Localized to delocalized spatial quantum correlation evolution in structured bright twin beams

Quantum correlations in the spatial domain hold great promise for applications in quantum imaging, quantum cryptography and quantum information processing, owing to the infinite dimensionality of the associated Hilbert space. Here, we present a theoretical investigation, complemented by experimental measurements, of the propagation dynamics of the spatial quantum correlations in bright structured twin beams generated via a four-wave mixing process in a double-$Λ$ configuration in atomic vapor. We derive an analytical expression describing the evolution of the spatial quantum correlation distribution from the near field to the far field. To qualitatively support the theoretical predictions, we perform experiments measuring intensity-difference noise between different spatial subregions of the twin beams as they propagate from the near field to the far field. The presence of quantum correlations is manifested as squeezing in the intensity difference noise measurement. With a Gaussian pump, we observe localized correlations in the near field and localized anti-correlations in the far field. In contrast, with a structured Laguerre-Gaussian pump, there is a transition from localized correlations in the near field to delocalized correlations in the far field. The present results offer valuable insights into the fundamental behavior of spatial quantum correlations and open possibilities for potential applications in quantum information, quantum imaging and sensing.

💡 Research Summary

This paper presents a comprehensive theoretical and experimental study of how spatial quantum correlations evolve during propagation in bright twin beams generated by four‑wave mixing (FWM) in a double‑Λ configuration of hot 85Rb vapor. The authors first develop an analytical model of the FWM process under the undepleted‑pump approximation. The interaction Hamiltonian includes the third‑order susceptibility χ^(3) and treats the pump as a classical field with a transverse profile E(ρ). By expressing the probe and conjugate fields in the momentum basis, they derive a two‑photon probability amplitude F(q_pr, q_c) that is the product of the pump’s transverse Fourier component E(q_pr+q_c) and a sinc function representing longitudinal phase‑matching.

Two pump configurations are considered: a simple Gaussian beam (azimuthal index l = 0) and a Laguerre‑Gaussian (LG) beam carrying orbital angular momentum (l ≠ 0). For the LG pump, the transverse field contains an e^{i l φ} phase factor and a ring‑shaped intensity profile, which leads to a distinct form of E(q) involving Bessel functions. The authors perform a Fourier transform to obtain the two‑photon amplitude in position space and then propagate it over a distance z using the Fresnel kernel K(ρ,ρ′;z). To keep the expressions tractable, the sinc term is approximated by a cosine‑Gaussian function, yielding a final propagated amplitude ˜F that depends on complex parameters α, β, and γ.

From ˜F they compute the joint position probability distribution P(ρ_pr, ρ_c; z)=|˜F|² and the conditional distribution P(ρ_c|ρ_pr). The theoretical results show that with a Gaussian pump the twin photons are strongly position‑correlated in the near field (z≈0), the correlation width broadens with propagation, and in the far field the photons become anti‑correlated (reflecting momentum conservation). With an LG pump, the near‑field distribution exhibits two bright lobes corresponding to the pump’s ring, and as the beams propagate these lobes expand and split into two off‑diagonal features, indicating a transition from localized to delocalized correlations.

Experimentally, the authors generate bright twin beams in a heated Rb cell and measure intensity‑difference noise between selected spatial sub‑regions at several propagation distances ranging from a few millimetres to over a metre. Quantum correlations manifest as squeezing of the intensity‑difference below the standard quantum limit. The measurements confirm the theoretical predictions: the Gaussian pump yields maximal squeezing in the near field and a reversal to anti‑squeezing in the far field, while the LG pump shows localized squeezing that spreads with distance, eventually producing squeezing over a larger spatial area.

The study demonstrates that the spatial structure of the pump beam directly engineers the spatial quantum correlation landscape of bright twin beams. By selecting pump modes, one can tailor the correlation from tightly localized (useful for high‑resolution quantum imaging) to broadly delocalized (advantageous for high‑dimensional quantum communication). Moreover, the work bridges the gap between single‑photon, SPDC‑based studies and macroscopic, bright‑beam regimes, showing that macroscopic twin beams retain rich multimode entanglement that can be harnessed for practical quantum technologies. The analytical framework provided offers a valuable tool for designing structured‑pump FWM sources tailored to specific quantum‑information applications.