Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

We study the transition between integrable and chaotic behaviour in dissipative open quantum systems, exemplified by a boundary driven quantum spin-chain. The repulsion between the complex eigenvalues of the corresponding Liouville operator in radial distance $s$ is used as a universal measure. The corresponding level spacing distribution is well fitted by that of a static two-dimensional Coulomb gas with harmonic potential at inverse temperature $\beta\in[0,2]$. Here, $\beta=0$ yields the two-dimensional Poisson distribution, matching the integrable limit of the system, and $\beta=2$ equals the distribution obtained from the complex Ginibre ensemble, describing the fully chaotic limit. Our findings generalise the results of Grobe, Haake and Sommers who derived a universal cubic level repulsion for small spacings $s$. We collect mathematical evidence for the universality of the full level spacing distribution in the fully chaotic limit at $\beta=2$. It holds for all three Ginibre ensembles of random matrices with independent real, complex or quaternion matrix elements.