2.1

Secure Key Rate Calculation

Here, we consider the BBM92 protocol,⁶ which is the most commonly considered protocol for entanglement-based QKD. For secure key rate calculations, we adapted the model from Ref. 7. Consider a continuous-wave pumped EPS that generates pairs of polarization-entangled photons with the rate G. Note, that this is the intrinsic pair generation rate, meaning that no loss – not even of the source optics – is yet considered. One photon of each generated pair is sent towards Alice and the other one towards Bob via their respective channels, which are characterized by their total channel efficiencies η_A and η_B.

Secure key generation is based on correlation of simultaneous detection events – coincidences – at Alice and Bob, and this correlation is provided by the quantum entanglement of the signal photon pairs. Coincidences are defined as events when a single-photon detection at one site happens within of a detection time at the other site, where τ is called coincidence window. Due to channel losses, many pairs do not make it to their respective detectors, in turn reducing the rate of desirable coincidences, called true coincidences. But what is even worse, is that some photons lose their partner photons while they alone make it to a detector. There is a chance that two such partner-less photons – one at each site – still produce simultaneous clicks, which are called accidental coincidences.

In order to estimate the secure key rate, the relationship between true and accidental coincidences need to be established. True coincidence rate is simply the pair generation rate reduced by the combined total loss of both channels,

Accidental coincidences, on the other hand, are determined by the single photons detected locally at each site. For high-loss scenarios, which certainly is the case for space links, accidental coincidence rate can be approximated by⁷

where and are the local single-photon detection rates at Alice and Bob, respectively. Since not only the signal but also noise contributes to the detected single-photon rates, we break them down as

where is the dark count rate, is the background noise rate, and is the rate of true single photon detection events that originate from the EPS, given by

We introduced i ∈ [A, B], which indicates Alice or Bob. Note, that these are total rates, therefore is a sum of dark count rates of all 4 detectors used at site i.

We may now combine true and accidental coincidences into the total detected coincidence rate,

where we introduced coincidence detection efficiency η_coin that only applies to coincidence and not to single-photon detection. It is given by

where is the total detection timing jitter at the ground station i. Strictly speaking, should in addition to detector jitters include also effects of dispersion and photon coherence times, however, these are often negligible compared to detector jitters. Some of the true coincidences may nevertheless cause erroneous (uncorrelated) outcomes. This is characterized by the baseline error rate e₀, which accounts for the source imperfections and imperfect alignment of polarization frame of reference between Alice and Bob. We calculate the rate of erroneous coincidences as

Note, that not all the accidental coincidences contribute to errors – some may completely by chance produce a correlated outcome, which also contributes to the secret key, hence the factor of 1/2. The quantum bit error rate (QBER) is then given by

Assuming that one of the two mutually unbiased measurement bases is selected with 50:50 probability, secure key rate is finally given by

where H₂ is the binary entropy function defined as

and f_EC is the error correction efficiency factor. In general, f_EC depends on E and the chosen error correction algorithm, but for simulations we may set it to a conservative fixed value, e.g. f_EC = 1.22.⁸ Note, that this is the asymptotic secure key rate, which assumes infinitely large blocks of raw key, before post-processing is applied to distil the final secret key. Finite key analysis is an important aspect of practical QKD – even more so for space links, where the satellite is available for only short periods of time – however, it is beyond the scope of this work.

2.2

Link Loss Model

As shown in Sec. 2.1, we need to know the total loss for each of the two channels to calculate the key rate. We now go step by step through the entire channel and identify each individual loss contribution. We start by defining the total efficiency of the channel i as

Here, is the source efficiency, which accounts for all the losses for the i-channel photons, before they leave the source module. All the remaining losses of the transmitter optics are combined in . The downlink efficiency is given by⁹

where the individual factors represent transmitter gain, pointing loss, free-space loss, atmospheric loss (absorption and scattering), beam spreading loss (due to atmospheric turbulence), and receiver gain, respectively. These factors can be calculated using well established formulas, which are listed in Appendix A, and depend on signal wavelength λⁱ, transmitted beam waist , transmitter pointing jitter , downlink path length zⁱ, receiver primary aperture diameter , atmospheric transmittance at zenith , and satellite altitude angle αⁱ, as seen from the ground station i.

The model for fiber coupling is fundamentally different for a single-mode (SM) or a multi-mode (MM) fibre. In literature, the Strehl ratio has been suggested as an estimator for SM fibre coupling efficiency.¹⁰ In the presence of atmospheric turbulence, the Strehl ratio can be approximated as¹¹

where r₀ is the Fried parameter that characterizes atmospheric turbulence. Factor γ accounts for the level of AO correction and is given by

Note, that only perfect correction is considered here, meaning that the tip-tilt correction completely removes the tip and the tilt contribution to the wavefront error, while full AO correction removes all the aberrations, resulting in a diffraction limited spot.

But even a perfectly corrected wavefront, having a flat phase, would not result in 100% coupling efficiency due to the unavoidable mismatch of the amplitude profiles. The mode of the SM fibre can be well approximated with a Gaussian function.¹² Since the aperture plane and the focal plane are related via a Fourier transform, perfect coupling could only be achieved with a matching Gaussian beam in the aperture of the coupling optics. The long-term average of the incoming light, however, is expected to have a flat-top amplitude profile, allowing at most 81% average coupling efficiency.¹³ In the case of MM coupling, we calculate the corresponding efficiency by integrating the focal plane intensity profile, as expected in the presence of atmospheric turbulence, over the area of the MM core (for derivation, see Appendix B). Assuming the coupling optics is chosen for the optimal coupling efficiency, we then estimate the coupling efficiency as

where

and the multi-mode fibre is characterized by its fibre core diameter d_MM and numerical aperture NA_MM.

Finally, is the detection efficiency of the detectors. Note, that some of the parameters above are very likely to be the same for Alice and for Bob channel, but for the sake of generality we made no such assumptions so far.

2.3

Fibre Coupling of Background Noise

We model the background noise as an incoherent Lambertian source with radiance B. The amount of noise power that eventually couples into a MM fibre is given by¹⁴

with

where M represents the number of guided modes. According to Ref. 14, Eq. (17) can be used even in the case of a SM fibre, where we simply take M = 2 (one fundamental mode for each of the two orthogonal polarizations).

The actual sky brightness is usually given in terms of spectral radiance B_λ, from which we calculate radiance as B = B_λΔλ, where Δλ is the bandwidth of the spectral filter employed at the receiver. With the energy of a single photon being hc/λ, we can now express the rate of detected background noise photons as

where h is the Planck constant and c is the speed of light in vacuum.

2.4

Signal Wavelength Considerations

When it comes to the choice of the signal wavelength for potential daylight QKD, there are several important aspects to consider. First of all, we need a source of high-fidelity entangled photons with high generation rates, and these are available only at certain wavelengths. Secondly, single-photon detectors with high detection efficiencies are required at the corresponding wavelengths. It is also crucial that that the Earth’s atmosphere has a high transmittance at these wavelengths. Based on these considerations, two wavelength ranges of interest have been established in the satellite-based QKD community: one around 800 nm and the other one at about 1550 nm.

The most obvious advantage of the roughly half smaller ∼800 nm wavelengths is considerably smaller diffraction, resulting in a much more concentrated photon flux at the OGS, improving the product approximately by a factor of 4, see Eqs. (20), (22), and (29). In turn, however, smaller transmitted beam cone results in higher pointing loss, Eq. (21), making the overall effect on the link loss much less trivial. Furthermore, atmosphere transmittance is about 10% lower at ~800 nm than it is at 1550 nm.⁴ Beam spreading loss due to atmospheric turbulence is in the case of downlinks substantially smaller compared to other losses,¹¹ therefore it does not need to be considered here. Atmospheric turbulence does, however, distort the wavefront of lower wavelengths more,¹¹ which would have an impact on single-mode fibre coupling or spatial filtering in general, unless corrected for.

All of the above is applicable to satellite-based QKD in general, but there are further considerations that are particularly relevant for daylight QKD. Spectral radiance of the daytime sky is generally about an order of magnitude higher at ~800 nm than at 1550 nm.⁹ By mimicking the actual tracking of a satellite across the sky with their telescope, Liao et al.³ measured the single photon counts at 1550 nm to be a factor of 22.5 larger than at 850 nm. Based solely on the argument of smaller noise, plus the argument of better compatibility with established fibre-optical communication infrastructure, they chose 1550 nm for their groundbreaking terrestrial daylight QKD experiments. Gruneisen et al.,⁴ on the other hand, took several more aspects including wavelength-dependent link loss into consideration. They conclude that at 780 nm signal-to-noise ratio would be about 1.8 higher and overall signal level about 3.6 times larger than at 1550 nm, finally choosing 780 nm for their terrestrial daylight QKD experiments.

Although we can narrow down the choice of wavelength to only two ranges, we can see that the final choice between the two is far from trivial. In the near-term, the community will therefore very likely to keep investigating and developing the relevant technologies in both domains.

2.5

Detection Technology

Another crucial consideration that can have an enormous impact on the achievable secure key rates is the choice of single-photon detection technology. Performance of single-photon detectors is typically characterized by their detection efficiency, dark count rate, and detection time jitter. Dark count rates, although critical for some other applications, are usually negligible compared to orders of magnitude higher background noise rates from the day-lit sky. Detection efficiency directly influences the rate of detected coincidences C_det in Eq. (9), therefore it is trivial to recognize its importance in achieving high secure key rates. The importance of low time jitters, however, is even more important but much less trivial to see. The entangled pair sources most commonly used for QKD are based on spontaneous parametric down-conversion (SPDC), which convert pump photons into pairs of entangled photons. To achieve a higher pair generation rate G, one may therefore simply increase the pump power. But even though higher G generally increase detected coincidences C_det and in turn secure key rate K, as can be seen from Eq. (9), there exists an optimal pair generation rate, which is dictated by the detector time jitter.

In practice, accidental coincidences can often be approximated by , ⁷ meaning that they scale quadratically with the pair generation rate G. True coincidences, which primarily contribute to the generation of the secure key, scale only linearly with G, as seen in Eq. (1). For this reason, increasing G would lead to higher QBER, eventually reaching the threshold value of about 10%, which is imposed by H₂ in Eq. (9), at which generation of positive secure key rate becomes impossible. What can also be appreciated from the approximation above is that we may decrease the rate of accidental coincidence by decreasing the coincidence window τ. This would allow for higher G before reaching the error threshold. Shorter τ also leads to better temporal filtering of the background noise, as discussed above, and is therefore particularly desired for daylight QKD. Through Eq. (6), however, time jitters of the detectors impose a lower bound on the choice of τ, since reducing τ below time jitter substantially reduces the coincidence detection efficiency. In conclusion, lower detector jitter allows for a shorter coincidence window, which in turn allows for a higher pair generation rate and reduces the background noise rate, finally leading to a higher secure key rate.

Here we focus on the two most popular yet fundamentally different families of devices: single-photon avalanche detectors (SPADs) and superconducting nanowire single-photon detectos (SNSPDs). Table 1 shows parameters for typical commercially available SPADs and SNSPDs for ~800 nm and 1550 nm, taken from the manufacturer data sheets. The efficiency of Si-based SPADs for ~800 nm has tremendously improved over the past years and is now almost reaching similar levels as SNSPDs. InGaS-based SPADs for 1550 nm, on the other hand, still suffer from very low detection efficiencies and high dark count rates. An order of magnitude lower timing jitter of SNSPDs is, however, expected to allow for considerably higher key rates even at ~800 nm, based on the arguments given above. While SNSPD is without a doubt a far superior technology in terms of performance, it is considerably more bulky and an order of magnitude more expensive, therefore making the more accessible SPAD technology still an option worth considering.

Table 1.

Typical parameters for commercially available single-photon detectors.

3.1

Satellite Orbit and Ground Station Selection

For all the simulations presented here, we chose the orbit of the Chinese Quantum Science Satellite – also known as Micius – which is a Sun-synchronous low Earth orbit. Micius was the first satellite to have ever demonstrated dual-downlink QKD with entangled photons.¹⁵ The orbit parameters were retrieved from CelesTrak online database^† in the form of a two-line element (TLE) set, shown in Tab. 2. For explanation of the TLE data format, please refer to CelesTrak documentation. We also chose transmitter aperture diameter of D_Tx = 180 mm (we assume w₀ = D_Tx/3) and pointing jitter of σ_p = 1 µrad according to Micius.¹⁶ For transmitter efficiency, we take the value of 75%, which roughly corresponds to 6 silver mirror surfaces.

Table 2.

Two-line element set of Micius used for simulations.

As the two ground station sites, we choose the European cities of Berlin and Munich, and their coordinates are shown in Tab. 3. The minimal surface distance (great-circle distance) between the two cities is about 500 km and an optical fibre connecting them would be even longer. QKD over such distances is infeasible due to exponential loss along optical fibres,² therefore the only alternative is a satellite link, making these two sites a suitable choice for our feasibility study. We assume both sites are equipped with a D_Rx = 800 mm telescope, having a central obstruction diameter of 328 mm (e.g. commercially available ASA AZ800). As the optical efficiency of the rest of the receiver system, we take the value of 60%.

Table 3.

Coordinates of the two ground sites used for simulations.

QKD can only be performed when the satellite can be seen from both ground sites simultaneously, i.e. its altitude angle at both sites needs to be positive. We further limit ourselves to satellite altitude angles that are larger than 30^°, since for smaller altitude angles the atmospheric models employed here are not valid anymore.⁹ Based on these considerations, we define a satellite pass as the time when the satellite altitude angle is above 30° at both ground sites simultaneously.

We chose all such passes in the year 2022 as our sample of possible satellite-Alice-Bob geometry dynamics for subsequent key calculations. Note, that about half of these passes actually happen around midnight and the other half around noon (ground station time), which is due to the fact that Micius is in a Sun-synchronous orbit. Nevertheless, the distribution of geometry dynamics is essentially the same for day and night passes. Although we are interested in investigating daylight QKD here, we use all these passes for the QKD simulations to have a larger and a more representative sample of link geometry dynamics.

Fig. 2 shows the distribution of this sample by the pass duration. In the year 2022, there is a total of 571 passes of Micius over Berlin and Munich simultaneously, with 271 happening during the day. The longest pass lasts 141 s and the mean pass duration is 102 s. Note, that these times would increase, should we allow lower satellite altitude angles. In Fig. 3, we show the link geometry dynamics for an example pass of which the duration roughly matches the median pass duration of 113 s.

Figure 2.

Distribution of simultaneous passes of Micius over Berlin and Munich in 2022 by their duration.

Figure 3.

Typical dynamics of link geometry parameters for a Micius pass over Berlin and Munich (14/07/2022).

3.2

Source and Detection Parameters

As discussed in Sec. 2.4, the choice between ~800 nm and 1550 nm as the signal wavelength is far from trivial and there remains a high interest in both within the community. Similarly, both SPAD and SNSPD detector technologies are worth considering for the reasons given in Sec. 2.5. We therefore perform simulations for all the four combinations of these wavelengths and detector technologies, using parameters from Tab. 1. As the coincidence window, we chose τ = 5σ_det, which corresponds to η_coin ≈ 92%, according to Eq. (6).

While the optimal source pair generation rate heavily depends on the detector parameters, we do not expect it to change considerably during a pass and between different passes. Moreover, optimizing it on the fly would be extremely difficult in practice. For this reason, we optimize the pair generation rate for a single satellite position for each of the four wavelength-detector combinations. The background noise was set to zero for the optimization. The resulting pair generation rates are shown in Tab. 4 and were used for all the simulations with the corresponding wavelength-detector combination.

Table 4.

Entangled pair source parameters used for simulations.

An example ~800 nm source from Ref. 17 can generate 6.25 Mpairs/s/mW with η_EPS = 80%. Note, that these values are calculated from the reported measured values, by taking into account the reported detector efficiency. To achieve the desired optimal pair generation rates, we would therefore need to pump such a source with less than 7 mW, if SPAD detectors are used, or with about 182 mW, if SNSPD detectors are used. At 1550 nm, an example source from Ref. 18 can generate 18.9 Mpairs/s/mW with η_EPS = 47.8%. We would therefore need to pump it with less than 4 mW in the case of SPADs or with about 36 mW in the case of SNSPDs. Pumping the source on board of a satellite with such powers should be feasible in practice.

For the spectral filter bandwidth, we took 3 nm, as this should be broad enough to work with the bandwidth of the example sources in Refs. 17 and 18, and such filters are readily available on the market. We set the baseline error rate to e₀ = 0.02, which is in line with the above mentioned sources and allows for some polarization basis alignment error.

3.3

Sky Brightness and Atmosphere Parameters

Sky spectral radiance B_λ depends on the relative geometry of the Sun, satellite and the corresponding ground station location. In general, it therefore differs for Alice and Bob, and varies as the the receiver tracks the satellite across the sky. Gruneisen et al.¹⁹ simulated B_λ for two typical LEO satellite passes. In the first case, B_λ varied between 15 and 23 Wm⁻²sr⁻¹µm⁻¹ throughout the pass, while in the second case, it varied roughly between 3 and 16 Wm⁻²sr⁻¹µm⁻¹. While including a dynamical simulation of B_λ in the present model would certainly be interesting, it is beyond the scope of this work. In our simulations, we kept a fixed B_λ for the entire path, and the values for Alice and Bob were the same.

We performed simulations for 4 different sky brightness scenarios. The first scenario, in which we set the back-ground noise to zero, served as a benchmark. Then we defined three daylight scenarios: dark day, bright day, very bright day. Under cloud-free conditions, sky spectral radiance from less than 1 to more than 100 Wm⁻²sr⁻¹µm⁻¹ can be expected during daytime.¹⁹ A MODTRAN simulation with the Sun being at α = 45° and the ob-server being at sea level and looking towards α = 50° estimates sky spectral radiance to be in the order of 100 Wm⁻²sr⁻¹µm⁻¹ at ~800 nm and an order of magnitude smaller at 1550 nm.⁹ Based on these arguments, we define the sky spectral brightness for the three daylight scenarios as shown in Tab. 5.

Table 5.

Sky spectral radiance used for simulations.

For atmospheric transmittance at zenith η_zen, we used the MODTRAN online tool^‡, arriving at values of 0.78 for 800 nm and 0.91 for 1550 nm. For calculating Fried parameters at each satellite position, the standard H-V₅/7¹¹ model was used.

3.4

Fibre Coupling and the Level of Wavefront Correction

It has often been argued in the community that full AO, together with some form of strong spatial filtering such as single-mode-fibre coupling, is the only way to go for daylight QKD. Full AO, however, incurs high additional costs and complexity to the receiver system. For this reason, we investigated whether a much simpler and affordable tip-tilt-only correction would, at least in principle, lead to acceptable amounts of secure key per satellite pass. Moreover, since tip-tilt correction usually suffices for efficient multi-mode fibre coupling, we investigated also coupling a MM fibre. We assumed a typical MM fibre, with its core and numerical aperture of 50 µm and 0.22, respectively. With the present model and for the parameters above, we observed no considerable improvement of MM fibre coupling efficiency by adding higher mode AO correction to the basic tip-tilt correction. And if full AO correction is available, going for MM fibre coupling does not seem to make much sense. For these reasons, we excluded simulations of MM fibre coupling with full AO.