# Universality classes of quantum chaotic dissipative systems

## TL;DR

This paper investigates the spectral statistics of a complex symmetric matrix ensemble relevant to dissipative quantum systems with time reversal symmetry, revealing unique eigenvalue repulsion behavior and validating findings with the quantum kicked rotor.

## Contribution

It introduces a new matrix ensemble modeling dissipation in time reversal invariant quantum systems and analyzes its spectral fluctuation statistics, showing weaker eigenvalue repulsion compared to Ginibre matrices.

## Key findings

- Eigenvalue spacing distribution follows $P(s) o -s^3 \log s$ at small s.
- Spectral statistics resemble those of dissipative chaotic systems with time reversal invariance.
- Quantum kicked rotor exhibits similar spacing distribution in the dissipative regime.

## Abstract

We study the ensemble of complex symmetric matrices. The ensemble is useful in the study of effect of dissipation on systems with time reversal invariance. We consider the nearest neighbor spacing distribution and spacing ratio to investigate the fluctuation statistics and show that these statistics are similar to that of dissipative chaotic systems with time reversal invariance. We show that, unlike cubic repulsion in eigenvalues of Ginibre matrices, these ensemble exhibits a weaker repulsion. The nearest neighbor spacing distribution exhibits $P(s) \propto -s^3 \log s$ for small spacings. We verify our results for quantum kicked rotor with time reversal invariance. We show that the rotor exhibits similar spacing distribution in dissipative regime. We also discuss a random matrix model for transition from time reversal invariant to broken case.