**Abstract**
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 [5].

[1] T. Niemczyk et al., Nature Physics 6, 772 (2010).

[2] P. Forn-Diaz et al., PRL 105, 237001 (2010).

[3] F. Yoshihara, T. Fuse, et al., Nature Phys. 13, 44 (2017).

[4] 布施 智子 et al., 日本物理学会誌 73, 21, (2018).

[5] F. Yoshihara et al., PRL 120, 183601 (2018).