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Quantum-Atom Optics Downunder (QAO)

Many-body effects in the dynamics of Bose-Einstein condensates

Thomas Ernst, Tobias Paul, Peter Schlagheck, Joachim Brand

Published Jan 1, 2007• 0 citations
Publisher

Abstract

We study the dynamics of repulsive and attractive Bose-Einstein condensates. While previous works used the Gross-Pitaevskii equation to calculate this process, we employ a kinetic dynamics approach similar to the famous Hartree-Fock-Bogoliubov equations. Therefore we are able to take into account excitations of the condensate as well as its depletion rate. We apply it to the transport of repulsive Bose-Einstein condensates and the scattering problem of a propagating soliton on an attractive potential, where previous works showed a sharp transition from total reflection to full transmission.

DOI
10.1364/qao.2007.qme15

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We study the dynamics of repulsive and attractive Bose-Einstein condensates. While previous works used the Gross-Pitaevskii equation to calculate this process, we employ a kinetic dynamics approach similar to the famous Hartree-Fock-Bogoliubov equations. Therefore we are able to take into account excitations of the condensate as well as its depletion rate. We apply it to the transport of repulsive Bose-Einstein condensates and the scattering problem of a propagating soliton on an attractive potential, where previous works showed a sharp transition from total reflection to full transmission.
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.
The existence and stability of solitonic states in one-dimensional repulsive Bose-Einstein condensates is investigated within a fully many-body framework by considering the limit of infinite repulsion (Tonks-Girardeau gas). A class of stationary, shape-invariant states propagating at constant velocity are explicitly found and compared to the known solution of the Gross-Pitaevskii equation. The typical features attributed to nonlinearity are thus recovered in a purely linear theory, provided the full many-particle physics is correctly accounted for. However, the formation dynamics predicted by the Gross-Pitaevskii approximation considerably differs from the exact many-body evolution.
This project aims to study the behavior of a cloud of Rubidium-85 atoms under an attractive interaction potential when used to create a Bose-Einstein condensate. Efforts to describe the dynamics of an attractive condensate using only the Gross-Pitaevskii equation have been unable to describe all of the features of such a system, and so the mathematical approach used is a perturbative method of corrections to the Gross-Pitaevskii equation. A system of analytical equations to describe a condensate is derived herein, which is then arranged into a form suitable for numerical solution. The actual calculation is not performed, but details of the computational procedure are provided, as well as an analysis of the difficulties inherent in the calculation.
This thesis investigates the rigorous derivation of non-linear effective equations describing the time evolution of Bose-Einstein condensation from the N-particle linear Schrödinger equation, focusing on the one-particle reduced density matrix. The first chapter outlines the mathematical framework of many-body quantum mechanics, including the Fock space formalism. The second chapter studies the mean-field regime, proving that the condensed state converges to the non-linear Hartree equation by analyzing fluctuation dynamics to quantify the convergence rate. The third chapter addresses the physically realistic but mathematically challenging Gross-Pitaevskii regime, characterized by rare but strong interactions. Here, Bogoliubov transformations are used to capture crucial microscopic short-scale correlations, allowing for a quantitative derivation of the cubic non-linear Schrödinger equation. While the preceding chapters focus exclusively on repulsive interactions, the fourth chapter explores the focusing case with attractive potentials, which can lead to finite-time blow-up. Operating in an intermediate scaling regime, the effective focusing cubic non-linear Schrödinger equation is rigorously derived. Since the negative energy cannot bound many-body fluctuations, convergence is proven by approximating the dynamics via a quadratic Hamiltonian, in the spirit of Bogoliubov theory, and avoiding Grönwall estimates through the analysis of a coupled system of one-body linear equations for the one-particle density matrices.
The evolution equation of a weakly interacting boson system is derived. The model includes the interaction between the atoms in the condensate and the surrounding gas of noncondensed particles. The Bogoliubov transformation is introduced in a full quantum context and the scattering kernel between dressed particles and the condensate phase is obtained. The final system is expressed by the Boltzmann evolution equation for noncondensed particles coupled to the Gross-Pitaevskii equation for the condensate. We consider an out-of-equilibrium situation that induces a fast production of condensed particles. We apply our model to study the condensation dynamics of positronium atoms by evaporation.