High-frequency suppression of inductive coupling between flux qubit and transmission line resonator

We perform theoretical calculations to investigate the naturally occurring high-frequency cutoff in a circuit comprising a flux qubit coupled inductively to a transmission line resonator (TLR). Our results generally agree with those of past studies that considered somewhat similar circuit designs. In particular, a decoupling occurs between the qubit and the high-frequency modes. As a result, the coupling strength between the qubit and resonator modes increases with mode frequency $omega$ as $sqrt{omega}$ at low frequencies and decreases as $1/sqrt{omega}$ at high frequencies. By avoiding the approximation of ignoring the qubit-TLR coupling in certain steps in the analysis, we obtain effects not captured in previous studies. We derive expressions for the mode frequencies, coupling strengths and Lamb shift in the qubit's characteristic frequency. We identify features in the spectrum of the system that can be used in future experiments to test and validate the theoretical model.