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Physical Review A

Unavoidability of nonclassicality loss in PT -symmetric systems

Jan Peřina, Adam Miranowicz, Joanna K. Kalaga, W. Leoński

Published Sep 15, 2023• 8 citations
Publisher

Abstract

The authors study the dynamics of two nonlinearly coupled oscillator modes in a $\mathcal{P}\phantom{\rule{0}{0ex}}\mathcal{T}$-symmetric system, in which gain and loss are provided by coupling a pair of bosonic modes to separate baths. The authors show that even when the model parameters are fine-tuned to achieve a balance between gain and loss in the semiclassical limit, the fluctuations of the bath lead to a gradual loss of nonclassicality and entanglement between the modes.

DOI
10.1103/physreva.108.033512

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Physical Review A•Nov 11, 2019•43 citations
The authors study the dynamics of two nonlinearly coupled oscillator modes in a \mathcal{P}\phantom{\rule{0}{0ex}}\mathcal{T}-symmetric system, in which gain and loss are provided by coupling a pair of bosonic modes to separate baths. The authors show that even when the model parameters are fine-tuned to achieve a balance between gain and loss in the semiclassical limit, the fluctuations of the bath lead to a gradual loss of nonclassicality and entanglement between the modes.
We show that the loss of nonclassicality (including quantum entanglement) cannot be compensated by the (incoherent) amplification of PT-symmetric systems. We address this problem by manipulating the quantum fluctuating forces in the Heisenberg-Langevin approach. Specifically, we analyze the dynamics of two nonlinearly coupled oscillator modes in a PT-symmetric system. An analytical solution allows us to separate the contribution of reservoir fluctuations from the evolution of quantum statistical properties of the modes. In general, as reservoir fluctuations act constantly, the complete loss of nonclassicality and entanglement is observed for long times. To elucidate the role of reservoir fluctuations in a long-time evolution of nonclassicality and entanglement, we consider and compare the predictions from two alternative models in which no fatal long-time detrimental effects on the nonclassicality and entanglement are observed. This is so as, in the first semiclassical model, no reservoir fluctuations are considered at all. This, however, violates the fluctuation-dissipation theorem. The second, more elaborated, model obeys the fluctuation-dissipation relations as it partly involves reservoir fluctuations. However, to prevent from the above long-time detrimental effects, the reservoir fluctuations have to be endowed with the nonphysical properties of a sink model. In both models, additional incorporation of the omitted reservoir fluctuations results in their physically consistent behavior. This behavior, however, predicts the gradual loss of the nonclassicality and entanglement. Thus the effects of reservoir fluctuations related to damping cannot be compensated by those related to amplification.
Quantum behavior of two oscillator modes, with mutually balanced gain and loss and coupled via linear coupling (including energy conserving as well as energy non-conserving terms) and nonlinear cross-Kerr effect, is investigated. Stationary states are found and their stability analysis is given. Exceptional points for PT -symmetric cases are identified. Quantum dynamics treated by the model of linear operator corrections to a classical solution reveals nonclassical properties of individual modes (squeezing) as well as their entanglement.
The Hamiltonian for a PT\mathcal{PT}PT-symmetric chain of coupled oscillators is constructed. It is shown that if the loss-gain parameter $\ensuremath{\gamma}$ is uniform for all oscillators, then as the number of oscillators increases, the region of unbroken PT\mathcal{PT}PT symmetry disappears entirely. However, if $\ensuremath{\gamma}$ is localized in the sense that it decreases for more distant oscillators, then the unbroken PT\mathcal{PT}PT-symmetric region persists even as the number of oscillators approaches infinity. In the continuum limit the oscillator system is described by a PT\mathcal{PT}PT-symmetric pair of wave equations, and a localized loss-gain impurity leads to a pseudobound state. It is also shown that a planar configuration of coupled oscillators can have multiple disconnected regions of unbroken PT\mathcal{PT}PT symmetry.
A pair of coupled quantum harmonic oscillators, one subject to a gain one to a loss, is a paradigmatic setup to implement PT-symmetric, non-Hermitian Hamiltonians in that one such Hamiltonian governs the mean-field dynamics for equal gain and loss strengths. Through a full quantum description (so as to account for quantum noise) here is shown that when the system starts in any two-mode coherent state, including vacuum, there appear quantum correlations (QCs) without entanglement, as measured by the Gaussian discord. When the loss rate is above a threshold, once generated QCs no more decay. This occurs in a wide region of parameters, significantly larger than that where the full quantum dynamics is stable. For equal gain and loss rates, in particular, QCs decay in the exact phase (including the exceptional point) and are stable in the broken phase.
Abstract We study the dynamics of correlations in a paradigmatic setup to observe PT -symmetric physics: a pair of coupled oscillators, one subject to a gain one to a loss. Starting from a coherent state, quantum correlations (QCs) are created, despite the system being driven only incoherently, and can survive indefinitely. Both total and QCs exhibit different scalings of their long-time behavior in the PT -broken/unbroken phase and at the exceptional point (EP). In particular, PT symmetry breaking is accompanied by non-zero stationary QCs. This is analytically shown and quantitatively explained in terms of entropy balance. The EP in particular stands out as the most classical configuration, as classical correlations diverge while QCs vanish.
A two-mode optical parity-time (PT\mathcal{PT}PT) symmetric system, with gain and damping, described by a quantum quadratic Hamiltonian with additional small Kerr-like nonlinear terms, is analyzed from the point of view of nonclassical-light generation. Two kinds of stationary states with different types of (in)stability are revealed. Properties of one of these are related to the presence of semiclassical exceptional points, i.e., exotic degeneracies of the non-Hermitian Hamiltonian describing the studied system without quantum jumps. The evolution of the logarithmic negativity, principal squeezing variances, and sub-shot-noise photon-number correlations, considered as entanglement and nonclassicality quantifiers, is analyzed in the approximation of linear-operator corrections to the classical solution. Suitable conditions for nonclassical-light generation are identified in the oscillatory regime, especially at and around exceptional points that considerably enhance the nonlinear interaction and, thus, the nonclassicality of the generated light. The role of quantum fluctuations, inevitably accompanying attenuation and amplification in the evolution of quantum states, is elucidated. The evolution of the system is analyzed for different initial conditions.