Ground state of open circuit QED systems in the deep-strong coupling regime

We investigate theoretically how the ground state of a qubit-resonator system in the deep-strong coupling regime is affected by the coupling to an environment. We show that the state of the qubit-resonator system strongly depends on the nature of the resonator-waveguide and resonator-qubit coupling, i.e., capacitive or inductive. When the resonator couples to the qubit and the environment in different ways (e.g., one is inductive and the other is capacitive), the system is almost unaffected by the resonator-waveguide coupling. In contrast, when the couplings are of the same type (e.g., both are inductive), the average number of virtual photons increases and the quantum superposition realized in the qubit-resonator entangled ground state is partially degraded due to the resonator-waveguide coupling. In this case, since the superposition becomes more fragile when the qubit-resonator coupling strength gets large, there exists an optimal strength of the qubit-resonator coupling to maximize the nonclassicality of the qubit-resonator system in a realistic setup.