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 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 ω as ω‾‾√ at low frequencies and decreases as 1/ω‾‾√ at high frequencies. We derive expressions for the multimode-resonator-induced Lamb shift in the qubit’s characteristic frequency. Because of the natural decoupling between the qubit and high-frequency modes, the Lamb-shift-renormalized qubit frequency remains finite.
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 (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 ωcutoff for the coupling between the FQ and the modes in the CPWR, where the coupling grows as ωn‾‾‾√ for ωn/ωcutoff≪1 and decreases as 1/ωn‾‾‾√ for ωn/ωcutoff≫1. Here ωn is the frequency of the nth mode. 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.
We report on experimentally measured light shifts of superconducting flux qubits deep-strongly-coupled to an LC oscillator, where the coupling constant is comparable to the qubit’stransition frequency and the oscillator’s resonance frequency. By using two-tone spectroscopy, the energies of the six-lowest levels of the coupled circuits are determined. We find a huge Lamb shift that exceeds 90% of the bare qubit frequencies and inversion of the qubits‘ ground and excited states when there is a finite number of photons in the oscillator. Our experimental results agree with theoretical predictions based on the quantum Rabi model.