Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

We study the transition between integrable and chaotic behavior 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 β∈[0,2]. Here, β=0 yields the two-dimensional Poisson distribution, matching the integrable limit of the system, and β=2 equals the distribution obtained from the complex Ginibre ensemble, describing the fully chaotic limit. Our findings generalize 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 β=2. It holds for all three Ginibre ensembles of random matrices with independent real, complex, or quaternion matrix elements.