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arXiv:physics/0105072 · Physical Review A

Dynamics of a two-mode Bose-Einstein condensate beyond mean-field theory

J. R. Anglin, Amichay Vardi

Published May 30, 2001• 183 citations
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Abstract

We study the dynamics of a two-mode Bose-Einstein condensate in the vicinity of a mean-field dynamical instability. Convergence to mean-field theory (MFT), with increasing total number of particles N, is shown to be logarithmically slow. Using a density-matrix formalism rather than the conventional wave-function methods, we derive an improved set of equations of motion for the mean-field plus the fluctuations, which goes beyond MFT and provides accurate predictions for the leading quantum corrections and the quantum break time. We show that the leading quantum corrections appear as decoherence of the reduced single-particle quantum state; we also compare this phenomenon to the effects of thermal noise. Using the rapid dephasing near an instability, we propose a method for the direct measurement of scattering lengths.

DOI
10.1103/physreva.64.013605

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We study the dynamics of a two-mode Bose-Einstein condensate in the vicinity of a mean-field dynamical instability. Convergence to mean-field theory (MFT), with increasing total number of particles N, is shown to be logarithmically slow. Using a density-matrix formalism rather than the conventional wave-function methods, we derive an improved set of equations of motion for the mean-field plus the fluctuations, which goes beyond MFT and provides accurate predictions for the leading quantum corrections and the quantum break time. We show that the leading quantum corrections appear as decoherence of the reduced single-particle quantum state; we also compare this phenomenon to the effects of thermal noise. Using the rapid dephasing near an instability, we propose a method for the direct measurement of scattering lengths.
We study the system of coupled atomic and molecular condensates within the two-mode model and beyond mean-field theory. Large-amplitude atom-molecule coherent oscillations are shown to be damped by the rapid growth of fluctuations near the dynamically unstable molecular mode. This result contradicts earlier predictions about the recovery of atom-molecule oscillations in the two-mode limit. The frequency of the damped oscillation is also shown to scale as N/lnN\sqrt{N}/\mathrm{ln}NN​/lnN with the total number of atoms N, rather than the expected pure N\sqrt{N}N​ scaling. Using a linearized model, we obtain analytical expressions for the initial depletion of the molecular condensate in the vicinity of the instability, and show that the important effect neglected by mean-field theory is an initially nonexponential ``spontaneous'' dissociation into the atomic vacuum. Starting with a small population in the atomic mode, the initial dissociation rate is sensitive to the exact atomic amplitudes, with the fastest (superexponential) rate observed for the entangled state formed by spontaneous dissociation.
The dynamics of the Bose–Einstein condensate (BEC) in a double-well potential is often investigated under the mean-field theory (MFT). This works successfully for large particle numbers with dynamical stability. But for dynamical instabilities, quantum corrections to the MFT becomes important [J. R. Anglin and A. Vardi, Phys. Rev. A64, 013605 (2001)]. Recently the adiabatic dynamics of the double-well BEC is investigated under the MFT in terms of a dark variable [C. Ottaviani et al., Phys. Rev. A81, 043621 (2010)], which generalizes the adiabatic passage techniques in quantum optics to the nonlinear matter-wave case. We give a fully quantized version of it using second-quantization and introduce new correction terms from higher order interactions beyond the on-site interaction, which are interactions between the tunneling particle and the particle in the well and interactions between the tunneling particles. If only the on-site interaction is considered, this reduces to the usual two-mode BEC.
A dynamical many-body theory is presented which systematically extends beyond mean-field and perturbative quantum-field theoretical procedures. It allows us to study the dynamics of strongly interacting quantum-degenerate atomic gases. The nonperturbative approximation scheme is based on a systematic expansion of the two-particle irreducible effective action in powers of the inverse number of field components. This yields dynamic equations which contain direct scattering, memory, and ``off-shell'' effects that are not captured by the Gross-Pitaevskii equation. This is relevant to account for the dynamics of, e.g., strongly interacting quantum gases atoms near a scattering resonance, or of one-dimensional Bose gases in the Tonks-Girardeau regime. We apply the theory to a homogeneous ultracold Bose gas in one spatial dimension. Considering the time evolution of an initial state far from equilibrium we show that it quickly evolves to a nonequilibrium quasistationary state and discuss the possibility to attribute an effective temperature to it. The approach to thermal equilibrium is found to be extremely slow.
The unstable mean-field dynamics of a Bose gas is shown to break down at time ${\ensuremath{\tau}}_{h}=({c}_{1}/\ensuremath{\gamma})lnN$, where $\ensuremath{\gamma}$ is the Lyapunov exponent of the mean-field theory, NNN is the number of bosons, and c1{c}_{1}c1​ is a system-dependent constant. The breakdown time ${\ensuremath{\tau}}_{h}$ is essentially the Ehrenfest time that characterizes the breakdown of the correspondence between classical and quantum dynamics. This breakdown can be well described by a quantum fidelity defined for one-particle reduced density matrices. Our results are obtained with the formalism in particle-number phase space and are illustrated with a triple-well model. The logarithmic quantum-classical correspondence time may be verified experimentally with Bose-Einstein condensates.
The fluctuations of a number of particles in the Bose-Einstein condensate are studied in the grand canonical ensemble with an effective single-mode Hamiltonian, which is derived from an assumption that the mode corresponding to the Bose-Einstein condensate does not asymptotically correlate with other modes. The fluctuations are evaluated in the dilute limit with a proposed simple method, which is beyond the mean-field approximation. The accuracy of the latter is estimated; it is shown that the mean-field scheme does not work for the single-mode Hamiltonian, while for the Hartree Hamiltonian it allows us to estimate the condensate fluctuations up to a numerical factor. As a hypothesis, a formula is proposed that relates the fluctuations in the canonical ensemble with that of the grand canonical one.