Extremely large Lamb shift in a deep-strongly coupled circuit QED system with a multimode resonator

Abstract

Abstract We report experimental and theoretical results on the extremely large Lamb shift in a multimode circuit quantum electrodynamics (QED) system in the deep-strong coupling (DSC) regime, where the qubit-resonator coupling strength is comparable to or larger than the qubit and resonator frequencies. The system comprises a superconducting flux qubit (FQ) and a quarter-wavelength coplanar waveguide resonator ( $$\lambda /4$$ λ / 4 CPWR) that are coupled inductively through a shared edge that contains a Josephson junction to achieve the DSC regime. Spectroscopy is performed around the frequency of the fundamental mode of the CPWR, and the spectrum is fitted by the single-mode quantum Rabi Hamiltonian to obtain the system parameters. Since the qubit is also coupled to a large number of higher modes in the resonator, the single-mode fitting does not provide the bare qubit energy but a value that incorporates the renormalization from all the other modes. We derive theoretical formulas for the Lamb shift in the multimode resonator system. As shown in previous studies, there is a cut-off frequency $$\omega _{\textrm{cutoff}}$$ ω cutoff for the coupling between the FQ and the modes in the CPWR, where the coupling grows as $$\sqrt{\omega _n}$$ ω n for $$\omega _n/\omega _{\textrm{cutoff}}\ll 1$$ ω n / ω cutoff ≪ 1 and decreases as $$1/\sqrt{\omega _n}$$ 1 / ω n for $$\omega _n/\omega _{\textrm{cutoff}}\gg 1$$ ω n / ω cutoff ≫ 1 . Here $$\omega _n$$ ω n is the frequency of the nth mode. The cut-off effect occurs because the qubit acts as an obstacle for the current in the resonator, which suppresses the current of the modes above $$\omega _{\textrm{cutoff}}$$ ω cutoff at the location of the qubit and results in a reduced coupling strength. Using our observed spectrum and theoretical formulas, we estimate that the Lamb shift from the fundamental mode is 82.3% and the total Lamb shift from all the modes is 96.5%. This result illustrates that the coupling to the large number of modes in a CPWR yields an extremely large Lamb shift but does not suppress the qubit energy to zero, which would happen in the absence of a high-frequency cut-off.