Nonlinear Quantum Electrodynamics of Epsilon-Near-Zero Nanocavities

TL;DR

This work addresses single-photon nonlinear optics in dispersive ENZ nanocavities by developing a rigorous quantum Langevin-noise framework within the Green's tensor formalism. It derives nonperturbative Kerr-type refractive-index changes and Kerr frequency shifts, obtaining closed-form expressions for sub-wavelength spherical ENZ cavities and validating them against quasi-normal-mode numerics, while extending the analysis to arbitrary shapes via QNM-based calculations. The results show observable single-photon Kerr phase shifts (e.g., ~3.6×10^-3 rad for a 10 nm sphere and ~10^-4 rad for 70 nm nonspherical cavities) and establish design guidelines (small V, moderate Q) to approach a ~0.5 rad shift, enabling on-chip QND detection, quantum sensing, and nanoscale photon blockade. The methodology is general to ENZ materials with Drude-Sommerfeld dispersion (including CdO and ITO) and provides a robust benchmark for nonlinear QED in lossy, dispersive nanostructures with broad implications for quantum technologies.

Abstract

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.

Figures (4)

• Figure 1: (a) Kerr-induced refractive index change $\Delta n_{\rm NL}$ and (b) Kerr-induced phase shift $\Delta\Phi_{\rm NL}$ as a function of the angular frequency of a single photon within the dipole mode of the sphere ($l=1$ and $m=1$) oriented along $x$. The blue curves represent the result evaluated using the RSs, and in red, the ones using the numerically evaluated QNMs. The dotted vertical black line identifies the real part of the complex resonant angular frequency calculated through the QNMs ($\omega_1\simeq 0.4138\,\omega_p$). In this figure, we consider $R=10\,{\rm nm}$.
• Figure 2: Comparison of the maximum value of the Kerr nonlinear phase shift $\Delta\Phi_{\rm NL}(\omega)$, as given by Eq. \ref{['deltaPhi']}, for both dipolar and quadrupolar RSs as a function of the radius $R$ of the ENZ nanosphere, and for the case, where a single photon is generated in a single RS, corresponding to the dipolar mode with $\ell=1, m=1$ (blue line), a quadrupolar mode with $\ell=2, m=0$ (green line), and $\ell=2, m=\{-1,\pm 2\}$ (cyan line). The red dots represent the results obtained from the QNMs analysis for the dipolar modes, each of which has been individually rotated to align along the $x$-axis.
• Figure 3: (a) Kerr-induced refractive index change $\Delta n_{\rm NL}$ and (b) Kerr-induced phase shift $\Delta\Phi_{\rm NL}$ as a function of the angular frequency of a single photon within the first(second) mode of the triangular cavity oriented along $x$. The dotted vertical lines identify the real part of the complex resonant angular frequencies of the first mode shown in (c) and the second mode shown in (d). In (c) and (d) is reported the normalized norm of the electric field, $|\tilde{\mathbf{E}}_n| / \sqrt{{\rm QN}_n}$. The side length of the equilateral triangle is $70\,{\rm nm}$, its thickness is $20\,{\rm nm}$, and the edge curvature radius is $8\,{\rm nm}$.
• Figure 4: (a) Kerr-induced refractive index change $\Delta n_{\rm NL}$ and (b) Kerr-induced phase shift $\Delta\Phi_{\rm NL}$ as a function of the angular frequency of a single photon within the first(second) mode of the nanodisk cavity oriented along $x$. The dotted vertical lines identify the real part of the complex resonant angular frequencies of the first mode shown in (c) and the second mode shown in (d). In (c) and (d) is reported the normalized norm of the electric field, $|\tilde{\mathbf{E}}_n| / \sqrt{{\rm QN}_n}$. The diameter of the nanodisk is $70\,{\rm nm}$, its thickness is $20\,{\rm nm}$, and the edge curvature radius is $8\,{\rm nm}$.