For a two-level system in a time-periodic field we show that in the non-Hermitian $\mathit{PT}$ case the level crossing is of two quasistationary states that have the same dynamical symmetry property. At the field's parameters where the two levels which have the same dynamical symmetry cross, the corresponding quasienergy states coalesce and a self-orthogonal state is obtained. This situation is very different from the Hermitian case where a crossing of two quasienergy levels happens only when the corresponding two quasistationary states have different dynamical symmetry properties and, unlike the situation in the non-Hermitian case, the spectrum remains complete also when the two levels cross.
For a two-level system in a time-periodic field we show that in the non-Hermitian PT case the level crossing is of two quasistationary states that have the same dynamical symmetry property. At the field's parameters where the two levels which have the same dynamical symmetry cross, the corresponding quasienergy states coalesce and a self-orthogonal state is obtained. This situation is very different from the Hermitian case where a crossing of two quasienergy levels happens only when the corresponding two quasistationary states have different dynamical symmetry properties and, unlike the situation in the non-Hermitian case, the spectrum remains complete also when the two levels cross.
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.
In recent years, there have been intensive studies on non-Hermitian physics and parity-time (PT) symmetry, due to their fundamental importance in theory and outstanding applications. A distinctive character in PT-symmetric systems is phase transition (spontaneous PT-symmetry breaking), where the energy spectrum changes from all real to complex 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. Generalization of PT-symmetric potentials to wider classes of non-PT-symmetric complex potentials with all-real spectra is a currently important endeavor. The simple PT-symmetric two-level Floquet quantum system is now being actively explored, because it holds potential for realization of non-unitary single-qubit quantum gate. However, studies on the evolution dynamics of non-PT-symmetric two-level non-Hermitian Floquet quantum system still remain 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. Based on these pseudo fixed points, we analytically construct the non-unitary evolution operator and then explore the dynamics 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 being non-parity-time-symmetric, still features a phase transition with the quasienergy spectrum changing from all real to complex, just as for PT symmetric systems. 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 respects fully the time-space symmetry (PT symmetry), while time-evolution of the quantum state (containing the phase) does not, due to the fact that time-evolution of the phases of the probability amplitudes on the two levels violates the PT symmetry requirement.
The dynamic effect of particles in a periodically driven two-level system is studied with the help of SU(2) symmetry. General closed-form solutions for the quasienergy and the Floquet states are obtained, from which it is found that the avoided quasienergy crossing is confined in the range (−ω/2, ω/2), where ω is the frequency of the driving fields. Of particular interest are high-frequency driving fields, of which it is demonstrated that the exact quasienergy crossings occur at certain values of amplitude and frequency of driving fields, and this phenomenon of dynamic localization should be determined by experiments.
In the standard (Hermitian) formalism of quantum mechanics, usually when a potential parameter is varied the crossing of energy levels with the same symmetry is avoided. However, within the framework of the non-Hermitian formalism it is possible that two (or even more) complex eigenvalues with the same symmetry will cross. At the crossing point the eigenvalues are degenerate and this is accompanied by the coalescence of the eigenfunctions (or eigenvectors). Therefore, we may term this special situation as a non-Hermitian degeneracy. This special situation is associated with a branch point in the complex energy plane which is commonly termed an “exceptional point” in the spectrum of the non-Hermitian Hamiltonian. With respect to the c-product defined for non-Hermitian operators (matrices) in Chapter 6, the degenerate eigenstate is self-orthogonal. Since a branch point in the spectrum is removed by any infinitesimally small external perturbation, it seems to be inaccessible experimentally and may be considered just as a mathematical object rather than a physical one. However, as we will show here, by varying the potential parameters the existence of a branch point is reflected in the measurement of the geometrical phases also known as Berry phases. It should be stressed here that while in our case the geometrical phase results from a coalescence of eigenfunctions of a non-Hermitian Hamiltonian, the so-called Berry phase phenomenon occurs also within the Hermitian formalism of quantum mechanics when the eigenvalues of the molecular Hamiltonian in the Born–Oppenheimer approximation are degenerate for specific geometry of the poly-atomic molecule.
We consider a two-level system such as a two-level atom, interacting with a cavity field mode in the rotating wave approximation, when the atomic transition frequency or the field mode frequency is periodically driven in time. We show that in both cases, for an appropriate choice of the modulation parameters, the state amplitudes in a generic n-excitation subspace obey the same equations of motion that can be obtained from a static non-Hermitian Jaynes-Cummings Hamiltonian with PT symmetry, that is with an imaginary coupling constant. This gives further support to recent results showing the possible physical interest of PT symmetric non-Hermitian Hamiltonians. We also generalize the well-known diagonalization of the Jaynes-Cummings Hamiltonian to the non-Hermitian case in terms of pseudobosons and pseudofermions, and discuss relevant mathematical and physical aspects.
The dynamics of open quantum systems is determined by avoided and true crossings of eigenvalue trajectories of a non-Hermitian Hamiltonian. The phases of the eigenfunctions are not rigid so that environmentally induced spectroscopic redistribution processes may take place and a dynamical phase transition may occur. Due to the formal equivalence between the quantum mechanical Schrödinger equation and the optical wave equation in PT symmetric lattices, the dynamics of the system is determined also in this case by avoided and true crossings of eigenvalue trajectories of the non-Hermitian Hamiltonian. In contrast to the eigenvalues characterizing an open quantum system, the eigenvalues describing the PT symmetric optical lattice are real as long as the influence of the environment (lattice) onto the optical wave equation is small. In the regime of avoided level crossings, the symmetry is destroyed, the eigenvalues become complex and a dynamical phase transition occurs similar as in the open quantum system. The redistribution processes in the regime of avoided level crossings allow to design systems with desired properties in a broad parameter range.