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Laser Physics

Quantum Dynamics of Trapped Bose-Einstein Condensates

M. K. Olsen, Lev I. Plimak

Published Jan 1, 2003• 1 citation
OpenAlex

Abstract

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.

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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 investigate the process of intracavity photoassociation of Bose-Einstein-condensed atoms to form a molecular condensate. As shown previously, this process can only be successfully described by a quantum treatment of all the interacting fields. We extend our previous work by representing the full quantum aspects of the problem using an extension of the positive P representation to model non-Wiener noises. This allows the mapping of a generalized Fokker-Planck equation with third-order derivatives onto a set of coupled stochastic difference equations. We also investigate parameter regimes not covered previously, as well as the effects of spontaneous dissociation of the condensed molecules.
We demonstrate that a photon blockade effect exists in the intracavity coherent photoassociation of an atomic Bose-Einstein condensate and that the dynamics of the coupled atomic and molecular condensates can only be successfully described by a quantum treatment of all the interacting fields. We show that the usual mean-field calculational approaches give answers that are qualitatively wrong, even for the mean fields. The quantization of the fields gives a degree of freedom that is not present in analogous nonlinear optical processes. The difference between the semiclassical and quantum predictions can actually increase as the three fields increase in size so that there is no obvious classical limit for this process.
Since dilute Bose gas condensates were first experimentally produced, the Gross-Pitaevskii equation has been successfully used as a descriptive tool. As a mean-field equation, it cannot by definition predict anything about the many-body quantum statistics of condensate. We show here that there are a class of dynamical systems where it cannot even make successful predictions about the mean-field behavior, starting with the process of evaporative cooling by which condensates are formed. Among others are parametric processes, such as photoassociation and dissociation of atomic and molecular condensates.
We explore the quantum dynamics of photoassociation of Bose-Einstein condensed atoms into molecules using an optical cavity field. Inside of an optical resonator, photoassociation of quantum degenerate atoms involves the interaction of three coupled quantum fields for the atoms, molecules, and the photons. The feedback created by a high-QQQ optical cavity causes the cavity field to become a dynamical quantity whose behavior is linked in a nonlinear manner to the atoms inside and where vacuum fluctuations have a more important role than in free space. We develop and compare several methods for calculating the dynamics of the atom-molecule conversion process with a coherently driven cavity field. We first present an alternate operator representation for the Hamiltonian from which we derive an improved form of mean field theory and an approximate solution of the Heisenberg-Langevin (HL) equations that properly accounts for quantum noise in the cavity field. It is shown that our improved mean field theory corrects several deficiencies in traditional mean field theory based on expectation values of annihilation and creation operators. Also, we show by direct comparison to numerical solutions of the density matrix equations that our approximate quantum solution of HL equations gives an accurate description of weakly or undriven cavities where mean field theories break down.
We study the photoassociation of Bose-Einstein condensed atoms into molecules using an optical cavity field. The driven cavity field introduces a dynamical degree of freedom into the photoassociation process, whose role in determining the stationary behavior has not previously been considered. The semiclassical stationary solutions for the atom and molecules as well as the intracavity field are found and their stability and scaling properties are determined in terms of experimentally controllable parameters including driving amplitude of the cavity and the nonlinear interactions between atoms and molecules. For weak cavity driving, we find a bifurcation in the atom and molecule number occurs that signals a transition from a stable steady state to nonlinear Rabi oscillations. For a strongly driven cavity, there exists bistability in the atom and molecule number.