A non-HermitianPT-symmetric kicked top

A non-HermitianPT-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 ofPT-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 asreversibledynamical systems. For large values of the kicking strength a strange attractor is observed that also persists ifPT-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 symmetrya(T)=a. 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. ThePT-symmetry of the system does not seem to influence the complex level spacings. Similar behaviour is also observed for the spectrum of aPT-symmetric extension of the triadic Baker map.