Optimal propagation distance for maximal biphoton entanglement through the weakly turbulent atmosphere

TL;DR

This paper develops a rigorous, continuous-variable model of SPDC biphoton propagation through weak atmospheric turbulence using the extended Huygens–Fresnel principle and Kolmogorov statistics. By retaining the full spatial structure in a position-basis density operator and incorporating thin-lens collimation, the authors derive analytical expressions for the two-photon field’s cross-spectral density and its evolution in turbulence, including the quantum-to-classical transition of spatial correlations. They show that turbulence reduces purity while preserving overall spatial correlations, and that conditional OAM distributions broaden with propagation, especially under stronger turbulence. Crucially, they identify a finite distance range where angle–OAM entanglement is maximized, with entanglement reviving after initial disappearance and persisting over several kilometers in weak turbulence, offering actionable guidance for optimizing free-space quantum communication links. The framework also opens pathways for extending to non-Kolmogorov turbulence, removing the double-Gaussian SPDC approximation, and integrating adaptive optics for turbulence mitigation.

Abstract

Understanding the influence of atmospheric turbulence on the propagation of entangled biphoton states is essential for free-space quantum communication protocols. Using the extended Huygens-Fresnel principle and the Kolmogorov turbulence model, we derive an analytical expression for the combined density operator of the signal and idler fields generated via SPDC, following propagation through separate turbulent channels. By expressing this density operator in the continuous position basis, we show how the spatial correlations between signal and idler persist through turbulence despite the loss of state purity, as they transition from being quantum to classical in nature. We further identify a finite range of propagation distances over which the angle-OAM entanglement is maximized, which provides valuable insights for designing free-space quantum communication links operating over several kilometers through the turbulent atmosphere.

Figures (5)

• Figure 1: Propagation of a two-photon field through turbulence. (a) Schematic illustrations of the SPDC process and the field propagation. (b) Joint two-photon position probability distribution function $P(x_s,x_i;z)$ at the output face of the crystal before collimation. (c) Joint two-photon angle probability distribution function $P(\theta_s,\theta_i;z)$ in the same plane, where $\theta$ is the azimuthal angle. The tight correlations in both the position and angle bases can be clearly seen.
• Figure 2: Purity $\gamma(z)$ and spatial correlation function $f_c(z)$ as functions of propagation distance z, under different turbulence strengths. The curves for the correlation function $f_c(z)$ overlap for all turbulence strengths. The parameters are chosen as: crystal length $L=1\,\mathrm{mm}$, focal length $f=50\,\mathrm{cm}$, pump waist $w_0=w(0)=507 \,\mu \mathrm{m}$, and pump wavelength $\lambda_p=355\,\mathrm{nm}$.
• Figure 3: The conditional OAM distribution at different propagation distances $z$. (a) $z=0.5$ km; (b) $z=1$ km; (c) $z=1.5$ km; (d) $z=2$ km. The spectrum broadens with propagation as more OAM modes are coupled together due to the effect of turbulence. The parameters are chosen as: crystal length: $L=1\,\mathrm{mm}$, focal length: $f=50\,\mathrm{cm}$, signal mode waist at z=0 m: $\sigma_s(0)=507 \,\mu \mathrm{m}$ and pump wavelength: $\lambda_p=355\,\mathrm{nm}$.
• Figure 4: The standard deviation of the conditional OAM distribution $P(\ell_s|\ell_i=0;z)$ increases more rapidly with propagation distance z under stronger turbulence. However, under weak turbulence strengths ($C_n^2 = 10^{-17} - 10^{-16} \text{ m}^{-2/3}$), the standard deviation increases only linearly with z and remains small over a long distance. The parameter values are the same as those used in figure \ref{['fig:oam_spectrum_z']}. Note that the OAM is expressed in units of $\hbar$.
• Figure 5: Conditional angle-OAM uncertainty product $\Delta (\theta_s | \theta_i=0;z) \cdot \Delta (\ell_s | \ell_i=0;z)$, and the right-hand side of the EPR criterion in the angle-OAM bases $0.5 \hbar \left[ 1 - 2\pi \cdot P(\theta_0 \mid \theta_i=0; z) \right]$ as a function of the propagation distance $z$. Entanglement in the angle-OAM bases exists when the blue curve falls below the orange curve. The range of distances indicated with yellow shading can be identified as the region of maximal entanglement. The parameter values are the same as those used in figure \ref{['fig:p_cor']}, and the turbulence strength is $C_n^2=10^{-16}\,\text{ m}^{-2/3}$. The conditional angle-OAM uncertainty product in the absence of turbulence is also shown, though note that the right-hand side of the EPR criterion is practically identical in both cases.