  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.
