Effects of an environment on the ground state of 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 (DSC) regime is affected by the coupling to an environment. We employ a superposition of coherent states displaced in the qubit-state-dependent directions as a variational ansatz for the ground state of the qubit-resonator-environment system. We show that the reduced density matrix of the qubit-resonator system strongly depends on types of the resonator-waveguide and resonator-qubit coupling, i.e., capacitive or inductive, because of the broken rotational symmetry of the eigenstates of the DSC system in the resonator phase space. When the resonator couples to the qubit and the environment in different ways (for instance, one is inductive and the other is capacitive), the system is almost unaffected by the resonator-waveguide coupling. In contrast, when the types of two couplings are the same (for instance, both are inductive), by increasing the resonator-waveguide coupling strength, the average number of virtual photons increases and the quantum superposition realized in the qubit-resonator entangled ground state is partially degraded. 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.