Quantum optics in the turbulent atmosphere: Fundamental issues and applications

TL;DR

This paper surveys the fundamental and practical issues of quantum-light propagation through turbulent air by focusing on the probability distribution of transmittance (PDT) as the central object linking atmospheric fluctuations to quantum-state evolution. It classifies and analyzes analytical PDT models—ranging from empirical log-normal and Beta distributions to physics-based beam-shape models (beam wandering, circular, and elliptic) and law-of-total-probability approaches—and validates them against phase-screen simulations under weak turbulence. A key advance is the explicit treatment of time correlations, introducing two-time PDT and correlation functions to assess when time averaging can substitute ensemble averaging and how to exploit or mitigate channel memory in protocols such as time-bin encoding and adaptive channel selection. The work provides practical guidance for model selection and parameter estimation from field-correlation data, while highlighting limitations and directions for developing more accurate analytical tools for free-space quantum communications and sensing.

Abstract

Quantum light propagation through turbulent atmosphere has become a subject of intensive research, spanning both theoretical and experimental studies. This interest is driven by its important applications in free-space quantum communication, remote quantum sensing, and environmental monitoring. At the same time, this phenomenon itself poses an intriguing fundamental problem. A consistent theoretical description typically makes explicit assumptions about the measurement scheme at the receiver station and/or the method of quantum-information encoding. A common and straightforward approach encodes the information in quantum states of a quasi-monochromatic mode, representing a pulsed Gaussian beam. Atmospheric turbulence induces random distortions of the pulse shape and, consequently, random fluctuations of the transmittance through the receiver aperture. These fluctuations, characterized by the probability distribution of transmittance (PDT), directly affect the quantum state of the received light. In this paper we examine various analytical models of the PDT, validate them through numerical simulations, and assess their range of applicability. Furthermore, we extend the analysis beyond the standard ensemble-averaging approach, recognizing that realistic experiments typically involve time averaging. This requires a detailed examination of the underlying random process, including the study of temporal correlations and their impact on nonclassical properties of electromagnetic radiation.

Figures (6)

• Figure 1: Typical behavior of $\eta(t)$ obtained from numerical simulations. Constant losses due to absorption, scattering, and imperfections of the optical system are not included.
• Figure 2: Log-normal and numerically simulated PDTs for a channel characterized by the following parameters: $C_n^2=1\times10^{-15}~\textrm{m}^{-2/3}$, $L_0=80~\textrm{m}$, $\ell_0=1~\textrm{mm}$, $W_0=2.78~\textrm{cm}$, $F_0=L$, $L=3~\textrm{km}$, and wavelength $\lambda=809~\textrm{nm}$. (a) Aperture radius $a =0.9~\textrm{cm}$, where the log-normal PDT provides a satisfactory fit. (b) Aperture radius $a =2.6~\textrm{cm}$, where the log-normal PDT deviates more from the numerical simulations. (c) KS statistic between the numerical data and the log-normal PDT as a function of the aperture radius. Dashed and dotted-dashed lines correspond to PTDs estimated with numerically simulated and analytical parameters, respectively.
• Figure 3: Beta and numerically simulated PDTs for a channel characterized by the following parameters: $C_n^2=1\times10^{-15}~\textrm{m}^{-2/3}$, $L_0=80~\textrm{m}$, $\ell_0=1~\textrm{mm}$, $W_0=2.78~\textrm{cm}$, $F_0=L$, $L=3~\textrm{km}$, and wavelength $\lambda=809~\textrm{nm}$ . (a) Aperture radius $a =0.9~\textrm{cm}$, where the Beta PDT provides a satisfactory fit. (b) Aperture radius $a =2.6~\textrm{cm}$, where the Beta PDT yields a poorer but still acceptable fit. (c) KS statistic between the numerical data and the Beta PDT as a function of the aperture radius. Dashed and dotted-dashed lines correspond to PTDs estimated with numerically simulated and analytical parameters, respectively.
• Figure 4: Beam-wandering and numerically simulated PDTs for a channel characterized by the following parameters: $C_n^2=1\times10^{-15}~\textrm{m}^{-2/3}$, $L_0=80~\textrm{m}$, $\ell_0=1~\textrm{mm}$, $W_0=2.78~\textrm{cm}$, $F_0=L$, $L=3~\textrm{km}$, and wavelength $\lambda=809~\textrm{nm}$. (a) Aperture radius $a =4.4~\textrm{cm}$, where the beam-wandering PDT provides a satisfactory fit. (b) Aperture radius $a =2.6~\textrm{cm}$, where the PDT reproduces the general shape but with a shifted mode. (c) KS statistic between the numerical data and the beam-wandering PDT as a function of aperture radius. Dashed and dotted-dashed lines correspond to PTDs estimated with numerically simulated and analytical parameters, respectively.
• Figure 5: Circular-beam and numerically simulated PDTs for a channel characterized by the following parameters: $C_n^2=1\times10^{-15}~\textrm{m}^{-2/3}$, $L_0=80~\textrm{m}$, $\ell_0=1~\textrm{mm}$, $W_0=2.78~\textrm{cm}$, $F_0=L$, $L=3~\textrm{km}$, wavelength $\lambda=809~\textrm{nm}$, and $a =2~\textrm{cm}$. (a) Parameters matched to the moments of $S$. (b) Parameters matched to the moments of $\eta$. (c) KS statistic between the numerical data and the circular-beam PDT as a function of the aperture radius. Green and violate colors correspond to matching by moments of $S$ and by moments of $\eta$, respectively. Dashed and dotted-dashed lines correspond to PTDs estimated with numerically simulated and analytical parameters, respectively.