Abstract The dynamics of a three-modes non-Hermitian quantum system with a periodical-driving are studied. The Stokes phenomenon is observed in the adiabatic approximation. It is shown that the multiplier of the exponentially dominant eigenstate in the evolving state may be negligible and the multiplier of the exponentially small eigenstate in the evolving state may appear or disappear and oscillates drastically. Such Stokes phenomenon is independent of whether or not the exceptional point is encircled. The origin of such Stokes phenomenon is from the imaginary part of the complex eigenenergies. The eigenstate with the maximum imaginary part of dynamical phase factor in the evolving state will inevitably mask the other eigenstates. We also estimate the range of driving-frequency for the adiabatic approximation condition. The contribution is primarily theoretical in nature.
Abstract The dynamics of a three-modes non-Hermitian quantum system with a periodical-driving are studied. The Stokes phenomenon is observed in the adiabatic approximation. It is shown that the multiplier of the exponentially dominant eigenstate in the evolving state may be negligible and the multiplier of the exponentially small eigenstate in the evolving state may appear or disappear and oscillates drastically. Such Stokes phenomenon is independent of whether or not the exceptional point is encircled. The origin of such Stokes phenomenon is from the imaginary part of the complex eigenenergies. The eigenstate with the maximum imaginary part of dynamical phase factor in the evolving state will inevitably mask the other eigenstates. We also estimate the range of driving-frequency for the adiabatic approximation condition. The contribution is primarily theoretical in nature.
The time evolution of periodically driven non-Hermitian systems is in general nonunitary but can be stable. It is hence of considerable interest to examine the adiabatic following dynamics in periodically driven non-Hermitian systems. We show in this work the possibility of piecewise adiabatic following interrupted by hopping between instantaneous system eigenstates. This phenomenon is first observed in a computational model and then theoretically explained, using an exactly solvable model, in terms of the Stokes phenomenon. In the latter case, the piecewise adiabatic following is shown to be a genuine critical behavior and the precise phase boundary in the parameter space is located. Interestingly, the critical boundary for piecewise adiabatic following is found to be unrelated to the domain for exceptional points. To characterize the adiabatic following dynamics, we also advocate a simple definition of the Aharonov-Anandan (AA) phase for nonunitary cyclic dynamics, which always yields real AA phases. In the slow driving limit, the AA phase reduces to the Berry phase if adiabatic following persists throughout the driving without hopping, but oscillates violently and does not approach any limit in cases of piecewise adiabatic following. This work exposes the rich features of nonunitary dynamics in cases of slow cycling and should stimulate future applications of nonunitary dynamics.
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 adiabatic quantum evolution of a two-state system without energy-level crossings is an example of the Stokes phenomenon. In the latter, a small (subdominant) exponential is an asymptotic expansion appears when a Stokes line is crossed; truncating the dominant asymptotic series at its least term causes the multiplier of the subdominant term to rise in a smooth, compact and universal manner across the Stokes line. In quantum evolution this corresponds to a smooth transition, universal in form, between 'superadiabatic' basis states (high-order WKB approximate solutions of the time-dependent Schrodinger equation). The authors give a numerical demonstration of this previously predicted universality by constructing, for two Hamiltonians, the superadiabatic quantum bases asymptotic to the actual evolving state. Universality when a Stokes line is crossed is seen in the changing probability that the system makes a transition away from the superadiabatic state, and occurs at that order of superadiabatic approximation corresponding to truncating the asymptotic series at its least term.
In many classical and quantum systems described by an effective non-Hermitian Hamiltonian, spectral phase transitions, from an entirely real-energy spectrum to a complex spectrum, can be observed as a non-Hermitian parameter in the system is increased above a critical value. A paradigmatic example is provided by systems possessing parity-time (PT) symmetry, where the energy spectrum remains entirely real in the unbroken PT phase while a transition to complex energies is observed in the broken PT phase. Such spectral phase transitions are universally sharp. However, when the system is slowly and periodically cycled, the phase transition can become smooth, i.e., imperfect, owing to the complex Berry phase associated to the cyclic adiabatic evolution of the system. This remarkable phenomenon is illustrated by considering the spectral phase transition of the Wannier-Stark ladders in a PT-symmetric class of two-band non-Hermitian lattices subjected to an external dc field, revealing that a nonvanishing imaginary part of the Zak phase---the Berry phase picked up by a Bloch eigenstate evolving across the entire Brillouin zone---is responsible for imperfect spectral phase transitions.
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.
Non-Hermitian Hamiltonians provide a simple picture for inspecting dissipative systems with natural or induced gain and loss. We investigate the Floquet dynamical phase transition in the dissipative periodically time driven XY and extended XY models, where the imaginary terms represent the physical gain and loss during the interacting processes with the environment. The time-independent effective Floquet non-Hermitian Hamiltonians disclose three regions by analyzing the non-Hermitian gap: pure real gap (real eigenvalues), pure imaginary gap, and complex gap. We show that each region of the system can be distinguished by the complex geometrical non-adiabatic phase. We have discovered that in the presence of dissipation, the Floquet dynamical phase transitions (FDPTs) still exist in the region where the time-independent effective Floquet non-Hermitian Hamiltonians reveal real eigenvalues. Opposed to expectations based on earlier works on quenched systems, our findings show that the existence of the non-Hermitian topological phase is not an essential condition for dissipative FDPTs (DFDPTs). We also demonstrate the range of driven frequency, over which the DFDPTs occur, narrows down by increasing the dissipation coupling and shrinks to a single point at the critical value of dissipation. Moreover, quantization and jumps of the dynamical geometric phase reveals the topological characteristic feature of DFDPTs in the real gap region where confined to exceptional points.