Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos

Abstract

We introduce a complex-plane generalization of the consecutive level-spacing ratio distribution used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of complex-valued ratios between nearest- and next-to-nearest-neighbor spacings. We show that this quantity can successfully detect the chaotic or regular nature of complex-valued spectra, which is done in two steps. First, we show that, if eigenvalues are uncorrelated, the distribution of complex spacing ratios is flat within the unit circle, whereas random matrices show a strong angular dependence in addition to the usual level repulsion. The universal fluctuations of Gaussian unitary and Ginibre unitary universality classes in the large-matrix-size limit are shown to be well described by Wigner-like surmises for small-size matrices with eigenvalues on the circle and on the two-torus, respectively. To study the latter case, we introduce the toric unitary ensemble, characterized by a flat joint eigenvalue distribution on the two-torus. Second, we study different physical situations where non-Hermitian matrices arise: dissipative quantum systems described by a Lindbladian, nonunitary quantum dynamics described by non-Hermitian Hamiltonians, and classical stochastic processes. We show that known integrable models have a flat distribution of complex spacing ratios, whereas generic cases, expected to be chaotic, conform to random matrix theory predictions. Specifically, we are able to clearly distinguish chaotic from integrable dynamics in boundary-driven dissipative spin-chain Liouvillians and in the classical asymmetric simple exclusion process and to differentiate localized from delocalized regimes in a non-Hermitian disordered many-body system.

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Plain Language Summary

The eventual success of quantum-based technologies hinges, in part, on understanding how these systems interact with their environment, which can easily destroy their fragile quantum states. Mathematical work in the field of quantum chaos has led to the establishment of several statistical properties for isolated chaotic quantum systems, but no one has yet attempted to generalize these findings to open systems in contact with an environment. Here, we introduce mathematical tools to help bridge this gap.

Hamiltonians of complex quantum systems behave in several aspects like a large random matrix. This finding links random-matrix theory and quantum chaos in a deep and intricate way. In open quantum systems, the matrices have complex eigenvalues, whose real components are energies and imaginary parts are decay rates.

In our work, we introduce ratios of spacings (or differences) of these eigenvalues, which are also complex numbers. Since the matrices are random, so are these ratios. We study their statistical distribution, which allows us to distinguish chaotic open systems from those whose dynamics is exactly solvable: Eigenvalues of chaotic systems repel each other whereas in solvable systems they attract.

By taking into account dissipation and decoherence via the imaginary parts of the eigenvalues (the decay rates), we adapt methods built for closed systems. Complex spacing ratios provide a simple and efficient tool to determine whether an open quantum system is chaotic or exactly solvable. The ubiquity of quantum dissipation and decoherence renders our findings of great interest to fields ranging from condensed matter to quantum optics and of potential technological impact in the fabrication of complex quantum structures and, ultimately, quantum computers.