Images of a Bose-Einstein condensate in position and momentum space

TL;DR

This paper explores the evolution of a Bose-Einstein condensate's quantum state under perturbations, revealing a Gaussian distribution for measurement outcomes in position and momentum space, and discusses related theoretical frameworks.

Contribution

It introduces a diagonal form of the dynamical Bogoliubov vacuum and provides Gaussian integral representations, enhancing understanding of condensate dynamics under perturbations.

Findings

Dynamical vacuum has a simple diagonal form in a time-dependent basis.

Density measurement outcomes follow a Gaussian distribution.

Relations with U(1) symmetry breaking are discussed.

Abstract

In the Bogoliubov theory a condensate initially prepared in its ground state described by stationary Bogoliubov vacuum and later perturbed by a time-dependent potential or interaction strength evolves into a time-dependent excited state which is dynamical Bogoliubov vacuum. The dynamical vacuum has a simple diagonal form in a time-dependent orthonormal basis of single particle modes. This diagonal representation leads to a gaussian probability distribution for possible outcomes of density measurements in position and momentum space. In these notes we also discuss relations with the U(1) symmetry breaking version of the Bogoliubov theory and give two equivalent gaussian integral representations of the dynamical vacuum state.