  We consider a non-Hermitian Hamiltonian in order to effectively describe a
two-level system coupled to a generic dissipative environment. The total
Hamiltonian of the model is obtained by adding a general anti-Hermitian part,
depending on four parameters, to the Hermitian Hamiltonian of a tunneling
two-level system. The time evolution is formulated and derived in terms of the
normalized density operator of the model, different types of decays are
revealed and analyzed. In particular, the population difference and coherence
are defined and calculated analytically. We have been able to mimic various
physical situations with different properties, such as dephasing, vanishing
population difference and purification.
