Self-consistent derivation of the modified Gross-Pitaevskii equation with Lee-Huang-Yang correction

TL;DR

This paper derives a modified Gross-Pitaevskii equation incorporating Lee-Huang-Yang corrections for dilute ultracold bosonic gases, advancing the theoretical understanding of quantum depletion effects in such systems.

Contribution

It presents a self-consistent derivation of the modified Gross-Pitaevskii equation with beyond-mean-field corrections using quantum field theory methods.

Findings

Derivation of a zero-temperature modified Gross-Pitaevskii equation with Lee-Huang-Yang correction.

Identification of key steps including semiclassical approximation and quantum depletion considerations.

Validation of the equation's consistency with the known equation of state in the uniform case.

Abstract

We consider a dilute and ultracold bosonic gas of weakly-interacting atoms. Within the framework of quantum field theory we derive a zero-temperature modified Gross-Pitaevskii equation with beyond-mean-field corrections due to quantum depletion and anomalous density. This result is obtained from the stationary equation of the Bose-Einstein order parameter coupled to the Bogoliubov-de Gennes equations of the out-of-condensate field operator. We show that, in the presence of a generic external trapping potential, the key steps to get the modified Gross-Pitaevskii equation are the semiclassical approximation for the Bogoliubov-de Gennes equations, a slowly-varying order parameter, and a small quantum depletion. In the uniform case, from the modified Gross-Pitaevskii equation we get the familiar equation of state with Lee-Huang-Yang correction.