N-particle Bogoliubov vacuum state

TL;DR

This paper analyzes the number-conserving Bogoliubov vacuum state, demonstrating its simple diagonal form in the particle basis, and compares its accuracy to exact ground states in a Bose-Hubbard model.

Contribution

It provides a simple diagonal representation of the Bogoliubov vacuum in the particle basis and evaluates its accuracy against exact solutions in a specific model.

Findings

Diagonal form of the vacuum can be obtained by diagonalizing the reduced single particle density matrix.

The Bogoliubov theory approximates the ground state well for finite N with constant depletion fraction.

Fails to accurately describe the stationary ground state as N approaches infinity with fixed depletion.

Abstract

We consider the Bogoliubov vacuum state in the number-conserving Bogoliubov theory proposed by Castin and Dum [Phys. Rev. A 57, 3008 (1998)]. We show that in the particle representation the vacuum can be written in a simple diagonal form. The vacuum state can describe the stationary N-particle ground state of a condensate in a trap, but it can also represent a dynamical state when, for example, a Bose-Einstein condensate initially prepared in the stationary ground state is subject to a time-dependent perturbation. In both cases the diagonal form of the Bogoliubov vacuum can be obtained by basically diagonalizing the reduced single particle density matrix of the vacuum. We compare N-body states obtained within the Bogoliubov theory with the exact ground states in a 3-site Bose-Hubbard model. In this example, the Bogoliubov theory fails to accurately describe the stationary ground state in the limit when N goes to infinity but a small fraction of depleted particles is kept constant.