Lindbladian operators, von Neumann entropy and energy conservation in time-dependent quantum open systems

The Lindblad equation is widely employed in studies of Markovian quantum open systems. Here, firstly, a simple result is presented on the time evolution of the non Neumann entropy under the Lindblad equation, which enables one to examine if the entropy increases/decreases. Then, secondly, the following question is posed: In a quantum open system with a time-dependent Hamiltonian, what is the corresponding Lindblad equation for the quantum state that keeps the internal energy of the system constant in time? This issue is of importance in realizing quasi-stationary states of open systems such as quantum circuits and batteries. As an illustrative example, the time-dependent harmonic oscillator is analyzed. The Lindbladian operator is uniquely determined with the help of a Lie-algebraic structure, and the time derivative of the von Neumann entropy is shown to be nonnegative depending on the time-dependence of the Hamiltonian.