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arXiv:2006.03557 · Physical Review A

Liouvillian exceptional points of any order in dissipative linear bosonic systems: Coherence functions and switching between PT and anti- PT symmetries

Ievgen I. Arkhipov, Adam Miranowicz, Fabrizio Minganti, Franco Nori

Published Sep 11, 2020• 73 citations
arXivHTMLPDF

Abstract

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 $n$, 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 ($\mathcal{PT}$) and anti-$\mathcal{PT}$-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.

DOI
10.1103/physreva.102.033715

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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≥nm\ensuremath{\ge}nm\ensuremath≥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.
Abstract Hamiltonian exceptional points (HEPs) are spectral degeneracies of non-Hermitian Hamiltonians describing classical and semiclassical open systems with losses and/or gain. However, this definition overlooks the occurrence of quantum jumps in the evolution of open quantum systems. These quantum effects are properly accounted for by considering quantum Liouvillians and their exceptional points (LEPs). Specifically, an LEP corresponds to the coalescence of two or more eigenvalues and the corresponding eigenmatrices of a given Liouvillian at critical values of external parameters (Minganti et al 2019 Phys. Rev. A 100 062131). Here, we explicitly describe how standard quantum process tomography, which reveals the dynamics of a quantum system, can be readily applied to detect and characterize quantum LEPs of quantum non-Hermitian systems. We conducted experiments on an IBM quantum processor to implement a prototype model with one-, two-, and three qubits simulating the decay of a single qubit through competing channels, resulting in LEPs but not HEPs. Subsequently, we performed tomographic reconstruction of the corresponding experimental Liouvillian and its LEPs using both single- and two-qubit operations. This example underscores the efficacy of process tomography in tuning and observing LEPs even in the absence of HEPs.
The open quantum system can be described by either a Lindblad master equation or a non-Hermitian Hamiltonian (NHH). However, these two descriptions usually have different exceptional points (EPs), associated with the degeneracies in the open quantum system. Here, considering a dissipative quantum Rabi model, we study the spectral features of EPs in these two descriptions and explore their connections. We find that, although the EPs in these two descriptions are usually different, the EPs of NHH will be consistent with the EPs of master equation in the weak coupling regime. Further, we find that the quantum Fisher information (QFI), which measures the statistical distance between quantum states, can be used as a signature for the appearance of EPs. Our study may give a theoretical guidance for exploring the properties of EPs in open quantum systems.
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.
Exceptional points (EPs), indicative of parity-time (PT) symmetry breaking, play a central role in non-Hermitian physics, yet most studies begin from deliberately engineered effective Hamiltonians whose parameters are tuned to exhibit exceptional behavior. In realistic open quantum systems, however, dynamics are governed by Lindblad superoperators whose spectral structure is high-dimensional, symmetry-constrained, and not obviously reducible to minimal non-Hermitian models. A general framework for discovering exceptional dynamics directly from microscopic dissipative models has been lacking. Here we introduce a symmetry-resolved approach for identifying and characterizing exceptional points directly from the full Liouvillian generator. Correlated dissipation induces graph symmetries that decompose Liouville space into low-dimensional invariant sectors, within which minimal non-Hermitian blocks govern the onset of EPs and PT-breaking behavior. We further introduce a numerical diagnostic - the exceptional-point strength E\mathcal{E}E - based on eigenvector conditioning, which quantifies proximity to defective dynamics without requiring analytic reduction. Applied to tight-binding models with correlated dephasing and relaxation, the method reproduces analytically predicted exceptional seams and reveals universal scaling of E\mathcal{E}E near EP manifolds. More broadly, the framework enables systematic discovery of hidden exceptional structure in complex or high-dimensional open systems and is naturally compatible with matrix-free and tensor-network implementations for scalable many-body applications.
Abstract The non-analyticity induced by exceptional points (EPs) has manifestations not only in non-Hermitian but also in Hermitian systems. In this work, we focus on a minimal Hermitian bosonic Kitaev model to reveal the dynamical demonstration of EPs in a Hermitian system. It is shown that the EPs separate the parameter space into four regions, in which the systems are characterized by different equivalent Hamiltonians, including the harmonic oscillator, the inverted harmonic oscillator, and their respective counterparts. We employ the second-order intensity correlation to characterize a nonequilibrium quantum phase transition by calculating the time evolution of a trivial initial state. The results indicate that the concept of the EP can be detected in a small Hermitian bosonic system.