The two-time Leggett-Garg inequalities of a superconducting qubit interacting with thermal photons in a cavity

In this paper, we study the two-time Leggett-Garg (LG) inequalities of a quantum optical model that appears in the Josephson-junction quantum bit (qubit) interacting with an external magnetic flux. This model is a natural extension of an exactly solvable model whose interaction between a qubit and single-mode photons is given by a product of the Pauli z operator of the qubit and a linear combination of annihilation and creation operators of the photons. By contrast, a photon's part of the interaction of our model is given by the square of the linear combination. Because our model is not solvable, we approximately investigate its time evolution up to the second-order perturbation. Our numerical calculations show that violation of the LG inequality diminishes as the temperature increases. Moreover, it exhibits power laws of the temperature, whose exponents vary depending on the coupling constant of the interaction between the qubit and photons. The violation of the LG inequality decreases and becomes less sensitive to the temperature as the coupling constant of the interaction gets larger.