Ground state of a circuit QED system in the deep-strong-coupling regime coupled to an environment

We investigate theoretically how the ground state of a qubit-resonator system in the deep-strong coupling regime is modified by the coupling to an environment. We employ the qubit-dependent coherent/squeezed-coherent state as the variational state for the ground state of the enlarged qubit-resonator-environment system and observe the following points. (i) The number of virtual photons increases by the coupling to an environment, proportionally to the square root of the loss rate κ of photons in the resonator. (ii) The ground state of the qubit-resonator system becomes mixed even at the zero temperature. The decrease in purity is proportional to the loss rate κ. (iii) There exists an optimal value of the qubit-resonator coupling to maximize the nonclassicality of the qubit-resonator system, which is quantified by the metrological power.