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 as a variational ansatz for the ground state of the qubit-resonator-environment system a superposition of coherent states displaced in qubit-state-dependent directions. We show that the reduced density matrix of the qubit-resonator system strongly depends on how the system is coupled to the environment, 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 two couplings are of the same type (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 with increasing the qubit-resonator coupling, there exists an optimal coupling strength to maximize the nonclassicality of the qubit-resonator system.
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