The chiral dynamics of three-mode non-Hermitian systems with a periodical driving are studied. An exact chiral behavior of the relative phases of the final states after encircling with opposite directions in parameter space is obtained analytically with the solutions of the dynamical equations. It is shown that the relative phases of the evolution states are opposite numbers for the clockwise and counterclockwise loops of the driving parameters. But the density distributions have no chirality during the evolution. We examine the dynamical behavior by a numerical simulation of the evolution equations, which agrees well with our analytical predictions for the adiabatic evolution. The final density distributions are found to be independent of initial states and always the same for the clockwise and counterclockwise encircling, while in most examples discussed in the literature the amplitudes show a nonsymmetric behavior when the parameter space circles are in opposite directions. It is also found that the dynamical behavior is steady for the long-time evolution. We attribute the results to the special symmetry of the model, which might encourage others to work on physical realizations of the model.
The chiral dynamics of three-mode non-Hermitian systems with a periodical driving are studied. An exact chiral behavior of the relative phases of the final states after encircling with opposite directions in parameter space is obtained analytically with the solutions of the dynamical equations. It is shown that the relative phases of the evolution states are opposite numbers for the clockwise and counterclockwise loops of the driving parameters. But the density distributions have no chirality during the evolution. We examine the dynamical behavior by a numerical simulation of the evolution equations, which agrees well with our analytical predictions for the adiabatic evolution. The final density distributions are found to be independent of initial states and always the same for the clockwise and counterclockwise encircling, while in most examples discussed in the literature the amplitudes show a nonsymmetric behavior when the parameter space circles are in opposite directions. It is also found that the dynamical behavior is steady for the long-time evolution. We attribute the results to the special symmetry of the model, which might encourage others to work on physical realizations of the model.
Non-Hermitian systems are widespread in both classical and quantum physics. The dynamics of such systems has recently become a focal point of research, showcasing surprising behaviors that include apparent violation of the adiabatic theorem and chiral topological conversion related to encircling exceptional points (EPs). These have both fundamental interest and potential practical applications. Yet the current literature features a number of apparently irreconcilable results. Here we develop a general theory for slow evolution of non-Hermitian systems and resolve these contradictions. We prove an analog of the adiabatic theorem for non-Hermitian systems and generalize it in the presence of uncontrolled environmental fluctuations (noise). The effect of noise turns out to be crucial due to inherent exponential instabilities present in non-Hermitian systems. Disproving common wisdom, the end state of the system is determined by the final Hamiltonian only, and is insensitive to other details of the evolution trajectory in parameter space. Our quantitative theory, leading to transparent physical intuition, is amenable to experimental tests. It provides efficient tools to predict the outcome of the system's evolution, avoiding the need to follow costly time-evolution simulations. Our approach may be useful for designing devices based on non-Hermitian physics and may stimulate analyses of classical and quantum non-Hermitian-Hamiltonian dynamics, as well as that of quantum Lindbladian and hybrid-Liouvillian systems.
The conventional time-dependent perturbation theory in quantum mechanics is established within the framework of Hermitian Hamiltonians, applicable for describing quantum transitions and associated energy level responses in such systems. However, this theory has fundamental limitations when applied to non-Hermitian systems. Consequently, researchers have systematically extended time-dependent perturbation theory to non-Hermitian systems, establishing a corresponding mature framework. Building on this foundation, this study extends the theory to investigate the transition dynamics induced by non-Hermitian interactions in non-Hermitian Hamiltonian systems. We employ a biorthogonal basis representation for a three-level non-Hermitian system. This work investigates a system comprising an unperturbed static non-Hermitian Hamiltonian and a periodically driven time-dependent perturbation Hamiltonian. Taking the three-level system as a concrete example, we combine analytical methods with numerical simulations to solve and analyze its dynamical evolution equations. These complementary approaches reveal that when system parameters complete a full cycle around an exceptional point, the transitional behavior exhibits specific evolutionary patterns. In this system, quantum transition probabilities exhibit significant asymmetry and non-conservation that depend on the initial and final states, revealing inherent directional characteristics in the dynamical process. Furthermore, for a three-level, periodically driven non-Hermitian system with time-dependent perturbations, this asymmetry is even more pronounced, manifesting as a distinct disparity between forward and reverse transition probabilities. The periodic driving actively amplifies the asymmetry in the transition process. By designing the perturbation spectrum, selective manipulation of specific quantum states can be achieved. Moreover, transition probabilities can be significantly enhanced under resonance conditions, while non-Hermiticity further breaks the system's inherent symmetry, leading to substantial amplification of transitions in a single direction. By precisely tuning the drive frequency, interactions between specific coupling channels can be selectively enhanced or suppressed. The amplification of channel asymmetry by non-Hermitian properties provides a novel mechanism for directional control of quantum states and opens new pathways for realizing related quantum technologies.
Exceptional points emerge in the complex eigenspectra of non-Hermitian systems, and give rise to rich critical behaviors. An outstanding example is the chiral state transfer, where states can swap under an adiabatic encircling around the exceptional point, but only along one direction. In dissipative quantum systems, such exceptional-point encirclings are often accompanied by decoherence, whose impact is beyond the description of non-Hermitian Hamiltonians. In this paper, we study in detail the effects of dephasing on the encircling dynamics, adopting the full Lindblad master equation. Introducing experimentally relevant quantum-jump processes that account for dephasing, we show that gaps emerge in the eigenspectra landscape of the corresponding Liouvillian superoperator. It follows that the chiral state transfer does not take place in the adiabatic limit, since the system always adiabatically follows the quasisteady state of the Liouvillian regardless of the encircling direction. Nevertheless, the chirality is restored at intermediate encircling times, where the dynamics is nonadiabatic in both encircling directions, distinct from the typical chiral state transfer in non-Hermitian systems. While our results are applicable to several recent experiments, we examine a recent cold-atom experiment in particular, and show that the observed long-time chirality is limited to the special encircling path therein. Our paper provides further insight into the chiral state transfer under experimental conditions, and is helpful for controlling open-system dynamics from the perspective of non-Hermitian physics.
Non-Hermitian systems have attracted much interest in recent decades, driven partly by the existence of exotic spectral singularities, known as exceptional points (EPs), where the dimensionality of the system evolution operator is reduced. Among various intriguing applications, the discovery of EPs has suggested the potential for implementing a symmetric mode switch, when encircling them in a system parameter space. However, subsequent theoretical and experimental works have revealed that dynamical encirclement of EPs invariably results in asymmetric mode conversion; namely, the mode switching depends only on the winding direction but not on the initial state. This chirality arises from the failure of adiabaticity due to the complex spectrum of non-Hermitian systems. Although the chirality revealed has undoubtedly made a significant impact in the field, a realization of the originally sought symmetric adiabatic passage in non-Hermitian systems with EPs has since been elusive. In this work, we bridge this gap and theoretically demonstrate that adiabaticity, and therefore a symmetric state transfer, is achievable when dynamically winding around an EP. This becomes feasible by specifically choosing a trajectory in the system parameter space along which the corresponding evolution operator attains a real spectrum. Our findings, thus, offer a promise for advancing various wave manipulation protocols in both quantum and classical domains.
In this paper, we study the time evolution of three-states spin-1 non-Hermitian systems. The state vector of the wave function for Schrödinger equation is obtained analytically by using the Cayley–Hamilton theorem. We note that the behavior of the non-Hermitian three-states system corresponds to the Hermitian three-state system. Furthermore, the fluctuation between three-state spin-1 is periodically oscillating when the eigenvalues of the non-Hermitian system are real. Moreover, if the eigenvalues are imaginary, we explore that from our analysis and analytical solution that the probabilities’ amplitudes always grow with time for all physical parameters in the three-state, i.e. including the [Formula: see text]-symmetry non-Hermitian system. It means that the Schrödinger equation becomes the growth exponential.
One intriguing property of non-Hermitian systems is the breakdown of adiabatic theorem and chiral state conversion as the system dynamically encircles exceptional points. However, the subtle dependence of the chiral dynamics on the loop geometry, the starting point, the encircling speed and especially the noise has not been studied systematically. Here we propose a non-chirality degree χc to measure the chirality quantitatively and analyze it in dynamics without noise by exact solution and dynamics with noise by numerical integration. The exact dynamics starting from the broken phase show chirality oscillations, which are extremely sensitive to noise when the speed is small. The encircling speed and the noise strength are found to compete with each other in determining χc, resulting in two distinguished limits, namely the noisy limit and the clean limit. The critical boundary between the two limits satisfies a simple scaling law, which could be explained in terms of first-order perturbation theory and the condition number of the transfer matrix. Our findings reveal the essential role played by noise in non-Hermitian dynamics and are relevant for both theoretical and experimental investigations.