Bell nonlocality in the turbulent atmosphere

Violations of Bell inequalities are better preserved by turbulent atmospheric channels than by comparable optical fibers in the scenario of copropagating entangled photons [A. A. Semenov and W. Vogel, Phys. Rev. A 81, 023835 (2010)]. Here we reexamine this result for the case of counterpropagation also considering the fact that each receiver registers so-called double-click events, which are caused by dark counts, stray light, and multiphoton entangled pairs. We show that advantages of the atmospheric links are feasible only for the copropagation scenario in the case of strong fluctuations of losses. For counterpropagation, the violations of Bell inequalities can be improved with an additional postselection procedure testing the channel transmittance.

Figure 1

Typical experimental setups for verification of Bell inequalities in atmospheric channels: (a) the copropagation scheme as used in Ref. [13]; (b) the counterpropagation scheme. For more details, see the text.

Figure 2

The Bell parameter, $\mathcal{B}$, vs the squeezing parameter, $\xi$, for the scenario of copropagation. The PDT is taken (a) for the strong-turbulence channel (cf. Appendix pp2), where the mean number of stray-light and dark counts is $\nu =1.7\times {10}^{-5}$, and (b) for the weak- to moderate-turbulence channel (cf. Appendix pp1), with $\nu =3\times {10}^{-3}$. For both cases, lines 1 and 2 correspond to fading channels without and with the incorporation of double-clicks events, respectively. Lines 3 and 4 correspond to the deterministic-loss channels with ${\eta }_0=\langle \eta \rangle$, without and with the incorporation of double-clicks events, respectively. The detection efficiency is ${\eta }_c=0.3$, which includes 3-dB losses at the beam splitter of the receiver. Here and in the following figures, the horizontal line $\mathcal{B}=2$ shows the local-realism bound.

Figure 3

The Bell parameter, $\mathcal{B}$, vs the squeezing parameter, $\xi$, for the scenario of counterpropagation. The PDT is taken (a) for the strong-turbulence channel (cf. Appendix pp2), where the mean number of stray-light and dark counts is $\nu =1.7\times {10}^{-5}$, and (b) for the weak- to moderate-turbulence channel (cf. Appendix pp1), with $\nu =3\times {10}^{-3}$. For both cases, lines 1 and 2 correspond to fading channels without and with the incorporation of double-clicks events, respectively. Lines 3 and 4 correspond to the deterministic-loss channels with ${\eta }_a={\eta }_b=\langle {\eta }_{\mathrm{A}}\rangle =\langle {\eta }_{\mathrm{B}}\rangle$, without and with the incorporation of double-clicks events, respectively. The detection efficiency is ${\eta }_c=0.6$.

Figure 4

Bell-inequality test with the postselection procedure proposed in Ref. [45]. Strong light pulses are sent in front of the nonclassical light pulses in order to test the transmittance of the channels. The time $\tau$ is much less than the typical time of atmospheric variations. One considers only those events for which on both sides the measured transmittance ${\eta }_{\mathrm{A}\left(\mathrm{B}\right)}\ge {\eta }_{\mathrm{ps}}$, where ${\eta }_{\mathrm{ps}}$ is the postselection threshold.

Figure 5

The Bell parameter, $\mathcal{B}$, vs the postselection efficiency, ${\eta }_{\mathrm{ps}}$ (solid line), and the feasibility [cf. Eq. (37)] of the postselection procedure, ${\overline{F}}_{\mathrm{AB}}$ (dashed line). Two scenarios are considered: (a) the strong-turbulence channel (cf. Appendix pp2), where the mean number of stray-light and dark counts is $\nu =4\times {10}^{-4}$ and $\xi =0.25$, and (b) the weak- to moderate-turbulence channel (cf. Appendix pp1), where $\nu =2\times {10}^{-2}$ and the squeezing parameter is $\xi =0.31$. The detection efficiency is ${\eta }_c=0.6$.