  One among the possible realizations of non-Hermitian systems is based on open
quantum systems by omitting quantum jumping terms in the master equation. This
is a good approximation at short times where the effects of quantum jumps can
be ignored. However, the jumps can affect the long time dynamics of the system,
motivating us to take the jumps into account in these studies. In this paper,
by treating the quantum jumps as perturbations, we examine the effect of the
quantum jumps on the non-Hermitian system. For this purpose, we first derive an
effective Hamiltonian to describe the dynamics of the open quantum system based
on the master equation, then expand the eigenstates and eigenenergies up to the
first and second order in the quantum jumps. Finally, we apply our theory to a
dissipative two-level system and dissipative fermionic superfluids. The effect
of quantum jump on the dynamics and the nonequilibrium phase transition is
demonstrated and discussed.
