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Physical Review E

Decay of Loschmidt echo in a Bose-Einstein condensate at a dynamical phase transition

Qiang Zheng, Wen-ge Wang, Pinquan Qin, Ping Wang, Xiaoping Zhang, Zhongzhou Ren

Published Jul 30, 2009• 14 citations
Publisher

Abstract

We study the quantum Loschmidt echo (LE) in a Bose-Einstein condensate (BEC) in a double-well potential. The BEC may undergo a dynamical phase transition between two phases: a tunneling phase and a self-trapping phase. For sufficiently weak perturbation, the LE has Gaussian decay in both phases. While, for relatively strong perturbation, the LE has a Gaussian decay in the self-trapping phase and has a stretched exponential decay in the tunneling phase. This qualitative difference in the decaying law of the LE in the two phases provides a characterization of the dynamical phase transition of the BEC. The semiclassical theory is used to explain the numerically observed behaviors of the LE decay.

DOI
10.1103/physreve.80.016214

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We study the quantum Loschmidt echo (LE) in a Bose-Einstein condensate (BEC) in a double-well potential. The BEC may undergo a dynamical phase transition between two phases: a tunneling phase and a self-trapping phase. For sufficiently weak perturbation, the LE has Gaussian decay in both phases. While, for relatively strong perturbation, the LE has a Gaussian decay in the self-trapping phase and has a stretched exponential decay in the tunneling phase. This qualitative difference in the decaying law of the LE in the two phases provides a characterization of the dynamical phase transition of the BEC. The semiclassical theory is used to explain the numerically observed behaviors of the LE decay.
The quantum coherence of a Bose-Einstein condensate is studied using the concept of quantum fidelity (Loschmidt echo). The condensate is confined in an elongated anharmonic trap and subjected to a small random potential such as that created by a laser speckle. Numerical experiments show that the quantum fidelity stays constant until a critical time, after which it drops abruptly over a single trap oscillation period. The critical time depends logarithmically on the number of condensed atoms and on the perturbation amplitude. This behavior may be observable by measuring the interference fringes of two condensates evolving in slightly different potentials.
By solving numerically the tunneling quantum dynamics between two trapped Bose-Einstein condensates (BEC's), we find two distinct time scales. On the short time scale, we recover semiclassical predictions such as ``macroscopic quantum self-trapping'' and the ``amplitude transition,'' first studied in the polaron context, and the $``\ensuremath{\pi}$ states,'' previously discovered in the context of BEC's. On a much longer time scale, quantum dynamics shows that self-trapping is destroyed in contrast to semiclassical behavior. However, this time scale increases exponentially with the total number of condensate atoms, indicating that self-trapping and $``\ensuremath{\pi}$ states'' would be relevant and experimentally observable in the tunneling of BEC's.
Identifying dynamical signatures of excited-state quantum phase transitions (ESQPTs) in experimentally realizable quantum many-body systems is helpful for understanding the dynamical effects of ESQPTs. In such systems, the highly controllable spinor Bose-Einstein condensates (BECs) offer an exceptional platform to study ESQPTs. In this work, we investigate the dynamical characteristics of the ESQPT in spin-1 BEC by means of the Loschmidt echo spectrum. The Loschmidt echo spectrum is an extension of the well-known Loschmidt echo and defined as the overlaps between the evolved state and the excited states of the initial Hamiltonian. We show that both the time-evolved and long-time-averaged Loschmidt echo spectrum undergo a remarkable change as the system passes through the critical point of the ESQPT. Moreover, the particular behavior exhibited by the Loschmidt echo spectrum at the critical point stands as a dynamical detector for probing the ESQPT. We further demonstrate how to capture the features of the ESQPT by using the energy distribution associated with the Loschmidt echo spectrum for time-evolved and long-time-averaged cases, respectively. Our findings contribute to a further verification of the usefulness of the Loschmidt echo spectrum for witnessing various quantum phase transitions in many-body systems and provide a different way to experimentally examine the dynamical consequences of ESQPTs.
We address the sensitivity of quantum mechanical time evolution by considering the time decay of the Loschmidt echo (LE) (or fidelity) for local perturbations of the Hamiltonian. Within a semiclassical approach we derive analytical expressions for the LE decay for chaotic systems for the whole range from weak to strong local perturbations and identify different decay regimes which complement those known for the case of global perturbations. For weak perturbations a Fermi-golden-rule (FGR) type behavior is recovered. For strong perturbations the escape-rate regime is reached, where the LE decays exponentially with a rate independent of the perturbation strength. The transition between the FGR regime and the escape-rate regime is non-monotonic, i.e. the rate of the exponential time-decay of the LE oscillates as a function of the perturbation strength. We further perform extensive quantum mechanical calculations of the LE based on numerical wave packet evolution which strongly support our semiclassical theory. Finally, we discuss in some detail possible experimental realizations for observing the predicted behavior of the LE.
The effect of decoherence on the phase transition of a Bose—Einstein condensate in a symmetric double-well potential is determined by the mean atom number difference. It still has two phases, the tunneling phase and the self-trapping phase, even under decoherence. The density matrix and the operator fidelity also show very different behaviors in the two phases. This suggests that operator fidelity can be used to characterize the phase transition of this Bose—Einstein condensate model, even under decoherence.
We consider a Bose-Einstein condensate in a double-well potential undergoing a dynamical transition from the regime of Josephson oscillations to the regime of self-trapping. We analyze the statistical properties of the ground state (or the highest excited state) of the Hamiltonian in these two regimes for attractive (repulsive) interactions. We demonstrate that it is impossible to describe the transition within the mean-field theory. In contrast, the transition proceeds through a strongly correlated delocalized state, with large quantum fluctuations, and spontaneous breaking of the symmetry.