Mid-Infrared Single-Photon Detection via Enhanced Cross-Phase Modulation in Topology-Optimized Epsilon-Near-Zero Dual-Wavelength Nanocavities

We use the Green's tensor quantization theory for open resonant nanostructures with absorption losses to study the cross-phase modulation (XPM) process at the single photon level in nanoscale Kerr-type epsilon-near-zero (ENZ) materials with an effective nonlinear susceptibility $\chi^{(3)}(\omega)$ integrated inside dual-wavelength nanocavities. We obtain general analytical formulas for the achievable XPM frequency shift in a hybrid nanocavity that simultaneously traps a classical probe (signal) beam at 1.5 $\mu$m and single photon pump at 3 $\mu$m wavelengths. By focusing on mid-infrared photon detection at room temperature, we present a comprehensive analysis of the fundamental limits for single photon detection in the quantum nondemolition modality for a nanoscale region of high mobility cadmium oxide (CdO) with ENZ-enhanced Kerr-type nonlinearity embedded in a surrounding silicon (Si) environment inverse designed by free-form topology optimization. We numerically implement our theoretical results using finite element simulations within the rigorous framework of quasi-normal modes, demonstrating a single photon XPM frequency shift $\Delta f_s \approx 18.4 \text{ GHz}$ with fractional shift (i.e., frequency pulling) $\Delta f_s / f_s \approx 9.23 \times 10^{-5}$ and addressing the feasibility of detection in the proposed hybrid Si-CdO dual-wavelength nanocavity, either with a classical probe beam or a squeezed probe state, beyond the traditional limitations from self-phase modulation noise, thermorefractive noise, shot noise, and electronic jitter effects. This work establishes a robust benchmark for the engineering of mid-infrared single-photon nonlinear devices such as nondemolition quantum detectors, sensors, and all-optical gates on a solid state photonic platform.