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

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 t...

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Veröffentlicht in:arXiv.org 2024-07
Hauptverfasser: Caraci, Cristina, Oldenburg, Jakob, Schlein, Benjamin
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Sprache:eng
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Zusammenfassung: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2308.11687