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arXiv:1506.04020 · Physical Review A

Ehrenfest breakdown of the mean-field dynamics of Bose gases

Xizhi Han, Biao Wu

Published Feb 12, 2016• 16 citations
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Abstract

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, $N$ is the number of bosons, and ${c}_{1}$ 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.

DOI
10.1103/physreva.93.023621

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The unstable mean-field dynamics of a Bose gas is shown to break down at time \ensuremathτh=(c1/\ensuremathγ)lnN{\ensuremath{\tau}}_{h}=({c}_{1}/\ensuremath{\gamma})lnN\ensuremathτh​=(c1​/\ensuremathγ)lnN, where \ensuremathγ\ensuremath{\gamma}\ensuremathγ 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 \ensuremathτh{\ensuremath{\tau}}_{h}\ensuremathτh​ 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.
In this work we numerically explore the quantum behavior of a classically unstable relativistic Bose-Einstein condensate (BEC). The main goal is to study the phenomenon of so-called quantum break time which amounts to a significant departure from a semiclassical mean-field description. It has been suggested previously that the existence of Lyapunov instability is crucial for a fast quantum breaking, chaos and scrambling. In order to clarify the issue, we work within the 2-PI effective action formalism and introduce a simple and very widely applicable dynamical criterion for identifying the timescale of quantum breaking. We indeed observe that the fast quantum break time is controlled by the Lyapunov exponent of the unstable BEC.
The mean-field limit of a bosonic quantum many-body system is described by (mostly) nonlinear equations of motion which may exhibit chaos very much in the spirit of classical particle chaos, i.e., by an exponential separation of trajectories in Hilbert space with a rate given by a positive Lyapunov exponent $\ensuremath{\lambda}$. The question now is whether $\ensuremath{\lambda}$ imprints itself onto measurable observables of the underlying quantum many-body system even at finite particle numbers. Using a Bose-Einstein condensate expanding in a shallow potential landscape as a paradigmatic example for a bosonic quantum many-body system, we show that the system loses its coherence at an exponentially fast rate. Furthermore, we show that the rate is given by the Lyapunov exponent associated with the chaotic mean-field dynamics. Finally, we demonstrate that this chaos-induced loss of coherence imprints itself onto the visibility of interference fringes in the total density after time of flight, thus, opening the possibility to measure $\ensuremath{\lambda}$ and with it the interplay between chaos and nonequilibrium quantum matter in a real experiment.
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.
In the evolution of distributions localized around classical equilibrium points, the quantum-classical correspondence breaks down at a time, the so-called quantum breaking, or Ehrenfest time, which is related to the minimal separation of the quantum levels in proximity of the classical equilibrium energy. By studying one-dimensional systems with single- and double-well polynomial potentials, we find that the Ehrenfest time diverges logarithmically with the inverse of the Planck constant whenever the equilibrium point is exponentially unstable. In all the other cases, we have a power law divergence with the exponent determined by the degree of the potential near the equilibrium point.
Describing partially-condensed Bose gases poses a long-standing theoretical challenge. We present exact stochastic Ehrenfest relations for the stochastic projected Gross-Pitaevskii equation, including both number and energy damping mechanisms, and all projector terms that arise from the energy cutoff separating system from reservoir. We test the theory by applying it to the center of mass fluctuations of a harmonically trapped prolate system, finding close agreement between c-field simulations and analytical results. The formalism lays the foundation to analytically explore experimentally accessible hot Bose-Einstein condensates.