Periodic Reporting for period 4 - CHAOS-PIQUANT (Universality and chaos in PT-symmetric quantum systems)
Reporting period: 2022-08-01 to 2023-01-31
The world of our daily experiences, described by classical physics, is built out of fundamental particles, governed by the laws of quantum mechanics. The striking difference between quantum and classical behaviour becomes most apparent in the realm of chaos, an extreme sensitivity to initial conditions, which is common in classical systems but impossible under quantum laws. The investigation of characteristic features of quantum systems whose classical counterparts are chaotic has illuminated foundational problems and led to a variety of technological applications. Traditional quantum theory focuses on the description of closed systems without losses. Every realistic system, however, contains unwanted losses and dissipation, but the idea to engineer them to generate desirable effects has recently come into the focus of scientific attention. The surprising properties of quantum systems with balanced gain and loss (PT-symmetric systems) have sparked much interest, and new experimental areas are rapidly emerging. Our theoretical understanding of PT-symmetric quantum systems, however, is still limited. One major shortcoming is that the emergence of chaos and universality in these systems is hitherto nearly unexplored. This project investigates PT-symmetric quantum chaos and aims to establish it as a new research area that will most likely overturn some common perceptions in the existing fields of PT-symmetry and quantum chaos. Ultimately this will lead to new experimental applications and quantum technologies. In particular, the main objectives are to a) identify spectral and dynamical features of chaos in PT-symmetric quantum systems, b) establish new universality classes, c) provide powerful semiclassical tools for the simulation of generic quantum systems, and d) facilitate experimental applications in microwave cavities and cold atoms.
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.
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.
The PT-symmetric kicked top is the first PT-symmetric quantum chaotic system for which the quantum-classical correspondence has been investigated in detail. This has the potential to open up a whole new line of research, and become a standard system such as the Hermitian kicked top has been in traditional quantum chaos. On the random matrix side, the real Ginibre ensemble remains notoriously little understood to this day. The new line of investigation linked to PT-symmetric features is a promising route and the initial numerical results for large matrices promise intriguing new behaviour. Also the results on the semiclassical side are ahead of the field, in leading a way towards new semiclassical propagation techniques for open quantum systems, as well as uncovering deep geometric connection. The newly uncovered classical structure in the time evolution of Husimi distributions for systems described by non-Hermitian Hamiltonians sheds a new light on non-Hermitian quantum dynamics, and I expect it to become a standard tool in the analysis of non-Hermitian dynamics.