  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.
