Circular-beam approximation for quantum channels in a turbulent atmosphere

TL;DR

This paper introduces a circular-beam approximation for modeling the probability distribution of transmittance in turbulent free-space quantum channels, offering a computationally efficient alternative to the elliptic-beam model. It defines a log-normal distribution for the fluctuating beam-spot radius and demonstrates two robust parameter-estimation strategies: transmittance-moment matching and S-moment matching, with a focus on reducing model misspecification bias. The approach is grounded in the phase-approximation of the Huygens–Kirchhoff method and is validated against phase-screen simulations, showing good agreement across a broad range of aperture sizes in weak turbulence. The framework is then applied to predict the transmission of nonclassical states, illustrating practical applicability for quantum communications and highlighting regimes where moment accuracy is critical for reliable predictions.

Abstract

The evolution of quantum states of light in free-space channels is strongly influenced by atmospheric turbulence, posing a significant challenge for quantum communication. The transmittance in such channels randomly fluctuates. This effect is commonly described by the probability distribution of transmittance (PDT). The elliptic-beam approximation provides an analytical model for the PDT, showing good agreement with experimental and simulation data within a specific range of channel parameters. In this work, we introduce the circular-beam approximation -- a simplified alternative that offers satisfactory accuracy while significantly reducing computational complexity. Our method naturally leads to a technique for determining the model parameters from the first two moments of the transmittance. This approach eliminates the model misspecification bias inherent in the elliptic-beam approximation and significantly extends the applicability range of the PDT model, providing a practical tool for characterizing atmospheric channels in quantum applications.

Figures (10)

• Figure 1: Comparison between the log-normal and simulated probability distributions of $S$. (a) Solid and dashed lines show the probability distribution functions $P(S)$ for the log-normal distribution (\ref{['Eq:lognorm']}) and the distribution estimated from the simulated data, respectively. (b) The Kolmogorov–Smirnov statistics (\ref{['Eq:KS']}) as a function of the Rytov parameter. Here $L$ ranges from $500$ to $5000$ m, $\lambda = 808$ nm, $W_0=\sqrt{L\lambda / \pi}$, $C_n^2=10^{-15}$ m$^{-2/3}$, $l_0=10^{-6}$ m, $L_0=5 \times 10^{3}$ m, $F_0=L$.
• Figure 2: Comparison of analytical results obtained using the Huygens-Kirchhoff method (solid blue lines) with simulated data based on the sparse-spectrum model of the phase-screen method (dashed orange lines) for (a) $\langle \eta \rangle$, (b) $\langle \eta^2 \rangle$, (c) $\langle \Delta\eta^2 \rangle$, and (d) $\sigma_{\mathrm{bw}}^2$ . The channel parameters are the same as in Fig. \ref{['Fig:distrS']}. The propagation distance $L$ ranges from $500$ m to $4000$ m. In plots (a)–(c), the aperture radii are $a_1=15$ mm, $a_2=12$ mm.
• Figure 3: Comparison of simulated PDT histograms with PDTs obtained using the circular-beam approximation, based on different matching methods and either analytical or numerically simulated parameters. The channel parameters are the same as in Fig. \ref{['Fig:distrS']}, with (a) $L = 2000$ m, $a = 12$ mm and (b) $L = 1000$ m, $a = 25$ mm. The green solid line (A) and the red solid line (B) represent the method of matching the moments of $S$ using analytical and numerically simulated values, respectively. The violet dashed line (C) and the orange dashed line (D) correspond to the transmittance–moment matching method, also using analytical and simulated values, respectively.
• Figure 4: Kolmogorov–Smirnov statistics quantifying the difference between numerically simulated PDTs and analytical PDTs for (a) $L = 2000$ m with $W_{\text{LT}} = 28$ mm and aperture radius $a$ ranging from 3 to 47 mm, and (b) $L = 1000$ m with $W_{\text{LT}} = 19$ mm and $a$ ranging from 3 to 32 mm. The lines correspond to the cases shown in Fig.\ref{['Fig:PDT_distrib']}. For the remaining channel parameters, see Fig.\ref{['Fig:distrS']}.
• Figure 5: Comparison of two approaches for incorporating constant losses into the transmittance-moment matching method. Solid lines F (violet) and E (brown) correspond to the direct rescaling of the PDT using analytical and numerically simulated moments of $\eta$, respectively. Dashed lines H (violet) and G (brown) represent the indirect incorporation of constant losses into the parameters of the log-normal distribution for $S$. Panel (a) shows analytical PDTs and corresponding numerically simulated histograms for $L=2000$ m, $a=12$ mm. Panel (b) presents the corresponding Kolmogorov–Smirnov statistics for $L=2000$ m, $W_{\text{LT}}=28$ mm, and aperture radius $a$ ranging from $3$ mm to $47$ mm. For the rest of the channel parameters, see Fig. \ref{['Fig:distrS']}. The values of constant losses are specified in the text.