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arXiv:1108.2768 · Physical Review A

Quantum phase transition of two-mode Bose-Einstein condensates with an entanglement order parameter

Wei Fan, Yan Xu, Bing Chen, Zhaoyang Chen, Xun‐Li Feng, C. H. Oh

Published Jan 31, 2012• 7 citations
arXivHTMLPDF

Abstract

The ground-state entanglement of the two-mode Bose-Einstein condensate is investigated through a quantum-phase-transition approach. The entanglement measure is taken as the order parameter and this is a nonlocal order parameter, which is different from the conventional order parameter of the Mott-insulator--superfluid-phase transitions. For this nonlocal order parameter, the scaling behavior corresponding to a continuous phase transition is obtained and a power-law divergence near the critical region follows it. This scaling behavior of quantum entanglement is analyzed by the finite-size scaling and the critical exponents are obtained as $\ensuremath{\nu}=1.01$ and $\ensuremath{\gamma}=0.86$. A close connection between quantum fluctuations and the phase transition of entanglement is also obtained.

DOI
10.1103/physreva.85.013645

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The ground-state entanglement of the two-mode Bose-Einstein condensate is investigated through a quantum-phase-transition approach. The entanglement measure is taken as the order parameter and this is a nonlocal order parameter, which is different from the conventional order parameter of the Mott-insulator--superfluid-phase transitions. For this nonlocal order parameter, the scaling behavior corresponding to a continuous phase transition is obtained and a power-law divergence near the critical region follows it. This scaling behavior of quantum entanglement is analyzed by the finite-size scaling and the critical exponents are obtained as \ensuremathν=1.01\ensuremath{\nu}=1.01\ensuremathν=1.01 and \ensuremathγ=0.86\ensuremath{\gamma}=0.86\ensuremathγ=0.86. A close connection between quantum fluctuations and the phase transition of entanglement is also obtained.
We investigate the nonlinear self-trapping phenomenon of a Bose-Einstein condensate (BEC) in a symmetric double well, emphasizing its underlying dynamical phase transition. As the nonlinear parameter characterizing the interaction between the degenerate atoms increases, the BEC becomes self-trapped, manifesting an asymmetric distribution of the atomic density profile. The essence of this phenomenon is revealed to be a continuous phase transition and the underlying critical behavior is studied analytically and found to follow a logarithmic scaling law. We then go beyond the mean-field treatment and extend our treatment to discuss the effect of many-body quantum fluctuations on the transition. It is found that the transition point is shifted and the scaling law is broken. In particular, the quantum phase transition is accompanied by a change of the entanglement entropy, which is found to reach a maximum at the transition point. The underlying physics is revealed.
In this paper, we exploit two-mode model to investigate the influence of quantum fluctuation on the self-trapping of the Bose-Einstein Condensate(BEC). We found that, even for the finite number of particles, the transition to self-trapping is still observed.In particular, because of the quantum fluctuation due to the finite number of particles, the critical phenomenon becomes fuzzily: the smaller the number of particles is, the more obviously fuzzily the critical phenomenon becomes. Quantum entanglement entropy is then introduced to depict the critical behavior near the transition point.
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In this paper we investigate the universality and scaling properties of the well-known quantities in classical statistical mechanics near the quantum phase transition point. We show that transverse susceptibility and derivatives of correlation functions with respect to the parameter that drives the quantum phase transitions, exhibit logarithmic divergence and finite size scaling the same as entanglement. In other words the non-analytic and finite size scaling behaviors of entanglement is not its intrinsic properties and inherit from the non-analytic and scaling behaviors of correlation functions and surveying at least the nearest neighbor correlation functions could specify the scaling and divergence properties of entanglement. However we show that the correlation functions could capture the quantum critical point without pre-assumed order parameters even for the cases where the two-body entanglement is absent.
This article aims to introduce the bosonic asymmetric quantum conditional mutual information (QCMI), IB(A;B∣C)I_{B}(A; B|C)IB​(A;B∣C), as a rigorously defined information-theoretic quantity for continuous-variable (bosonic) systems. After defining IB(A;B∣C)I_{B}(A; B|C)IB​(A;B∣C) through quantum instruments on infinite-dimensional Fock spaces, we prove that Bose-Einstein condensation (BEC) induces a logarithmic divergence $ I_{B}(A; B|C) \geq α\log_2 (N_{0}^C+1)$, which further establishes IB(A;B∣C)I_{B}(A; B|C)IB​(A;B∣C) as a quantum-information order parameter for BEC with power-law scaling near the critical point. We provide the complete five-step proof of this result, connecting measurement back-action, off-diagonal long-range order, and Jensen's inequality. In the condensed phase, the effective causal propagation velocity is set by the Bogoliubov sound speed csc_scs​ in the long-wavelength limit, and vanishes continuously at the superfluid--Mott-insulator transition. Applications to composite bosons, the BEC--BCS crossover, and a quantum-gas-microscope measurement protocol are presented. Finally, we demonstrate how the reduced bosonic QCMI serves as a finite-size scaling probe analogous to the Binder cumulant.
Summary form only given. Phase transitions are among the most dramatic phenomena in nature. Minute variations in the conditions controlling a system trigger a fundamental change of its properties. In the critical regime near a second-order phase transition point the fluctuations of the order parameter are so dominant that they completely govern the behaviour of the system on all length scales. The correlation length of the ultracold gas measures the characteristic length scale of these fluctuations and diverges at the critical temperature. Using cold atomic gases, we have directly observed critical fluctuations of the order parameter near the phase transition of Bose-Einstein condensation. From the divergence of the correlation length of the order parameter versus temperature we have determined its critical exponent v. Despite our densities being ten orders of magnitude smaller and our interactions considerably weaker, we find a remarkable similarity to case of the lambda-transition of helium-4.