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A solvable dilation model of PT -symmetric systems

Minyi Huang, Ray‐Kuang Lee, Guoqiang Zhang, Junde Wu

Published Apr 11, 2021• 1 citation
OpenAlex

Abstract

The dilation method is an important and useful way in experimentally simulating non-Hermitian, especially $\cal PT$-symmetric systems. However, the time dependent dilation problem cannot be explicitly solved in general. In this paper, we consider a special two dimensional time dependent $\cal PT$-symmetric system, which is initially set in the unbroken $\cal PT$-symmetric phase and later goes across the exceptional point and enters the broken $\cal PT$-symmetric phase. For this system, the dilation Hamiltonian and the evolution of $\cal PT$-symmetric system are analytically worked out.

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arXiv:1801.05828•Journal of Physics A Mathematical and Theoretical•May 17, 2018•30 citations
The dilation method is an important and useful way in experimentally simulating non-Hermitian, especially PT\cal PTPT-symmetric systems. However, the time dependent dilation problem cannot be explicitly solved in general. In this paper, we consider a special two dimensional time dependent PT\cal PTPT-symmetric system, which is initially set in the unbroken PT\cal PTPT-symmetric phase and later goes across the exceptional point and enters the broken PT\cal PTPT-symmetric phase. For this system, the dilation Hamiltonian and the evolution of PT\cal PTPT-symmetric system are analytically worked out.
The dilation method is a practical way to experimentally simulate non-Hermitian, especially PT\mathcal{PT}PT-symmetric, quantum systems. However, the time-dependent dilation problem cannot be explicitly solved in general. In this paper, we present a simple yet nontrivial exactly solvable dilation problem for two-dimensional time-dependent PT\mathcal{PT}PT-symmetric Hamiltonians. Our system is initially set in the unbroken PT\mathcal{PT}PT-symmetric phase, then goes across the so-called exceptional point, and ends in the broken PT\mathcal{PT}PT-symmetric phase. For this system, the dilated Hamiltonian and the evolution of PT\mathcal{PT}PT-symmetric system are analytically worked out. By investigating the large-time behaviors, we give an effective method to choose and adjust the dilation parameters. Our result also shows that the exceptional points do not have much physical relevance in a time-dependent system.
A conceptual framework extending (time-independent) PT\mathcal{PT}PT-symmetric quantum mechanics into the time-dependent domain is presented. It is built upon a nontrivial time-dependent metric operator identified here and works for generic finite-dimensional non-Hermitian systems. All the ingredients of our framework, such as the time-dependent Hilbert space, the observable, and the measurement postulate, can be ``realized'' by means of dilating and reinterpreting the non-Hermitian system in question as a part of a larger Hermitian system. Aided by our metric operator, we formulate the concepts of stable and unstable phases for generic non-Hermitian systems and argue that they, respectively, generalize the notions of unbroken and broken phases in time-independent PT\mathcal{PT}PT-symmetric systems. Possible applications of our framework are illustrated with well-known examples in quantum thermodynamics.
In the past years, many efforts have been made to study various noteworthy phenomena in both parity-time (PT)(\mathcal{PT})(PT) and anti-parity-time (APT)(\mathcal{APT})(APT) symmetric systems. However, entanglement dynamics in APT\mathcal{APT}APT-symmetric systems has not previously been investigated in both theory and experiments. Here, we investigate the entanglement evolution of two qubits in an APT\mathcal{APT}APT-symmetric system. In the APT\mathcal{APT}APT-symmetric unbroken regime, our theoretical simulations demonstrate the periodic oscillations of entanglement when each qubit evolves identically, while the nonperiodic oscillations of entanglement when each qubit evolves differently. In particular, when each qubit evolves near the exceptional point in the APT\mathcal{APT}APT-symmetric unbroken regime, there exist entanglement sudden vanishing and revival. Moreover, our simulations demonstrate rapid decay and delayed death of entanglement provided one qubit evolves in the APT\mathcal{APT}APT-symmetric broken regime. In this work, we also perform an experiment with a linear optical setup. The experimental results agree well with our theoretical simulation results. Our findings reveal novel phenomena of entanglement evolution in the APT\mathcal{APT}APT-symmetric system and opens a new direction for future studies on the dynamics of quantum entanglement in multiqubit APT\mathcal{APT}APT-symmetric systems or other non-Hermitian quantum systems.
To effectively realize a PT\mathcal{P}\mathcal{T}PT-symmetric system, one can dilate a PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonian to some global Hermitian one and simulate its evolution in the dilated Hermitian system. However, with only a global Hermitian Hamiltonian, how do we know whether it is a dilation and is useful for simulation? To answer this question, we consider the problem of how to extract the internal nonlocality in the Hermitian dilation. We unveil that the internal nonlocality brings nontrivial correlations between the subsystems. By evaluating the correlations with local measurements in three different pictures, the resulting different expectations of the Bell operator reveal the distinction of the internal nonlocality. When the simulated PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonian approaches its exceptional point, such a distinction tends to be most significant. Our results clearly make a distinction between the Hermitian dilation and other global Hamiltonians without internal nonlocality. They also provide the figure of merit to test the reliability of the simulation, as well as to verify a PT\mathcal{P}\mathcal{T}PT-symmetric (sub)system.
Non-Hermitian, mathcalPT\\mathcal{PT}mathcalPT -symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, mathcalPT\\mathcal{PT}mathcalPT potentials for a non-relativistic particle confined in a circular geometry. We show that the mathcalPT\\mathcal{PT}mathcalPT -symmetry threshold can be tuned by introducing a second gain-loss potential or its Hermitian counterpart. Our results explicitly demonstrate that mathcalPT\\mathcal{PT}mathcalPT breaking in two dimensions has a rich phase diagram, with multiple re-entrant mathcalPT\\mathcal{PT}mathcalPT -symmetric phases.
Abstract We provide exact analytical solutions for a two-dimensional explicitly time-dependent non-Hermitian quantum system. While the time-independent variant of the model studied is in the broken PT-symmetric phase for the entire range of the model parameters, and has therefore a partially complex energy eigenspectrum, its time-dependent version has real energy expectation values at all times. In our solution procedure we compare the two equivalent approaches of directly solving the time-dependent Dyson equation with one employing the Lewis–Riesenfeld method of invariants. We conclude that the latter approach simplifies the solution procedure due to the fact that the invariants of the non-Hermitian and Hermitian system are related to each other in a pseudo-Hermitian fashion, which in turn does not hold for their corresponding time-dependent Hamiltonians. Thus constructing invariants and subsequently using the pseudo-Hermiticity relation between them allows to compute the Dyson map and to solve the Dyson equation indirectly. In this way one can bypass to solve nonlinear differential equations, such as the dissipative Ermakov–Pinney equation emerging in our and many other systems.