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

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.

💡 Research Summary

The authors present a unified theoretical‑numerical framework for predicting far‑field two‑photon correlations generated by arbitrary loss‑free dielectric scatterers. Building on macroscopic quantum electrodynamics, they formulate the input‑output relation of the quantized electromagnetic field as a multi‑channel scattering problem. Incident photons are described in a complete set of orthonormal free‑space modes (plane‑wave basis), which are unitarily transformed into an incoming spherical‑wave basis. The scatterer is characterized by a unitary scattering matrix S that maps these incoming channels to outgoing ones.

Only two single‑photon wave packets, injected into two selected input channels, are considered. Because the normally ordered detection operators annihilate vacuum contributions, the analysis reduces to the two populated channels. Under a narrow‑band approximation the positive‑frequency field operators at the detectors become linear combinations of the two input annihilation operators with complex transfer coefficients T_{aj} and T_{bj}. These coefficients encode the full electromagnetic response of the object, including propagation, scattering, and polarization projection, and are extracted directly from a fast‑Fourier‑transform (FFT) accelerated volume integral equation (VIE) solver. The VIE formulation uses dyadic Green’s functions that automatically satisfy the radiation condition, eliminating the need for perfectly matched layers or separate near‑to‑far‑field transformations.

Substituting the linear field expressions into the definition of the normalized second‑order correlation function g^{(2)} yields a compact expression (Eq. 18) that involves the classical far‑field amplitudes of the two photons at the two detection points. The numerator contains a sum of a direct term, an exchange term, and a cross term that originates from quantum indistinguishability. When the photons are perfectly indistinguishable, the cross term maximally interferes destructively, driving g^{(2)} to zero—this is the Hong‑Ou‑Mandel (HOM) dip. If the photons are distinguishable, the cross term vanishes and g^{(2)} reduces to an incoherent sum of intensities, approaching unity. Thus the framework directly links classical scattering amplitudes to quantum interference visibility.

To connect with experimentally measured coincidence counts, the authors extend the standard beam‑splitter model to an arbitrary loss‑free scattering environment. By introducing a controllable temporal delay δτ on one of the input wave packets, they derive the time‑dependent second‑order correlation G^{(2)}(τ;δτ) and integrate over the detection window to obtain the coincidence number N_c(δτ). The overlap function h(τ)=∫ϕ_1^*(ω)ϕ_2(ω)e^{-iωτ}dω governs the shape of the HOM dip; for identical Gaussian spectra the dip follows a Gaussian profile with width set by the photon bandwidth.

Numerically, the VIE‑FFT solver computes the induced polarization currents inside the dielectric body and evaluates the far‑field radiation via the dyadic Green’s function. FFT‑based matrix‑vector multiplication and block preconditioning enable simulations of structures with hundreds of thousands of unknowns on modest computational resources. The method is validated against analytical Mie theory for a homogeneous dielectric sphere, showing perfect agreement in both amplitude and phase of the transfer coefficients and consequently in the predicted g^{(2)}.

A second demonstration involves a polarization‑converting Pancharatnam‑Berry (PB) metasurface. The metasurface imposes a spatially varying geometric phase that rotates the incident linear polarization. By scanning the detection angles, the authors reveal a strong angular dependence of the HOM visibility: certain directions exhibit near‑perfect dip (g^{(2)}≈0), while others show reduced interference due to incomplete overlap of the two polarization components. This example illustrates how engineered scattering can be used to tailor quantum interference patterns, opening pathways for scatterer‑enabled quantum state engineering.

The key advantages of the proposed approach are: (1) avoidance of heavy normal‑mode expansions; (2) exact enforcement of radiation boundary conditions via Green’s functions, which preserves phase accuracy essential for interference; (3) computational efficiency and scalability thanks to FFT acceleration; (4) a transparent physical picture where classical far‑field amplitudes directly determine quantum correlation functions; and (5) a natural bridge between frequency‑domain g^{(2)} and time‑domain coincidence curves, facilitating direct comparison with experiments.

In conclusion, the paper delivers a robust, physically intuitive, and computationally efficient tool for analyzing structure‑dependent two‑photon interference in the far field. It paves the way for inverse design of dielectric nanostructures that produce desired quantum‑optical functionalities, such as customized HOM dip shapes, entanglement generation, or quantum‑enhanced sensing. Future extensions may incorporate material loss, nonlinearities, or active modulation, further broadening the applicability of this framework to emerging quantum photonic technologies.