Generation of long-lived $W$ states via reservoir engineering in dissipatively coupled systems

Very recently, dissipative coupling was discovered, which develops and broadens methods for controlling and utilizing light-matter interactions. Here, we propose a scheme to generate the tripartite $W$ state in a dissipatively coupled system, where one qubit and two resonators simultaneously interact with a common reservoir. With appropriate parameters, we find the $W$ state is a dark state of the system. By driving the qubit, the dissipatively coupled system will evolve from the ground state to the tripartite $W$ state. Because the initial state is the ground state of the system and no measurement is required, our scheme is easy to implement in experiments. Moreover, the $W$ state decouples from the common reservoir and thus has a very long lifetime. This scheme is applicable to a wide class of dissipatively coupled systems, and we specifically illustrate how to prepare the $W$ state in a hybrid qubit-photon-magnon system by using this scheme.