Exploring Quantum–Classical Correspondence in the Ultrastrong Coupling Regime by Nonlinear Terahertz Spectroscopy

The ultrastrong coupling (USC) of light and matter arises when the coupling strength (g) becomes a significant fraction of the bare light and matter frequencies. In a quantum description, in the USC regime, not only the counter-rotating terms but also the A² terms (A, vector potential of light) dominate. The A² terms supply a positive energy to the system and blueshift the bare photon frequency ω_0, a hallmark of USC. In a classical description, this blueshift can be modeled as a cavity-renormalized polarizability of the matter. Recent theoretical works showed that the quantum and classical descriptions yield exactly the same linear spectra, including the blueshift. Examining such correspondence beyond linear regime would identify new approaches for unraveling uniquely quantum effects. Here, we performed nonlinear terahertz (THz) spectroscopy experiments in the USC regime to help illuminate this issue. We utilized two-dimensional electrons in GaAs quantum wells coupled with ultra-subwavelength THz nanoslots. At low THz field strengths, we observed an extremely large blueshift (0.35 THz) of the cavity mode, corresponding to ~40% of ω₀, signaling USC. As the THz field strength increased, the amount of blueshift decreased, implying that a strong THz field induces a nonlinear reduction of g. We model this reduction, both quantum mechanically and classically, by introducing a mechanism involving a ponderomotive potential that repels electrons from the cavity and compare the two scenarios quantitatively.