Quantum Fluctuations of Many-Body Dynamics around the Gross-Pitaevskii Equation

TL;DR

This paper analyzes the quantum fluctuations in many-body bosonic systems around the Gross-Pitaevskii equation, constructing a quasi-free approximation that converges to the true dynamics and establishing a central limit theorem for fluctuations.

Contribution

It introduces a quasi-free approximation for the many-body dynamics in the Gross-Pitaevskii regime and derives a Bogoliubov dynamics for excitations, with convergence results.

Findings

Convergence of the quasi-free approximation to the true many-body dynamics.

Derivation of a Bogoliubov dynamics for excitations.

A central limit theorem for fluctuations of observables.

Abstract

We consider the evolution of a gas of $N$ bosons in the three-dimensional Gross-Pitaevskii regime (in which particles are initially trapped in a volume of order one and interact through a repulsive potential with scattering length of the order $1/N$). We construct a quasi-free approximation of the many-body dynamics, whose distance to the solution of the Schr\"odinger equation converges to zero, as $N \to \infty$, in the $L^2 (\mathbb{R}^{3N})$-norm. To achieve this goal, we let the Bose-Einstein condensate evolve according to a time-dependent Gross-Pitaevskii equation. After factoring out the microscopic correlation structure, the evolution of the orthogonal excitations of the condensate is governed instead by a Bogoliubov dynamics, with a time-dependent generator quadratic in creation and annihilation operators. As an application, we show a central limit theorem for fluctuations of bounded observables around their expectation with respect to the Gross-Pitaevskii dynamics.