Dynamical formation of correlations in a Bose-Einstein condensate

Abstract

We consider the evolution of $N$ bosons interacting with a repulsive short range pair potential in three dimensions. The potential is scaled according to the Gross-Pitaevskii scaling, i.e. it is given by $N^{2}V(N(x_i-x_j))$. We monitor the behavior of the solution to the $N$-particle Schrödinger equation in a spatial window where two particles are close to each other. We prove that within this window a short scale interparticle structure emerges dynamically. The local correlation between the particles is given by the two-body zero energy scattering mode. This is the characteristic structure that was expected to form within a very short initial time layer and to persist for all later times, on the basis of the validity of the Gross-Pitaevskii equation for the evolution of the Bose-Einstein condensate. The zero energy scattering mode emerges after an initial time layer where all higher energy modes disperse out of the spatial window. We can prove the persistence of this structure up to sufficiently small times before three-particle correlations could develop.

36 pages, latex file

References in corpus (2)

• Proof of Bose-Einstein Condensation for Dilute Trapped Gases
• Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional

Cited by in corpus (6)

• Quantitative Derivation of the Gross-Pitaevskii Equation
• Gross-Pitaevskii Dynamics for Bose-Einstein Condensates
• Complete Bose-Einstein condensation in the Gross-Pitaevskii regime
• Quantum many-body fluctuations around nonlinear Schrödinger dynamics
• Kinetic Energy Estimates for the Accuracy of the Time-Dependent Hartree-Fock Approximation with Coulomb Interaction
• Effective non-linear spinor dynamics in a spin-1 Bose-Einstein condensate