Ultrastrong photon-photon coupling

Correlations between photons are a key feature of nonclassical states of light. Recent studies suggest that the ground state of a cavity quantum electrodynamics system can have light-matter correlations in the form of a squeezed vacuum state in thermal equilibrium when the matter ultrastrongly couples with cavity photons. This raises a question whether different photonic modes can also be correlated in the ground state via ultrastrong light-matter coupling. Here we demonstrate ultrastrong coupling between photonic modes of a multi-mode three-dimensional terahertz photonic-crystal cavity that is mediated by their simultaneous ultrastrong coupling with the cyclotron resonance of a two-dimensional electron gas in GaAs. Terahertz spectroscopy measurements of Landau polaritons showed excellent agreement with our calculations based on a microscopic quantum model. Despite the lack of nonlinearity in the matter system, the model shows significant correlations between the photonic modes in the ground state of the system, which can be controlled by changing the matter and photon frequencies and the spatial overlap of their mode profiles. We propose a detuning-independent figure of merit to quantify all possible couplings in multi-mode systems. These findings pave the way for creating multi-mode nonclassical states and exploring the many-body regime of quantum optics with vacuum fields.