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

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.

Revision of earlier manuscript [3] correcting an error in the proof of local existence

Cited by in corpus (4)

• The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
• Gaussian Quantum Trajectories for the Variational Simulation of Open Quantum-Optical Systems
• Beyond Bogoliubov Dynamics
• Uniform in $N$ Global Well-posedness of the Time-Dependent Hartree-Fock-Bogoliubov Equations in ${\mathbb{R}}^{1+1}$