Correlated Quantum Airy Photons: An Analytical Approach

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.

💡 Research Summary

The paper presents a comprehensive analytical study of generating correlated photon pairs via spontaneous parametric down‑conversion (SPDC) when the pump is a finite‑energy Airy beam, a non‑diffracting, self‑accelerating optical field. Starting from the interaction Hamiltonian for a χ^(2) crystal, the authors treat the pump classically as a superposition of plane waves and quantize the signal and idler fields. Using first‑order perturbation theory they obtain the two‑photon state |ψ⟩ = |0⟩ + ∫∫F(k₁,k₂) a†₁ a†₂ |0⟩, where the spectral function factorises into an energy‑conserving delta and a joint amplitude ϕ(k₁,k₂). The joint amplitude contains the Fourier transform of the pump profile, f_Ep(κ₁+κ₂), multiplied by a sinc term that encodes longitudinal phase‑matching through Δ = k_pz – k₁z – k₂z. Because the Airy pump’s Fourier spectrum carries the characteristic Airy phase and amplitude, ϕ(k₁,k₂) is intrinsically non‑separable in the transverse momenta κ₁ and κ₂, leading to a genuinely entangled biphoton in both position and momentum space.

The authors then introduce an optical system function G_j(k_j,x_j) that propagates the field from the crystal output plane to the detection planes, including free‑space propagation and a thin lens. This formalism allows them to write the first‑order correlation G^(1) and the second‑order coincidence rate G^(2) in terms of integrals over the joint amplitude and the system functions. Two detection regimes are examined:

1. Near‑field (image plane): Here the system function essentially reproduces the transverse field distribution of the pump. Consequently, the coincidence probability reproduces the finite‑energy Airy intensity pattern in the spatial correlation of the two photons. The authors show that the Airy beam can be “reconstructed” from the biphoton correlation, confirming that the non‑diffracting nature of the pump survives the down‑conversion process.

2. Far‑field (Fourier plane): In this configuration the system function performs a Fourier transform, so the coincidence map reflects the momentum‑space joint amplitude. The authors find a clear anti‑correlation κ₁ ≈ –κ₂, but modulated by the Airy‑specific oscillatory structure, which differs from the smooth Gaussian case. This demonstrates a direct mapping of the pump’s transverse momentum distribution onto the biphoton momentum correlations.

A central part of the work is the spatial Schmidt decomposition of the biphoton state. By varying the crystal length L, they show that longer crystals tighten the longitudinal phase‑matching, reducing the width of the real‑space correlation while increasing the number of Schmidt modes (higher Schmidt number). Thus, longer crystals yield tighter position correlations but richer angular (momentum) entanglement, whereas shorter crystals produce broader position correlations and fewer angular modes. The authors quantify the Schmidt number and entanglement entropy, providing a clear trade‑off between spatial resolution and dimensionality of the entangled Hilbert space.

Practical considerations are addressed: the Airy pump is generated by imposing a cubic phase on a Gaussian beam and truncating it with an exponential apodization parameter a; the crystal is assumed to be a negative uniaxial birefringent medium (type‑I e → o + o) with walk‑off vector M_p taken into account. The paper lists realistic parameters (pump wavelength ~400 nm, crystal length 1–10 mm, χ^(2)≈10 pm/V, focal lengths of the imaging lens) and estimates detection efficiencies and background noise. The analysis predicts that with current technology the Airy‑based biphoton source can achieve higher-dimensional entanglement (effective mode numbers >30) while preserving the non‑diffracting propagation over several centimeters, making it attractive for low‑loss quantum communication, high‑dimensional quantum key distribution, and quantum imaging schemes that benefit from the self‑accelerating trajectory of Airy beams.

In conclusion, the paper establishes that using a finite‑energy Airy pump in SPDC not only preserves the distinctive non‑diffracting, self‑accelerating properties of the classical beam but also imprints them onto the quantum correlations of the generated photon pairs. This results in a biphoton state with controllable spatial and momentum entanglement, tunable via crystal length and pump truncation. The analytical framework provided can guide experimental implementations and opens avenues for exploiting Airy‑structured light in advanced quantum information protocols.