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Universality and chaos in PT-symmetric quantum systems

Periodic Reporting for period 2 - CHAOS-PIQUANT (Universality and chaos in PT-symmetric quantum systems)

Reporting period: 2019-08-01 to 2021-01-31

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.
The work has so far focused on three main strands of research: 1) The detailed investigation of PT-symmetric model systems; 2) The study of PT-symmetric random matrix models; and 3) Semiclassical methods for open quantum systems. Details of the work performed in each of these strands are briefly outlined below.

1) PT-symmetric model systems. The work in this direction has evolved around three different types of models. The first are 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 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. The second type of models are 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. A small version of the system has also been used in our studies of the spectral statistics of PT-symmetric chaotic systems. The third type of models are quantum billiards. We are in the process of implementing stable numerical schemes using finite element methods to investigate nontrivial geometries with interesting gain and loss patterns. In parallel we are studying a number of analytically solvable (integrable) billiards.

2) PT-symmetric random matrix models. Here we have focused on the split-complex Gaussian ensemble, which is isospectral to the real Ginibre ensemble. We have made progress in using the split-Hermiticity to rederive known results for the real Ginibre ensemble in a potentially more versatile way, that could be generalised to yet unknown properties. This work is currently underway. In a collaboration with Prof Joshua Feinberg and Dr Roman Riser of the University of Haifa, Israel, we have further 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.

3) Semiclassical methods for open quantum systems. This activity is divided into Lindblad dynamics and dynamics generated by non-Hermitian Hamiltonians. Semiclassical methods are indispensable for the interpretation and simulation of quantum phenomena for closed quantum systems. In collaboration with Roman Schubert from the University of Bristol and one of his PhD students we have derived a general structure for the semiclassical limit of open quantum systems described by the Lindblad equation. 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. On the non-Hermitian side work focuses 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.
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 connections. By the end of the project it is expected that the spectral results for the real Ginibre ensemble will be combined with and applied to the spectra of model systems such as the kicked top and the kicked rotor to identify new universal features of these models. Further, it is expected that the semiclassical branch will lead to new propagators for quantum dynamical applications. Finally, the theoretical results will by the end of the project be ready to be verified in direct experimental applications.
Husimi Schur representation of the quantum states with growing norm for a PT-symmetric kicked top