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Physical Review A

Dynamical transition between synchronization and antisynchronization with exceptional points

Kun-Jie Zhou, Jian Zou, Bin Shao

Published Oct 10, 2023• 6 citations
Publisher

Abstract

In this paper we consider a spin chain locally coupled to dissipative environments with unbalanced gain and loss and explore the relation between the emergence of synchronization and exceptional points. For our model in the Liouvillian formalism the matrix of the Liouvillian superoperator is block diagonal and the different blocks correspond to the dynamics of different observables. We find that for ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}^{z}$ corresponding to the block having a single Liouvillian exceptional point (LEP), when the coupling strength between the nearest spins is beyond the LEP, the synchronization or antisynchronization will occur. For ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}^{x}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}^{y}$ corresponding to the block having two LEPs, when the coupling strength between the nearest spins is beyond both LEPs, the synchronization or antisynchronization will appear, and as time evolves the dynamical transition between synchronization and antisynchronization can occur. Moreover, we also find that the initial state of the system does not affect the emergence of synchronization or antisynchronization, but it determines whether the system is synchronized or antisynchronized and affects the time at which the dynamical transition between synchronization and antisynchronization occurs. Furthermore, by postselecting the quantum trajectories without quantum jump, the system can exhibit Hamiltonian exceptional points, and we find that quantum jumps play an important role in the synchronization of the system.

DOI
10.1103/physreva.108.042206

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In this paper we consider a spin chain locally coupled to dissipative environments with unbalanced gain and loss and explore the relation between the emergence of synchronization and exceptional points. For our model in the Liouvillian formalism the matrix of the Liouvillian superoperator is block diagonal and the different blocks correspond to the dynamics of different observables. We find that for \ensuremathσ\ifmmode^\elseˆ\fiz{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}^{z}\ensuremathσ\ifmmode^\elseˆ\fi​z corresponding to the block having a single Liouvillian exceptional point (LEP), when the coupling strength between the nearest spins is beyond the LEP, the synchronization or antisynchronization will occur. For \ensuremathσ\ifmmode^\elseˆ\fix{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}^{x}\ensuremathσ\ifmmode^\elseˆ\fi​x and \ensuremathσ\ifmmode^\elseˆ\fiy{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}^{y}\ensuremathσ\ifmmode^\elseˆ\fi​y corresponding to the block having two LEPs, when the coupling strength between the nearest spins is beyond both LEPs, the synchronization or antisynchronization will appear, and as time evolves the dynamical transition between synchronization and antisynchronization can occur. Moreover, we also find that the initial state of the system does not affect the emergence of synchronization or antisynchronization, but it determines whether the system is synchronized or antisynchronized and affects the time at which the dynamical transition between synchronization and antisynchronization occurs. Furthermore, by postselecting the quantum trajectories without quantum jump, the system can exhibit Hamiltonian exceptional points, and we find that quantum jumps play an important role in the synchronization of the system.
Open quantum systems can be approximately described by non-Hermitian Hamiltonians (NHHs) and Liouvillian superoperators. The two approaches differ by quantum jump terms corresponding to a measurement of the system by its environment. We analyze the emergence of exceptional points (EPs) in NHHs and Liouvillian superoperators. In particular, we show how EPs in NHHs relate to a novel type of EPs -- multi-block EPs -- in the no-jump Liouvillian, i.e. the Liouvillian superoperator in absence of quantum jump terms. We further analyze how quantum jump terms modify the multi-block structure. To illustrate our general findings, we present two prime examples: qubits and qutrits coupled to additional ground state levels that serve as sinks of the population. In those examples, we can navigate through the EP block structure by a variation of physical parameters. We analyze how the dynamics of the population of the states is affected by the order of the EPs. Additionally, we demonstrate that the quantum geometric tensor serves as a sensitive indicator of EPs of different kinds.
Usually, when investigating exceptional points (EPs) of an open Markovian bosonic system, one deals with spectral degeneracies of a non-Hermitian Hamiltonian (NHH), which can correctly describe the system dynamics only in the semiclassical regime. A recently proposed quantum Liouvillian framework [Minganti et al., Phys. Rev. A 100, 062131 (2019)] enables the complete determination of the dynamical properties of such systems and their EPs (referred to as Liouvillian EPs, or LEPs) in the quantum regime by taking into account the effects of quantum jumps, which are ignored in the NHH formalism. Moreover, the symmetry and eigenfrequency spectrum of the NHH become a part of much larger Liouvillian eigenspace. As such, the EPs of an NHH form a subspace of the LEPs. Here we show that once an NHH of a dissipative linear bosonic system exhibits an EP of a certain finite order nnn, it immediately implies that the corresponding LEP can become of any higher order $m\ensuremath{\ge}n$ defined in the infinite Hilbert space. These higher-order LEPs can be identified by the coherence and spectral functions at the steady state. The coherence functions can offer a convenient tool to probe extreme system sensitivity to external perturbations in the vicinity of higher-order LEPs. As an example, we study a linear bosonic system of a bimodal cavity with incoherent mode coupling to reveal its higher-order LEPs; particularly, of second and third order via first- and second-order coherence functions, respectively. Accordingly, these LEPs can be additionally revealed by squared and cubic Lorentzian spectral lineshapes in the power and intensity-fluctuation spectra. Moreover, we demonstrate that these EPs can also be associated with spontaneous parity-time (PT\mathcal{PT}PT) and anti-PT\mathcal{PT}PT-symmetry breaking in the system studied. These symmetries can be switched in the output fields (the so-called supermodes) of an additional linear coupler with a properly chosen coupling strength. Thus, we show that the initial loss-loss dynamics for the supermodes can be equivalent to the balanced gain-loss evolution.
Exceptional points (EPs) are degeneracies of classical and quantum open systems, which are studied in many areas of physics including optics, optoelectronics, plasmonics, and condensed matter physics. In the semiclassical regime, open systems can be described by phenomenological effective non-Hermitian Hamiltonians (NHHs) capturing the effects of gain and loss in terms of imaginary fields. The EPs that characterize the spectra of such Hamiltonians (HEPs) describe the time evolution of a system without quantum jumps. It is well known that a full quantum treatment describing more generic dynamics must crucially take into account such quantum jumps. In a recent paper [F. Minganti et al., Phys. Rev. A 100, 062131 (2019)], we generalized the notion of EPs to the spectra of Liouvillian superoperators governing open system dynamics described by Lindblad master equations. Intriguingly, we found that in situations where a classical-to-quantum correspondence exists, the two types of dynamics can yield different EPs. In a recent experimental work [M. Naghiloo et al., Nat. Phys. 15, 1232 (2019)], it was shown that one can engineer a non-Hermitian Hamiltonian in the quantum limit by postselecting on certain quantum jump trajectories. This raises an interesting question concerning the relation between Hamiltonian and Lindbladian EPs, and quantum trajectories. We discuss these connections by introducing a hybrid-Liouvillian superoperator, capable of describing the passage from an NHH (when one postselects only those trajectories without quantum jumps) to a true Liouvillian including quantum jumps (without postselection). Beyond its fundamental interest, our approach allows to intuitively relate the effects of postselection and finite-efficiency detectors.
The coupling between self-sustained oscillators, such as lasers or polariton condensates, leads to different steady-states in different regimes. Weak coupling allows the oscillators to behave independently, with different frequencies and uncorrelated phases, while stronger coupling can produce synchronization. In classical or semiclassical theories transitions between these steady-states correspond to changes in the topology of the phase space attractors and occur at generalized exceptional points. We investigate the corresponding changes in a quantum theory of coupled lasers or condensates. We show how the different steady-state regimes of the classical limit give rise to distinct forms for the spectra and eigenmatrices of the slow modes of the Liouvillian. By following the spectral flow between the different regimes we show that the synchronization transition is controlled by cascades of exceptional points in these slow modes, protected by a generalized PT symmetry. Our results show how singularities of the quantum Liouvillian control the different dynamical regimes, and give rise to experimental signatures of the synchronization transition which remain well-defined in the few-particle regime dominated by quantum effects.
Open quantum systems interacting with an environment exhibit dynamics described by the combination of dissipation and coherent Hamiltonian evolution. Taken together, these effects are captured by a Liouvillian superoperator. The degeneracies of the (generically non-Hermitian) Liouvillian are exceptional points, which are associated with critical dynamics as the system approaches steady state. We use a superconducting transmon circuit coupled to an engineered environment to observe two different types of Liouvillian exceptional points that arise either from the interplay of energy loss and decoherence or purely due to decoherence. By dynamically tuning the Liouvillian superoperators in real time we observe a non-Hermiticity-induced chiral state transfer. Our study motivates a new look at open quantum system dynamics from the vantage of Liouvillian exceptional points, enabling applications of non-Hermitian dynamics in the understanding and control of open quantum systems.
Exceptional points in non-Hermitian quantum systems give rise to novel genuine quantum phenomena. Recent explorations of exceptional-point-induced quantum phase transitions have extended from discrete-variable to continuous-variable-encoded quantum systems. However, quantum phase transitions driven by Liouvillian exceptional points (LEPs) in continuous-variable platforms remain largely unexplored. Here, we construct and investigate a Liouvillian exceptional structure based on a driven-dissipative Kerr-cat qubit. Through numerical simulations, we reveal a quantum phase transition occurring at the LEP characterized by a sudden change in dynamical behavior from underdamped oscillations to overdamped relaxations as visualized via Wigner functions and Bloch sphere trajectories. Notably the negativity of the Wigner function serves as a direct signature of genuine quantum coherence unattainable in conventional single-qubit non-Hermitian systems. Furthermore, we introduce the phase difference between the off-diagonal elements of the Liouvillian eigenmatrices as a novel parameter to quantify the transition. Our results establish the Kerr-cat qubit as a novel continuous-variable setting for exploring dissipative quantum criticality and intrinsic non-Hermitian physics.