Simple determination of dark states in a general multi-level system

Abstract In a multi-level energy system with energy transitions, dark states are eigenstates of a Hamiltonian that consist entirely of ground states, with zero amplitude in the excited states. We present several criteria which allows one to deduce the presence of dark states in a general multi-level system based on the submatrices of the Hamiltonian. The dark states can be shown to be the right-singular vectors of the submatrix that connect the ground states to the excited states. Furthermore, we show a simple way of finding the dark state involving the determinant of a matrix constructed from the same submatrix.