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We present a theoretical framework for the dynamics of bosonic Bogoliubov quasiparticles. We call it Lorentz quantum mechanics because the dynamics is a continuous complex Lorentz transformation in complex Minkowski space. In contrast, in usual quantum mechanics, the dynamics is the unitary transformation in Hilbert space. In our Lorentz quantum mechanics, three types of state exist: space-like, light-like and time-like. Fundamental aspects are explored in parallel to the usual quantum mechanics, such as a matrix form of a Lorentz transformation, and the construction of Pauli-like matrices for spinors. We also investigate the adiabatic evolution in these mechanics, as well as the associated Berry curvature and Chern number. Three typical physical systems, where bosonic Bogoliubov quasi-particles and their Lorentz quantum dynamics can arise, are presented. They are a one-dimensional fermion gas, Bose-Einstein condensate (or superfluid), and one-dimensional antiferromagnet.

A non-Hermitian PT-symmetric version of the kicked top is introduced to study the interplay of quantum chaos with balanced loss and gain. The classical dynamics arising from the quantum dynamics of the angular momentum expectation values are derived. It is demonstrated that the presence of PT-symmetry can lead to 'stable' mixed regular chaotic behaviour without sinks or sources for subcritical values of the gain-loss parameter. This is an example of what is known in classical dynamical systems as reversible dynamical systems. For large values of the kicking strength a strange attractor is observed that also persists if PT-symmetry is broken. The intensity dynamics of the classical map is investigated, and found to provide the main structure for the Husimi distributions of the subspaces of the quantum system belonging to certain ranges of the imaginary parts of the quasienergies. Classical structures are also identified in the quantum dynamics. Finally, the statistics of the eigenvalues of the quantum system are analysed and it is shown that if most of the eigenvalues are complex (which is the case already for fairly small non-Hermiticity parameters) the nearest-neighbour distances of the (unfolded) quasienergies follow a two-dimensional Posisson distribution when the classical dynamics is regular. In the chaotic regime, on the other hand they are in line with recently identified universal complex level spacing distributions for non-Hermitian systems, with transpose symmetry ÂT = Â. It is demonstrated how breaking this symmetry (by introducing an extra term in the Hamiltonian) recovers the more familiar universality class for non-Hermitian systems given by the complex Ginibre ensemble. Both universality classes display cubic level repulsion. The PT-symmetry of the system does not seem to influence the complex level spacings. Similar behaviour is also observed for the spectrum of a PT-symmetric extension of the triadic Baker map.

We investigate the fourth-order generalized Ginzburg–Landau equation and the nonlinear modes modulated by PT -symmetric potentials. By means of Hirota method, we obtained the bilinear form of the equation and further derived the analytic soliton solution. Dynamic behaviors of the solitons under the modulation of near PT -symmetric potentials were studied by numerical simulation: The nonlinear modes tend to be unstable when the potential is closer to conventional PT -symmetric potential, and the amplitude of the nonlinear modes oscillates periodically when the imaginary part of the PT -symmetric potentials is sufficiently large. Moreover, we obtained new nonlinear modes that are different from the above analytic soliton solutions by numerical excitation and tested their stability. These new findings of nonlinear modes in the generalized Ginzburg–Landau model can be potentially applied to hydrodynamics, optics and matter waves in Bose–Einstein condensates.

We investigate the formation and propagation of multipole solitons (MSs) in a cold atomic gas with a parity-time (PT) symmetric potential. A ∧ -type three-level configuration is considered in the electromagnetically induced transparency arrangement. The PT potential is introduced through an effective magnetic field. Various types of MS solutions are studied theoretically, and the stability analysis of the solutions is discussed numerically, using the modified square-operator method. It is shown that the characteristics of MSs can be easily controlled and manipulated via adjusting the nonlinearity and the external magnetic field parameters.