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arXiv:1702.05625 · Analysis & PDE

Gross–Pitaevskii dynamics for Bose–Einstein condensates

Christian Brennecke, Benjamin Schlein

Published Feb 7, 2019• 93 citations
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Abstract

We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of \cite{LNS}, where fluctuations around the Hartree dynamics for $N$-particle initial data in the mean-field regime have been analyzed, with ideas from \cite{BDS}, where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.

DOI
10.2140/apde.2019.12.1513

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We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of \cite{LNS}, where fluctuations around the Hartree dynamics for NNN-particle initial data in the mean-field regime have been analyzed, with ideas from \cite{BDS}, where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.
Abstract We revisit the time evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. We show that the system continues to exhibit BEC once the trap has been released and that the dynamics of the condensate is described by the time-dependent Gross-Pitaevskii equation. Like the recent work [15], we obtain optimal bounds on the number of excitations orthogonal to the condensate state. In contrast to [15], however, whose main strategy consists of controlling the number of excitations with regard to a suitable fluctuation dynamics t↦e−Bte−iHNtt\mapsto e^{-B_t} e^{-iH_Nt}t↦e−Bt​e−iHN​t with renormalized generator, our proof is based on controlling renormalized excitation number operators directly with regards to the Schrödinger dynamics t↦e−iHNtt\mapsto e^{-iH_Nt}t↦e−iHN​t .
Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N\mathit{N}N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as N−1/2\mathit{N}^{-1/2}N−1/2 in the limit of large N\mathit{N}N.
We study the Bose-Einstein condensates of trapped Bose gases in the Gross-Pitaevskii regime. We show that the ground state energy and ground states of the many-body quantum system are correctly described by the Gross-Pitaevskii equation in the large particle number limit, and provide the optimal convergence rate. Our work extends the previous results of Lieb, Seiringer and Yngvason on the leading order convergence, and of Boccato, Brennecke, Cenatiempo and Schlein on the homogeneous gas. Our method relies on the idea of 'completing the square', inspired by recent works of Brietzke, Fournais and Solovej on the Lee-Huang-Yang formula, and a general estimate for Bogoliubov quadratic Hamiltonians on Fock space.
In experiments, Bose-Einstein condensates are prepared by cooling a dilute Bose gas in a trap. After the phase transition has been reached, the trap is switched off and the evolution of the condensate observed. The evolution is macroscopically described by the Gross-Pitaevskii equation. On the microscopic level, the dynamics of Bose gases are described by the NNN-body Schrödinger equation. We review our article [BdS12] in which we construct a class of initial data in Fock space which are energetically close to the ground state and prove that their evolution approximately follows the Gross-Pitaevskii equation. The key idea is to model two-particle correlations with a Bogoliubov transformation.
Starting from first‐principle many‐body quantum dynamics, we show that the dynamics of Bose‐Einstein condensates can be approximated by the time‐dependent nonlinear Gross‐Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one‐particle reduced density, the form of the initial data is preserved by the many‐body evolution, up to a small error that vanishes as N−1/2 in the limit of large N.© 2015 Wiley Periodicals, Inc.
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 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.