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arXiv:0805.0531 · Physical Review Letters

Differences between Mean-Field Dynamics and N -Particle Quantum Dynamics as a Signature of Entanglement

Christoph Weiß, Niklas Teichmann

Published Apr 10, 2008• 80 citations
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Abstract

A Bose-Einstein condensate in a tilted double-well potential under the influence of time-periodic potential differences is investigated in the regime where the mean-field (Gross-Pitaevskii) dynamics become chaotic. For some parameters near stable regions, even averaging over several condensate oscillations does not remove the differences between mean-field and N-particle results. While introducing decoherence via piecewise deterministic processes reduces those differences, they are due to the emergence of mesoscopic entangled states in the chaotic regime.

DOI
10.1103/physrevlett.100.140408

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A Bose-Einstein condensate in a tilted double-well potential under the influence of time-periodic potential differences is investigated in the regime where the mean-field (Gross-Pitaevskii) dynamics become chaotic. For some parameters near stable regions, even averaging over several condensate oscillations does not remove the differences between mean-field and N-particle results. While introducing decoherence via piecewise deterministic processes reduces those differences, they are due to the emergence of mesoscopic entangled states in the chaotic regime.
For a Bose–Einstein condensate in a double-well potential, time-periodic shaking can lead to the emergence of non-classical mesoscopic superpositions on very short timescales. It is suggested to experimentally detect these deviations from mean-field (Gross–Pitaevskii) behaviour by investigating both the variance of the particle number and the interference pattern. The disappearance of the interference pattern followed by its reappearance a short time later could serve as an experimental signature of mesoscopic entanglement.
In a parameter regime for which the mean-field (Gross-Pitaevskii) dynamics becomes chaotic, mesoscopic quantum superpositions in phase space can occur in a double-well potential, which is shaken periodically. For experimentally realistic initial states, such as the ground state of some 100 atoms, the emergence of mesoscopic quantum superpositions in phase space is investigated numerically. It is shown to be reproducible, even if the initial conditions change slightly. Although the final state is not a perfect superposition of two distinct phase states, the superposition is reached an order of magnitude faster than in the case of the collapse-and-revival phenomenon. Furthermore, a generator of entanglement is identified.
A recent publication [Phys. Rev. Lett. 100, 140408 (2008)] shows that there is a relation between mean-field chaos and multi-particle entanglement for BECs in a periodically shaken double well. “Schrödingercat” like mesoscopic superpositions in phase-space occur for conditions for which the system displays mean-field chaos. In the present manuscript, more general highly-entangled states are investigated. Mean-field chaos accelerates the emergence of multi-particle entanglement; the boundaries of stable regions are particularly suited for entanglement generation.
For the dynamics of Bose-Einstein condensates (BECs), differences between mean-field (Gross-Pitaevskii) physics and NNN-particle quantum physics often disappear if the BEC becomes larger and larger. In particular, the time scale for which both dynamics agree should thus become larger if the particle number increases. For BECs in a double-well potential, we find both examples for which this is the case and examples for which differences remain even for huge BECs on experimentally realistic short time scales. By using a combination of numerical and analytical methods, we show that the differences remain visible on the level of expectation values even beyond the largest possible numbers realized experimentally for BECs with ultracold atoms.
Dynamical phase transition of Bose-Einstein condensates (BEC) in double-well trap with external periodic modulation applied to the interaction term is investigated in this paper. Particularly, phase transition to chaos is studied. It is found that, when resonance between the modulation frequency and the natural frequency of the system happens, the system turns to instability (chaos) at certain values of the parameters. In the chaotic region, distribution of particles is random, and the average of population difference fluctuates around zero. Especially, chaos can be manifested by the entanglement entropy. In the chaotic zone, both of the average entanglement entropies approach to their maxima.
We discuss the dynamics of Bose-Einstein condensates in a double-well potential subject to decoherence (or particle loss). Starting from the full many-body dynamics described by the master equation, an effective Gross-Pitaevskii-like equation is derived in the mean-field approximation. By numerically solving the GP equation, we find that macroscopic quantum self-trapping disappears for strong decoherence, while generalized self-trapping occurs under weak decoherence. The fixed points have been calculated, and we find that an abrupt change from elliptic to an attractor and a repeller occurs, reflecting the metastable behavior of the system around these points.