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

Cristina Caraci, Jakob Oldenburg, Benjamin Schlein

Cristina Caraci, Jakob Oldenburg, Benjamin Schlein

**22/8/23**Published in : arXiv:2308.11687

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 Schrödinger 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.