Satellite-mediated quantum atmospheric links

The establishment of quantum communication links over a global scale is enabled by satellite nodes. We examine the influence of the Earth's atmosphere on the performance of quantum optical communication channels with emphasis on the downlink scenario. We derive the geometrical path length between a moving low Earth orbit satellite and an optical ground station as a function of the ground observer's zenith angle, his geographical latitude, and the meridian inclination angle of the satellite. We show that the signal distortions due to regular atmospheric refraction, atmospheric absorption, and turbulence have a strong dependence on the zenith angle. The observed saturation of transmittance fluctuations for large zenith angles is explained. The probability distribution of the transmittance for slant propagation paths is derived, which enables us to perform the security analysis of decoy-state protocols implemented via satellite-mediated links.

Figure 1

Typical communication scenarios of optical ground stations (OGS) with satellites $S$ via uplink (d) and downlinks (a)–(c). The subscript $i=1,2,3$ in the notation $S_i$ refers to the successive positions of the satellite. After the satellite makes one orbital revolution along its orbit, (i) its new position $S_3$ as well as orbit (ii) appear inclined for the ground observer ${\mathrm{OGS}}_2$. This apparent change in the orbit inclination is due to the Earth's rotation.

Figure 2

Geometry of the communication configuration shown in the plane that passes through the Earth's center $C$, the observer's location $O$, and the satellite position $S$ (a). The same communication link $OS$ is shown in its relative position to the observer's meridian plane $ABOC$ (b). The cross section of the trajectory with the cone of angle $2Z$ and side $SO$ yields the associated link $O{S}^{\prime }$ of the same length.

Figure 3

Optical path length elongation factor due to atmospheric refraction as a function of observer's apparent $Z_a$ and true $Z$ zenith angles for several LEO satellite orbits with altitudes $H$.

Figure 4

Aperture-averaged scintillation index as a function of the zenith angle for several inclination angles of the satellite orbit. The dotted intervals indicate the minimal zenith angles for inclined orbits [cf. Eq. (2)]. The dotted curve corresponds to the asymptotic value (27).

Figure 5

Intensity scintillation index of LEO-ground downlink at 847 nm. Experimental data are given with confidence intervals (shaded area) and mean value (dashed line) calculated from the individual measurements. The theoretical curve (solid) is calculated using Eq. (49).

Figure 6

Variation of the mean short-term divergence angle $W_{\mathrm{ST}}/L_r$ and the standard deviation of the angle-of-arrival fluctuations ${\sigma }_{\mathrm{BW}}/L_r$, with the zenith angle. The results are presented for several inclination angles of the satellite orbit $\mathrm{\Delta }\iota$. Dotted lines indicate the minimal zenith angles (2) for inclined orbits.

Figure 7

Quantum bit error rates (a) and lower bounds of averaged secure key rates (b) as functions of the communication time for several inclination angles.

Figure 8

Geometric representation of the surface and geocentric coordinate systems. The Earth's surface coordinate system for the observer showing the azimuth $A$ and the zenith $Z$ angles. The geocentric coordinate system is defined in terms of the declination angle $\delta$ and the inclination angle $\mathrm{\Delta }\iota$. The geographical position of the observer is given by the latitude $\mathrm{\Psi }$ while the parallax correction caused by the finiteness of the satellite altitude $H$ is determined by the angle $\phi$.

Figure 9

Geometry of a typical satellite-ground communication. Initially, the ground observer $\tilde{O}$ establishes the communication link with the satellite whose trajectory lies within the meridian plane of the observer $\tilde{O}$. At a later time, due to Earth's rotation, the observer is translated to the point $O$ while the satellite trajectory is inclined by an angle $\mathrm{\Delta }\iota$ relative to the new observer meridian plane (or fictitious orbit plane). Circles $\tilde{\mathrm{H}}$ and $\mathrm{H}$ denote the local horizons of the observers $\tilde{O}$ and $O$, respectively. The communication between the observer $O$ and the satellite $S$ can be established provided the satellite trajectory $S^{\prime }S$ lies above the horizon $\mathrm{H}$.

Figure 10

Temperature, pressure, and refractive index as functions of altitude. Dots represent the standard atmosphere values and lines show the linear approximation of temperature and refractive-index curves used in this article, as well as the functional dependence (B3), (B4) for the pressure.

Figure 11

The influence of atmospheric refraction on the elongation of the optical ray trajectory (a). The geometrical path of length $OS$ deforms into the curved path $O{P}_1{P}_2...P_{n}S$ while the true zenith angle $Z$ changes to the apparent zenith angle $Z_a$. The magnified upper part of the ray path shows the relevant refraction angles (b).

Figure 12

Typical example of the mean meridional (solid line) and the zonal (dashed) winds is shown together with the corresponding vertical wind shear components. The HWM93 empirical wind model [115] for the northern hemisphere (${48}^{\circ }$ N, $11.5^{\circ }$ E) summer (day 236) at midnight has been used for the calculation.

Figure 13

Profiles of the refractive-index structure function: AFGL, WK, and Hufnagel models are compared to the thermosonde data adopted from Ref. [119]. The inset shows the refractive-index structure parameter within the boundary layer with respect to the inversion height.