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arXiv:1406.5967 · Physical Review A

Systems of coupled PT -symmetric oscillators

Carl M. Bender, Mariagiovanna Gianfreda, S. P. Klevansky

Published Aug 19, 2014• 46 citations
arXivHTMLPDF

Abstract

The Hamiltonian for a $\mathcal{PT}$-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 $\mathcal{PT}$ symmetry disappears entirely. However, if $\ensuremath{\gamma}$ is localized in the sense that it decreases for more distant oscillators, then the unbroken $\mathcal{PT}$-symmetric region persists even as the number of oscillators approaches infinity. In the continuum limit the oscillator system is described by a $\mathcal{PT}$-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 $\mathcal{PT}$ symmetry.

DOI
10.1103/physreva.90.022114

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The Hamiltonian for a PT\mathcal{PT}PT-symmetric chain of coupled oscillators is constructed. It is shown that if the loss-gain parameter \ensuremathγ\ensuremath{\gamma}\ensuremathγ 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 \ensuremathγ\ensuremath{\gamma}\ensuremathγ 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.
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.
All of the PT\mathcal{P}\mathcal{T}PT-symmetric potentials that have been studied so far have been local. In this paper, nonlocal PT\mathcal{P}\mathcal{T}PT-symmetric separable potentials of the form $V(x,y)=i\ensuremath{\epsilon}[U(x)U(y)\ensuremath{-}U(\ensuremath{-}x)U(\ensuremath{-}y)]$, where U(x)U(x)U(x) is real, are examined. Two specific models are examined. In each case, it is shown that there is a parametric region of the coupling strength $\ensuremath{\epsilon}$ for which the PT\mathcal{P}\mathcal{T}PT symmetry of the Hamiltonian is unbroken and the bound-state energies are real. The critical values of $\ensuremath{\epsilon}$ that bound this region are calculated.
We investigate the parity- and time-reversal (PT\mathcal{PT}PT-) symmetry breaking in lattice models in the presence of long-ranged, non-Hermitian, PT\mathcal{PT}PT-symmetric potentials that remain finite or become divergent in the continuum limit. By scaling analysis of the fragile PT\mathcal{PT}PT threshold for an open finite lattice, we show that continuum loss-gain potentials ${V}_{\ensuremath{\alpha}}(x)\ensuremath{\propto}i{|x|}^{\ensuremath{\alpha}}\mathrm{sgn}(x)$ have a positive PT\mathcal{PT}PT-breaking threshold for $\ensuremath{\alpha}>\ensuremath{-}2$, and a zero threshold for $\ensuremath{\alpha}\ensuremath{\le}\ensuremath{-}2$. When $\ensuremath{\alpha}<0$ localized states with complex (conjugate) energies in the continuum energy band occur at higher loss-gain strengths. We investigate the signatures of PT\mathcal{PT}PT-symmetry breaking in coupled waveguides, and show that the emergence of localized states dramatically shortens the relevant time scale in the PT\mathcal{PT}PT-symmetry broken region.
More than a decade ago, it was shown that non-Hermitian Hamiltonians with combined parity (P) and time-reversal (T ) symmetry exhibit real eigenvalues over a range of parameters. Since then, the field of PT symmetry has seen rapid progress on both the theoretical and experimental fronts. These effective Hamiltonians are excellent candidates for describing open quantum systems with balanced gain and loss. Nature seems to be replete with examples of PT -symmetric systems; in fact, recent experimental investigations have observed the effects of PT symmetry breaking in systems as diverse as coupled mechanical pendula, coupled optical waveguides, and coupled electrical circuits. Recently, PT -symmetric Hamiltonians for tight-binding lattice models have been extensively investigated. Lattice models, in general, have been widely used in physics due to their analytical and numerical tractability. Perhaps one of the best systems for experimentally observing the effects of PT symmetry breaking in a one-dimensional lattice with tunable hopping is an array of evanescently-coupled optical waveguides. The tunneling between adjacent waveguides is tuned by adjusting the width of the barrier between them, and the imaginary part of the local refractive index provides the loss or gain in the respective waveguide. Calculating the time evolution of a wave packet on a lattice is relatively straightforward in the tight-binding model, allowing us to make predictions about the behavior of light propagating down an array of PT -symmetric waveguides. In this thesis, I investigate the the strength of the PT -symmetric phase (the region over which the eigenvalues are purely real) in lattices with a variety of PT - symmetric potentials. In Chapter 1, I begin with a brief review of the postulates of quantum mechanics, followed by an outline of the fundamental principles of PT - symmetric systems. Chapter 2 focuses on one-dimensional uniform lattices with a pair of PT -symmetric impurities in the case of open boundary conditions. I find that the PT phase is algebraically fragile except in the case of closest impurities, where the PT phase remains nonzero. In Chapter 3, I examine the case of periodic boundary conditions in uniform lattices, finding that the PT phase is not only nonzero, but also independent of the impurity spacing on the lattice. In addition, I explore the time evolution of a single-particle wave packet initially localized at a site. I find that in the case of periodic boundary conditions, the wave packet undergoes a preferential clockwise or counterclockwise motion around the ring. This behavior is quantified by a discrete momentum operator which assumes a maximum value at the PT -symmetry- breaking threshold. In Chapter 4, I investigate nonuniform lattices where the parity-symmetric hop- ping between neighboring sites can be tuned. I find that the PT phase remains strong in the case of closest impurities and fragile elsewhere. Chapter 5 explores the effects of the competition between localized and extended PT potentials on a lattice. I show that when the short-range impurities are maximally separated on the lattice, the PT phase is strengthened by adding short-range loss in the broad-loss region. Consequently, I predict that a broken PT symmetry can be restored by increasing the strength of the short-range impurities. Lastly, Chapter 6 summarizes my salient results and discusses areas which can be further developed in future research.
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.