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arXiv:1208.0373 · Communications on Pure and Applied Mathematics

Quantitative Derivation of the Gross‐Pitaevskii Equation

Niels Benedikter, Gustavo de Oliveira, Benjamin Schlein

Published Sep 29, 2014• 96 citations
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Abstract

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.

DOI
10.1002/cpa.21542

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Abstract Consider a system of N bosons on the three‐dimensional unit torus interacting via a pair potential N2V(N(xi − xj)) where x = (x1, …, xN) denotes the positions of the particles. Suppose that the initial data ψN, 0 satisfies the condition where HN is the Hamiltonian of the Bose system. This condition is satisfied if ψN, 0 = WNϕN, 0 where WN is an approximate ground state to HN and ϕN, 0 is regular. Let ψN, t denote the solution to the Schrödinger equation with Hamiltonian HN. Gross and Pitaevskii proposed to model the dynamics of such a system by a nonlinear Schrödinger equation, the Gross‐Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k‐particle density matrices ⊗k|ut〉 〈ut| solves the GP hierarchy. We prove that as N → ∞ the limit points of the k‐particle density matrices of ψN, t are solutions of the GP hierarchy. Our analysis requires that the N‐boson dynamics be described by a modified Hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance. Our proof can be extended to a modified Hamiltonian that only forbids at least n particles from coming close together for any fixed n. © 2006 Wiley Periodicals, Inc.

arXiv:math-ph/0410005•Communications on Pure and Applied Mathematics•Feb 2, 2006•171 citations
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.
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.
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.
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.
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.
In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.
Abstract Consider a system of N bosons on the three‐dimensional unit torus interacting via a pair potential N2V(N(xi − xj)) where x = (x1, …, xN) denotes the positions of the particles. Suppose that the initial data ψN, 0 satisfies the condition where HN is the Hamiltonian of the Bose system. This condition is satisfied if ψN, 0 = WNϕN, 0 where WN is an approximate ground state to HN and ϕN, 0 is regular. Let ψN, t denote the solution to the Schrödinger equation with Hamiltonian HN. Gross and Pitaevskii proposed to model the dynamics of such a system by a nonlinear Schrödinger equation, the Gross‐Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k‐particle density matrices ⊗k|ut〉 〈ut| solves the GP hierarchy. We prove that as N → ∞ the limit points of the k‐particle density matrices of ψN, t are solutions of the GP hierarchy. Our analysis requires that the N‐boson dynamics be described by a modified Hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance. Our proof can be extended to a modified Hamiltonian that only forbids at least n particles from coming close together for any fixed n. © 2006 Wiley Periodicals, Inc.