Unraveling Integrability and Quantum Chaos in Open Quantum System

We study the quantum and semiclassical dynamics of the dissipative SU(3) BoseHubbard trimer model. By setup non-cyclic and cyclic limits, we could study thegapped and gapless Liouvilian spectrum and the dynamics of steady states. Employingexact diagonalization on the quantum Liouvillian superoperator and steady-statedensity matrix, we characterize quantum chaos through the level statistics of theireigenvalues.

The gapped case exhibits a unique well-defined steady state, while thegapless one possesses multiple steady states with regular, limit cycles and chaotictrajectories. The spacing level statistics of the density matrix associated with a unique steady stateor regular trajectories is a Poisson, while for chaotic multiples steady states thedistribution is Gaussian Unitary Ensemble.

From the semi-classical point of view, we obtained the equations of motion by astandard Keldysh path integral method together with the proper stochastic Langevinequation. We show robust evidence pointing out a deep connection between thequantum-level statistics of the density matrix and the distribution of Lyapunovexponents associated with classical trajectories for long times.