Beyond mean field: condensate coupled with pair excitations

TL;DR

This paper proves existence results for a coupled system of PDEs modeling a dilute Bose gas at zero temperature, incorporating condensate and pair excitations with a variational approach and iterative solution construction.

Contribution

It introduces a rigorous mathematical framework for a coupled PDE system describing condensate and pair excitations, extending previous physical models.

Findings

Existence of solutions for the coupled PDE system.

Inclusion of non-condensate contributions in the condensate equation.

Development of an iterative method for solution construction.

Abstract

We prove existence results for a system of partial differential equations describing the approximate condensate wavefunction and pair-excitation kernel of a dilute (T=0) Bose gas in the stationary setting, in the presence of a trapping potential and repulsive pairwise atomic interactions. Notably, the Hartree-type equation for the condensate in this system contains contributions from non-condensate particles, and the pair excitation kernel satisfies a nonlinear operator equation. The equations studied here are inspired by the work of Griffin, who derived this system in the study of finite temperature condensates. The techniques employed include a variational principle, which exploits the connection between unitary Bogoliubov rotations and a nonlinear operator equation for the pair excitation kernel. An iterative procedure for constructing solutions is also included.