  The interaction of light and matter is often described by the exchange of
single excitations. When the coupling strength is a significant fraction of the
system frequencies, the number of excitations are no longer preserved and that
simple picture breaks down. This regime is known as the ultrastrong coupling
regime and is characterized by non-trivial light-matter eigenstates and complex
dynamics. In this work, we propose to use a an array Josephson junctions to
increase the impedance of the light mode enabling ultrastrong coupling to a
transmon qubit. We show that the resulting dynamics can be generated and probed
by taking advantage of the multi-mode structure of the junction array. This
proposal relies on the frequency tunability of the transmon and, crucially, on
the use of a low frequency mode of the array, which allows for non-adiabatic
changes of the ground state.
