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.
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
(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.
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.
Exceptional points (EPs) have attracted extensive research interest due to their intriguing properties. One of the hallmarks of EP physics is that dynamically encircling the EPs induces chiral mode switching, arising from the breakdown of adiabaticity due to the presence of a complex spectrum in the system's Hamiltonian. While such chiral mode behavior has been widely observed experimentally, achieving truly adiabatic, and thus symmetric, state transfer, regardless of the winding direction, in time-modulated non-Hermitian systems has remained elusive. In this work, we demonstrate that this long-sought adiabatic state dynamics can indeed be restored. By steering a two-mode photonic setup along specifically designed trajectories in parameter space, we realize conditions where the associated non-Hermitian evolution operator acquires a purely real spectrum. Moreover, our experimental platform enables controlled switching between symmetric (adiabatic) and chiral (non-adiabatic) state-transfer regimes for the same set of initial modes, thus effectively implementing a universal symmetric-asymmetric two-mode switch. Our results therefore open new avenues for harnessing unique topological spectral properties of non-Hermitian systems, paving the way for the practical design of versatile optical wave-manipulation devices and for advancing both classical and quantum information technologies.
Nontrivial spectral properties of non-Hermitian systems can give rise to intriguing effects that lack counterparts in Hermitian systems. For instance, when dynamically varying system parameters along a path enclosing an exceptional point (EP), chiral mode conversion occurs. A recent study [Phys. Rev. Lett. 133, 113802 (2024)] demonstrates the achievability of pure adiabatic state transfer by specifically selecting a trajectory in the system parameter space where the corresponding evolution operator exhibits a real spectrum while winding around an EP. However, the intended adiabatic state transfer becomes fragile when taking into account the effect of the nonadiabatic transition. In this work, we propose a scheme for achieving robust and rapid adiabatic state transfer in time-modulated two-level non-Hermitian systems by appropriately modulating the system Hamiltonian and time-evolution trajectory. Numerical simulations confirm that a complete adiabatic transfer can always be achieved even under nonadiabatic conditions after one period for different initialized adiabatic states, and the scheme remains insensitive to moderate fluctuations in control parameters. Therefore, this scheme offers alternative approaches for quantum-state engineering in non-Hermitian systems.
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
Tcr at which these transitions develop. Using a
2×2 Hamiltonian with circular parameter trajectories, we derive
Tcr=Gln(1/∣Δ∣) in closed form for non-encircling loops, phase-shifted loops, offset loops, and loops encircling exceptional points, where
G is a geometry-dependent growth factor and
Δ is the instability seed. This formula sharply separates the regime where the system remains in the averagely dominant eigenstate (
T<Tcr) from the superadiabatic regime where the instantaneous dominant eigenstate takes over (
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∝mlnβ, linear in the number
m of digits in base
β (bits for
β=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
Tcr remains linear in
m. 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,
Tcr additionally determines the onset of chirality: the dynamics is non-chiral for
T<Tcr and chiral for
T>Tcr.
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.