Spatial correlations in four-wave mixing with structured light

TL;DR

This work develops a quantized paraxial theory of four-wave mixing with structured light and demonstrates the multi-spatial-mode character of the resulting biphoton state. By treating the pump classically and projecting onto Laguerre-Gaussian modes, it connects position- and momentum-space descriptions and shows a direct transfer of the pump angular spectrum to the spatial coincidence profile, analogous to spontaneous parametric down-conversion. The analysis covers how the medium geometry (cold-atom clouds vs vapor cells) and the Gaussian versus arbitrary pump structures shape the phase-matching and mode composition, and it provides explicit expressions for the coincidence amplitudes and entanglement measures such as spiral bandwidth and Schmidt rank. These results establish a theoretical framework to design and interpret structured-light experiments in nonlinear and quantum optics, with implications for controllable spatial correlations and high-dimensional entanglement.

Abstract

We present a detailed theoretical treatment of four-wave mixing (FWM) in a quantized paraxial framework, capturing the multi-spatial-mode nature of the biphoton state generated in the process. By analyzing the biphoton state both in position and momentum representations, we identify the conditions under which these descriptions become equivalent. We also highlight formal and physical similarities between FWM and spontaneous parametric down-conversion (PDC), showing that the transfer of pump structure to the spatial coincidence profile, an important and well-known characteristic of the biphoton state, carries over naturally to FWM. In addition, our treatment captures the transition from position correlations in the near field to momentum correlations in the far field, reflecting the underlying spatial entanglement. The measures of entanglement, including the spiral bandwidth and the Schmidt rank, are discussed. Our work consolidates known and new results on spatial correlations in FWM and provides a theoretical framework that may support future studies in nonlinear and quantum optics with structured light.

Figures (8)

• Figure 1: Four-wave mixing configuration. A strong optical pump field with spatial profile $\mathcal{V}_\mathrm{p}(\mathbf{r})$ shines on the third-order nonlinear medium. The pump may carry Gaussian or non-Gaussian spatial distributions, as indicated in the leftmost inset. As a result of the nonlinear interactions, two photons from the pump are absorbed, and two photons are generated on the signal and probe channels. The pair of generated beams are symmetrically distributed, at an angle $\theta$, with respect to the pump propagation direction, which is parallel to the $\mathbf{e}_z$ direction. The central inset represents the conservation of energy and momentum in the process. The rightmost inset illustrates the inherent phase-mismatch associated with the frequency degeneracy condition.
• Figure 2: (a) Representation of the spatial distribution of atomic density in the case of a sample of cold atoms, $N_\mathrm{cloud}(\mathbf{r})$, as given by Eq. (\ref{['eq:mu_cloud']}). The cloud characteristic lengths $\mathcal{R}$ and $\mathcal{L}$ may be comparable to the optical waist $w_0$. (b) Depiction of an optical field incident on a glass cell containing a heated sample of atomic vapor, with a radius much larger than the beam waist parameter, $R\gg w_0$, in such a way that we may consider a uniform density of atoms along the cell length $L$.
• Figure 3: (a) Distribution of coincidence amplitudes, $|C^{\ell_{\mathrm{pr}},\ell_{\mathrm{s}}}_{p_{\mathrm{pr}},p_{\mathrm{s}}}|^2$, in the case of a Gaussian pump with a beam waist $w_0=1\,\mathrm{mm}$ exciting a thin nonlinear sample (the medium length does not significantly influence the calculations) of uniform transverse profile. (b) OAM distribution $P_{\ell,-\ell}$ of the biphoton state distribution shown in (a) evidencing a finite SBW $\Delta\ell\approx2$. (c) Variation of the SBW (left) and of the associated entanglement entropy (right) with the total OAM pumped into the system $\ell_\mathrm{T}$. The results were calculated using Eq. (\ref{['eq:C_T']}) considering the subspace of spatial modes $\mathbb{S}(2,4)$.
• Figure 4: Purity of the partially traced biphoton state considering a Gaussian pump exciting a medium of length $L=5\,\mathrm{cm}$ restricted to subspaces $\mathbb{S}(l_\mathrm{max},p_\mathrm{max})$ of different sizes as a function of the waist $w_0$ (left), and the corresponding estimated Schmidt ranks in each subspace (right).
• Figure 5: (a) Configuration for the detection of the spatial coincidence profiles in the case where we introduce spatial resolution by placing tight pinholes in front of detectors with a single translational degree of freedom. C corresponds to a coincidence counting system. (b) Evolution of the resulting coincidence profiles $g^{(2)}(X_\mathrm{pr},X_\mathrm{s})$ from the medium exit, $z=L/2$, up to $z=z_R$.