Quantum Communication over Atmospheric Channels: A Framework for Optimizing Wavelength and Filtering

Despite quantum networking concepts, designs, and hardware becoming increasingly mature, there is no consensus on the optimal wavelength for free-space systems. We present an in-depth analysis of a daytime free-space quantum channel as a function of wavelength and atmospheric spatial coherence (the Fried coherence length). We choose decoy-state quantum key distribution bit yield as a performance metric in order to reveal the ideal wavelength choice for an actual qubit-based protocol under realistic atmospheric conditions. Our analysis represents a rigorous framework to analyze requirements for spatial, spectral, and temporal filtering. These results will help guide the development of free-space quantum communication and networking systems. In particular, our results suggest that shorter wavelengths in the optical band should be considered for free-space quantum communication systems. Our results are also interpreted in the context of atmospheric compensation by higher-order adaptive optics.

Figure 1

(a) A schematic illustrating the concept of spatially filtering sky-noise photons created by atmospheric scattering. An entrance pupil ($D_R$) with focal length $f$ and a field stop with diameter $d$ determine the solid-angle FOV (${\mathrm{\Omega }}_{\mathrm{FOV}}$). Reduction of the FOV of the receiver can reduce the number of sky-noise photons transmitted to the spectral filter ($\mathrm{\Delta }\lambda$). (b) A schematic illustrating the effect of focusing in the presence of turbulence; the aberrated wave front results in a broadened turbulence-limited spot (TL spot) as compared to the diffraction-limited spot (DL spot).

Figure 2

The site-dependent (a) atmospheric transmission ${\eta }_{\mathrm{trans}}$ and (b) spectral radiance $H_b$ for a receiver pointed at zenith on the winter solstice at 1:00 PM with 50-km visibility.

Figure 3

The solid-angle FOV ${\mathrm{\Omega }}_{\mathrm{FOV}}^{(i)}$ for 1550, 781, and 431 nm (black, red, and blue, respectively), plotted as a function of $r_0$. The solid and dashed curves indicate the DL and TL FOVs, respectively. The vertical lines indicate higher-order AO system performance. The line at 5 cm corresponds to no AO correction, whereas the lines at $r_0^{(\mathrm{CL})}=37$, 50, and 74 cm correspond to full AO with 130-, 200-, and 500-Hz closed-loop bandwidths, respectively.

Figure 4

The (a) DL strategy field-stop transmission efficiency ${\eta }_{\mathrm{FS}}^{(\mathrm{DL})}$ and (b) total system efficiency ${\eta }^{(\mathrm{DL})}$ for 1550, 781, and 431 nm (black, red, and blue, respectively), plotted as a function of $r_0$. The solid and dashed curves indicate the DL and TL strategies, respectively. The vertical line at 5 cm corresponds to no AO correction and the line at 50 cm corresponds to the effective $r_0$ of a $f_c=200\mathrm{-Hz}$ AO system.

Figure 5

(a) The wavelength dependence of the number of background photons for the DL strategy $N_b^{(\mathrm{DL})}$. (b) The $r_0$ dependence of $N_b^{(i)}$ for 1550, 781, and 431 nm (black, red, and blue, respectively), plotted as a function of $r_0$. In (b), the solid and dashed curves indicate the DL and TL strategies, respectively. The vertical line at 5 cm corresponds to no AO correction and the line at 50 cm corresponds to the effective $r_0$ of a $f_c=200\mathrm{-Hz}$ AO system.

Figure 6

The $r_0$ dependence of $E_r^{(i)}$ for 1550, 781, and 431 nm (black, red, and blue, respectively). The solid and dashed curves indicate the DL and TL strategies, respectively. In (a), the vertical line at 5 cm corresponds to no AO correction and the line at 50 cm corresponds to the effective $r_0$ of a $f_c=200\mathrm{-Hz}$ AO system. In (b), we plot the range of $r_0$ representative of no AO correction. The dashed lines are nearly totally obscured by the solid lines.

Figure 7

The (a) $r_0$ and (b) $S$ dependence of the SNR in Eq. (14) for 1550, 781, and 431 nm (black, red, and blue, respectively). The solid and dashed curves indicate the DL and TL strategies, respectively. In (a), the vertical line at 5 cm corresponds to no AO correction and the line at 50 cm corresponds to the effective $r_0$ of a $f_c=200\mathrm{-Hz}$ AO system. In (b), the $\lambda$ dependence of $S$ produces six vertical lines. The grouping near $S=0$ corresponds to no AO and the vertical lines at $S=0.71$, 0.37, and 0.14 correspond to a $f_c=200\mathrm{-Hz}$ AO system operating at 1550, 781, and 431 nm, respectively. The dashed lines are nearly totally obscured by the solid lines.

Figure 8

The (a) $r_0$ and (b) $S$ dependence of the key-bit rate $R_{\mathrm{KB}}^{(i)}$ for 1550, 781, and 431 nm (black, red, and blue, respectively). The solid and dashed curves indicate the DL and TL strategies, respectively. In (a), the vertical line at 5 cm corresponds to no AO correction and the line at 50 cm corresponds to the effective $r_0$ of a $f_c=200\mathrm{-Hz}$ AO system. In (b), the $\lambda$ dependence of $S$ produces three vertical lines indicating AO correction.

Figure 9

The ratio $Q_{\mu }^{(\mathrm{TL})}/Q_{\mu }^{(\mathrm{DL})}$ for 1550, 781, and 431 nm (black, red, and blue, respectively), plotted as a function of $r_0$. The vertical line at 5 cm corresponds to no AO correction and the line at 50 cm corresponds to the effective $r_0$ of a $f_c=200\mathrm{-Hz}$ AO system.

Figure 10

The closed-loop bandwidth $f_c$ dependence of the key-bit rate $R_{\mathrm{KB}}^{(i)}$ for 1550, 781, and 431 nm (black, red, and blue, respectively). The solid and dashed curves indicate the DL and TL strategies, respectively. The vertical line at $f_c=130\mathrm{-Hz}$ indicates the closed-loop bandwidth of the system built for our field experiment [26], whereas $f_c=200\mathrm{Hz}$ and 500 Hz indicate bandwidths assumed in previous numerical simulations [21, 22]. The horizontal lines and color-coded markers indicate the maximum key-bit rate for each wavelength; that is, the key-bit rate for the receiver operating at the diffraction limit.

Figure 11

The (a),(b) atmospheric transmission ${\eta }_{\mathrm{trans}}$ and spectral radiance $H_b$ and (c),(d) the key-bit rate $R_{\mathrm{KB}}^{(i)}$ for the winter solstice with medium visibility (23 km), plotted as a function of $r_0$. In (e) and (f), we choose 1550, 781, and 431 nm (black, red, and blue, respectively) and the solid and dashed curves indicate the DL and TL strategies, respectively. In (e), the vertical line at 5 cm corresponds to no AO correction and the line at 50 cm corresponds to the effective $r_0$ of a $f_c=200\text{−}\mathrm{Hz}$ AO system. In (f), the $\lambda$ dependence of $S$ produces three vertical lines indicating AO correction.

Figure 12

The (a),(b) atmospheric transmission ${\eta }_{\mathrm{trans}}$ and spectral radiance $H_b$ and (c),(d) the key-bit rate $R_{\mathrm{KB}}^{(i)}$ for the winter solstice with low visibility (5 km), plotted as a function of $r_0$. In (e) and (f), we choose 1550, 781, and 431 nm (black, red, and blue, respectively) and the solid and dashed curves indicate the DL and TL strategies, respectively. In (e), the vertical line at 5 cm corresponds to no AO correction and the line at 50 cm corresponds to the effective $r_0$ of a $f_c=200\text{−}\mathrm{Hz}$ AO system. In (f), the $\lambda$ dependence of $S$ produces three vertical lines indicating AO correction.

Figure 13

The (a),(b) atmospheric transmission ${\eta }_{\mathrm{trans}}$ and spectral radiance $H_b$ and (c),(d) the key-bit rate $R_{\mathrm{KB}}^{(i)}$ for the summer solstice with high visibility (50 km), plotted as a function of $r_0$. In this case, we narrow the spectral filter width to $\mathrm{\Delta }\lambda =0.05\text{−}\mathrm{nm}$. In (e) and (f), we choose 1550, 781, and 405 nm (black, red, and blue, respectively) and the solid and dashed curves indicate the DL and TL strategies, respectively. In (e), the vertical line at 5 cm corresponds to no AO correction and the line at 50 cm corresponds to the effective $r_0$ of a $f_c=200\text{−}\mathrm{Hz}$ AO system. In (f), the $\lambda$ dependence of $S$ produces three vertical lines indicating AO correction.

Figure 14

The (a),(b) atmospheric transmission ${\eta }_{\mathrm{trans}}$ and spectral radiance $H_b$ and (c),(d) the key-bit rate $R_{\mathrm{KB}}^{(i)}$ for the summer solstice with medium visibility (23 km), plotted as a function of $r_0$. In this case, we narrow the spectral filter to $\mathrm{\Delta }\lambda =0.05\text{−}\mathrm{nm}$. In (e) and (f), we choose 1550, 781, and 405 nm (black, red, and blue, respectively) and the solid and dashed curves indicate the DL and TL strategies, respectively. In (e), the vertical line at 5 cm corresponds to no AO correction and the line at 50 cm corresponds to the effective $r_0$ of a $f_c=200\text{−}\mathrm{Hz}$ AO system. In (f), the $\lambda$ dependence of $S$ produces three vertical lines indicating AO correction.

Figure 15

The system Strehl ratio for 1550, 781, and 431 nm (black, red, and blue, respectively), plotted as a function of ${\sigma }_{\mathrm{OPD}}$. The vertical lines at 144 nm and 980 nm indicate the ${\sigma }_{\mathrm{OPD}}$ of a closed- and open-loop 200-Hz AO system correcting for an atmosphere characterized by $r_0=5\mathrm{cm}$, respectively. The horizontal lines indicate the closed-loop Strehl ratio achieved at each wavelength.

Figure 16

The linear-angle FOV for 1550, 781, and 431 nm (black, red, and blue, respectively), plotted as a function of ${\sigma }_{\mathrm{OPD}}$ for (a) 0–500 nm and (b) 500–1000 nm. The solid and dashed curves indicate the DL and TL FOVs, respectively. The line at ${\sigma }_{\mathrm{OPD}}=980\mathrm{nm}$ corresponds to no AO correction, whereas the lines at ${\sigma }_{\mathrm{OPD}}=184,144$, and 104 nm correspond to full AO with 130-, 200-, and 500-Hz closed-loop bandwidths, respectively.

Figure 17

The key-bit rate $R_{\mathrm{KB}}^{(\mathrm{TL})}$ for the winter solstice with high visibility and $r_0=50\mathrm{cm}$. (a),(b) The key-bit rate for a 1-nm (0.05-nm) filter in gray (black). (c),(d) Enlargements that reveal a range of optimal wavelengths for a 1-nm and 0.05-nm filter, respectively.