Effective Quantization of Lossy Nonlinear Epsilon-Near-Zero Media
In epsilon-near-zero (ENZ) materials, the vanishing real linear permittivity results in the leading-order contribution to the displacement field being nonlinear in the electric field, making conventional canonical quantization approaches highly non-trivial. Existing treatments generally quantize the linear modes first and introduce nonlinear interactions subsequently, an ordering that becomes inadequate in the ENZ regime. Here, we provide, to our knowledge, the first effective single-excitation quantization in which the near-zero linear response, leading nonlinearity, and loss jointly determine the elementary excitation. Using a solvable microscopic atomic system as a theoretical scaffold, we find that this excitation is a polariton: a dressed quasiparticle with partly field and partly material excitations whose coefficients can be parameterized by macroscopic susceptibilities. While this work considers a leading-order $χ^{(3)}$ nonlinearity, the framework could be generalized to include other nonlinear corrections, including a controllable second-order susceptibility $χ^{(2)}$ and systematic higher-order contributions $χ^{(n)}$, $n>2$. Our findings pave the way toward practical applications in quantum photonics, such as single-photon non-demolition detection, by leveraging the strong nonlinearities intrinsic to zero-index materials.