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arXiv:2104.07349 · Physical Review A

PT phase transition in open quantum systems with Lindblad dynamics

Yuma Nakanishi, Tomohiro Sasamoto

Published Feb 25, 2022• 33 citations
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Abstract

We investigate parity-time ($\mathcal{PT}$) phase transitions in open quantum systems and discuss a criterion of Liouvillian $\mathcal{PT}$ symmetry proposed recently by Huber et al. [SciPost Phys. 9, 52 (2020)]. Using the third quantization, which is a general method to solve the Lindblad equation for open quadratic systems, we show, with a proposed criterion of $\mathcal{PT}$ symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the $\mathcal{PT}$-symmetry-breaking point for an open two-spin model with exactly balanced gain and loss if the total spin is large. In particular, in a $\mathcal{PT}$-unbroken phase, some eigenvalues are pure imaginary numbers, while in a $\mathcal{PT}$-broken phase, all the eigenvalues are real. From this result, it is analytically shown for an open quantum system including quantum jumps that the dynamics in the long-time limit changes from an oscillatory to an overdamped behavior at the proposed $\mathcal{PT}$-symmetry-breaking point. Furthermore, we show a direct relation between the criterion of Huber et al. of Liouvillian $\mathcal{PT}$ symmetry and the dynamics of the physical quantities for quadratic bosonic systems. Our results support the validity of the proposed criterion of Liouvillian $\mathcal{PT}$ symmetry.

DOI
10.1103/physreva.105.022219

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We investigate parity-time (PT\mathcal{PT}PT) phase transitions in open quantum systems and discuss a criterion of Liouvillian PT\mathcal{PT}PT symmetry proposed recently by Huber et al. [SciPost Phys. 9, 52 (2020)]. Using the third quantization, which is a general method to solve the Lindblad equation for open quadratic systems, we show, with a proposed criterion of PT\mathcal{PT}PT symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the PT\mathcal{PT}PT-symmetry-breaking point for an open two-spin model with exactly balanced gain and loss if the total spin is large. In particular, in a PT\mathcal{PT}PT-unbroken phase, some eigenvalues are pure imaginary numbers, while in a PT\mathcal{PT}PT-broken phase, all the eigenvalues are real. From this result, it is analytically shown for an open quantum system including quantum jumps that the dynamics in the long-time limit changes from an oscillatory to an overdamped behavior at the proposed PT\mathcal{PT}PT-symmetry-breaking point. Furthermore, we show a direct relation between the criterion of Huber et al. of Liouvillian PT\mathcal{PT}PT symmetry and the dynamics of the physical quantities for quadratic bosonic systems. Our results support the validity of the proposed criterion of Liouvillian PT\mathcal{PT}PT symmetry.
We investigate parity-time reversal (PT) phase transitions in open quantum systems and discuss a criterion of Liouvillian PT symmetry proposed recently by J. Huber, P. Kirton, S. Rotter and P. Rabl. Using the third quantization, which is a general method to solve the Lindblad equation for open quadratic systems, we show, with a proposed criterion of PT symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the PT symmetry breaking point for an open 2-spin model with exactly balanced gain and loss if the total spin is large. Specially, in a PT unbroken phase, some eigenvalues are pure imaginary numbers while in a PT broken phase, all the eigenvalues are real. From this result, it is analytically shown for open quantum system including quantum jumps that the dynamics in the long time limit changes from an oscillatory to an overdamped behavior at the proposed PT symmetry breaking point. Furthermore, we show a direct relation between the criterion of Huber et al. of Liouvillian PT symmetry and dynamics of the physical quantities for quadratic bosonic systems. Our results support validity of the proposed criterion of Liouvillian PT symmetry.
A parity-time (PT) transition is a spectral transition characteristic of non-Hermitian generators; it typically occurs at an exceptional point, where multiple eigenvectors coalesce. The concept of a PT transition has been extended to Markovian open quantum systems, which are described by the GKSL equation. Interestingly, the PT transition in many-body Markovian open quantum systems, the so-called \textit{Lindbladian PT (L-PT) phase transition}, is closely related to two classes of exotic nonequilibrium many-body phenomena: \textit{continuous-time crystals} and \textit{non-reciprocal phase transitions}. In this review, we describe the recent advances in the study of L-PT phase transitions. First, we define PT symmetry in three distinct contexts: non-Hermitian systems, nonlinear dynamical systems, and Markovian open quantum systems, highlighting the interconnections between these frameworks. Second, we develop mean-field theories of L-PT phase transitions for collective-spin systems and for bipartite bosonic systems with particle-number conservation. Within these classes of models, we show that L-PT symmetry can induce a breaking of continuous time-translation symmetry down to a discrete one, leading to persistent periodic dynamics. We further demonstrate that the L-PT phase transition point is typically \textit{a critical exceptional point}, where multiple collective excitation modes with zero excitation spectrum coalesce. These findings establish an explicit connection to continuous-time crystals and non-reciprocal phase transitions. Third, going beyond the mean-field theory, we analyze statistical and quantum properties, such as purity and quantum entanglement indicators of time-independent steady states for several specific models with the L-PT symmetry. Finally, we discuss future research directions for L-PT phase transitions.
The effect of \mathcal{PT} 𝒫 𝒯 -symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. However, it is still an unsolved problem how to generalize the concept of \mathcal{PT} 𝒫 𝒯 symmetry to the quantum domain, where the conventional definition in terms of non-Hermitian Hamiltonians is not applicable. Here we introduce a symmetry relation for Liouville operators that describe the dissipative evolution of arbitrary open quantum systems. Specifically, we show that the invariance of the Liouvillian under this symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected \mathcal{PT} 𝒫 𝒯 -symmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe \mathcal{PT} 𝒫 𝒯 -symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes.
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.
We outline two general classes of examples of PT\mathbb{P}\mathbb{T}PT-symmetric quantum Liouvillian dynamics of open many-qubit systems, namely, interacting hard-core bosons (or more general XYZXYZXYZ-type spin-1/2 systems) having either (i) pure dephasing noise or (ii) solely single-particle (spin) injection (absorption) incoherent processes. The concept of PT\mathbb{P}\mathbb{T}PT symmetry is defined following Prosen [Phys. Rev. Lett. 109, 090404 (2012)] as a formal quantum Liouville-space analog of the parity-time PT\mathcal{PT}PT symmetry used in nonconservative classical systems with symmetrically distributed gain and loss.
Continuous time-translation symmetry is often spontaneously broken in open quantum systems, and the condition for their emergence has been actively investigated. However, there are only a few cases in which its condition for appearance has been fully elucidated. In this Letter, we show that a Lindbladian parity-time (PT) symmetry can generically produce persistent periodic oscillations in a wide class of systems. This includes one-collective spin models, which have been studied thoroughly in the context of dissipative continuous time crystals, and spatially extended bipartite bosonic systems with a conserved particle number. By making an analogy to nonreciprocal phase transitions, we demonstrate that a transition point from the dynamical phase is associated with spontaneous PT symmetry breaking that typically corresponds to a critical exceptional point. Interestingly, the periodic orbits in the PT symmetric phase are found to be center type, implying an initial state dependence. These results are established by proving that the Lindbladian PT symmetry at the microscopic level implies a nonlinear PT symmetry and by performing a linear stability analysis near the transition point. This research will further our understanding of novel nonequilibrium phases of matter and phase transitions with spontaneous antiunitary symmetry breaking.