A detailed study on ergodic to non-ergodic phase transition in the dissipative anisotropic Dicke model

Abstract: Ergodic-non-ergodic phase transition is one of the paramount features of the anisotropic Dicke model. Here, we have thoroughly examined the effect of the dissipation by analyzing both the eigenvalue and eigenvector properties of the Liouvillian with the aid of the scaling of Liouvilian gap, and the average participation ratio. We show that the properties of the eigenvectors of Liouvillian are consistent with those of the eigenvalues, revealing a phase diagram, which has similarities to the non-ergodic to ergodic transition in the closed undriven system. We also uncover that the Liouvillian gap is independent of system size in the non-ergodic phase whereas in the ergodic phase, it scales with the atom number as: $N^{-z}$ where $0<z<1$. Moreover, we extend our analysis to the driven case where a Thue-Morse quasi-periodic sequence is applied and observe that the boson dissipation plays a pivotal role in stabilizing the prethermal plateau. Our investigation indicates that a non-ergodic phase is more favorable than the ergodic phase in the presence of bosonic dissipation.