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

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 (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 R space lead to state conversion to the same instantaneous eigenstate) or asymmetric (CW and CCW encirclements in 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.