Light-matter interaction is one of the most important topics in physics. The simplest platform to study light-matter interaction consists of a two-level atom and a harmonic oscillator, interacting to each other. Compared to natural atoms, superconducting qubits, having micro-meter size, have allowed the realization of much stronger coupling to a harmonic oscillator. There have been an ongoing debate about what is the fundamental limit of the coupling strength. So far, the maximum ratio of
the coupling strength g to the oscillator's frequency omega is g/omega ~ 0.1 [1, 2]. This regime is called ultrastrong coupling regime, where the rotating wave approximation breaks down. To have larger g/omega values, we have fabricated circuits where a superconducting flux qubit and an LC oscillator are inductively coupled via Josephson junctions [3, 4, 5]. From the spectroscopic measurement, the circuit can be well fitted by the Hamiltonian of the quantum Rabi model, and the g/omega ratio is obtained to be larger than one. This regime is called deep-strong-coupling regime. In the deep-strong-coupling regime, light shifts of a flux qubit is quite large. We find Lamb shift over 90 % of the bare qubit frequency and inversions of the qubit's ground and excited states when there are a finite number of photons in the oscillator .
 T. Niemczyk et al., Nature Physics 6, 772 (2010).
 P. Forn-Diaz et al., PRL 105, 237001 (2010).
 F. Yoshihara, T. Fuse, et al., Nature Phys. 13, 44 (2017).
 布施 智子 et al., 日本物理学会誌 73, 21, (2018).
 F. Yoshihara et al., PRL 120, 183601 (2018).