Nonlinear Quantum Electrodynamics of Epsilon-Near-Zero Media

We investigate vacuum-induced nonlinearity and single-photon nonlinear optical responses of Epsilon-Near-Zero (ENZ) media and of sub-wavelength nanocavities. We apply the rigorous quantum Langevin-noise approach in the framework of the Green's tensor quantization method to ENZ materials and resonators with causal dispersion and we derive generally valid closed-form analytical solutions for the nonlinear phase shifts achievable in cavities with spherical geometry. This is achieved by employing a fully nonperturbative methodology for the analysis of open quantum systems with single-photon Kerr-type nonlinearity in dielectric environments with absorption losses and dispersion that satisfy the Kramers-Kronig relations. Our findings are relevant for the design of quantum optical devices and are of importance to emerging quantum technology applications, including on-chip single-photon nondemolition detection, quantum sensing, and controlled quantum gates driven by enhanced photon blockade effects at the nanoscale.