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

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 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.

59 pages; revised version

References in corpus (9)

• Rigorous Derivation of the Gross-Pitaevskii Equation
• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• Bose-Einstein Condensation with Optimal Rate for Trapped Bosons in the Gross-Pitaevskii Regime
• Excitation Spectrum for Bose Gases beyond the Gross-Pitaevskii Regime
• A second order upper bound for the ground state energy of a hard-sphere gas in the Gross-Pitaevskii regime
• The excitation spectrum of two dimensional Bose gases in the Gross-Pitaevskii regime
• The Bose gas in a box with Neumann boundary conditions
• Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit
• The free energy of dilute Bose gases at low temperatures

Cited by in corpus (3)

• The excitation spectrum of a Bose gas with an impurity in the Gross-Pitaevskii regime
• Validity of the Fröhlich model for a mobile impurity in a Bose-Einstein condensate
• Derivation of the Gross-Pitaevskii Dynamics through Renormalized Excitation Number Operators