  The interaction between an atom and an electromagnetic mode of a resonator is
of both fundamental interest and is ubiquitous in quantum technologies. Most
prior work studies a linear light-matter coupling of the form $g
\widehat{\sigma}_x (\widehat{a} + \widehat{a}^\dagger)$, where $g$ measured
relative to photonic ($\omega_a$) and atomic ($\omega_b$) mode frequencies can
reach the ultrastrong regime ($g/\omega_{a}\!>\!10^{-1}$). In contrast, a
nonlinear light-matter coupling of the form $\frac{\chi}{2} \widehat{\sigma}_z
\widehat{a}^\dagger \widehat{a}$ has the advantage of commuting with the atomic
$\widehat{\sigma}_z$ and photonic $\widehat{a}^\dagger\widehat{a}$ Hamiltonian,
allowing for fundamental operations such as quantum-non-demolition measurement.
However, due to the perturbative nature of nonlinear coupling, the
state-of-the-art $\chi/\text{max}(\omega_a, \omega_b)$ is limited to
$\!<\!10^{-2}$. Here, we use a superconducting circuit architecture featuring a
quarton coupler to experimentally demonstrate, for the first time, a
near-ultrastrong $\chi/\text{max}(\omega_a, \omega_b)=
(4.852\pm0.006)\times10^{-2}$ nonlinear coupling of a superconducting
artificial atom and a nearly-linear resonator. We also show signatures of
light-light nonlinear coupling
($\chi\widehat{a}^\dagger\widehat{a}\widehat{b}^\dagger\widehat{b}$), and
$\chi/2\pi = 580.3 \pm 0.4 $ MHz matter-matter nonlinear coupling
($\frac{\chi}{4}\widehat{\sigma}_{z,a}\widehat{\sigma}_{z,b}$) which represents
the largest reported $ZZ$ interaction between two coherent qubits. Such
advances in the nonlinear coupling strength of light, matter modes enable new
physical regimes and could lead to applications such as orders of magnitude
faster qubit readout and gates.
