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Physical Review Research

Universal state conversion in discrete and slowly varying non-Hermitian cyclic systems: An analytic proof and exactly solvable examples

Nicholas S. Nye

Published Jul 26, 2023• 6 citations
Publisher

Abstract

In this paper, we formally prove how, by cyclically varying the parameters of a generalized two-level discrete and non-Hermitian Hamiltonian, the respective state vector converts to the instantaneous eigenstate of the system in the adiabatic limit, irrespective of how the state was initially prepared or of the form of the cyclic trajectory followed within the parameter space $(\mathsc{R})$ or of whether a singularity/exceptional point is encircled or not. Our proof also applies to continuous configurations in the limit of infinitesimally small discrete time steps. The observed mode switching behavior, which is a clear signature of the irreversible nature of non-Hermitian arrangements, can be either of symmetric (clockwise [CW] and counterclockwise [CCW] encirclements in $\mathsc{R}$ space lead to state conversion to the same instantaneous eigenstate) or asymmetric (CW and CCW encirclements in $\mathsc{R}$ space lead to state conversion to different instantaneous eigenstates). As a specific example, we investigate the case of rhombic parametric trajectories for a discrete parity-time $(\mathcal{P}\phantom{\rule{0.16em}{0ex}}\mathcal{T}\phantom{\rule{0.28em}{0ex}})$-symmetric-like setting. Exact analytical solutions are retrieved in terms of Weber/parabolic cylinder (large rhombic loops) and Airy functions (small rhombic loops), with their asymptotic behavior dictated by the Stokes phenomenon. Both analytical derivations and numerical computations indicate that the observed mode conversion is primarily an artifact of the adiabatic character of the state vector evolution, while the encirclement or not of a singularity/exceptional point affects only the magnitude of the required adiabaticity rate for mode switching to take place. Overall, our results provide a deeper theoretical insight on slowly varying discrete non-Hermitian Hamiltonian systems, and pave the way towards exploring the dynamics underpinning the traversal of higher-dimensional cyclic parametric trajectories in the vicinity (or not) of higher-order spectral singularities for both continuous and discrete settings under linear or nonlinear conditions.

DOI
10.1103/physrevresearch.5.033053

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In this paper, we formally prove how, by cyclically varying the parameters of a generalized two-level discrete and non-Hermitian Hamiltonian, the respective state vector converts to the instantaneous eigenstate of the system in the adiabatic limit, irrespective of how the state was initially prepared or of the form of the cyclic trajectory followed within the parameter space (\mathscR)(\mathsc{R})(\mathscR) or of whether a singularity/exceptional point is encircled or not. Our proof also applies to continuous configurations in the limit of infinitesimally small discrete time steps. The observed mode switching behavior, which is a clear signature of the irreversible nature of non-Hermitian arrangements, can be either of symmetric (clockwise [CW] and counterclockwise [CCW] encirclements in \mathscR\mathsc{R}\mathscR space lead to state conversion to the same instantaneous eigenstate) or asymmetric (CW and CCW encirclements in \mathscR\mathsc{R}\mathscR space lead to state conversion to different instantaneous eigenstates). As a specific example, we investigate the case of rhombic parametric trajectories for a discrete parity-time (PT)(\mathcal{P}\phantom{\rule{0.16em}{0ex}}\mathcal{T}\phantom{\rule{0.28em}{0ex}})(PT)-symmetric-like setting. Exact analytical solutions are retrieved in terms of Weber/parabolic cylinder (large rhombic loops) and Airy functions (small rhombic loops), with their asymptotic behavior dictated by the Stokes phenomenon. Both analytical derivations and numerical computations indicate that the observed mode conversion is primarily an artifact of the adiabatic character of the state vector evolution, while the encirclement or not of a singularity/exceptional point affects only the magnitude of the required adiabaticity rate for mode switching to take place. Overall, our results provide a deeper theoretical insight on slowly varying discrete non-Hermitian Hamiltonian systems, and pave the way towards exploring the dynamics underpinning the traversal of higher-dimensional cyclic parametric trajectories in the vicinity (or not) of higher-order spectral singularities for both continuous and discrete settings under linear or nonlinear conditions.
In the present paper, we aim to firmly establish the adiabatic properties of two-level non-Hermitian quantum structures evolving along generalized (open/acyclic or closed/cyclic) paths in parameter space. Analytical solutions in terms of Airy and modified Bessel functions have been retrieved for linear and hyperbolic temporal dependencies in parity-time-symmetric-like systems, which were subsequently studied in the slowly varying limit to show conversion to one of the instantaneous eigenstates. Such a mode switching behavior is found to be an identifying feature of dissipative quantum settings, whether they evolve along cyclic or acyclic trajectories, and this has been proven in our paper by separately analyzing the dynamics of (i) the ratio of the state vector components, via a variant of the Möbius transformation, and (ii) the complex probability amplitudes, through a systematic inspection of the mode population equations. In the latter instance, it was furthermore shown that the identity of the eigenstate, to which the quantum arrangement transitions, depends highly on the magnitude of the adiabatic rate β. Along these lines, the concepts of the instantaneous(D) and averagely(Dav)dominant eigenstates are brought forth, while a reconfigurable photonic switch is also proposed, which can convert either to the D or to the Dav modes based on the total period of evolution. Finally, we apply our findings in the case of closed parametric paths to demystify the recently reported symmetric and asymmetric state conversion effects and additionally demonstrate that operation at or near exceptional points does not qualitatively affect the conclusions of the current investigation.
Non-Hermitian systems driven along slow parametric loops undergo non-adiabatic transitions whose outcome depends sensitively on the driving speed, yet no explicit formula has been available for the critical timescale TcrT_{\mathrm{cr}}Tcr​ at which these transitions develop. Using a 2×22\times 22×2 Hamiltonian with circular parameter trajectories, we derive Tcr=G ln⁡(1/∣Δ∣)T_{\mathrm{cr}} = \mathcal{G}\,\ln(1/|Δ|)Tcr​=Gln(1/∣Δ∣) in closed form for non-encircling loops, phase-shifted loops, offset loops, and loops encircling exceptional points, where G\mathcal{G}G is a geometry-dependent growth factor and ΔΔΔ is the instability seed. This formula sharply separates the regime where the system remains in the averagely dominant eigenstate (T<TcrT< T_{\mathrm{cr}}T<Tcr​) from the superadiabatic regime where the instantaneous dominant eigenstate takes over (T>TcrT> T_{\mathrm{cr}}T>Tcr​), resolving the apparent tension between the previous literature. We identify two competing seeds: a geometric Stokes multiplier and the finite-precision floor. When the geometric seed vanishes, precision alone governs the transition, yielding Tcr∝mln⁡βT_{\mathrm{cr}} \propto m\lnβTcr​∝mlnβ, linear in the number mmm of digits in base βββ (bits for β=2β=2β=2). For loops encircling an exceptional point, the Stokes seed only sets an intermediate plateau; the final transition is again seeded by the precision floor, and TcrT_{\mathrm{cr}}Tcr​ remains linear in mmm. This provides a purely forward-evolution manifestation of precision-induced irreversibility [Phys. Rev. A 114, 012207 (2026)], demonstrating that the fundamental limit identified through echo protocols also controls the outcome of slow non-Hermitian dynamics without requiring time reversal. For PT-symmetric energy spectra, TcrT_{\mathrm{cr}}Tcr​ additionally determines the onset of chirality: the dynamics is non-chiral for T<TcrT< T_{\mathrm{cr}}T<Tcr​ and chiral for T>TcrT> T_{\mathrm{cr}}T>Tcr​.
Abstract We study state conversion in a parity-time (PT) symmetry broken non-Hermitian two level system. We construct a theory and explain the underlying mechanism for state conversion and define adiabatic evolutions in non-Hermitian systems. The adiabatic theorem can be used if the initial state is an instantaneous eigenstate in Hermitian systems. We show that the system can adiabatically follow the modulationally stable instantaneous eigenstate sooner or later, regardless of the initial state if the system parameters are varied slowly in non-Hermitian systems. We discuss the topological feature of dynamical circling in the parameter space.
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.
The dynamics of a periodically driven system whose time evolution is governed by the Schr\"odinger equation with non-Hermitian Hamiltonians can be perfectly stable. This finding was only obtained very recently and will be enhanced by many exact solutions discovered in this work. The main concern of this study is to investigate the adiabatic following dynamics in such non-Hermitian systems stabilized by periodic driving. We focus on the peculiar behavior of stable cyclic (Floquet) states in the slow-driving limit. It is found that the stable cyclic states can either behave as intuitively expected by following instantaneous eigenstates, or exhibit piecewise adiabatic following by sudden switching between instantaneous eigenstates. We aim to cover broad categories of non-Hermitian systems under a variety of different driving scenarios. We systematically analyze the sudden-switch behavior by a universal route. That is, the sign change of the critical exponent in our asymptotic analysis of the solutions is always found to be the underlying mechanism to determine if the adiabatic following dynamics is trivial or piecewise. This work thus considerably extends our early study on the same topic [Gong and Wang, Phys. Rev. A 97, 052126 (2018)] and shall motivate more interest in non-Hermitian systems.
Dynamically varying system parameters along a path enclosing an exceptional point is known to lead to chiral mode conversion. But is it necessary to include this non-Hermitian degeneracy inside the contour for this process to take place? We show that a sufficiently slow variation of parameters, even away from the system's exceptional point, can also lead to a robust asymmetric state exchange. To study this process, we consider a prototypical two-level non-Hermitian Hamiltonian with a constant coupling between elements. Closed-form solutions are obtained when the amplification or attenuation coefficients in this arrangement are varied in conjunction with the resonance detuning along a circular contour. Using asymptotic expansions, this input-independent mode conversion is theoretically proven to take place irrespective of whether the exceptional point is enclosed upon encirclement. Our results significantly broaden the range of parameter space required for the experimental realization of such chiral mode conversion processes.