A continuous transformation between non-Hermitian Hamiltonian and Lindbladian evolution

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.