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Conference on Lasers and Electro-Optics

Quantum dynamics of a degenerate Bose gas: a P-representation approach

K. G. Petrosyan, Li You

Published Nov 27, 2002• 0 citations
Publisher

Abstract

Summary form only given.The experimental realization of Bose-Einstein condensation has stimulated much interest in investigating the rich physics of these trapped weakly interacting Bose gases. Although much has been learned within the last several years, more subtle aspects of the condensation process and the quantum dynamics await further elucidation. In this paper, we use the well-established phase space techniques, in particular, the P-representation approach, to study the quantum dynamics of trapped degenerate Bose gases. The system under consideration is described by a Hamiltonian and a functional P-representation quasiprobability distribution is derived. We have developed a self-consistent approximation scheme, which allows us to solve the corresponding stochastic differential equations. Various results are reported.

DOI
10.1109/iqec.1998.680132

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Summary form only given.The experimental realization of Bose-Einstein condensation has stimulated much interest in investigating the rich physics of these trapped weakly interacting Bose gases. Although much has been learned within the last several years, more subtle aspects of the condensation process and the quantum dynamics await further elucidation. In this paper, we use the well-established phase space techniques, in particular, the P-representation approach, to study the quantum dynamics of trapped degenerate Bose gases. The system under consideration is described by a Hamiltonian and a functional P-representation quasiprobability distribution is derived. We have developed a self-consistent approximation scheme, which allows us to solve the corresponding stochastic differential equations. Various results are reported.
We introduce the study of dynamical quantum noise in Bose-Einstein condensates through numerical simulation of stochastic partial differential equations obtained using phase-space representations. We derive evolution equations for a single trapped condensate in both the positive-PPP and Wigner representations and perform simulations to compare the predictions of the two methods. The positive-PPP approach is found to be highly susceptible to the stability problems that have been observed in other strongly nonlinear, weakly damped systems. Using the Wigner representation, we examine the evolution of several quantities of interest using from a variety of choices of initial state for the condensate and compare results to those for single-mode models.
Summary form only given. With the realisation of Bose-Einstein condensation in dilute gases, it has been possible to study the dynamics of quantum fields at finite temperatures. The Gross-Pitaevskii equation (GPE) has proved to be a very useful tool for describing Bose-Einstein condensates (BECs) and has shown very good agreement with experimental results at low temperatures. In the present paper we apply the mean-field approximation to describe the quantum state of a Bose-Einstein condensate, using a phase-space representation to visualise such a state.
Abstract Recent experimental progress has enabled cold atomic gases to be studied at nanokelvin temperatures, creating new states of matter where quantum degeneracy occurs—Bose–Einstein condensates and degenerate Fermi gases. Such quantum states are of macroscopic dimensions. As its title suggests, this book presents the phase space theory approach to treating the physics of degenerate quantum gases, an approach already widely used in quantum optics. However, degenerate quantum gases involve massive bosonic and fermionic atoms, not massless photons. The book begins with a review of Fock states for systems of identical atoms, where large numbers of atoms occupy the various single-particle states or modes. First, separate modes are considered, and here the quantum density operator is represented by a phase space distribution function of phase space variables which replace mode annihilation and creation operators, the dynamical equation for the density operator determines a Fokker–Planck equation for the distribution function, and measurable quantities such as quantum correlation functions are given as phase space integrals. Finally, the phase space variables are replaced by time-dependent stochastic variables satisfying Langevin stochastic equations obtained from the Fokker–Planck equation, with stochastic averages giving the measurable quantities. Second, a quantum field approach is then treated, the density operator being represented by a distribution functional of field functions which replace field annihilation and creation operators, the distribution functional satisfying a functional Fokker–Planck equation, etc. A novel feature of this book is that the phase space variables for fermions are Grassmann variables, not c-numbers. However, the book shows that Grassmann distribution functions and functionals still provide equations for obtaining both analytic and numerical solutions.
First principles simulations of the quantum dynamics of interacting Bose gases using the stochastic gauge representation are analysed. In a companion paper, we showed how the positive-P representation can be applied to these problems using stochastic differential equations. That method, however, is limited by increased sampling error as time evolves. Here, we show how the sampling error can be greatly reduced and the simulation time significantly extended using stochastic gauges. In particular, local stochastic gauges (a subset) are investigated. Improvements are confirmed in numerical calculations of single-, double- and multi-mode systems in the weak-mode coupling regime. Convergence issues are investigated, including the recognition of two modes by which stochastic equations produced by phase-space methods in general can diverge: movable singularities and a noise-weight relationship. The example calculated here displays wave-like behaviour in spatial correlation functions propagating in a uniform 1D gas after a sudden change in the coupling constant. This could in principle be tested experimentally using Feshbach resonance methods.
In quantum optics, the most interesting processes occur due to nonlinear interactions between the electromagnetic field and various media. The parametric processes which produce nonclassical states of the electromagnetic field have analogs in the processes of photoassociation and dissociation of atomic and molecular Bose-Einstein condensates. Just as the optical processes need a full quantum theoretical treatment in some regimes, we show that photoassociation and dissociation require full quantum treatments to be confident of the theoretical predictions we can make. If the processes take place inside an optical cavity, we show that we may also need to consider the quantum nature of the intracavity electromagnetic field. As the process of intracavity photoassociation that we analyze has no exact mapping onto a Fokker-Planck equation, we show how quantum-field-theoretical techniques can be used to write stochastic difference equations in a generalized positive-P representation, which may be solved numerically.
We review some modelling and numerical aspects in quantum kinetic theory for a gas of interacting bosons and we try to explain what makes Bose-Einstein condensation in a dilute gas mathematically interesting and numerically challenging. Particular care is devoted to the development of efficient numerical schemes for the quantum Boltzmann equation that preserve the main physical features of the continuous problem, namely conservation of mass and energy, the entropy inequality and generalized Bose-Einstein distributions as steady states. These properties are essential in order to develop numerical methods that are able to capture the challenging phenomenon of bosons condensation. We also show that the resulting schemes can be evaluated with the use of fast algorithms. In order to study the evolution of the condensate wave function the Gross-Pitaevskii equation is presented together with some schemes for its efficient numerical solution. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.