Beyond Bogoliubov Dynamics

TL;DR

This paper develops a method to accurately approximate the true dynamics of interacting bosons in the mean-field regime by constructing corrections to Bogoliubov dynamics, enabling precise calculation of correlation functions and reduced densities.

Contribution

It introduces N-independent correction terms based on Bogoliubov and Hartree equations that improve the approximation of N-body dynamics to arbitrary precision.

Findings

Achieves arbitrary accuracy in approximating N-body dynamics

Provides explicit formulas for n-point correlation functions

Reduces complex many-body problem to solving PDEs for Hartree and Bogoliubov functions

Abstract

We consider a system of N interacting bosons in the mean-field scaling regime and construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to arbitrary precision. The N-independent corrections are given in terms of the solutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick's theorem. We determine the n-point correlation functions of the excitations around the condensate, as well as the reduced densities of the N-body system, to arbitrary accuracy, given only the knowledge of the two-point functions of a quasi-free state and the solution of the Hartree equation. In this way, the complex problem of computing all n-point correlation functions for an interacting N-body system is essentially reduced to the problem of solving the Hartree equation and the PDEs for the Bogoliubov two-point functions.