Near-ultrastrong nonlinear light-matter coupling in superconducting circuits

Light-matter interaction between an atom and an electromagnetic resonator is ubiquitous in quantum technologies. Although linear light-matter coupling $$g{\hat{\sigma }}_{x}(\hat{a}+{\hat{a}}^{{{\dagger}} })$$ g σ ̂ x ( a ̂ + a ̂ † ) can reach the ultrastrong regime g/ω > 10−1, nonlinear light-matter coupling $$\frac{\chi }{2}{\hat{\sigma }}_{z}{\hat{a}}^{{{\dagger}} }\hat{a}$$ χ 2 σ ̂ z a ̂ † a ̂ is typically perturbative and limited to χ/ω