General dark-state theory for arbitrary multilevel quantum systems

The dark-state effect, caused by destructive quantum interference, is an important physical effect in atomic physics and quantum optics. It not only deepens the understanding of light-atom interactions, but also has wide applications in quantum physics and quantum information. Therefore, how to efficiently and conveniently determine the number and form of the dark states in multilevel quantum systems with complex transitions is an important and interesting topic in this field. In this work, we present a general theory for determining the dark states in multilevel quantum systems with any coupling configuration using the arrowhead-matrix method. To confirm the dark states in a multilevel system, we first define the upper- and lower-state subspaces, and then diagonalize the Hamiltonians restricted within the two subspaces to obtain the dressed upper and lower states. By further expressing the transitions between the dressed upper and lower states, we can map the multilevel system to a bipartite-graph network, in which the nodes and links are acted by the dressed states and transitions, respectively. Based on the coupling configurations of the network, we can determine the lower dark states with respect to the upper-state subspace. As examples, we analyze the dark states in three-, four-, and five-level quantum systems, for all possible configurations through the classification of the numbers of upper and lower states. Furthermore, we extend the framework to multilevel quantum systems and discuss the existence of dark states in some typical configurations. We also recover the results of the dark-state polaritons in driven three-level systems with the arrowhead-matrix method. Our theory paves the way for manipulating and utilizing the dark states of multilevel quantum systems in atomic physics and quantum optics.