Theory of correlations between ultra-cold bosons released from an optical lattice

TL;DR

This paper develops a comprehensive theoretical framework to analyze correlations in ultra-cold bosons after release from an optical lattice, incorporating various approximations and examining effects like quantum depletion and pairing.

Contribution

It introduces a unified formalism for calculating correlations in inhomogeneous lattice systems, including effects of quantum depletion and pairing, validated against direct calculations.

Findings

Correlation functions depend on quantum depletion and pairing effects.

The Gaussian approximation effectively models the correlations in realistic regimes.

The formalism is adaptable for numerical simulations of experimental setups.

Abstract

In this paper we develop a theoretical description of the correlations between ultra-cold bosons after free expansion from confinement in an optical lattice. We consider the system evolution during expansion and give criteria for a far field regime. We develop expressions for first and second order two-point correlations based on a variety of commonly used approximations to the many-body state of the system including Bogoliubov, meanfield decoupling, and particle-hole perturbative solution about the perfect Mott-insulator state. Using these approaches we examine the effects of quantum depletion and pairing on the system correlations. Comparison with the directly calculated correlation functions is used to justify a Gaussian form of our theory from which we develop a general three-dimensional formalism for inhomogeneous lattice systems suitable for numerical calculations of realistic experimental regimes.