons

2016/02/16 by Volker Bach, Bach, Volker, Sébastien Breteaux +7
Computer Science · Physics and Astronomy · #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Optical properties and cooling technologies in crystalline materials #Quantum Information and Cryptography

paper · doi:10.48550/arxiv.1602.05171

openalex publication_date 2016/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we use quasifree reduction to derive the time-dependent Hartree-Fock-Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate in \mathbb R^d. We prove global well-posedness for the HFB equations for sufficiently regular pair interaction potentials, and establish key conservation laws. Moreover, we show that the solutions to the HFB equations exhibit a symplectic structure, and have a form reminiscent of a Hamiltonian system. In particular, this is used to relate the HFB equations to the HFB eigenvalue equations encountered in the physics literature. Furthermore, we construct the Gibbs states at positive temperature associated with the HFB equations, and establish criteria for the emergence of Bose-Einstein condensation.

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