Multivariate Central Limit Theorem in Quantum Dynamics

TL;DR

This paper establishes a multivariate central limit theorem for quantum fluctuations of bosonic systems in the mean field regime, showing convergence to a Gaussian measure with explicit convergence rates.

Contribution

It provides the first rigorous derivation of a multivariate CLT for quantum many-body dynamics, including explicit bounds on convergence rates.

Findings

Expectations of averaged observables converge to Gaussian expectations as N→∞.

The covariance matrix is explicitly expressed via a Bogoliubov transformation.

Results include Berry-Esseen type bounds on the convergence rate.

Abstract

We consider the time evolution of $N$ bosons in the mean field regime for factorized initial data. In the limit of large $N$, the many body evolution can be approximated by the non-linear Hartree equation. In this paper we are interested in the fluctuations around the Hartree dynamics. We choose $k$ self-adjoint one-particle operators $O_1, \dots, O_k$ on $L^2 (\R^3)$, and we average their action over the $N$-particles. We show that, for every fixed $t \in \R$, expectations of products of functions of the averaged observables approach, as $N \to \infty$, expectations with respect to a complex Gaussian measure, whose covariance matrix can be expressed in terms of a Bogoliubov transformation describing the dynamics of quantum fluctuations around the mean field Hartree evolution. If the operators $O_1, \dots, O_k$ commute, the Gaussian measure is real and positive, and we recover a "classical" multivariate central limit theorem. All our results give explicit bounds on the rate of the convergence (we obtain therefore Berry-Ess{\'e}en type central limit theorems).