  Non-Hermitian Hamiltonians and Lindblad operators are some of the most
important generators of dynamics for describing quantum systems interacting
with different kinds of environments. The first type differs from conservative
evolution by an anti-Hermitian term that causes particle decay, while the
second type differs by a dissipation operator in Lindblad form that allows
energy exchange with a bath. However, although under some conditions the two
types of maps can be used to describe the same observable, they form a disjoint
set. In this work, we propose a generalized generator of dynamics of the form
$L_\text{mixed}(z,\rho_S) = -i[H,\rho_S] + \sum_i
\left(\frac{\Gamma_{c,i}}{z+\Gamma_{c,i}}F_i\rho_S F_i^{\dagger} -\frac{1}{2}
\{F_i^{\dagger} F_i,\rho_S \}_+\right)$ that depends on a general energy $z$,
and has a tunable parameter $\Gamma_c$ that determines the degree of particle
density lost. It has as its limits non-Hermitian ($\Gamma_c \to 0$) and
Lindbladian dynamics ($\Gamma_c \to \infty$). The intermediate regime evolves
density matrices such that $0 \leq \text{Tr} (\rho_S) \leq 1$. We derive our
generator with the help of an ancillary continuum manifold acting as a sink for
particle density. The evolution describes a system that can exchange both
particle density and energy with its environment. We illustrate its features
for a two level system and a five $M$ level system with a coherent population
trapping point.
