Classification of dark states in multilevel dissipative systems

Dark states are eigenstates or steady states of a system that are decoupled from the radiation. Their use, along with associated techniques such as stimulated Raman adiabatic passage, has extended from atomic physics, where it is an essential cooling mechanism, to more recent versions in the condensed phase where it can increase the coherence times of qubits. These states are often discussed in the context of unitary evolution and found with elegant methods exploiting symmetries, or via the Morris-Shore transformation. However, the link with dissipative systems is not always transparent, and distinctions between classes of coherent population trapping are not always clear. We present a detailed overview of the arguments to find stationary dark states in dissipative systems, and examine their dependence on the Hamiltonian parameters, their multiplicity, and purity. We evidence the class of dark states that depends not only on the detunings of the lasers but also on their relative intensities and phases. We illustrate the criteria with the more complex physical system of the hyperfine transitions of Rb-87 and show how a knowledge of the dark-state manifold can guide the preparation of pure states.