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arXiv:math-ph/0603055 · Communications in Mathematical Physics

Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons

Jürg Martin Fröhlich, S. Graffi, Simon Schwarz

Published Feb 21, 2007• 84 citations
arXivHTMLPDF
DOI
10.1007/s00220-007-0207-5

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We consider a class of many-body Hamiltonians composed of a free (kinetic) part and a multi-particle (potential) interaction with a compactness assumption on the latter part. We investigate the mean field limit of such quantum systems following the Wigner measures approach. We prove in particular the propagation of these measures along the flow of a nonlinear (Hartree) field equation. This enhances and complements some previous results of the same type shown in Z. Ammari and F. Nier and Fröhlich et al. [“Mean field limit for bosons and propagation of Wigner measures,” J. Math. Phys. 50(4), 042107 (2009); Z. Ammari and F. Nier and Fröhlich et al., “Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states,” J. Math. Pures Appl. 95(6), 585–626 (2011); Z. Ammari and F. Nier and Fröhlich et al., “Mean-field- and classical limit of many-body Schrödinger dynamics for bosons,” Commun. Math. Phys. 271(3), 681–697 (2007)].
Mean-field limits for bosonic systems explain how an effective one-particle nonlinear equationemerges from an N-body Schrödinger dynamics. In the standard mean-field setting, themicroscopic Hamiltonian contains a scaled pair interaction, and the limiting dynamics is aHartree equation. This paper considers a closely related but structurally different situation: inaddition to the usual pair interaction, the microscopic system carries a phase drive which isgenerated by the evolving N-body state itself.The distinction is important. The phase field considered here is not a prescribed timedependent potential and is not imported from the limiting Hartree trajectory. At finite N, it iscomputed from the normalized one-particle marginal density
The many-body Schr\"odinger dynamics of a one-dimensional bosonic Josephson junction is investigated for up to 10 000 bosons and long times. The initial states are fully condensed and the interaction strength is weak. We report on a universal fragmentation dynamics on the many-body level: systems consisting of different numbers of particles fragment to the same value at constant mean-field interaction strength. The phenomenon manifests itself in observables such as the correlation functions of the system. We explain this universal fragmentation dynamics analytically based on the Bose-Hubbard model. We thereby show that the extent to which many-body effects become important at later times depends crucially on the initial state. Even for arbitrarily large particle numbers and arbitrarily weak interaction strength the dynamics is many-body in nature and the fragmentation universal. There is no weakly interacting limit where the Gross-Pitaevskii mean field is valid for long times.