Lindbladian dynamics of the Sachdev-Ye-Kitaev model

We study the open quantum dynamics of the Sachdev-Ye-Kitaev (SYK) model, where the SYK system is coupled to a Markovian bath. The dynamics is described by the Lindblad master equation and the models we consider have jump operators that are either linear or quadratic in the Majorana fermions. In the limit of large number of Majorana fermions, these models become analytically tractable. We thus compute various steady-state and dynamical properties of the models. In particular we compute the steady-state Green's functions and associated decay rates. For the quadratic model, the Green's functions exhibit an underdamped to overdamped transition, while the decay rates show evidence of the quantum Zeno effect. To study the open quantum dynamics, we compute the dissipative form factor, an open quantum generalization of the Loschmidt echo, defined as the average overlap between the initial and time-evolved density matrices. The dissipative form factor exhibits first and second-order dynamical phase transitions.