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Europhysics Letters (EPL)

Quantum-dynamical phase transition and Fisher information in a non-Hermitian Bose-Hubbard dimer

Yixiao Huang, Heng-Na Xiong, Wei Zhong, Zheng-Da Hu, En-Jia Ye

Published Apr 1, 2016• 11 citations
Publisher

Abstract

We study the dynamical properties in a non-Hermitian Bose-Hubbard dimer and show that the steady state consists mostly of the eigenstate whose eigenvalue has the largest imaginary part. A sharp phase transition occurs in the steady state and the phase transition occurs for a finite particle number, not for infinite ones, in contrast to the Hermitian system. We also investigate the quantum Fisher information and entanglement in two different phases. The results show that the steady state in the Josephson oscillation regime is fully N -particle entangled and the corresponding parameter sensitivity approaches the Heisenberg limit. In the self-trapping regime, the parameter sensitivity just scales as the shot-noise limit. Moreover, quantum Fisher information of the steady state is robust to the initial state, which indicates that the non-Hermitian dynamics takes more advantage than the Hermitian one for quantum metrology.

DOI
10.1209/0295-5075/114/20002

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We study the dynamical properties in a non-Hermitian Bose-Hubbard dimer and show that the steady state consists mostly of the eigenstate whose eigenvalue has the largest imaginary part. A sharp phase transition occurs in the steady state and the phase transition occurs for a finite particle number, not for infinite ones, in contrast to the Hermitian system. We also investigate the quantum Fisher information and entanglement in two different phases. The results show that the steady state in the Josephson oscillation regime is fully N -particle entangled and the corresponding parameter sensitivity approaches the Heisenberg limit. In the self-trapping regime, the parameter sensitivity just scales as the shot-noise limit. Moreover, quantum Fisher information of the steady state is robust to the initial state, which indicates that the non-Hermitian dynamics takes more advantage than the Hermitian one for quantum metrology.
Quantum Fisher information, which plays a central role in parameter estimation theory, is a useful quantity for metrological applications. We use quantum Fisher information to distinguish and characterize behaviors of the evolved state for Bose-Einstein condensates in a symmetric double well, which display a classical bifurcation and a transition from Josephson oscillation to self-trapping. We find that the precision of the parameter estimation on the evolved state in the unstable regime is much higher than that in the stable regime, which lies at the shot-noise level. We also show that the parameter sensitivity of the evolved state approaches the Heisenberg limit in the Josephson oscillation regime, while it is just scaled as the shot-noise limit in the self-trapping regime. Finally, we discuss the relation between the parameter sensitivity of the evolved state and the entanglement entropy.
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We investigate an N-particle Bose-Hubbard dimer with an additional effective decay term in one of the sites. A mean-field approximation for this non-Hermitian many-particle system is derived, based on a coherent state approximation. The resulting nonlinear, non-Hermitian two-level dynamics, in particular, the fixed point structures showing characteristic modifications of the self-trapping transition, are analyzed. The mean-field dynamics is found to be in reasonable agreement with the full many-particle evolution.
In this paper we consider a bosonic Josephson junction described by a two-mode Bose-Hubbard model, and we thoroughly analyze a quantum phase transition occurring in the system in the limit of infinite bosonic population. We discuss the relation between this quantum phase transition and the dynamical bifurcation occurring in the spectrum of the discrete self-trapping equations describing the system at the semiclassical level. In particular, we identify five regimes depending on the strength of the effective interaction among bosons, and study the finite-size effects arising from the finiteness of the bosonic population. We devote special attention to the critical regime which reduces to the dynamical bifurcation point in the thermodynamic limit of infinite bosonic population. Specifically, we highlight an anomalous scaling in the population imbalance between the two wells of the trapping potential, as well as in two quantities borrowed from quantum information theory, i.e., the entropy of entanglement and the ground-state fidelity. Our analysis is not limited to the zero-temperature case, but considers thermal effects as well.
Quantum Fisher information bounds the ultimate precision limit in the parameter estimation and has important applications in quantum metrology. In recent years, the theoretical and experimental studies of non-Hermitian Hamiltonians realized in quantum systems have attracted wide attention. Here, the parameter estimation based on eigenstates of non-Hermitian Hamiltonians is investigated, and the corresponding quantum Fisher information and quantum Cramér-Rao bound for the single-parameter and two-parameter estimations are given. In particular, the quantum Fisher information about estimating intrinsic momentum and external parameters in the non-reciprocal and gain-and-loss Su-Schrieffer-Heeger models, and non-Hermitian versions of the quantum Ising chain, Chern-insulator model and two-level system are calculated and analyzed. For these non-Hermitian models, the results show that in the case of single-parameter estimation in these non-Hermitian models, the quantum Fisher information increases significantly in the gapless regime and near the exceptional points, which can improve the accuracy limit of parameter estimation. For the two-parameter estimation, the determinant of the quantum Fisher information matrix also increases obviously near the gapless and exceptional points. In addition, a higher overall accuracy can be achieved in the topological regime than in the trivial regime, and the topological bound in two-parameter estimation can be determined by the Chern number.
Dynamical quantum phase transitions in non-Hermitian systems pose fundamental challenges due to the intrinsic biorthogonality of their eigenstates. In this work, we extend a biorthogonal framework to investigate dynamical quantum phase transitions in non-Hermitian topological superconductors. Taking the non-Hermitian Kitaev chain as a prototypical model, we construct an associated-state formalism and reformulate the Loschmidt rate function, dynamical topological order parameter, and dynamical Fisher zeros. Within this framework, we find that the critical times at which dynamical quantum phase transitions occur differ from those based on the conventional self-normal approaches. We further analyze momentum-resolved subsystems at critical momenta and demonstrate the robustness of the biorthogonal framework. Our work highlights the essential role of biorthogonality in nonequilibrium dynamics and establishes a consistent theoretical framework for dynamical quantum phase transitions in non-Hermitian topological superconductors.