The Dynamical Invariant of Open Quantum System

The dynamical invariant, whose expectation value is constant, is generalized to open quantum system. The evolution equation of dynamical invariant (the dynamical invariant condition) is presented for Markovian dynamics. Different with the dynamical invariant for the closed quantum system, the evolution of the dynamical invariant for the open quantum system is no longer unitary, and the eigenvalues of it are time-dependent. Since any hermitian operator fulfilling dynamical invariant condition is a dynamical invariant, we propose a sort of special dynamical invariant (decoherence free dynamical invariant) in which a part of eigenvalues are still constant. The dynamical invariant in the subspace spanned by the corresponding eigenstates evolves unitarily. Via the dynamical invariant condition, the results demonstrate that this dynamical invariant exists under the circumstances of emergence of decoherence free subspaces.