Diagonalization of the effective Hamiltonian describing an open quantum system is the usual method of tracking its exceptional points (EPs). Although such a method is successful for tracking EPs in Markovian systems, it may be problematic in non-Markovian systems where a closed expression of the effective Hamiltonian describing the open system may not exist. In this work we provide an alternative method of tracking EPs in open quantum systems, using an experimentally measurable quantity, namely, the effective decay rate of a qubit. The quantum system under consideration consists of two nonidentical interacting qubits, one of which is coupled to an external environment. We develop a theoretical framework in terms of the time-dependent Schr\"odinger equations of motion, which provides analytical closed-form solutions of the Laplace transforms of the qubit amplitudes, enabling the study of various cases of environmental spectral densities. The link between the peaked structure of the effective decay rate of the qubit that interacts indirectly with the environment and the onset of the quantum Zeno effect is investigated, revealing the connections between the latter and the presence of exceptional points. Our treatment and results in addition reveal an intricate interplay between non-Markovian dynamics, the quantum Zeno effect, and non-Hermitian physics.
Diagonalization of the effective Hamiltonian describing an open quantum system is the usual method of tracking its exceptional points (EPs). Although such a method is successful for tracking EPs in Markovian systems, it may be problematic in non-Markovian systems where a closed expression of the effective Hamiltonian describing the open system may not exist. In this work we provide an alternative method of tracking EPs in open quantum systems, using an experimentally measurable quantity, namely, the effective decay rate of a qubit. The quantum system under consideration consists of two nonidentical interacting qubits, one of which is coupled to an external environment. We develop a theoretical framework in terms of the time-dependent Schr"odinger equations of motion, which provides analytical closed-form solutions of the Laplace transforms of the qubit amplitudes, enabling the study of various cases of environmental spectral densities. The link between the peaked structure of the effective decay rate of the qubit that interacts indirectly with the environment and the onset of the quantum Zeno effect is investigated, revealing the connections between the latter and the presence of exceptional points. Our treatment and results in addition reveal an intricate interplay between non-Markovian dynamics, the quantum Zeno effect, and non-Hermitian physics.
Exceptional points (EPs) are singularities in the spectra of non-Hermitian operators where eigenvalues and eigenvectors coalesce. Open quantum systems have recently been explored as EP testbeds due to their non-Hermitian nature. However, most studies focus on the Markovian limit, leaving a gap in understanding EPs in the non-Markovian regime. This work addresses this gap by proposing a general framework based on two numerically exact descriptions of non-Markovian dynamics: the pseudomode equation of motion (PMEOM) and the hierarchical equations of motion (HEOM). The PMEOM is particularly useful due to its Lindblad-type structure, aligning with previous studies in the Markovian regime while offering deeper insights into EP identification. This framework incorporates non-Markovian effects through auxiliary degrees of freedom, enabling the discovery of additional or higher-order EPs that are inaccessible in the Markovian regime. We demonstrate the utility of this approach using the spin-boson model and linear bosonic systems. The full description of exceptional points in the non-Markovian regime is complicated by the unclear applicability of standard techniques such as spectral analysis. Here, the authors fill this gap by proposing a general framework that combines the pseudomode equation of motion and the hierarchical equations of motion.
One of the most remarkable features that distinguish open systems from closed ones is the presence of exceptional points (EPs), where two or more eigenvectors of a non-Hermitian operator coalesce, accompanying the convergence of the corresponding eigenvalues. So far, EPs have been demonstrated on a number of platforms, ranging from classical optical systems to fully quantum-mechanical spin-boson models. In these demonstrations, the reservoir that induced the non-Hermiticity was treated as a Markovian one, without considering its memory effect. We here present the first experimental demonstration of non-Markovian quantum EPs, engineered by coupling a Josephson-junction-based qubit to a leaky electromagnetic resonator, which acts as a non-Markovian reservoir. We map out the spectrum of the extended Liouvillian superoperator by observing the quantum state evolution of the qubit and the pseudomode, in which the memory of the reservoir is encoded. We identify a twofold second-order EP and a third-order EP in the Liouvillian spectrum, which cannot be realized with a Markovian reservoir. Our results pave the way for experimental exploration of exotic phenomena associated with non-Markovian quantum EPs.
The decay of an unstable quantum state can be inhibited or enhanced by tailored measurements, known as quantum Zeno effect (QZE) or anti Zeno effect (QAZE). QZE(QAZE) has been intensively explored in terms of the cases of various system-environment couplings, where the time evolution can be affected either by the projective measurements, or through the dissipative couplings to the environment. A general relation between QZE(QAZE) and the dissipation, for arbitrary dissipation strength and periodicity, is yet to be developed. In this paper, we show a framework to unify the QZE(QAZE) and the parity-time (PT) symmetry breaking transition, where the pure dissipative Hamiltonian is mapped onto a PT symmetric non-Hermitian Hamiltonian. This method uses the PT symmetry transitions to distinguish QZE and QAZE, and can be applied to analyze the crossover behavior between these two effects. Using a heuristic example of a two-level system which is coupled to environment by periodical dissipation, we show the relation diagram between the QZE(QAZE) and the PT symmetry breaking transition, in which the QZE appears at an exceptional point that separates the PT symmetric (PTS) phase and the PT symmetry broken (PTSB) phase, and ends at the resonance point of the maximum PT symmetry breaking; after that, QAZE exists at the rest of the PTSB phase and remains in the next PTS phase. This interesting finding reveals a hidden relation between the QZE--QAZE and PTS-PTSB phases in a non-Hermitian quantum system.
Spontaneous emission of a quantum emitter, such as an excited atom, is a fundamental process in quantum electrodynamics (QED), typically associated with exponential decay to the ground state accompanied by irreversible photon emission. This simple Markovian picture, however, is profoundly modified in the presence of time-delayed feedback, structured continua, or cooperative emission, as occurs when an emitter radiates in front of a mirror, when several emitters radiate collectively, or in the case of a giant atom. In such regimes, strong non-Markovian dynamics arise from photon reabsorption and interference effects, leading to pronounced deviations from exponential decay. Here we demonstrate the emergence of exceptional points (EPs) in these highly non-Markovian waveguide-QED environments, i.e., non-Markovian EPs. These EPs appear directly in the relaxation dynamics as sharp transitions to oscillatory behavior, manifested by the appearance of real zeros in the excited-state amplitude. We analyze in detail the spontaneous emission of giant atoms with two or more coupling points, highlighting the mechanisms leading to non-Markovian EPs, and show that similar phenomena arise in other waveguide-QED settings, such as the collective spontaneous emission of spatially separated point-like emitters. Our results reveal waveguide-QED systems as experimentally accessible platforms for realizing and exploring non-Markovian EP physics.
We suggest to employ the dissipative nature of open quantum systems for the purpose of parameter estimation: the dynamics of open quantum systems is typically described by a quantum dynamical semigroup generator The eigenvalues of are complex, reflecting unitary as well as dissipative dynamics. For certain values of parameters defining non-Hermitian degeneracies emerge, i.e. exceptional points (EP). The dynamical signature of these EPs corresponds to a unique time evolution. This unique feature can be employed experimentally to locate the EPs and thereby to determine the intrinsic system parameters with a high accuracy. This way we turn the disadvantage of the dissipation into an advantage. We demonstrate this method in the open system dynamics of a two-level system described by the Bloch equation, which has become the paradigm of diverse fields in physics, from NMR to quantum information and elementary particles.
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.