1) PT-symmetric model systems: a) PT-symmetric generalisations of one-dimensional quantum maps. In particular for a PT-symmetric kicked top a detailed investigation was conducted analysing features of both the quantum and the classical model. This revealed a rich structure way beyond what has been observed in either Hermitian or fully open versions of the kicked top studied before. The statistics of the quasi energy nearest neighbour spacings follow a recently identified universality class. The result have been published in a paper in New Journal of Physics. The study of a PT-symmetric kicked rotor has led to new insights into the quantum phase-space features of lossy systems and their relation to the classical systems, which have revealed an intriguing connection between the Schur vectors of the quantum Hamiltonian and the classical norm as a function of an initial phase space point in the long-time limit. The results have been submitted to Physical Review Letters. b) Cold atoms in lossy lattices. The first system that was investigated was a lattice with loss form every other lattice site, in the presence of particle interactions, driven by a static external force. We have investigated Bloch oscillations in this system and unravelled a rich structure of new dynamical phenomena. The results have been published in a paper in Journal of Physics B. This study led to a spin off study on Landau-Zener transitions through exceptional points, as these occurred during the Bloch oscillations. We have published a paper on Landau-Zener transitions through a pair of exceptional points, as we have encountered in the Bloch oscillations of cold atoms in a lattice with loss from every other site, in Physical Review A. This led to a further study on Landau-Zener transitions through a pair of higher order exceptional points, which was published in Physical Review A. A small version of the system has also been used in our studies of the spectral statistics of PT-symmetric chaotic systems. c) Quantum billiards. We have throughout the project investigated geometries with interesting gain and loss patterns, and have obtained a number of interesting results.
2) PT-symmetric random matrix models: In a collaboration with Prof Joshua Feinberg and Dr Roman Riser of the University of Haifa, we have investigated the spectral statistics of a real Ginibre ensemble conditioned on having few real eigenvalues. We have implemented an effective Metropolis Monte Carlo sampling scheme to numerically sample these ensembles. Further we have used saddle-point techniques to make progress on analytical results, the problem turns out to be a very hard one, and no final results have been obtained yet. The collaboration is ongoing.
3) Semiclassical methods for open quantum systems: Semiclassical methods are indispensable for the interpretation and simulation of quantum phenomena for closed quantum systems. a) Lindblad systems: In collaboration with Roman Schubert from the University of Bristol we have derived a general structure for the semiclassical limit of open quantum systems described by the Lindblad equation, published in Journal of Physics A. This structure provides a starting point for the development of semiclassical propagators for simulating the full quantum dynamics of Markovian open systems or exploring the relationship between non-Hermitian and Lindblad dynamics. We have further investigated Gaussian wave packet propagation and related Hagedorn propagation in stochastic unravellings of the Lindblad equation, also in collaboration with Dr Roman Schubert, published in Journal of Physical A. b) Non-Hermitian systems: work initially focussed on the classical and semiclassical limit on non-flat phase spaces, such as the Bloch sphere for SU(2) systems, which are of importance for example in models for cold atoms in optical lattices. In a collaboration with Sven Gnutzmann from the University of Nottingham we have made great progress on the classical limit arising from non-Hermitian Hamiltonians, and are in the process of preparing the results for publication. As a spin-off of these ideas, we have re-visited the dynamics for systems on flat spaces, analysing the evolution of the Husimi distribution in the semiclassical limit. This has revealed a rich underlying classical structure of complexified classical trajectories following coherent state dynamics, and an additional norm landscape. The results have been published Physical Review Letters.
