Observation of Extremely Large Lamb Shift in a Multi-mode Circuit QED System in Deep-strong Coupling Regime*

We report experimental results on an extremely large Lamb shift observed in a multi-mode circuit QED system in the deep-strong coupling (DSC) regime, where the qubit-resonator coupling strength exceeds the qubit energy and the resonator frequency. The system comprises a superconducting flux qubit (FQ) and a quarter-wavelength coplanar waveguide resonator (CPWR) which are coupled inductively through a shared edge that contains a Josephson junction to achieve the DSC regime [1,2]. The FQ is placed at the open end of the CPWR to couple with every mode in the CPWR. Spectroscopy results around the frequency of the fundamental mode of the CPWR is measured and fitted by the single-mode quantum Rabi model Hamiltonian as an approximation to obtain the parameters of the system. The renormalized qubit energy is calculated through the difference between the frequencies of the 03, from the ground to the third excited state, and 13, from the first excited to the third excited state, transitions. The fundamental mode contributes an 85.4% Lamb-shift suppression of the qubit energy. Since the qubit is also coupled to an infinite number of higher modes in the resonator, the single-mode fitting does not provide the bare qubit energy. If we use the designed qubit energy as reference, we find that the Lamb shift is 99.7% of the bare qubit energy. This result shows that the coupling to an infinite number of modes induces a huge Lamb shift to the qubit energy. We also provide theoretical formulas for the Lamb shift in our circuit, which is consistent with previous results in the literature [3,4] despite differences in the circuit design and qubit-resonator coupling mechanism.