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arXiv:cond-mat/0106427 · Physical Review A

Quantum field effects in coupled atomic and molecular Bose-Einstein condensates

J. J. Hope

Published Oct 8, 2001• 20 citations
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Abstract

This paper examines the parameter regimes in which coupled atomic and molecular Bose-Einstein condensates do not obey the Gross-Pitaevskii equation. Stochastic field equations for coupled atomic and molecular condensates are derived using the functional positive-$P$ representation. These equations describe the full quantum state of the coupled condensates and include the commonly used Gross-Pitaevskii equation as the noiseless limit. The model includes all interactions between the particles, background gas losses, two-body losses, and the numerical simulations are performed in three dimensions. It is found that it is possible to differentiate the quantum and semiclassical behavior when the particle density is sufficiently low and the coupling is sufficiently strong.

DOI
10.1103/physreva.64.053608

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This paper examines the parameter regimes in which coupled atomic and molecular Bose-Einstein condensates do not obey the Gross-Pitaevskii equation. Stochastic field equations for coupled atomic and molecular condensates are derived using the functional positive-PPP representation. These equations describe the full quantum state of the coupled condensates and include the commonly used Gross-Pitaevskii equation as the noiseless limit. The model includes all interactions between the particles, background gas losses, two-body losses, and the numerical simulations are performed in three dimensions. It is found that it is possible to differentiate the quantum and semiclassical behavior when the particle density is sufficiently low and the coupling is sufficiently strong.
We show that in certain parameter regimes there is a macroscopic dynamical breakdown of the Gross-Pitaevskii equation. Stochastic field equations for coupled atomic and molecular condensates are derived using the functional positive- P representation. These equations describe the full quantum state of the coupled condensates and include the commonly used Gross-Pitaevskii equation as the noiseless limit. The full quantum theory includes the spontaneous processes which will become significant when the atomic population is low. The experimental signature of the quantum effects will be the time scale of the revival of the atomic population after a near total conversion to the molecular condensate.
We use numerical integration of coupled differential equations in a truncated Wigner representation to show that the initial quantum state of an atomic Bose-Einstein condensate plays an important role in the process of Raman photoassociation to form a molecular condensate. Comparing the predictions for different initial quantum states, we find that, even when the stochastic prediction for an initial coherent state is close to the Gross-Pitaevskii prediction, other possible states may result in very different dynamics. The mean-field dynamics can be appreciably different, depending on the initial conditions.
We study the dynamics of the atomic and molecular Bose-Einstein condensates (BECs) of 87Rb^{87}\text{R}\text{b}87Rb in a spherically symmetric trap coupled by stimulated Raman photoassociation process. Considering the higher order nonlinearity in the atom-atom interaction we analyze the dynamics of the system using coupled modified Gross-Pitaevskii (MGP) equations and compare it with mean-field coupled Gross-Pitaevskii (GP) dynamics. Considerable differences in the dynamics are obtained in these two approaches at large scattering length, i.e., for large values of peak-gas parameter ${x}_{pk}\ensuremath{\ge}{10}^{\ensuremath{-}3}$. We show how the dynamics of the coupled system is affected when the atom-molecule and molecule-molecule interactions are considered together with the atom-atom interaction and also when the strengths of these three interactions are increased. The effect of detuning on the efficiency of conversion of atomic fractions into molecules is demonstrated and the feasibility of maximum molecular BEC formation by varying the Raman detuning parameter at different values of time is explored. Thus by varying the Raman detuning and the scattering length for atom-atom interaction one can control the dynamics of the coupled atomic-molecular BEC system. We have also solved coupled Gross-Pitaevskii equations for atomic to molecular condensate formation through magnetic Feshbach resonance in a BEC of 85Rb^{85}\text{R}\text{b}85Rb. We found similar features for oscillations between atomic and molecular condensates noted in previous theoretical study and obtained fairly good agreement with the evolution of total atomic condensate observed experimentally.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.
An atomic Bose-Einstein condensate (BEC) is often described as a macroscopic object which can be approximated by a coherent state. This, on the surface, would appear to indicate that its behavior should be close to being classical. In this paper, we clarify the extent of how "classical" a BEC is by exploring the semiclassical equations for BECs under the mean field Gaussian approximation. Such equations describe the dynamics of a condensate in the classical limit in terms of the variables < x > and < p > as well as their respective variances. We compare the semiclassical solution with the full quantum solution based on the Gross-Pitaevskii Equation (GPE) and find that the interatomic interactions which generate nonlinearity make the system less "classical." On the other hand, many qualitative features are captured by the semiclassical equations, and the equations to be solved are far less computationally intensive than solving the GPE which make them ideal for providing quick diagnostics, and for obtaining new intuitive insight.
Recent experiments have been moving ever closer to the goal of creating cold quantum-degenerate Bose gases in one and two dimensional systems. In this thesis we investigate such systems by adapting the mean-field theories used so successfully to describe Bose-Einstein condensates in the three dimensional case. We find that the key to describing the low dimensional systems lies in an accurate description of the interactions within those systems. In three dimensions the interactions may be well approximated by a simple form in which atoms are assumed to interact in a vacuum. Many-body effects on the scattering then lead to only a small perturbation. However, in one and two dimensions we find that the inclusion of many-body effects on the scattering is crucial, even for a first order calculation. We use a many-body T-matrix to describe the interactions, and develop a simple approximation in the zero-temperature limit that involves the off-shell two-body T-matrix evaluated at a shifted interaction energy. We therefore derive expressions for the off-shell two-body T-matrix in one, two, and three dimensions, and so find expressions for the coupling parameters which describe interactions in one and two dimensional Bose gases. These lead to interaction terms in the one and two dimensional Gross-Pitaevskii equations which differ significantly from the three dimensional case. We present numerical solutions of the Gross-Pitaevskii equation in both one and two dimensions, and in one dimension we find that the results are in agreement with exact results in the extreme limits of the interaction strength. We also investigate higher order theories in the one-dimensional case, with a view to describing the cross-over from a weakly-interacting Bose-Einstein condensate to a quasicondensate as the interactions become stronger. Finally, we study the Bogoliubov-de Gennes excitation spectrum for a one dimensional condensate containing solitons. We show that it has an anomalous mode with a negative energy, and that this represents the collective motion of the soliton relative to the background condensate. (author)