Measurement of critical photon number of the 2^nd excited state of a Transmon qubit

Two level quantum states in a transmon qubit can be distinguished by their interaction with a dispersive coupled cavity. When the interaction between the cavity and the qubit is relatively smaller than exchange energy of cavity photon and qubit excitation, the combined cavity-qubit system can be assumed as a linear system and perturbation. Non-linear contribution at higher Fock numbers and higher order qubit state becomes more relevant in the interaction Hamiltonian, H=g(a^†b+ab^†), as the photon number in the cavity increases. The resonance frequency of cavity shifts gradually with increasing power and splits finally when exceeding the critical point. These shift and splitting, non-linear behavior, can be understood by the non-linearity of interaction Hamiltonian. In addition to previous research[1], we study the power-dependent dynamics of ground, primary, and secondary excited states of a transmon qubit in the dispersive regimes up to punch-out. We try to obtain the critical photon number of nonlinear behavior and explain how it is determined.