  We study the feasibility of reaching the ultrastrong (USC) and deep-strong
coupling (DSC) regimes of light-matter interaction, in particular at resonance
condition, with a superconducting charge qubit, also known as Cooper-Pair box
(CPB). We numerically show that by shunting the charge qubit with a
high-impedance LC-circuit, one can maximally reach both USC and DSC regimes
exceeding the classical upper bound $|g|\leq \sqrt{\omega_q\omega_r}/2$ between
two harmonic systems with frequencies $\omega_q$ and $\omega_r$. As an
application, we propose a hybrid system consisting of two CPBs ultrastrongly
coupled to an LC-oscillator as a mediator device that catalyzes a quantum state
transfer protocol between a pair of transmon qubits, with all the parties
subjected to local thermal noise. We demonstrate that the QST protocol
maximizes its efficiency when the mediator operates in the USC regime,
exhibiting comparable times with proposals relying on highly coherent and
controllable mediators requiring state-preparation or post-selection. This work
opens the door for studying light-matter interactions beyond the quantum Rabi
model at extreme coupling strengths, providing a new building block for
applications within quantum computation and quantum information processing.
