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arXiv:0806.3143 · Journal of Physics Conference Series

Evolution of fragmented states

Gheorghe Sorin Paraoanu

Published Feb 1, 2009• 0 citations
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Abstract

We consider the problem of evolution of the many-body state of a weakly interacting system of bosons in an initially fragmented (Fock) state. We show that the state at any time can be expressed as a continuous superposition of an infinite number of Gross-Pitaevskii states. PACS:03.75.-b, 03.65.-w, 03.75.Kk
DOI
10.1088/1742-6596/150/3/032079

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Systems of interest in physics are usually composed by a very large number of interacting particles. At equilibrium, these systems are described by stationary states of the many-body Hamiltonian (at zero temperature, by the ground state). The reaction to perturbations, for example to a change of the external fields, is governed by the time-dependent many-body Schroedinger equation. Since it is typically very difficult to extract useful information from the Schroedinger equation, one of the main goals of non-equilibrium statistical mechanics is the derivation of effective evolution equations which can be used to predict the macroscopic behavior of the system. In these notes, we are going to consider systems of interacting bosons in the so called Gross-Pitaevskii regime, and we are going to show how coherent states and Bogoliubov transformations can be used to approximate the many body dynamics.
Abstract Fragmented Bose–Einstein condensates are large systems of identical bosons displaying multiple macroscopic occupations of one-body states, in a suitable sense. The quest for an effective dynamics of the fragmented condensate at the leading order in the number of particles, in analogy to the much more controlled scenario for complete condensation in one single state, is deceptive both because characterising fragmentation solely in terms of reduced density matrices is unsatisfactory and ambiguous, and because as soon as the time evolution starts the rank of the reduced marginals generically passes from finite to infinite, which is a signature of a transfer of occupations on infinitely many more one-body states. In this work we review these difficulties, we refine previous characterisations of fragmented condensates in terms of marginals, and we provide a quantitative rate of convergence to the leading effective dynamics in the double limit of infinitely many particles and infinite energy gap.
The dynamics of attractive ultracold bosonic clouds in one dimension is studied by solving the many-particle time-dependent Schrödinger equation. The initially coherent wave packet can dynamically dissociate into two parts when its energy exceeds a threshold value. Noticeably, the time-dependent Gross-Pitaevskii theory does not show up the splitting. We call the split object fragmenton. It possesses remarkable properties; in particular, it is macroscopically fragmented. A simple static model predicts the existence of fragmented states responsible for the formation and dynamics of fragmentons.
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.