Recent years have seen a surge of interest in exceptional points in open quantum systems. The natural approach in this area has been the use of Markovian master equations. While the resulting Liouvillian EPs have been seen in a variety of systems and have been associated to numerous exotic effects, it is an open question whether such degeneracies and their peculiarities can persist beyond the validity of master equations. In this work, taking the example of a dissipative double-quantum-dot system, we show that exact Heisenberg equations governing system and bath dynamics exhibit the same EPs as the corresponding master equations. To highlight the importance of this finding, we prove that the paradigmatic property associated to EPs – critical damping, persists well beyond the validity of master equations. Our results demonstrate that Liouvillian EPs can arise from underlying fundamental exact principles, rather than merely as a consequence of approximations involved in deriving master equations.
Recent years have seen a surge of interest in exceptional points in open quantum systems. The natural approach in this area has been the use of Markovian master equations. While the resulting Liouvillian EPs have been seen in a variety of systems and have been associated to numerous exotic effects, it is an open question whether such degeneracies and their peculiarities can persist beyond the validity of master equations. In this work, taking the example of a dissipative double-quantum-dot system, we show that exact Heisenberg equations governing system and bath dynamics exhibit the same EPs as the corresponding master equations. To highlight the importance of this finding, we prove that the paradigmatic property associated to EPs – critical damping, persists well beyond the validity of master equations. Our results demonstrate that Liouvillian EPs can arise from underlying fundamental exact principles, rather than merely as a consequence of approximations involved in deriving master equations.
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 primary pathway to the investigation of exceptional points (EPs) in open quantum systems has been the use of Markovian master equations. However, recently it has been realised that they can arise in settings that show non-Markovian behaviour. The main approach in this direction has been the use of pseudomodes, which allow the influence of a structured environment to be replaced by new damped auxiliary modes. In this work, we use a Heisenberg-equation approach which relies on the reaction-coordinate formalism to obtain exact analytical dynamics. First, we show how our method considerably simplifies the analysis compared to the use of master equations. Second, we utilise it to discuss non-Markovian revivals in systems with second and higher-order EPs, showing that the notion that EPs demarcate the onset of revivals is not generally true. We then utilise our results to discuss the dynamical effects of higher-order non-Markovian EPs, deriving bounds on the number of revivals and investigating enhanced asymptotic relaxation near higher-order EPs. Finally, we investigate non-Markovian EPs with engineered reservoirs and show how they can be utilised to control the rate of irreversible information loss.
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.
Abstract The dynamics of open quantum systems described by the Lindblad master equation follows according to non-Hermitian operators. As a result, such systems can host non-Hermitian degeneracies called Liouvillian exceptional points (EPs). In this work, we show that Newton polygons and tropical geometric approach allow identification and characterization of Liouvillian EPs. We use two models—dissipative spin − 1 / 2 system and dissipative superconducting qubit system—to illustrate our method. We demonstrate that our approach captures the anisotropy and order of the Liouvillian EPs, while also revealing the subtle dependence on the form of the perturbation. Our analytical analysis is supplemented by direct numerical calculations of the scaling and exchange of eigenvalues around Liouvillian EPs. Our analytical approach could be useful in understanding and designing Liouvillian EPs of desired order.
Quantum Mpemba effect describes an anomalous phenomenon of accelerated relaxation which is of fundamental interest in the field of nonequilibrium thermodynamics. Conventional theories on this phenomenon strongly rely on the Born-Markovian approximation resulting in a Lindblad-type master equation whose evolution is governed by a Liouvillian superoperator. It has been demonstrated that exceptional points of the Liouvillian superoperator can induce the Mpemba effect in Markovian regimes. Moving beyond this Markovian limit, we here propose a mechanism for realizing the quantum Mpemba effect in a general non-Markovian relaxation process by means of non-Markovian exceptional points. We verify the feasibility of this mechanism within a dissipative quantum harmonic oscillator model, which is exactly solvable and experimentally practical. Providing new insight into the interesting non-equilibrium dynamics, our work paves a way to accelerate the transfer of energy and information in quantum systems.
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 (PT) and anti-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.