Universality classes of quantum chaotic dissipative systems

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.