Correlated Quantum Airy Photons: An Analytical Approach

TL;DR

This work analyzes spontaneous parametric down-conversion driven by a finite-energy Airy pump to generate spatially entangled photon pairs. It develops an analytical framework that couples the SPDC spectral function $F(\mathbf{k}_1,\mathbf{k}_2)$ with an optical-system description to predict detection statistics in near-field and far-field setups, including the effects of spatial walk-off. The study reveals how crystal length and Airy-pump properties control entanglement: long crystals yield tight real-space correlations but highly multimode momentum correlations, while short crystals produce broader position correlations with fewer momentum modes, enabling tunable, high-dimensional entanglement. The results point to robust Airy-beam-based quantum imaging and communication schemes with tailored spatial-mode structures and resilience to propagation losses.

Abstract

We describe the generation of correlated photon pairs by means of spontaneous parametric down-conversion of an optical pump in the form of a finite energy Airy beam. The optical system function, which contributes to the propagation of the down-converted beam before being registered by the detectors, is computed. The spectral function is utilized to calculate the biphoton amplitude for finding the coincidence count of the inbound Airy photons in both far-field and near-field configurations. We report the reconstruction of the finite energy Airy beam in the spatial correlation of the down-converted beams in near field scenario. In far field, the coincidence counts resembles the probability density of the biphoton in momentum space, revealing a direct mapping of the anti-correlation of the biphoton momentum. By examining the spatial Schmidt modes, we also demonstrate that longer crystals have tighter real-space correlations, but higher-dimensional angular correlations, whereas shorter crystals have fewer modes in momentum space and broader multimode correlations in position space.

Figures (13)

• Figure 1: Schematic diagram of the photon pairs generation. A pump beam at frequency $\omega_p$ and wave vector $\bm{K}_p$ travels through a non--linear crystal of length $L$, producing pairs of correlated photons at $(\omega_1, \bm{k}_1)$ and $(\omega_2, \bm{k}_2)$. A lens at plane $\bm{\rho}_s$ is placed at distance $d_1$ from the crystal output plane $\bm{\rho}_x$, and $d_2$ from the detection planes $\bm{\rho}_j$. O.A. indicates the crystal optical axes, which has component along $\hat{z}$ and $\hat{y}$, the extraordinary direction, while the $\hat{x}$ is the ordinary one.
• Figure 2: Extraordinary beam propagation inside a negative uniaxial bi-refringent crystal. The pump wave vector and optic axis (OA) make an angle $\theta$ and $\bm{\Theta}_{OA}$, with respect to the propagation direction (z-axis), respectively. The Poynting vector $\bm{S}$ is tilted at an angle $\delta$ relative to $\bm{k}$. $\bm{M}_p$ denotes the spatial walk-off vector directed towards $\bm{S}$ from $\bm{k}$.
• Figure 3: 1D Airy pump beam propagation for three values of the truncation parameter $w = (0.02,\,0.1,\,0.5)$, panel (a), (b) and (c), respectively. Panel (a) shows a diffracting-free propagation as anticipated in Ref. berry1979nonspreading, while panel (c) shows prominent diffraction. Other parameters for the beam propagation are $l = 100µm$, $\lambda = 0.5µm$Siviloglou:07. In panel (b) dotted and dashed lines mark the propagation distances at which the 2D profiles in Fig. \ref{['fig:pump_beam_2D']} are evaluated.
• Figure 4: 2D Airy pump beam profile for three values of the propagation distance $z = (0,\,25,\,50)\,cm$, panel (a), (b) and (c), respectively. They correspond to the propagation distances marked by dotted and dashed lines in panel (b) of Fig. \ref{['fig:pump_beam_1D']}.
• Figure 5: Probability density for the crystal lengths $L = (0.1,\,1,\,10)\,mm$ with truncation parameter $w=0.1$, for the ordinary and extraordinary components, panels (a--c) and (d--f), respectively.