Defect production due to time-dependent coupling to environment in the Lindblad equation

Recently defect production was investigated during non-unitary dynamics due to non-Hermitian Hamiltonian. By ramping up the coupling linearly in time through an exceptional point, defects are produced much same way as approaching a Hermitian critical point. A generalized Kibble--Zurek scaling accounted for ensuing of density terms speed drive and corresponding exponents. Here we extend this setting by adding recycling term considering full Lindbladian evolution problem with quantum jumps. We find that environmental time, going beyond steady-state solution Liouvillian, scales all cases. This is unaffected presence points which can show transient states. using variant adiabatic perturbation theory, determined exactly from set algebraic equations. Our study indicates distinct sensitivity steady states