We show that the analog of the geometric phase for non-Hermitian coupled waveguides with $\mathcal{PT}$ symmetry and at least one periodically varying parameter can be purely imaginary, is hence no longer a true phase and instead a real multiplier, and will consequently result in the amplification of Floquet sidebands in the system. The sideband peaks seen in the spectrum of the system's eigenstates after evolution along the waveguides can be directly mapped to the spectrum of the derivative of the geometric function. The sidebands are magnified (becoming virtually unstable) as the exceptional point of the system is approached, and nonadiabatic effects begin to appear. Because the system cannot evolve adiabatically in the vicinity of the exceptional point, $\mathcal{PT}$ symmetry will be observed breaking earlier than theoretically predicted.
We show that the analog of the geometric phase for non-Hermitian coupled waveguides with PT symmetry and at least one periodically varying parameter can be purely imaginary, is hence no longer a true phase and instead a real multiplier, and will consequently result in the amplification of Floquet sidebands in the system. The sideband peaks seen in the spectrum of the system's eigenstates after evolution along the waveguides can be directly mapped to the spectrum of the derivative of the geometric function. The sidebands are magnified (becoming virtually unstable) as the exceptional point of the system is approached, and nonadiabatic effects begin to appear. Because the system cannot evolve adiabatically in the vicinity of the exceptional point, PT symmetry will be observed breaking earlier than theoretically predicted.
A non-Hermitian coupled waveguide system with periodically varying parameters, in which the Berry curvature is analogous to a hyperbolic magnetic monopole or antimonopole, is investigated. It is shown to have a purely imaginary Berry connection and is consequently influenced by a geometric multiplier. It is possible for this multiplier to induce net gain or loss in the system, corresponding to the existence of the antimonopole or monopole in parameter space, respectively. For the right choice of parameters, the system will display an apparent nonadiabatic change in behavior, which implies a switch between the dominant eigenstate in the waveguides, leading to a change in parameter space analogous to a charge reversal of the hyperbolic magnetic monopole.
The geometrical Berry phase is key to understanding the behaviour of quantum states under cyclic adiabatic evolution. When generalised to non-Hermitian systems with gain and loss, the Berry phase can become complex, and should modify not only the phase but also the amplitude of the state. Here, we perform the first experimental measurements of the adiabatic non-Hermitian Berry phase, exploring a minimal two-site PT-symmetric Hamiltonian that is inspired by the Hatano-Nelson model. We realise this non-Hermitian model experimentally by mapping its dynamics to that of a pair of classical oscillators coupled by real-time measurement-based feedback. As we verify experimentally, the adiabatic non-Hermitian Berry phase is a purely geometrical effect that leads to significant amplification and damping of the amplitude also for non-cyclical paths within the parameter space even when all eigenenergies are real. We further observe a non-Hermitian analog of the Aharonov--Bohm solenoid effect, observing amplification and attenuation when encircling a region of broken PT symmetry that serves as a source of imaginary flux. This experiment demonstrates the importance of geometrical effects that are unique to non-Hermitian systems and paves the way towards the further studies of non-Hermitian and topological physics in synthetic metamaterials.
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.
This work presents a comprehensive study of geometric transport in non-Hermitian Floquet lattices. We derive the non-Hermitian Berry curvature using the biorthogonal formalism and prove an exact algebraic no-go theorem: perfectly anti-correlated driving (φ=0) yields zero net displacement for any wavepacket. When driving symmetry is broken (φ=π/2), a Gaussian wavepacket acquires finite directional displacement per cycle. Using adaptive peak-centered quadrature with analytic derivatives and direct time-dependent wavepacket simulations on finite chains, we demonstrate transport enhancement near the exceptional-point manifold, accompanied by non-monotonic behavior due to moving curvature hotspots. The paper includes full analytical characterization of the EP manifold topology, explicit convergence tests, finite-size scaling, and clear separation between geometric pumping and non-Hermitian skin effect. All Python scripts, raw outputs, and high-resolution figures are provided for full reproducibility. Keywords: non-Hermitian Floquet systems, exceptional points, geometric transport, Berry curvature, no-go theorem, adiabaticity, wavepacket dynamics
In recent years, there have been intensive studies of non-Hermitian physics and parity–time (PT) symmetry due to their fundamental importance in theory and outstanding applications. A distinctive character in PT-symmetric system is phase transition (spontaneous PT-symmetry breaking), i.e. an all-real energy spectrum changes into an all-complex one when the non-Hermitian parameter exceeds a certain threshold. However, the conditions for PT-symmetric system with real energy spectrum to occur are rather restrictive. The generalization of PT-symmetric potentials to wider classes of non-PT-symmetric complex potentials with all-real energy spectra is a currently important endeavor. A simple PT-symmetric two-level Floquet quantum system is now being actively explored, because it holds potential for the realization of non-unitary single-qubit quantum gate. However, studies of the evolution dynamics of non-PT-symmetric two-level non-Hermitian Floquet quantum system are still relatively rare. In this paper, we investigate the non-Hermitian physics of a periodically driven non-PT-symmetric two-level quantum system. By phase-space analysis, we find that there exist so-called pseudo fixed points in phase space representing the Floquet solutions with fixed population difference and a time-dependent relative phase between the two levels. According to these pseudo fixed points, we analytically construct a non-unitary evolution operator and then explore the dynamic behaviors of the non-PT-symmetric two-level quantum system in different parameter regions. We confirm both analytically and numerically that the two-level non-Hermitian Floquet quantum system, although it is non-parity-time-symmetric, still features a phase transition with the quasienergy spectrum changing from all-real to all-complex energy spectrum, just like the PT symmetric system. Furthermore, we reveal that a novel phenomenon called quasi-PT symmetric dynamics occurs in the time evolution process. The quasi-PT symmetric dynamics is so named in our paper, in the sense that the time-evolution of population probabilities in the non-PT-symmetric two-level system satisfies fully the time-space symmetry (PT symmetry), while time-evolution of the quantum state (containing the phase) does not meet such a PT symmetry, due to the fact that time-evolution of the phases of the probability amplitudes on the two levels violates the requirement for the PT symmetry.
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 nonadiabatic 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.