Theory of atmospheric quantum channels based on the law of total probability

The atmospheric turbulence is the main factor that influences quantum properties of propagating optical signals and may sufficiently degrade the performance of quantum communication protocols. The probability distribution of transmittance (PDT) for free-space channels is the main characteristic of the atmospheric links. Applying the law of total probability, we derive the PDT by separating the contributions from turbulence-induced beam wandering and beam-spot distortions. As a result, the obtained PDT varies from log-negative Weibull to truncated log-normal distributions depending on the channel characteristics. Moreover, we show that the method allows one to consistently describe beam tracking, a procedure which is typically used in practical long-distance free-space quantum communication. We analyze the security of decoy-state quantum key exchange through the turbulent atmosphere and show that beam tracking does not always improve quantum communication.

Figure 1

Transmitted beam profile shown relative to the aperture opening of radius $a$. The beam centroid is displaced relative to the aperture center due to beam wandering and its position is given by ${\mathit{r}}_0$. The tracking coordinate system $(x^{\prime },y^{\prime })$ is connected with the fluctuating position of the beam centroid.

Figure 2

The PDTs for the light beam transmitted through the atmospheric turbulence and collected by the circular aperture. The solid line shows the PDT for an atmospheric channel of 1 km length with the refractive index structure constant $C_n^2=4\times {10}^{-14}{\text{m}}^{-2/3}$ (Rytov parameter ${\sigma }_R^2=1.7$). The dashed line corresponds to a channel of 2 km length with $C_n^2=3\times {10}^{-15}{\text{m}}^{-2/3}$ (${\sigma }_R^2=0.46$). The dash-dotted line represents the PDT for a 3-km channel with $C_n^2=3\times {10}^{-15}{\text{m}}^{-2/3}$ (${\sigma }_R^2=0.96$). The additional extinction losses of 1 dB/km are also included according to [64].

Figure 3

The PDTs for a light beam propagating through the atmospheric turbulence with the fixed refractive index structure constant $C_n^2=4\times {10}^{-14}{\text{m}}^{-2/3}$ and various propagation lengths: 1 km, solid line (${\sigma }_R^2=1.72$); 1.1 km, dashed line (${\sigma }_R^2=2.05$); 1.2 km, dash-dotted line (${\sigma }_R^2=2.41$); 1.5 km, dotted line (${\sigma }_R^2=3.60$); and 2 km, long-dashed line (${\sigma }_R^2=6.14$). The other parameters are the same as in Fig. 2.

Figure 4

Influence of the beam-tracking procedure on (a) the PDT and (b) the corresponding exceedance function, shown for increasing values of the beam-tracking parameter: ${\sigma }_{\mathrm{tr}}=0$ (solid line), ${\sigma }_{\mathrm{tr}}=0.25{\sigma }_{\mathrm{bw}}$ (dashed line), ${\sigma }_{\mathrm{tr}}=0.5{\sigma }_{\mathrm{bw}}$ (dash-dotted line), and ${\sigma }_{\mathrm{tr}}={\sigma }_{\mathrm{bw}}$ (dotted line). The increase of ${\sigma }_{\mathrm{tr}}$ results in a shift of the maximum of the PDT or of the tail of the exceedance function to larger values of the transmittance $\eta$. This signifies the importance of beam tracking for protocols that utilize a postselection of transmission events with a high transmittance. The case ${\sigma }_{\mathrm{tr}}={\sigma }_{\mathrm{bw}}$ corresponds to perfect beam tracking. The parameters of the beam and atmosphere correspond to those for the solid curve in Fig. 3.

Figure 5

Transmitted value of squeezing as a function of a postselection threshold ${\eta }_{\min }$. The input light is squeezed to $-2.4$ dB and sent through a 1-km optical channel with ${\sigma }_R^2=1.72$. It is detected by the receiver with the application of both tracking and postselection procedures. The solid line corresponds to the squeezing transmission without beam tracking. The corresponding PDTs and exceedance functions are shown in Fig. 4.

Figure 6

Averaged secure key rate for the two-decoy-state quantum key distribution in an atmospheric quantum channel as a function of mean channel losses without (solid line) and with beam tracking (dashed line). The signal with the mean photon number ${\mu }_s=0.27$ at $\lambda =800$ nm is mixed with a weak decoy state (${\mu }_d=0.39$) and an empty decoy state (${\mu }_v=0$). It is sent through atmospheric links of various lengths but equal refractive-index structure function. Other beam and aperture parameters are the same as in Fig. 2. The additional 1-dB/km losses due to absorption are also taken into account. The inset shows the relative improvement given in percent [cf. Eq. (49)] of the average secure rate by applying the beam-tracking procedure.