Homodyne detection for atmosphere channels

We give a systematic theoretical description of homodyne detection in the case where both the signal and the local oscillator pass through the turbulent atmosphere. Imperfect knowledge of the local-oscillator amplitude is effectively included in a noisy density operator, leading to postprocessing noise. Alternatively, we propose a technique with monitored transmission coefficient of the atmosphere, which is free of postprocessing noise.

Figure 1

Scheme of homodyne detection for quantum light passing through the turbulent atmosphere as reported in Ref. [12]. The signal and the local oscillator are sent in two orthogonally polarized modes. After passing through the atmospheric channel and collection by a telescope, homodyne detection is realized with the half-wave plate $\mathrm{HWP}$, the polarization beam splitter $\mathrm{PBS}$, and the detectors ${\mathrm{D}}_1$ and ${\mathrm{D}}_2$.

Figure 2

Photon-number distribution (diagonal elements of the effective density operator) for the SPATS with $n_{\mathrm{th}}=1.11$ (such as realized in Ref. [21]) disturbed by beam wandering for the scheme with unmonitored transmission coefficient. In the corresponding turbulence model, cf. the appendix, the beam-spot radius, $W$, on the receiver aperture of radius $a$ is $W=0.9a$, and the standard deviations of beam deflection are (a) $\sigma =a$ (strong turbulence) and (b) $\sigma =0.5a$ (weak turbulence). The corresponding reference transmission coefficient is $T_{\mathrm{ref}}=\sqrt{\langle T^2\rangle }\approx 0.586$ and $T_{\mathrm{ref}}=\sqrt{\langle T^2\rangle }\approx 0.822$, respectively. For both cases, we use the detection efficiency $\eta =0.5$. Negative probabilities demonstrate fake effects caused by imperfect postprocessing.

Figure 3

Homodyne detection of quantum light passing through the turbulent atmosphere with monitoring the transmission coefficient. The signal and the local oscillator are sent in two orthogonally polarized modes. After passing through the atmospheric channel and collection by a telescope, the signal and the local oscillator are split by the polarization beam splitter $\mathrm{PBS}$. A part of the local oscillator transmitted through the beam-splitter ${\mathrm{BS}}_2$ is used for monitoring the transmission coefficient with the detector ${\mathrm{D}}_3$. Another part, after setting the needed phase by the phase modulator $\mathrm{PM}$ and conversion of the polarization direction by the half-wave plate $\mathrm{HWP}$, is combined with the signal on a standard homodyne detector, which consists of the $50:50$ beam splitter ${\mathrm{BS}}_2$ and two photodetectors ${\mathrm{D}}_1$ and ${\mathrm{D}}_2$.

Figure 4

$P$ function of the displaced SPATS for $\mathrm{Im}\alpha =0$, with $n_{\mathrm{th}}=1.11$ (such as realized in Ref. [21]) and different values of $\gamma$, disturbed by beam wandering and detected with monitored transmission coefficient. In the corresponding turbulence model (cf. the appendix), the beam-spot radius, $W$, on the receiver aperture of radius $a$ is $W=0.9a$, and the standard deviation of beam-deflection is $\sigma =10a$. The detection efficiency is $\eta =0.5$.

Figure 5

Product of noisy variances, ${\langle \Delta {\widehat{x}}_1^2\rangle }_{\mathrm{noisy}}{\langle \Delta {\widehat{x}}_2^2\rangle }_{\mathrm{noisy}}$, vs the reference transmission coefficient, $T_{\mathrm{ref}}$, for the scheme with unmonitored turbulence. The solid (dashed) line corresponds to the vacuum state (8-dB squeezed vacuum state) at the transmitter. We suppose that fluctuating losses are caused by beam wandering with the standard deviation of beam deflection $\sigma =40a$ and with the beam-spot radius $W=0.95a$, which leads to 35-dB mean losses. The detection efficiency is $\eta =0.5$. The shaded area corresponds to the violation of the Heisenberg uncertainty relation.