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arXiv:2104.05039 · Physical Review A

Solvable dilation model of time-dependent PT -symmetric systems

Minyi Huang, Ray‐Kuang Lee, Qinghai Wang, Guo-Qiang Zhang, Junde Wu

Published Jun 7, 2022• 10 citations
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Abstract

The dilation method is a practical way to experimentally simulate non-Hermitian, especially $\mathcal{PT}$-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 $\mathcal{PT}$-symmetric Hamiltonians. Our system is initially set in the unbroken $\mathcal{PT}$-symmetric phase, then goes across the so-called exceptional point, and ends in the broken $\mathcal{PT}$-symmetric phase. For this system, the dilated Hamiltonian and the evolution of $\mathcal{PT}$-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.

DOI
10.1103/physreva.105.062205

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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.
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.
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.
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.
In this work, we explore the PT\mathcal{PT}PT-symmetric quantum Rabi model, which describes a PT\mathcal{PT}PT-symmetric qubit coupled to a quantized light field. By employing the adiabatic approximation (AA), we are able to solve this model analytically in the parameter regime of interest and analyze various physical aspects. We investigate the static and dynamic properties of the model, using both the AA and numerical diagonalization. Our analysis reveals a multitude of exceptional points (EPs) that are closely connected with the exactly solvable points in the Hermitian counterpart of the model. Intriguingly, these EPs vanish and revive depending on the light-matter coupling strength. Furthermore, we discuss the time evolution of physical observables under the non-Hermitian Hamiltonian. Rich and exotic behaviors are observed in both strong and ultrastrong coupling regimes. Our work extends the theory of PT\mathcal{PT}PT symmetry into the full quantum light-matter interaction regime and provides insights that can be readily enlarged to a broad class of quantum optical 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.
The internal nonlocality in PT\cal PTPT-symmetric systems
In this paper, we investigate the properties of PT\cal PTPT-symmetric quantum systems, proposing a new way to illustrate its novelty and measure its departure from classical and Hermitian quantum systems. We argue that there exists some internal nonlocal structure when simulating PT\cal PTPT-symmetric quantum systems. This is done by unveiling its merit in a concrete scenario. In addition, by using the dilation method, it is clearly shown that there exists some bounds of the correlation expectations. Moreover, it is shown that the exceptional point is some extreme point in this case. When the parameters are adjusted and the Hamiltonian tends to the exceptional point, it also tends to a largest departure from the usual Hermitian systems. This also provides us with a new perspective to discuss the properties of the exceptional point.