Edinburgh Research Explorer Towards combined quantum bit detection and spatial tracking using an arrayed single-photon sensor

: Experimental quantum key distribution through free-space channels requires accurate 13 pointing-and-tracking to co-align telescopes for efficient transmission. The hardware requirements for the sender and receiver could be drastically reduced by combining the detection 15 of quantum bits and spatial tracking signal using two-dimensional single-photon detector arrays. Here, we apply a two-dimensional CMOS single-photon avalanche diode detector array to 17 measure and monitor the single-photon level interference of a free-space time-bin receiver 18 interferometer while simultaneously tracking the spatial position of the single-photon level signal. 19 We verify an angular field-of-view of 1.28°, and demonstrate a post-processing technique to 20 reduce background noise. The experimental results show a promising future for two-dimensional 21 single-photon detectors in low-light level free-space communications, such as quantum 22 communications. 23


26
The exchange of quantum bits via optical fiber or free-space links offers the potential of 27 unconditionally verifiably secure quantum key distribution (QKD) for sharing encryption keys 28 between two [1,2]

30
State-of-the-art QKD using point-to-point fiber links has been demonstrated over distances of 31 several hundreds of kilometers [9][10][11][12][13] in laboratory settings, and over metropolitan distances in 32 deployed dark-fiber networks [14,15]. Optical loss in fibers leads to an exponential decay in secret 33 key rates. Expanding terrestrial fiber QKD beyond point-to-point would therefore require trusted 34 quantum repeater nodes [16,17]. Since quantum repeaters are still far from a mature technology, 35 the fastest route to global quantum networks will be to connect separate metropolitan networks 36 via long distance free-space channels and trusted quantum satellite nodes [18].

37
Line-of-sight free-space QKD has been developed in parallel to fiber-based systems, with 38 initially static terrestrial links over increasing distances [19,20] and development towards day-39 time operation [21,22]. In addition, demonstrations and system verification tests have been carried 40 out with mobile platforms, for example ground vehicles [23], aerial platforms [24][25][26], and a range 41 of in-orbit satellites [27][28][29][30][31][32][33][34]. Due to the unguided and turbulent nature of free-space channels, 42 optical beacons and spatial position sensors are required to actively co-align the transmitter and 43 receiver telescopes during communications, in order to minimize losses in the optical link [35].
Here we demonstrate the feasibility of simultaneous detection of the temporal information 45 encoded in single-photon per clock-cycle level optical pulses as well as their spatial position for 46 telescope pointing-and-tracking. We implement a free-space time-bin receiver interferometer with an optical relay to address wave-front distortion [36] and employ a two-dimensional (2D) silicon-48 complementary metal-oxide-semiconductor (CMOS) single-photon avalanche diode (SPAD) 49 array with picosecond time-of-arrival resolution for signal detection. Combined signal detection 50 with a single device reduces system complexity and relaxes the size, weight, and power (SWAP) 51 demands on single-photon level receivers, such as those used in QKD. Satellites in particular have 52 a severely restricted SWAP allowance [37] and could therefore greatly benefit from a rollout of 53 future-generation CMOS SPAD arrays. In free-space QKD, polarization is typically used for the encoding due to the robustness of 56 polarization to atmospheric transmission [38], and the relative ease of implementation. A 57 drawback is the need to actively maintain a common polarization basis between a quickly moving 58 sender and stationary receiver. Time-bin QKD protocols do not have this requirement, and they 59 have advantages in terms of SWAP, especially for the transmitter, which can be integrated onto a 60 chip [39]. A number of time-bin configurations have been implemented [40][41][42][43], but in the most 61 common scenario sender and receiver encode and decode their time-bin qubits with a matched 62 unbalanced (or asymmetric) interferometers. Free-space transmission causes optical wave-front 63 and phase distortions due to atmospheric turbulence. In free-space unbalanced interferometers, 64 these distortions will lead to a reduction in the measured interferometric visibility if no correction 65 optics are used [36]. If the interferometry is performed using single-mode fiber, large and varying 66 coupling losses will occur due to the distortions if high performance adaptive optics are not 67 implemented [44,45].   The system requirements for the beacon are dependent on the accuracy required for effectively 79 aligning the quantum signal to single-photon detectors. The small dimensions of single-pixel 80 single-photon detectors (SPDs) and core-diameter of multimode fiber couplers, typically less than 81 200 µm, impose stringent requirements for fine pointing-and-tracking in free-space 82 communications. Large area single-pixel SPDs (up to and greater than 500 µm) would relax 83 requirements; however, will suffer from greater detector dark noise from the detector itself and 84 ambient light contamination due to the wider FoV. An array of smaller single-pixel SPDs, rather 85 than a larger single pixel, could counteract the increase in noise through spatial post-processing, 86 proving to be an ideal solution for photon-starved free-space communications. Arrayed 87 superconducting nanowire SPD technology is already being implemented in photon-starved 88 optical communication applications, such as deep-space communications [47].

89
Due to those recent advances and the inclusion of picosecond resolution time-correlated 90 single-photon counting (TCSPC) capabilities, there is now the realistic possibility of performing 91 pointing-and-tracking with single-photon per bit level optical beacons. With the dual capability of 92 the 2D SPAD array, the need for fine-tracking sensors and high power optical beacons could be 93 relaxed or negated in free-space QKD modules, allowing a reduction in system complexity. In addition, as free-space links are a versatile channel in quantum networks, if the optical power 95 emitted from the free-space modules can be reduced to or be less than laser safety category Class 96 1, the number of viable locations for QKD transmitter and receiver modules would substantially 97 increase due to compliance with safety considerations air corridors.

98
Although the 2D CMOS SPAD detector array technology is promising, there are two main 99 challenges when using the weak optical signal as the optical beacon: the dark count rate of the 100 SPAD pixels; and the detection efficiency of the 2D SPAD arrays.

101
A high dark count rate reduces the effectiveness of the pointing-and-tracking due to the signal 102 to noise of the image created. In QKD the maximum achievable channel loss is determined by the 103 quantum bit error rate (QBER) of the implementation, to which the dark count rate of the detector 104 will make a contribution, particularly in a high channel loss regime. Single-pixel silicon SPADs 105 are Peltier cooled and typically have dark count rates in the order of 1×10 -3 Hz/μm 2 when operated 106 at -20°C [48]. In order for 2D SPAD arrays to perform at the equivalent channel loss, individual 107 pixels within the detector array will require comparable dark count rates. Improvements to dark 108 count rate have been proposed and demonstrated through the use of different fabrication 109 techniques [49], active cooling of the SPAD pixels [50,51], and the removal of hot-pixels by post-110 processing [52]. Ito et al. [53] have demonstrated 2D SPAD array pixels with dark count rates 111 3×10 -2 Hz/μm 2 at room temperature, only an order of magnitude different from commercial single-112 pixel SPAD devices when cooled to -20°C.

113
The detection efficiency places similar constraints on the pointing-and-tracking and QKD     decoding at the receiver was performed with an asymmetric MI, which interfered the time-bin 136 signal to attain intensity information while the 2D SPAD array was used to resolve the time-bins.

137
The interference occurs between the two laser pulses: |short from MZI which takes the long path 138 in MI and the |long from MZI which takes the short MI path.

139
The optical transmitter signal was generated from a pulsed Picoquant laser source, which 140 provided < 70 ps full-width at half-maximum (FWHM) optical pulses at a wavelength of 852 nm.

141
The laser driver was externally triggered by the 2D SPAD array electronic board with a repetition 142 rate of 5 MHz. The transmitter MZI was constructed of polarization-maintaining optical fiber, Figure 1 (a), implementing a temporal delay between |short and |long of 1.33 ns. That time 144 difference was chosen to be greater than the FWHM timing-jitter of the 2D SPAD array (see 145 Appendix A). The output of the transmitter interferometer was attenuated to the single-photon per 146 clock-cycle level using a fiber attenuator, simulating a qubit state that would be sent in QKD.

147
The output of the transmitter was coupled into free-space through an additional optical fiber 148 and a variable fiber-to-free-space collimation package. The additional optical fiber was used to 149 simulate varying degrees of turbulent channel, similar to how previous experiments simulated 150 multimodal signals [36,46]. A single-mode fiber was used to simulate a non-turbulent free-space 151 channel, and provided the benchmark for performance. To simulate turbulent channel 152 (multimodal) measurement capability, two independent multimode optical fibers, of core-153 diameters 10 µm and 25 µm, were used to test of the optical receiver at two turbulent levels. The Mach-Zehnder interferometer. The output of the transmitter was coupled into free-space through 164 interchangeable optical fibers, which were used to simulate different level of turbulence. (b) The 165 quantum optical receiver was a free-space asymmetric Michelson interferometer. The asymmetric 166 arm was constructed with relay optical elements to balance the spatial movement of the two optical 167 paths. The optical output and resultant interferometric visibility were measured using a free-space 168 coupled 2D single-photon avalanche diode (SPAD) detector array. The 2D SPAD array control 169 board was used to electrically trigger the pulsed laser source.

170
The free-space channel between the transmitter and receiver was 30 cm. The environment was 171 an air-conditioned and temperature stabilized laboratory, which meant the turbulent effects were 172 directly related to the multimode optical fiber transmission. Two tip-tilt alignment mirrors were 173 used to set the initial 0° alignment into the receiver asymmetric interferometer, and a precision 174 dial was used to set known misalignment of the AoI for experiments.

175
The time-bin interferometer receiver was an asymmetric free-space MI, Figure 1 (b). The short 176 arm of the receiver MI, denoted ll in Figure 1 (b), was 12 cm in length one way, to allow for a 177 mirror mounted on a piezoelectric controlled z-translation stage to be included in the optical set 178 up, which allowed for active control of the optical phase shift. The optical time delay between the 179 two optical paths of the receiver was set to 1.33 ns, which corresponded to 40 cm of air. Similar 180 to a previous demonstration [36], the optical path of the optical time delay was constructed from 181 optical relay lens elements, to compensate for the difference in spatial movement of the 182 asymmetric interferometer arms. The relay lens optical system had one-way optical length of 4 183 times the focal length, f, Figure 1 (b). As the interferometer reflects the light back through the relay system, the total path length was 8f. It was the doubling back through the relay lens system, 185 which enabled the compensation of spatial mode. To match the time difference between the 186 successive optical pulses and the free-space interferometer, the focal length of each relay lens was 187 chosen to be 5 cm. The long arm was a total length of 32 cm, one-way, considering 20 cm from 188 the relay lens system and the additional length of the short arm, 12 cm.

189
At the receiver interferometer's output, a 30 mm focal length convergent lens focused the 190 beam onto the 2D SPAD array plane, which created a focal spot of ~60 µm in diameter for the 191 single-mode, non-turbulent, channel. The 2D SPAD array was aligned to the optical system and

206
The use of the 2D SPAD array and the 8f asymmetric interferometer enables wide FoV 207 interferometry for photon-starved optical communication applications, such as time-bin QKD.

208
The ability to do both temporal and spatial filtering is not something that can be achieved with a 209 single-pixel single-photon detector. The use of spatial filtering is of particular benefit in wide FoV 210 applications where scattering due to the atmosphere and ambient light will result in an inherently 211 higher noise level.

224
FWHM from an independent instrumental response function (IRF) measurement) of all operational pixels in the 2D SPAD array when measured using the laser source described above, 226 was 245.8 ± 39.5 ps. See Appendix A for more details on the IRF measurement. Figure 2 227 highlights that the overall 2D SPAD array response was longer than the IRF time-jitter, this is due 228 to optical reflections within the optical receiver and inter-symbol interference.

229
The electrical readout board and control software allowed acquisition of a 25 ns long event

246
The detector readout provided accumulated information from all individual pixels, which were 247 activated for a measurement. That meant that individual pixels from the larger accumulation could

253
In order to perform a measurement with the 2D SPAD array, a set of pixels were selected and 254 then activated for a pre-determined acquisition time and number of frames. In order to measure 255 the spatial information, the 2D SPAD array was split into grid sections of a defined size. Individual 256 sections of the predefined size of the SPAD array were then activated sequentially, scanning across 257 the entire 2D SPAD array area, similar to a raster scan.

258
The detector area (672 × 672 μm) and focal length of the asymmetric interferometer receiver

289
Without the relay optics, the visibility decreases rapidly with increasing AoI, Figure 3. Even 290 at an AoI of 0°, the visibility is significantly lower without the relay lenses when compared to the 291 other case. Based on theory described in [60], the visibility versus AoI for the single-mode 292 (Gaussian beam profile) channel, is plotted in Figure 3 as the solid black line. The theory takes 293 into account the asymmetry of the interferometer, the optical beam diameter, and AoI. When using 294 the single-mode fiber input, the visibility measurements without relay lenses was generally in 295 agreement with the model proposed in [60]. As expected, the other multimode inputs (10 µm and 296 25 µm diameter) also showed poor visibility response and a sharp drop in visibility as the AoI was 297 increased.

298
With relay optics in place, the interference visibility remains high for all three different optical 299 channels, as expected from previous demonstrations [36]. There is a visibility drop-off at large 300 angles of incidence, which is due to fine optical misalignment of the relay optical elements in the 301 receiver. It is clear that the optical relay significantly improves performance for all values of AoI 302 and for different input fibers.

303
The results in Figure 3 demonstrate that a 2D SPAD array can be used to monitor the visibility 304 evolution of an asymmetric interferometer at the single-photon level over a large AoI, an essential 305 step towards permitting the measurement of quantum bit information in a QKD protocol.

307
A potential benefit of using a 2D SPAD array over the single-pixel counterpart is the ability to 308 measure both the TCSPC events as well as spatial information of the detected photon, which 309 corresponds to the unpredictable AoI in a free-space QKD receiver in the presence of atmospheric 310 turbulence. Due to the read-out of the 2D SPAD array used in this experiment, the 2D SPAD array 311 was scanned electronically to capture spatial information. This electronic scanning of read-out information simulates how the spatial information would be acquired if the 2D SPAD array 313 utilised a read-out architecture capable of processing each SPAD pixel independently.

314
To demonstrate the measurement of spatial information, the 2D SPAD array was set into four 315 configurations. Each configuration set a different spatial resolution of the 2D SPAD array. The 316 four configurations split the 2D SPAD array into 1 × 1, 2 × 2, 4 × 4, and 8 × 8 grids, which divided 317 the 1024 individual pixels equally amongst each section of the grids. The sections were chosen to 318 simulate a large single pixel SPAD detector, a quadrature detector, and two higher resolution 2D 319 array detectors.

320
All measurements of spatial information were carried out in two steps. The first step was a 321 background measurement, to record the average background counts resulting from detector dark 322 events in the 2D SPAD array and contributions from other ambient light sources, for each of the 323 specified pixel sections. The second step was a measurement with the quantum optical signal 324 incident on the 2D SPAD array. The experiment was carried out with the same optical system 325 outlined in Figure 1, including the interference measurements. However, the interferometers were 326 stabilized to ensure consistent signal power. The single-mode, non-turbulent, channel was used 327 over the multimode channel, but the same results are expected independent of channel. Each 328 spatial resolution measurement was a full scan of the 2D SPAD array for a specified configuration.

329
The sections were activated for 5 frames, each frame having an acquisition time of 100 µs. There 330 was no dead-time between the frames of measurement, however there was a read-out dead-time 331 (4.6 μs) between the sections due to read-out of information from the chip to the FPGA. The 332 background noise measurement was subtracted from the signal measurement during post-333 processing, to enable easy identification of the signal. Figure 4 shows the resultant measurements identifying the spatial position of the optical beam 335 for a 1 × 1 (a), 2 × 2 (b), 4 × 4 (c), and 8 × 8 (d) grids. As can be seen in Figure 4, as the grid 336 resolution increases, the precise location of the optical beam can be identified more accurately.

337
However, a spatial resolution as low as 2 × 2 section is sufficient for identifying the approximate

347
The focal spot diameter of the optical beam on the 2D SPAD array was designed to be 60 µm, 348 based on the initial beam width for the single-mode channel and focal length of the final lens.

349
With a single-pixel SPAD detector, verifying the single-photon level spot size would require 350 precise spatial positioning mechanisms, and a time-consuming process to move the SPAD into 351 place. Here, direct measurement of the focal spot size using the 2D SPAD array can be performed 352 by taking advantage of the spatial measurement. The 8 × 8 grid, where each section is composed 353 of a 16 pixel square, has a section dimension of 80 × 80 µm. It can be seen from Figure 4 (d), that 354 the beam is focused within one section, demonstrating that the 2D SPAD array was well positioned 355 at the focal point of the lens, and that the spot size was less than 80 µm.

356
To verify the angular FoV of the detector and the angular movement set for experimental 357 measurements, the spatial movement with AoI was measured using the 2D SPAD array. Figure 5 358 shows the spatial movement for an 8 × 8 scan as the AoI was changed. It can be seen that as the 359 AoI changes, so does the position of the spatial position on the 2D SPAD array. The spatial 360 position was measured for set angles of incidence 0° (a), 0.107° (b), 0.215° (c), and 0.322° (d).

361
The angular FoV of the 2D SPAD array was calculated to be 1.28°, based on the detector 362 square dimensions (672 µm) and the focal length of the final lens (30 mm). The angular setting 363 ranged from 0° to 0.322°, which is ~25 % of the angular FoV. As the AoI increases from 0° to

368
As the fill-factor of the 2D SPAD array used was not 100%, it is inevitable that light will fall 369 between the active pixels while performing the scan, leading to a reduction in the measured signal when light falls between the pixels. This intensity fluctuation is most evident in Figure 5, due to 371 the AoI changes. By increasing the fill-factor of the SPAD detector array, we hope to reduce these 372 intensity fluctuations.

373
The measurements from this section demonstrate that spatial position of a single-photon level 374 intensity optical signal can be captured using 2D SPAD array technology. The measurements also

384
Visibility is an essential measurement for the security of time-bin QKD protocols [41]. Low 385 visibility performance in time-bin protocols will lead to an increase in key rate reduction cause by 386 post-processing algorithms, or even protocol failure [2]. Although this paper has highlighted the 387 benefits of using 2D SPAD array technology, the use of multiple, and many, SPAD pixels 388 inevitably leads to an increase in the detector background noise, which will increase the quantum 389 bit error rate (QBER) of a QKD protocol, reducing protocol performance.

390
The 2D SPAD array enables a wide FoV receiver due to the larger detector dimensions.

391
However, because the optical beam is focused on the 2D SPAD array, there are pixels in the array 392 that are not illuminated by the single-photon level signal but contribute to the background noise.

393
Post-selecting pixels, which contribute to the final single-photon level measurement, could enable 394 better performance from the same SPAD array. This experiment investigates the improvement in 395 measured visibility by reducing the number of pixels included in acquisition of the measurement.
For each AoI, the visibility was measured, using the same methodology as Section 3.1, for various 397 pixel grid sizes, defined in Section 3.2. 398 Figure 6 shows the interferometric visibility versus AoI for different pixel grid sizes. As can 399 be seen in Figure 6, as the number of active SPAD pixels incorporated in the visibility 400 measurement decreases, the visibility measured increases. The optical beam from the 401 interferometer is focused down to 60 µm, meaning the majority of the 2D SPAD array area is not 402 receiving photons for the visibility measurement. Those excess pixels add additional, and 403 unwanted, noise to the visibility measurement, essentially reducing the visibility.

404
The result highlights that incorporating all the pixels in the 2D SPAD array for one single-405 photon level channel would not be beneficial, unless the noise associated with each pixel was 406 negligible. The wide FoV of the detector would also lead to additional background noise from the 407 channel, which would further reduce the visibility. While using this protocol, the whole detector 408 array could be used to collect single-photon level bits, taking advantage of the wide FoV, allowing 409 the beam to wander across the array during measurements. The data, which will have the spatial

418
The experiment shows that the reconfigurable flexibility of 2D SPAD array enables the 419 optimization between two extreme conditions to obtain high visibility single-photon measurement 420 while still maintaining a wide FoV spatial measurement. However, in practice, that functionality 421 would require independent read-out circuitry for each pixel to enable the post-processing. Here

425
This paper set out to demonstrate the feasibility of combining the measurements of single-426 photon level bit information and optical spatial position using an arrayed single-photon sensor.

427
The combined detection capability is beneficial for a free-space single-photon level 428 communications, such as QKD, alleviating the requirement for high-power optical alignment 429 beacons, as the fine pointing and tracking detection could be performed using the single-photon 430 level signal and a 2D SPAD array (or more generally a 2D SPD array). We conducted three 431 experimental demonstrations to show the practical feasibility, implementing a 32 × 32 pixel Si-432 CMOS 2D SPAD array and a time-bin receiver interferometer.
In the first experiment, the capability to measure and monitor single-photon level visibility of 434 a time-bin receiver interferometer, a critical measurement for time-bin QKD protocols, was 435 demonstrated. All measurements required the spatial information to be tracked as the AoI was 436 changed. The robustness of the interferometer design to changes in AoI and wave-front distortions 437 was also demonstrated.

438
The single-photon level spatial position tracking was demonstrated in the second experiment.

439
We showed that a 2D SPAD array can be configured for various pixel resolutions, creating higher 440 or lower spatial resolution detectors for spatial tracking. The 2D SPAD was also used to verify 441 the angular FoV of the optical system (1.28°) and to track known spatial misalignment.

442
In the final experiment, we showed that the signal-to-noise of a quantum measurement can be 443 improved by processing the spatial position of the single-photon level optical signal and then 444 down-selecting the number of individual 2D SPAD array pixels used for the visibility 445 measurement. This can be performed by post-processing the measurement data, provided the 2D 446 SPAD array has the ability to independently read-out each pixel.

447
The experiments from this paper have demonstrated the promising future for SPAD detectors, 448 and more generally SPD, array technology in photon-starved free-space optical communications, 449 such as QKD. With many teams working on SPD array technology, continuing advances will 450 further improve the prospects for SPD detector arrays in photon-starved free-space optical 451 communications. Key challenges to enable applications in free-space QKD are already being 452 addressed, such as fabrication of 2D SPAD arrays with inherently low dark count rate per pixel, 453 even before active cooling [49,51,53,57], and increasing the detection efficiency [49,[53][54][55][56][57]. With 454 the reduction in dark count rate per pixel and improvements in detection efficiency for specific 455 wavelengths, applications in QKD for 2D SPAD arrays will be more realistic, especially with the 456 inclusion of specialized electronics to enable QKD processing.

458
Appendix A: 2D single-photon avalanche diode array characterization

459
This appendix provides additional details on the 2D single-photon avalanche diode (SPAD) 460 detector array, which can be considered a digital silicon photomultiplier, implemented in the main 461 paper. Understanding the measurement capabilities of the 2D SPAD array was a vital aspect when 462 designing quantum key distribution (QKD) receiver's asymmetric interferometer [42]. The

474
To measure the dark count rate per second, the dark count per 1 ms was multiplied to give 1s 475 acquisition. To reduce potential light background ambient light levels further, the detector was 476 housed inside a lightproof container.

477
The timing-jitter of the 2D SPAD array was measured in the following way. A pulsed laser 478 source, providing < 70 ps FWHM optical pulses, at a wavelength of 852 nm, was used to 479 illuminate the 2D SPAD array through an engineered diffusor in free-space. The engineered 480 diffusor created a circular top-hat intensity profile, and the detector was placed at a distance where 481 the intensity profile was relatively even across the whole 2D SPAD array. The laser source was 482 electrically pulsed at an operational clocking frequency of 5 MHz. The histogram data generated 483 for each pixel, example in Figure 7, was post-processed to identify the peak intensity and the 484 FWHM timing-jitter using a Gaussian fitting function.   The dark count rate for each pixel is shown in Figure 8 (a), highlighting the spatial distribution 491 of the dark count rate over the 2D array. It can be seen that the dark count rate per pixel is randomly 492 distributed throughout the 2D array. The minimum dark count rate was found to be 493 4.05 × 10 4 ± 6.18 × 10 3 counts per second, while the maximum was 2.64 × 10 7 ± 3.34 × 10 5 counts per second. The average over all 1024 pixels was found to be 1.39 × 10 6 ± 7.56 × 10 4 counts per 495 second. The three orders of magnitude difference in minimum and maximum is due to the presence 496 of hot-pixels, which have a substantially higher dark count rates than other pixels [52].

497
Reducing the dark count rate is essential for QKD applications, as excess noise contributes to 498 the quantum bit error rate (QBER) of the protocol, limiting the loss budget available and reducing 499 the secret key rate. The limited loss budget is analogous to reducing the achievable transmission 500 distances. It should be noted that the average dark count rate could be reduced if cooling was 501 applied to the detector chip. Even when cooling the 2D SPAD array chip, hot-pixels are still an 502 issue. Identifying hot-pixels within the 2D array is essential to identifying and removing those 503 troublesome pixels. The authors of [52] note that up to 20% of a 2D SPAD array could be hot-504 pixels.

505
The dark count rate measurements were re-analyzed to identify hot pixels in the 2D SPAD 506 array, based on the methods described in [52]. A histogram of number of pixels versus dark count 507 rate with a fitted log-normal distribution, Figure 8 (b), shows that the distribution is elongated, as 508 there are a number of pixels that have a significantly higher dark count rate. Reference [52] 509 identifies a pixel as hot if it has a dark count rate larger than 3σ of the distribution average. Using 510 that definition, it was found that 7 % of the pixels in the 2D SPAD array characterized were termed 511 "hot". These pixels were not removed or post-selected out during experiments within the main 512 paper, as it was deemed more representative of the overall performance to include all pixels in 513 measurements for this feasibility study.  in the array (a), the dark count rate distribution to identify hot pixel (b), the full-width at half-518 maximum (FWHM) timing-jitter per pixel in the array (c), and the distribution of timing-jitter (d).

519
Figure 8 (c) shows the timing-jitter measured for each pixel in the 2D array at the position in 520 the 2D SPAD array. As can be seen, and unlike the dark count rate measurements, there is a clear 521 split in measured timing-jitter between the top and bottom halves of the chip. The splitting is also evident in Figure 8 (d). The split in timing-jitter is due to signal routing of the array pixels to the 523 time-to-digital converter (TDC). There was a shorter path from one-half of the chip to the TDC, 524 leading to a shorter timing-jitter. The longer path would cause slower edges on the response, 525 leading to an increased spread in timing. The FWHM timing-jitter values range from 151 ps to 526 393 ps, and the distribution can be seen in Figure 8 (d). As well as spatial position, reference [58] 527 highlights that the timing-jitter response is also dependent on the number of pixels activated for a 528 measurement. However, that effect on the quantum bit measurement was not extensively studied.

529
The average timing-jitter was found to be 245.8 ± 39.5 ps. As the laser pulse duration was <70 ps, 530 the stated time-jitter values were dominated by the effects of the detector itself.

531
During the post-processing of the timing jitter, the central peak position per pixel was analyzed 532 using the information from the Gaussian fitting function in MATLAB. It was found that 88% of 533 the central positions occurred in the same time bin (width 94.69 ps), while 10% occurred in the 534 bin before and 2% in the bin after. This highlights that there was a relative time delay between a 535 small number of the pixels in the array. In this paper, the time interval between the optical pulses 536 was much larger than the relative time delay, which means the effects on these experiments are 537 negligible. However, this time-delay is fixed and can be characterized for any 2D SPAD array 538 fabricated, meaning post-processing corrections can be made routinely to individual detector 539 timings as they are in single-photon LIDAR measurement using arrayed detectors [63], reducing 540 any impact on time-correlation measurements in QKD.

541
The timing-jitter measurements were critical for the design of the quantum optical receiver.

542
Due to the extended timing-jitter tail of single-photon detectors [62], the optical time difference 543 needs to be greater than the measured timing-jitter to reduce the QBER. In the case of the 2D 544 SPAD array used for this paper, the average full width at 10 % maximum was found to be ~1 ns.

545
This set the minimum time difference to be ≥ 1 ns, which is ~30 cm propagation in air. The optical 546 system for the main paper was designed with a time delay of 1.33 ns due to this detector 547 characteristic.

549
The interferometric visibility was calculated using the following method after data was acquired 550 using the method described in 3.1. The data output for analysis was 1000 frames (histogram plots), 551 each acquisition being 100 µs long. The central, interfering, peak in the histogram was first 552 identified by summing all 1000 histogram frames together, followed by a peak finder function in 553 MATLAB to identify the central peak. The peak finder function was verified visually. After the 554 central peak identification, a time-gating window of 500 ps was applied to each histogram, the 555 center of the time-gate was positioned at the middle of the interfering peak. For each frame, the 556 number of photon events within the time-gated window were summed together to give a count 557 rate (CR). As the interferometer was actively tuned to capture multiple interference fringes, over 558 the 1000 frames the CR reaches a maximum (CRMax) and minimum value (CRMin). The

562
The data associated with this work can be downloaded from the Heriot-Watt University data 563 archive at XXXXXXXXXXX.