Nonlinear Quantum Electrodynamics of Epsilon-Near-Zero Nanocavities

We investigate single-photon nonlinear refractive index change and frequency shift of Epsilon-Near-Zero (ENZ) sub-wavelength nanocavities. We apply the rigorous quantum Langevin-noise approach in the framework of Green's tensor quantization method to realistic ENZ materials with causal dispersion and derive closed-form analytical solutions for 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. The analytical results are validated numerically using the established quasi-normal mode expansion method and extended to nonspherical nanocavity geometries that can be experimentally fabricated using state-of-the-art electron lithography. Our findings establish a rigorous benchmark for understanding single-photon nonlinear optical effects in Kerr-type ENZ nanostructures with losses 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.