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