We derive the analytical solution of the model of a two-state system interacting with an external coherent field, in which the Hamiltonian is pseudo-Hermitian. We describe in detail the non-Hermitian generalization of the famed Landau-Zener-St\"uckelberg-Majorana model, but similar generalizations can be derived in a very simple fashion for the other analytically soluble two-state models. The analytical solutions possess a non-Hermitian dynamical invariant, which replaces the probability conservation condition in the Hermitian case. Implementations in waveguide optics and nonlinear frequency conversion are suggested.
We study nonlinear Landau-Zener-St\"uckelberg-Majorana (LZSM) tunneling with a
PT-symmetric potential, which can be converted to a non-Hermitian nonlinear two-level model by a two-mode approximation. Then, we construct an equivalent Hermitian model through a transformation, verify that the tunneling probabilities obtained by the two models are consistent, and comprehensively analyze the effect of the gain-dissipation term on tunneling. In addition, due to the gain-dissipation properties, the total population density of the non-Hermitian model is no longer a conserved quantity, which will increase or decay to a fixed value as the model evolves. Finally, we prove that the spatial inversion symmetry breaking of the Hamiltonian leads to nonreciprocal tunneling, and the numerical simulation results confirm our analysis. These studies provide alternative ideas to study the nonlinear dynamics of non-Hermitian systems in the future.
We systematically investigate the non-Hermitian generalizations of the Landau-Zener (LZ) transition and the Landau-Zener-St\"uckelberg (LZS) interferometry. The LZ transition probabilities, or band populations, are calculated for a generic non-Hermitian model and their asymptotic behavior is analyzed. We then focus on non-Hermitian systems with a real adiabatic parameter and study the LZS interferometry formed out of two identical avoided level crossings. Four distinctive cases of interferometry are identified and the analytic formulas for the transition probabilities are calculated for each case. The differences and similarities between the non-Hermitian case and its Hermitian counterpart are emphasized. In particular, the geometrical phase originated from the sign change of the mass term at the two level crossings is still present in the non-Hermitian system, indicating its robustness against the non-Hermiticity. We further apply our non-Hermitian LZS theory to describing the Bloch oscillation in one-dimensional parity-time
(PT) reversal symmetric non-Hermitian Su-Schrieffer-Heeger model and propose an experimental scheme to simulate such dynamics using photonic waveguide arrays. The Landau-Zener transition, as well as the LZS interferometry, can be visualized through the beam intensity profile and the transition probabilities measured by the center of mass of the profile.
Motivated by recent advances in modeling pseudo-Hermitian Hamiltonian (pHH) systems using superconducting qubits, we analyze their quantum dynamics subject to a small time-dependent perturbation. In particular, we develop the linear response theory formulation suitable for application to various pHH systems and compare it to the ones available in the literature. We derive analytical expressions for the generalized temporal quantum-mechanical correlation function
C(t) and the time-dependent dynamic susceptibility $\ensuremath{\chi}(t)\ensuremath{\propto}\text{Im}\phantom{\rule{0.16em}{0ex}}C(t)$. We apply our results to two
PT-symmetric non-Hermitian quantum systems: a single qubit and two unbiased/biased qubits coupled by the exchange interaction. For both systems, we obtain the eigenvalues and eigenfunctions of the Hamiltonian and identify
PT-symmetry unbroken and broken quantum phases and quantum phase transitions between them. The temporal oscillations of the dynamic susceptibility of the qubits polarization (
z projection of the total spin), $\ensuremath{\chi}(t)$, relate to
ac-induced transitions between different eigenstates and we analyze the dependencies of the oscillation frequency and the amplitude on the gain/loss parameter $\ensuremath{\gamma}$ and the interaction strength
g. Studying the time dependence of $\ensuremath{\chi}(t)$, we observe different types of oscillations, i.e., undamped, heavily damped, and amplified ones, related to the transitions between eigenstates with broken (unbroken)
PT symmetry. These predictions can be verified in the microwave transmission experiments allowing controlled simulation of the pHH systems.
Abstract In this work, we have studied the non-Hermitian nonlinear Landau–Zener–Stückelberg–Majorana (LZSM) interferometry in a non-Hermitian N -body interacting boson system in which the non-Hermiticity is from the nonreciprocal tunnelings between the bosons. By using the mean-field approximation and projective Hilbert space, the effect of nonreciprocity and nonlinearity on the energy spectrum, the dynamics, and the formation of the interference fringes have been studied. The different symmetries and the impact of the two different types of reciprocity, i.e. the in-phase tunneling and anti-phase tunneling, on the energy spectrum and the phase transition between the Josephson oscillations and the self-trapping have been investigated. For the LZSM interferometry, the strength of the nonreciprocity is found to take an essential role in the population of the projective state and the strengths of the interference patterns in the projective space. While the conditions of destructive and constructive interference under the weak-coupling approximation still only depend on the strength of nonlinearity. Our result provides an application of the nonlinear non-Hermitian LZSM interferometry in studying the parameters of a two-level system which related to the nonlinearity and the non-Hermiticity.
Abstract In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo P T symmetry, i.e. the non-Hermitian Hamiltonian H is related to its adjoint H † via the relation, H † = P T H P T . We propose a derivation of pseudo P T symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new symmetry operator η ˜ ( t ) = P T η ( t ) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1, 1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new symmetry operator and discuss a concrete physical applications of our results. Dedicated to the memory of my mother Djabou Zoulikha and to Djabi Smail