Imaging Walk-Off Driven Distortions in EPR Photon Pair Correlations

TL;DR

The paper investigates how birefringence-induced transverse walk-off couples the sum and difference transverse coordinates in spontaneous parametric down-conversion, revealing propagation-dependent, wedge-shaped distortions in the near-field two-photon spatial correlations that persist even for relatively thin crystals. By combining heuristic analysis, Wigner-function propagation, and a phase-curvature ansatz, the authors show that finite walk-off introduces a non-separable coupling in the phase-matching term, leading to an anti-correlation width $oldsymbol{ riangle x_-}$ that depends on the sum coordinate $oldsymbol{x_+}$ and grows with propagation distance $oldsymbol{z}$. The experimental demonstrations using a 1 mm BBO crystal, a pulsed 405 nm pump, and an event-based camera confirm the predicted wedge-shaped patterns and the enhanced effects with structured (OAM) pumping, while highlighting limitations of simple separable models. These results refine the understanding of spatial entanglement in birefringent media and have implications for spatial-mode quantum information processing and spatially resolved quantum imaging, including the need to account for propagation-induced coupling in state reconstruction and measurement.

Abstract

Spontaneous parametric down-conversion is the primary source of position-correlated and momentum-anticorrelated photon pairs that form the canonical Einstein-Podolsky-Rosen (EPR) state. Their transverse spatial correlations are usually analyzed within the thin-crystal approximation, where the two-photon wavefunction is assumed to factorize into independent functions of the sum and difference coordinates. In practice, however, birefringence-induced transverse walk-off breaks this factorization and couples these degrees of freedom. Here, we show that this coupling persists even for nominally thin crystals once the free-space propagation of the joint spatial intensity is taken into account. This sum-difference coordinate coupling leads to a distinctive tapering of the transverse correlations near the crystal image plane-an effect that standard factorized models cannot capture. Numerical simulations and experimental data clearly confirm this novel behavior. Our findings provide a more complete description of photon-pair generation in birefringent nonlinear media and clarify fundamental limits on spatially resolved quantum imaging and spatial-mode quantum information processing with EPR states.

Figures (4)

• Figure 1: Birefringence-induced transverse walk-off effects. (a) Schematic illustration of birefringence-induced transverse walk-off in a nonlinear crystal. A pump beam (purple) with a polarization state along extraordinary axes of the crystal undergoes anomalous refraction, causing its centroid to drift laterally along the optic-axis direction as it propagates through the crystal. Consequently, photon pairs generated at different longitudinal positions emerge from laterally displaced locations, leading to a position-dependent width $\Delta x_-$ of the transverse anti-correlations. HF denotes a high-pass spatial filter. (b) Anti-correlation width $\Delta x_-$ as a function of propagation distance $z$ and sum coordinate $x_+$. The inset shows the normalized ratio $\Delta x_-(z,x_+)/\Delta x_-(z,x_+=0)$, highlighting that the dependence on $x_+$ is strongest near the crystal image plane. (c) Numerical simulations of the free-space evolution of the transverse spatial correlations in the presence of transverse walk-off, obtained from the propagation of the Wigner function expressed in the transformed sum--difference coordinate frame. (d) Corresponding simulations obtained using the angular spectrum propagation method, starting from an initial two-photon wavefunction given by the ansatz in Eq. \ref{['eq:ansatz']}.
• Figure 2: Experimental results. (a) Schematic of the experimental setup. BBO: 1-mm-thick type-I $\beta$-barium borate crystal; HF: high-pass filter; L: lens; HWP: half-wave plate; PBS: polarizing beamsplitter; M: mirror. (b) Experimentally reconstructed transverse spatial correlations along the $x$ coordinate. (c) Corresponding simulation results obtained using the ansatz in Eq. \ref{['eq:ansatz']}. (d,e) Detailed analysis of transverse walk-off effects, highlighting the dependence of the correlation width on the sum coordinate $x_+$. The one-dimensional cuts show best fits of the coincidence distributions for fixed values of $x_i - x_s$, sampled in steps of $233~\mu$m, illustrating how the spatial correlations evolve along this direction due to transverse walk-off. (f) Same one-dimensional analysis as in (d,e), shown for all measured propagation planes. When stacked together, these reconstructed profiles map out the free-space propagation of the phase-matching function.
• Figure 3: Transverse walk-off effects with OAM pumping. (a) Measured transverse spatial correlations at a propagation distance of $z=20$ mm when the crystal is pumped with a beam carrying orbital angular momentum $\ell = 6$. Transverse walk-off effects are particularly evident in the $x$-correlations, where they introduce pronounced asymmetries. (b) Coincidence images obtained by post-selecting on spatially correlated photon pairs satisfying $x_i = x_s + c$. In the absence of transverse walk-off, this procedure would directly reconstruct the pump intensity profile. When a walk-off is present, however, the reconstructed intensity becomes dependent on the post-selection condition, as highlighted in the insets. The rectangular bars indicate the different values of $c$ used for post-selection.
• Figure S1: Fits of sections of measured correlation patterns. Full comparison between experimental data (black circles) and best fits (red plots) that was shown in Fig. \ref{['fig:exp']}-(c)-(e) for all propagation planes and sections considered.