Quantum Gross-Pitaevskii Equation

TL;DR

This paper develops a non-commutative, quantum generalization of the Gross-Pitaevskii equation for 1D quantum gases, capturing entanglement and correlations beyond mean-field theory, and derives related excitation equations.

Contribution

It introduces a novel quantum extension of the Gross-Pitaevskii equation using matrix product states, enabling full quantum many-body analysis beyond mean-field approximations.

Findings

Derived a quantum Gross-Pitaevskii equation for 1D systems.

Extended the Bogoliubov-de Gennes equations to the quantum regime.

Applied the framework to analyze steady state responses to periodic perturbations.

Abstract

We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system ---including entanglement and correlations--- and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi) one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov -- de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.