Modeling of Far-Field Quantum Coherence by Dielectric Bodies Based on the Volume Integral Equation Method

TL;DR

This work addresses the challenge of predicting angle-resolved far-field two-photon correlations in structured dielectric environments. It introduces a unified theory–numerics framework that couples quantum photodetection formalism with FFT-accelerated volume integral equation solvers to map two input channels to two detection modes via a compact $2\times2$ transfer, enabling closed-form expressions for $g^{(2)}$ and the time-domain coincidence $ ilde{N}_c$. Key contributions include deriving the frequency-domain $g^{(2)}$ under, and extending to, arbitrary lossless scatterers, transforming these results into the Hong–Ou–Mandel envelope in time, and validating the approach on a dielectric sphere and a Pancharatnam–Berry metasurface. The method provides an efficient, boundary-condition-friendly route to structure-dependent quantum interference, with potential for quantum state engineering and quantum inverse design in complex nanostructures; future work will extend to lossy or dispersive media and quantum inverse problems.

Abstract

The Hong-Ou-Mandel (HOM) effect is a hallmark of nonclassical two-photon interference. This paper develops a unified theory-numerics framework to compute angle-resolved far-field two-photon correlations from arbitrary lossless dielectric scatterers. We describe the input-output relation using a multi-channel scattering formulation that maps two populated incident channels to two selected far-field detection modes, yielding a compact two-channel transfer relation for second-order correlation function and time-domain coincidence counts. The required transfer coefficients are extracted from classical far-field complex amplitudes computed by an fast Fourier transform-accelerated volume integral equation solver, avoiding perfectly matched layers and near-to-far-field post-processing. The method is validated against analytical results for dielectric spheres and demonstrated on a polarization-converting Pancharatnam-Berry-phase metasurface, revealing strong angular dependence of quantum interference and its direct impact on HOM-dip visibility. The framework provides an efficient and physically transparent tool for structure-dependent quantum-correlation analysis, with potential applications in scatterers-enabled quantum state engineering and quantum inverse design.

Figures (7)

• Figure 1: The schematic of the photon detection setup, where two detectors are placed at positions $\mathbf{r}_1$ and $\mathbf{r}_2$, with polarization detection directions $\mathbf{e}_a$ and $\mathbf{e}_b$, respectively.
• Figure 2: (a) The schematic illustrating the two-photon state and the positions of detectors with respect to the dielectric sphere. (b) Normalized second-order correlation function $g_{z z}^{(2)}\left(\phi_1, \phi_2\right)$ for the sphere at $r=60$ nm as a function of the azimuthal positions of the two detectors fixed at $\theta_1=45^{\circ},\theta_2=135^{\circ}$. (c) Normalized second-order correlation function $g_{z z}^{(2)}\left(\phi_1, \phi_2=135^{\circ}\right)$ for spheres with different radii $r$ = 60 nm, 110 nm, 123 nm.
• Figure 3: Angular maps of normalized fourth-order field correlations. (a) Classical intensity-product correlation $P^{(2)}_{zz}$ (normalized). (b) Quantum normalized second-order correlation $g^{(2)}_{zz}$.
• Figure 4: (a) Four representative detection configurations on the angular map. (b) Normalized coincidence counts $\tilde{N}_c$ versus the controlled input delay $\delta \tau$ for these configurations.
• Figure 5: (a) The geometric schematic of the metasurface unit cell. (b) The schematic of the two-photon state and the positions of detectors with respect to the PBP metasurface.