Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality

Tap-proof long-distance quantum communication requires a deep understanding of the strong losses in transmission channels. Here we provide a rigorous treatment of the effects of beam wandering, one of the leading disturbances in atmospheric channels, on the quantum properties of light. From first principles we derive the probability distribution of the beam transmissivity, with the aim to completely characterize the quantum state of light. It turns out that beam wandering may preserve nonclassical effects, such as entanglement, quadrature and photon number squeezing, much better than a standard attenuating channel of the same losses.

Figure 1

Transmission efficiency, $T^2$, vs the beam-deflection distance, $r$, for different values of the beam-spot radius $W$. Solid and dashed lines correspond to the numerically calculated integral (3) and the analytical approximation (4), respectively. The inset schematically shows the aperture of radius $a$ and the beam-spot of radius $W$.

Figure 2

The exceedance (tail distribution) cf. Eq. (10), is shown in the logarithmic scale for different values of the beam-spot radius $W$, and standard deviation of beam deflection $\sigma$. Solid and dashed lines correspond to $d=0$ and $d=\sigma$, respectively.

Figure 3

The Bell parameter is shown in dependence of the squeezing parameter of a parametric down-conversion source. The solid line shows the situation for beam wandering, with a standard deviation $\sigma =28.5a$ of beam deflection and a beam-spot radius of $W=1.1a$, leading to a mean loss of 32 dB. The receiver-module losses are supposed to be 9 dB. The dashed line shows the behavior for constant losses of the same size. The mean number of stray-light and dark counts is assumed to be ${10}^{-5}$. The shaded area indicates the violation of the Bell inequality.

Figure 4

The reachable values of quadrature and photon-number squeezing are shown as a function of the corresponding exceedance. The input light is amplitude-squeezed by 6 dB and coherently displaced by an amplitude of 10. The mean losses are 32 dB with the same parameters of beam wandering as in Fig. 3. The dependence of $T_{\min }$ on the exceedance is also shown.