Quantum interference effects in two-photon scattering by a macroscopic lossy sphere

TL;DR

The paper develops a macroscopic quantum electrodynamics framework based on the Modified Langevin Noise Formalism to study two-photon scattering by a finite, lossy sphere. It shows that matter losses create three outgoing polariton channels ($ss$, $se$, $ee$) and, thanks to non-collinear input directions, enable interference between direct and exchange quantum paths, yielding Hong-Ou-Mandel–like effects in selected geometries. By modeling the sphere with Drude–Lorentz dispersion, it demonstrates that classical Mie resonances produce sharp, Fano-like spectral features that strongly modulate both coincidence detection and total scattering probabilities, with the interference pattern highly sensitive to the spectral symmetry of the input wavepackets. The work suggests loss-assisted spectral techniques to identify entanglement and extends multi-port interference concepts beyond traditional lossless beam splitters, leveraging sphere symmetry and resonance phenomena for robust quantum control observations.

Abstract

We investigate the quantum optical scattering of two-photon wavepackets by a macroscopic lossy sphere by means of macroscopic quantum electrodynamics in the form of modified Langevin noise formalism. The two ingoing photons with arbitrary frequency-polarization spectrum impinge onto the sphere along two different directions and, as consequence of matter losses, their scattering involves the three independent processes where two, one and zero outgoing photons survive. Non-collinearity of ingoing photons causes the existence of two different quantum paths they can follow upon scattering, this producing interference effects in the detection of the above three processes which is governed by the wavepacket spectral symmetry. By exploiting rotational invariance, we show that different classes of scattering geometries exist such that the coincidence detection of the scattered photons shows perfect constructive or destructive (Hong-Ou-Mandel) interference, both for symmetric and antisymmetric wavepackets. To assess the impact of matter dispersion/losses on quantum interference effects accompanying photons detection, we analyze the scattering of narrow band two-photon wavepackets by high-index dielectric lossy spheres. We show that classical Mie resonance peaks, due to their Fano-like traits, yield very strong constructive and destructive interference effects, occurring when the wavepacket carrier frequency matches the resonance frequency and side Fano dip frequency, respectively. In addition we consider the overall scattering probabilities of two, one and zero photons and we prove that, at the Mie resonance frequencies, they exhibit quantum interference effects which are extremely sensitive to the spectral symmetry of the input wavepacket, thus suggesting an efficient spectral technique assisted by matter losses to identify entanglement.

Figures (4)

• Figure 1: (a) Geometry of two directional $s$ polaritons scattering by a lossy object $V$ of finite size. At $t = -\infty$ (red) the radiation-matter quantum state contains only the pairs $ss$ of ingoing $s$-polaritons (circles) whose directions lies within the two non overlapping cones of solid angles $\Delta o \ll 1$ around the unit vectors ${\bf m}^1$ and ${\bf m}^2$. At $t = +\infty$ (green) the state contains the three pairs $ss$, $se$ and $ee$ of outgoing $s$ (circles) and $e$ (squares) polaritons. (b) Diagramatic representation of the three processes involved in the scattering of the two ingoing $s$ polaritons. (c) Graphical representation of the two possible quantum paths connecting the ingoing $s$ polaritons to the outgoing ones in the three processes.
• Figure 2: Sample classes of scattering geometries leading to perfect constructive or destructive interference in the polariton coincidence detection. The ingoing polariton triads $\left\{ {{\bf{m}}^1 ,{\bf{e}}_{{\bf{m}}^1 1} ,{\bf{e}}_{{\bf{m}}^1 2} } \right\}$ and $\left\{ {{\bf{m}}^2 ,{\bf{e}}_{{\bf{m}}^2 1} ,{\bf{e}}_{{\bf{m}}^2 2} } \right\}$, representing the input ports through which light is launched, are plotted with colors outlining the action of the rotation ${\cal R}$ and the reflection ${\cal W}$ whereas the outgoing polariton triads $\left\{ {{\bf{N}}^1 ,{\bf{e}}_{{\bf{N}}^1 1} ,{\bf{e}}_{{\bf{N}}^1 2} } \right\}$ and $\left\{ {{\bf{N}}^2 ,{\bf{e}}_{{\bf{N}}^2 1} ,{\bf{e}}_{{\bf{N}}^2 2} } \right\}$, representing the detectors spatial placements and polarization directions, are plotted in green.
• Figure 3: Selected scattering geometry for discussing the spectral features of quantum interference effects.
• Figure 4: Quantum optical scattering by a subwavelength-sized lossy sphere with high-index dielectric behavior. (a) Real and imaginary parts of the dielectric permitivity. (b) Square moduli $\left| {a_{n\omega } } \right|^2$, $\left| {b_{n\omega } } \right|^2$ and combinations ${\mathop{\rm Re}\nolimits} \left( {a_{n\omega } } \right) - \left| {a_{n\omega } } \right|^2$, ${\mathop{\rm Re}\nolimits} \left( {b_{n\omega } } \right) - \left| {b_{n\omega } } \right|^2$ of the Mie coefficients. (c) Plots of the relevant quantities ruling the probability density of polariton coincidence detection of Eq.(\ref{['Disp_Coin']}). (d) Plots of the relevant quantities yielding the polariton scattering probabilities of Eqs.(\ref{['Disp_SurvProb']}).