In order to present a feasible implementation, it must be demonstrated that enough data can be collected within the lifetime of the satellite to confirm, or refute, the hypothesised signal to a pre-specified confidence. In this particular proposal, there are four key issues that must be addressed before feasibility can be demonstrated: stability of the interferometer, loss from the source to the detectors, noise from extraneous and atmospheric photons, and mitigation of any other degrees of freedom besides the Shapiro delay that might introduce distinguishability (in particular, photon dispersion). We first outline a feasible setup in Section 3.1, before presenting an analysis of these key issues.

### 3.1 Optical setup and components

A detailed illustration of the interferometer setup, with all the key optical components, can be seen in Figure 4.

The choice of operational parameters is ultimately a compromise between what is feasible to implement and what is necessary to recover the desired data. As such, there is no single optimal set of components; relaxing the constraints on their performance may be acceptable if they become more durable and the satellite can operate for longer and take more data, for example (we aim to mitigate large pulse dispersion, noise and random fluctuations in signal by integrating counts over long time periods; hence, durability of components is critical). Therefore, the following choices of components and parameters are not intended to be a fixed specification for the payload, but to demonstrate one setup that could feasibly produce meaningful results.

We highlight the following optical elements, whose operational parameters are crucial to the success of the experiment.

*Single photon source*: An off-the-shelf, mode-locked, pulsed laser with operational wavelength of 1550 nm and 1 GHz repetition rate is attenuated to a mean photon number of 0.1 photons per pulse. While an attenuated laser is a straightforward option, true, heralded single photon sources have been space-qualified and operated on the QUESS satellite [19]. However, we are not aware of a true, heralded source of single photons that meets the requirements of repetition rate and pulse width presented here. The reason for this particular level of attenuation is the tradeoff between lowering multiphoton emission and maintaining a high number of total counts - this attenuation reduces the probability of multiphoton emission to 5% and results in a single photon rate of 100 MHz, which matches the resolution of the single photon detectors. The chosen wavelength guarantees high atmospheric transmission and the utilisation of well-developed telecommunication technology. The functional dependence of the fringe contrast on the Shapiro delay Δ*τ* and the pulse width \(\sqrt{\sigma}\) (see [10], Eq. 13) demands the utilisation of ultra-short (\(<{1}~ps\)) single photon pulses to measure a noticeable effect (we note single photon sources with a width as short as 65 fs have been demonstrated in [20]). We take \(\sqrt {\sigma} = 150~\mbox{fs}\).

*Classical reference laser*: A 1300 nm multi-purpose laser, operating in continuous wave mode, is led alongside the single photons through the interferometer. Its purpose is to provide reference data to estimate phase fluctuations and systematic errors. For precise corrections, an operational wavelength as close as possible to the single photon source is required, but overlap with the bandwidth of the single photon pulses has to be avoided. The chosen wavelength is a reasonable compromise. Frequency stability of both lasers is paramount; see Section 3.2. Additionally, a fraction of the incident laser power is separated at the ground station and used for wavefront reconstruction and compensation of polarisation changes due to satellite movement.

*Fibres*: We expect the delay fibres must be stabilised to relative length changes of 10^{−10} for observing the predicted interference effects, which includes a thermal stabilisation of \({\pm} 10^{-5}~\mbox{K}\). Moreover, the fibre length must change dynamically to recover full contrast of the interference fringes. Active length corrections are carried out by a piezo fibre-stretcher. Fibres of length \(l = 60~\mbox{km}\) and refractive index \(n = 1.5\) (glass) are assumed.

*Transmission telescope*: The onboard emitting telescope is used to focus the beam from the two sources. The aperture needs to be large enough to emit a strong signal but is limited by size and weight; a reasonable choice is a 60 cm aperture, 1 m long, 6 *μ*rad field of view, Cassegrain reflector telescope. Material choices, such as Beryllium mirrors, would also minimise weight.

*Single photon detectors*: Since the goal of the detectors is to collect single photons that have travelled large distances through free space, it is extremely beneficial to minimise dark counts and maximise efficiency. We consider single-nanowire single photon detectors (SNSPDs), to benefit from the superior dark count over conventional avalanche photodiodes. While these detectors aren’t as commonplace, the technology has already been established in a similar setting in the LLCD mission (NASA) [21].

*Timing*: Emission of the single photon pulses are tagged with timestamps by a Rb-clock (chosen primarily for its small size, weight and commercial availability). Synchronisation with an identical clock on the ground will be used to exclude background noise at the data post-processing stage and to track the path of the satellite. Precision timing is also necessary to modulate the action of time-gate filters in front of the detectors.

### 3.2 Random fluctuation and stability

As the effect we wish to measure is a minute change in optical path length in the interferometer due to the Shapiro delay, exposing the interferometer to other sources of instability can swamp the desired signal. We find that optical path and phase changes in the atmosphere due to temperature fluctuations are negligible compared to fluctuations in the length of fibre; given the thermal properties of fused silica we calculate necessary temperature stabilisation of the fibres of less than \(10^{-5}~\mbox{K}\) to ensure a relative path length stability of 10^{−10}. Passive insulation and active heat distribution can mitigate a large fraction of the thermal instability, but both the satellite and the ground station include a feedback loop of a frequency stabilised reference laser in combination with a piezo fibre stretcher, to allow for fibre noise reduction. Also, some thermal fluctuation can be erased in post-processing by referring to data from the reference laser.

In addition, continued operation of the experiment requires protection of the reference laser from frequency fluctuations and drift. Given the magnitude of the path length change, we estimate a necessary relative frequency stability of the reference laser less than 10^{−11}. Long term accuracy of the reference laser within required precision is most feasible via frequency comb stabilisation [22]. This method will increase further complications and costs, and could be circumvented by improvements in the area of stabilisation by using atomic absorption lines, which today are close to reaching comparable relative accuracies [23].

### 3.3 Systematic transmission errors and dispersion

We highlight three systematic errors present in the experiment, mitigation of which are critical: dispersion, both in the fibre and the atmosphere; Doppler shift due to the velocity of the satellite; and changes in optical path length due to the ellipticity of the orbit.

Dispersion is prevalent both in the optical fibre and in the atmosphere. We estimate a requirement for fibre dispersion of \({<}5~\mbox{fs}/\mbox{km}/\mbox{nm}\), ensuring a broadening of the pulse width in the fibres of \({<}0.5\%\) per km; this is a stringent enough requirement to fix dispersion as the primary hurdle facing this experiment. The current state-of-the-art for dispersion-limited fibre is a factor of ten worse than this: 50 fs/km/nm [24]. Current technologies are capable of dispersion compensation in optical fibres of 0.5 ps/km/nm, which has to be improved by about a factor of ten to make the scientific requirement feasible. However, the utilisation of telecommunication fibre wavelengths ensure ongoing research in that scientific area. For example dispersion-free transmission of 610 fs laser pulses over a 160 km fibre has been demonstrated, but dispersion-free transmission techniques typically suffer from large losses [25, 26]. On the other hand active compensation of *atmospheric* chromatic dispersion of a 250 fs pulse propagating horizontally through the atmosphere over 200 km has been demonstrated to a uncertainty of \({\pm}10~\mbox{fs}\) [27]. This indicates that control of atmospheric dispersion of comparably short pulses in our proposal is well within reach, especially considering the pulses considered here are travelling vertically, on a much shorter trajectory through the atmosphere than in [27].

Doppler shift, conversely, presents much less of a problem. This is straightforward to calculate based on the velocity of the satellite relative to the Earth. For the worst possible case (at apogee), the Doppler redshift *ν* is calculated using the formula:

$$ \Delta\nu_{\mathrm{Doppler}} =f_{p}\times\frac{c}{c-v_{0}}, $$

(3)

where *c* is the speed of light, \(f_{p}\) is the frequency of the single photon source and \(v_{0}\) is the maximum velocity of the satellite at the lowest sampling point. This gives a shift in photon frequency of 2.48 GHz. Considering the bandwidth size and that the photon frequency exceeds 193 THz, this is considered negligible.

Finally, the radial motion of the satellite causes constant variation in optical path length. To keep the path lengths of the interferometer equal this variation must be continuously compensated for. The time spent in the fibre loop by the part of the photon state in the upper arm, before it is emitted from the satellite, is given by:

$$ \Delta t = \frac{\ell}{c n} \approx10^{-4}~\mbox{s}, $$

(4)

where \(\ell= 60~\mbox{km}\) is the fibre length and \(n=1.5\) is the refractive index. Multiplying this quantity by the radial velocity gives the change in path length between parts of the superposition due to the motion of the satellite; this change in path is plotted in Figure 5. However, this effect can be precisely calculated in advance and so either active compensation with optical components or passive compensation at the post-processing stage can be built into the experiment beforehand.

### 3.4 Loss

Given the repetition rate of the laser and strength of attenuation, we can calculate the expected rate of signal photons registering at the detectors. The analogous discussion on noise photons reaching the detectors is deferred to Section 3.5. There are three major sources of signal attenuation: divergence of the beam from the transmitting telescope to the ground; loss due to atmospheric irradiance and interference; and loss within the fibres.

First, we examine the effects of atmospheric transmission. A wavelength of 1550 nm is deemed suitable as it is standardised for telecoms use, maximises transmittivity in the fibre and is not readily absorbed by the atmosphere. The high frequency of the emitted photon allows us to ignore ionospheric effects on polarisation and is easily distinguishable from auroral activity over the ground stations [28].

Assuming homogeneity, using a variant of the Beer-Lambert Law we can estimate the probability of a single photon passing through the atmosphere as:

$$ \mathrm{P}(\mathrm{transmission}) = \exp \biggl( \frac{-\tau_{0}}{\eta_{0}} \biggr), $$

(5)

where \(\tau_{0}\) is the optical depth of the atmosphere, and \(\eta_{0}\) is the angle of incidence of the beam. The optical depth varies over time and depends on myriad dynamic factors such as aerosol content, atmospheric mixing and Raman and Rayleigh scattering. Modelling these effects is essential, so one might consider optical depth readings from MODIS, MISR or future Sentinel satellites. A numerical weather prediction model might then be produced detailing daily optical depth over ground stations. An alternative might be to model turbulence transfer functions as in [29]. As shown in Figure 6, we can allow for an optical depth of 0.5 before the satellite signal is severely attenuated. Combining these factors produces an estimate for the atmospheric transmission loss to be \({\simeq}{-}2.3~\mbox{dB}\).

As for beam divergence, a downlink direction is chosen as accentuated divergence would then occur only in the last 12 km of travel. This downlink beam divergence can be computed for a photon of wavelength *λ* by:

$$ \theta_{\mathrm{div}} = \frac{\lambda}{\pi w_{0}} = 1.6~\mu\mbox{rad}, $$

(6)

where \(w_{0} = 30~\mbox{cm}\) corresponds to the radius of the transmitting telescope on the satellite. Consequently, the beam diameter on the ground at maximum and minimum altitude is given by \(D_{\mathrm{ground}} \simeq 2h\theta\), which implies that \(58~\mbox{m}\leq D_{\mathrm{ground}}\leq105~\mbox{m}\), taking bounding values for the altitude as \(18\text{,}000~\mbox{m} \leq h \leq32\text{,}000~\mbox{m}\).

Atmospheric turbulence must also be considered. As the atmospheric parameters used for post-processing are extracted from data from the reference laser, this turbulence must not radically change in the time between measurement of the reference laser and of the single photon source. Assuming that the gap between reference and single photon measurement is half the repetition rate of the laser, this delay comes to 5 ns. Assuming a wind speed of \(10~m~s^{-1}\), an air parcel would move by about 50 nm in some direction between the single photon measurement and reference laser measurement. This is considered negligible compared to other atmospheric effects.

Atmospheric transmission loss, turbulence and beam divergence can be compiled into a ‘link budget’ that calculates the full transmission loss from satellite to ground station. We can modify and simplify the Friis Transmission Equation to give the following link budget equation [30]:

$$ \operatorname{Loss}(\mathrm{dB}) =10 \log \biggl[ \biggl(\frac{\pi D_{T} D_{R}}{4 \lambda h} \biggr)^{2} L_{p} L_{t} \biggr], $$

(7)

where here \(D_{T}\) is the transmitter diameter, \(D_{R}\) is the receiver diameter, *λ* is the wavelength of the single photon source, *h* is the satellite altitude, \(L_{p}\) is the pointing loss (taken to be 0.63 [31]), and \(L_{t}\) the atmospheric transmission loss as calculated above.

Assuming an apogee of 32,000 km and a lowest altitude bin of 18,000 km, a ground receiver diameter of 3 m, a satellite transmitter diameter of 0.6 m, and superconducting detector efficiencies of about 90% [32]; we calculate the baseline signal gain from the link budget to be \(-30~\mbox{dB}\) at perigee, and \(-35~\mbox{dB}\) at apogee.

Further factoring in losses from the optical fibre of \(-15.5~\mbox{dB}\) and fibre blackening from radiation of \(-1.7~\mbox{dB}\), the link budget achieves a total attenuation of \(-52.2~\mbox{dB}\) in the worst case, towards the end of mission lifetime. Of course, to further reduce signal attenuation, one could increase the aperture diameters of the receiver and transmitter. However, this causes a loss in manoeuvrability and a significant cost increase for rapidly decreasing returns.

### 3.5 Noise

A feasibility case must also ensure a signal-to-noise ratio (SNR) great enough to produce enough meaningful data for statistical analysis. It must be stressed that the signal received on Earth is fixed by the link budget above, however since we are trying to detect a single photon from a plethora of solar and planetary photon noise, optimising the SNR is crucial.

The effects of noise on space to ground quantum channels has previously been explored [33]. The noise power received by the ground telescope (\(P_{\mathrm{b}}\)) can be expressed as

$$ P_{\mathrm{b}} = \frac{1}{4} H_{\mathrm{b}} \Omega_{\mathrm{fov}} \pi D_{R}^{2} \Delta\nu\Delta t_{d} , $$

(8)

where \(H_{\mathrm{b}}\) is the brightness of the sky in units of \(\mbox{W}~\mbox{m}^{-2}~\mbox{sr}^{-1}~{\mu}\mbox{m}^{-1}\), \(\Omega_{\mathrm{fov}}\) is the field of view, Δ*ν* is the bandwidth and \(\Delta t_{d}\) is the detection time.

Typical sky brightness for quantum cryptography applications is discussed in [33], using data from [34]. However, the data therein must be modified in light of the experiment proposed here. Firstly, the data presented in [33, 34] is for a frequency band just below 1550 nm, which is subject to much less noise than at 1550 nm itself. Conversely, the primary source of noise photons at 1550 nm is hydroxyl airglow, the strength of which is strongly dependent on ambient temperature. The data from [34] assumes a receiving station in the tropics at Mauna Kea, whereas we propose an arctic station at Svalbard. Utilising instead polar sky brightness data from [35], we take a sky brightness of \(2\times10^{-5}~\mbox{W}~\mbox{m}^{-2}~\mbox{sr}^{-1}~{\mu}\mbox{m}^{-1}\) at our operational frequency. The received signal band is filtered to a bandwidth of 15 nm, corresponding to twice the bandwidth of the single photon pulses (twice the full width at half maximum of the Lorentzian pulse). The field of view of the receiving telescope is assumed to be \(10~{\mu}\mbox{rad}\), but the effective field of view is further reduced by a factor of 10 with a variable iris diaphragm [36]. Noise power is further reduced with a 50 ps time gate filter leading to an available detection time of 50 ms per second. Reflections from the satellite or its black body radiation do not significantly contribute to the background noise [33]. Besides received background photons, total noise power also depends on the dark count rate of the single photon detectors. Superconducting detectors have negligible intrinsic dark count rate but as a worst case estimate the detector system dark count rate is assumed to be 1 kHz (although a large fraction of these counts are neglected due to time gating) [32]. The rate of detected signal photons is calculated as 590 Hz (using the repetition rate of the laser, attenuation and loss). Combining this figure with the expected noise from ambient photons and system dark count gives an \(\mathrm{SNR} \approx9.0 = 9.6~\mbox{dB}\).