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Brazilian Journal of Physics

Quantum atom optics with trapped Bose-Einstein condensates

M. K. Olsen

Published Dec 1, 2004• 2 citations
Publisher

Abstract

Both Bose-Einstein condensates and optical fields are composed of bosons, so that the majority of the processes which have long been studied in quantum and nonlinear optics have equivalents in the field of Bose-Einsten condensation. However, due to the masses of the condensed atoms, the confining potentials and the huge collisional nonlinearities, the simpler theoretical approaches common to quantum optics can sometimes give misleading answers when applied to condensates. In this work we describe some of the areas where simplified treatments can be misleading, and compare and contrast the predictions of quantum many-body treatments with those of the single-mode type treatments which have been so successful in quantum optics.

DOI
10.1590/s0103-97332004000700030

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Both Bose-Einstein condensates and optical fields are composed of bosons, so that the majority of the processes which have long been studied in quantum and nonlinear optics have equivalents in the field of Bose-Einsten condensation. However, due to the masses of the condensed atoms, the confining potentials and the huge collisional nonlinearities, the simpler theoretical approaches common to quantum optics can sometimes give misleading answers when applied to condensates. In this work we describe some of the areas where simplified treatments can be misleading, and compare and contrast the predictions of quantum many-body treatments with those of the single-mode type treatments which have been so successful in quantum optics.
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.
In this tutorial we present an introduction to some theoretical methods of quantum field theory applied to the description of a trapped Bose–Einstein condensate. First of all, we give a brief account of the main characteristics of the phenomenon of condensation and present the many-body Hamiltonian of the system. We outline some of the most important approaches used in the characterization of a condensed Bose gas, including the mean-field theory and the Hartree–Fock–Bogoliubov method. Finally we illustrate the use of these techniques addressing some important issues in quantum atom optics. We characterize the quantum state of a Bose–Einstein condensate (BEC) at zero temperature. We also describe a process of Beliaev coupling between quasiparticles using a method that includes terms beyond the usual Bogoliubov approach.
Summary form only given. Since the experimental realization of Bose-Einstein condensation in the magnetically trapped quantum atomic gases, there have now been an enormous quantity of theory devoted to the properties of the trapped Bose-Einstein condensates (BEC). Motivated by the existing experiments, the bulk of the theoretical work has assumed a harmonic trap. Although recently a few papers have considered the consequences of exposing a BEC to a periodic optical potential, BEC confined in an optical lattice is still a new topic in this area. In the work, we develop theory to study the stationary and propagating properties of a Bose-Einstein condensate in an optical lattice. Instead of using an infinite optical lattice, we study the realistic case where the BEC is formed in a finite trapping potential and then is exposed to an infinite periodic potential created by off-resonance multiple laser beams.