Open Access
Add to BibSonomy
Abstract
Over the last decade, free-space quantum key distribution (QKD), a secure key sharing protocol, has risen in popularity due the adaptable nature of free-space networking and the near-term potential to share quantum-secure encryption keys over a global scale. While the literature has primarily focused on polarization based-protocols for free-space transmission, there are benefits to implementing other protocols, particularly when operating at fast clock-rates, such as in the GHz. In this paper, we experimentally demonstrate a time-bin QKD system, implementing the coherent one-way (COW) at 1 GHz clock frequency, utilizing a free-space channel and receiver. We demonstrate the receiver’s robustness to atmospheric turbulence, maintaining an operational visibility of 92%, by utilizing a lab-based turbulence simulator. With a fixed channel loss of 16 dB, discounting turbulence, we obtain secret key rate (SKR) of 6.4 kbps, 3.4 kbps, and 270 bps for three increasing levels of turbulence. Our results highlight that turbulence must be better accounted for in free-space QKD modelling due to the additional induced loss.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Quantum communications is a growing research field that seeks to take advantage of quantum phenomena to create unconditionally secure communications protocols [1–5]. Quantum key distribution (QKD), an encryption key sharing protocol, is the most mature quantum communications technology [6,7]. QKD has been extensively experimentally investigated on point-to-point optical fiber links and dark fiber networks [8–13] and is growing in line-of-sight free-space and underwater implementations [14–18]. While many experiments typically utilize qubits, investigations with higher dimension quantum states, so-called qudits, are growing due to their higher noise tolerance and information capacity [19,20].
In line-of-sight demonstrations, QKD has been dominated by polarization based protocols due to the relative simplicity of the implementation, particularly for low repetition rates [21–23]. Utilizing other QKD protocols in line-of-sight applications based on other degrees of freedom, such as time-bin or phase, is seen as challenging, due to the need to perform asymmetric interferometry of a multimode signal [24–26]. The challenge can be met in two ways; using adaptive optics (AO) to couple light into a single mode fiber [27,28]; or passive free-space optical design which enables multimodal response [29–32]. The passive methods can operate over a range of bandwidths, however, there is a need to optimize and choose the best solution due to the implication on size, stability, and practicality [33]. Beyond feasibility studies, time-bin based protocols in free-space channels have been demonstrated using time-bin entangled qubits over 1.2 km [34] and a reference-frame-independent protocol over 2 km [35]. While the use of a free-space channel enables the experimentation with real atmospheric conditions, for example turbulence, demonstrations are limited to the atmospheric conditions on the days of measurement, limiting examination over a broad range. Lab-based equipment that manipulates the wavefront of light, such as spatial light modulators (SLM), can be utilized to simulate atmospheric turbulence [36], enabling a broader range of testing in addition to real channel testing, beneficial for capturing the full range of performance.
In this paper, we utilize a fully free-space time-bin optical receiver with passive relay-lens asymmetric interferometer to demonstrate a coherent one-way (COW) protocol, [26,37], at a pattern frequency rate of 500 MHz (1 GHz clock frequency). We incorporate a SLM in the transmission channel to simulate three levels of atmospheric turbulence, without needing to use a long-distance range. Our demonstrations show the interferometric visibility of 92% is maintained for all levels of turbulence. For a channel loss of 16 dB, discounting losses associated with turbulence, we show secure key rates of 6.4 kbps, 3.4 kbps, and 270 bps for low, medium, and high turbulence levels, respectively, demonstrating that loss from atmospheric turbulence is a non-negligible parameter for system modelling. The work highlights that lab-based equipment can be used to simulate real-world atmospheric effects, enabling broader testing parameters to be tested on a faster time-scale.
2. Turbulence simulation and implementation
Atmospheric turbulence is known to vary with time due to disparities in both the pressure and temperature of the atmosphere along the propagation path. Consequently, this causes variations in the index of refraction, resulting in both phase and amplitude fluctuations of the propagating light beam, leading to fading and reduced signal to noise ratio [38]. Throughout the day usually three turbulence regimes are defined: low, medium and strong [39,40].
To simulate turbulences in the lab the atmosphere is usually divided in several layers or “phase screens”. However, for some cases using a single phase screen is a good enough approximation, this happens when the turbulent path is not too long or it is only a small portion of the whole path [41,42]. Therefore, we made use of the ‘thin screen approximation’ procedure [41,43,44]. Atmospheric propagation can be understood as a plane wave accumulating a random phase distortion due to the different properties in the channel that different areas of the wave see. The wavefront observed can be described as:
where ${f}$ are the spatial frequency coordinates ${f} = [{{f}_{x}}{,}{{f}_{y}}]$, $W({\cdot} )$ is the pupil function and $\phi ({\cdot} )$ is the phase distortion function. This distortion function will be modelled making use of the Zernike polynomials approximation. Zernike polynomials are a set of polynomials orthogonal among themselves, commonly used to describe wavefront aberrations. The distortion function was generated as: where $R$ and ${\rho }$ are the polar transformation of $f$, being $R$ the radius of the receiver aperture, so the usual radius coordinate $r$ can be scaled to $r = R\rho $ with $0 \le \rho \le 1$. Therefore $R{\rho} = {[R\rho ,\theta ]^T}$ and the function is normalized to the radius of the aperture. ${Z_j}$ is the jth Zernike polynomial, and ${a_j}$ are random number following a normal distribution and correlated among each other [43], according to the formula: where $C$ is a correlation matrix, $b$ is vector of random numbers, ${r_0}$ is the Fried parameter and $D$ is the diameter of the aperture.The Fried parameter is a measure of the quality of optical transmission through the atmosphere due to fluctuations in the refractive index of the atmosphere (i.e., turbulence), with higher values for better visibilities, which is given as:
where k is the wavenumber, z is the elevation angle, measured with respect to the vertical, h is the height, $L$ is the turbulence path-length, set to 10 km for this work, which is the effective value for the atmosphere in a satellite-to-ground scenario [18,45] and $C_n^2$ is the refractive index structure. $C_n^2$ is dependent on the altitude, and in this work it has been modelled according to the widely used Hufnagel-Valley model. To define different atmospheric conditions, it is a standard procedure to modify the $C_n^2$ value at ground level [46]. We have chosen $C_{n0}^2 = 1.7 \times {10^{ - 14}}$m−2/3 for low, $C_{n0}^2 = 9 \times {10^{ - 14}}$ m−2/3 for medium, and $C_{n0}^2 = 7.8 \times {10^{ - 13}}$ m−2/3 for strong turbulence conditions.The normalized atmospheric turbulence strength is given as the relation between the receiver aperture and the Fried parameter (${D / {{r_0}}}$). We used $D = 0.4$ m (a modest optical telescope receiver size), giving a ${r_0}$ of 9, 3 and 1 cm, and a ${D / {{r_0}}}$ relation of 4, 10 and 36 for low, medium and strong turbulence respectively for our operational wavelength of $\lambda = 852$ nm.
The turbulence values are in accordance with other works [40]. The chosen wavelength is based on the availability of light sources, moderate and cost effective single-photon detector technologies [47], as well as the relatively high free-space transmission.
With this method, six different turbulence patterns were generated for the experimental work. First, three different average turbulence patterns (low, medium and strong) were created as the ponderation of 1000 different patterns, to represent the power distribution on the detector, Fig. 1(a), (b), and (c) respectively. The average images have most of the power concentrated in the center, giving a shape close to a Gaussian, this is an intended feature to simulate how a system would perform in a long run experiment. However, it does not demonstrate the capabilities of the system to deal with multimode beams, a critical property for multimode time-bin receivers. In order to prove this point, three different single snapshots for the same three levels of turbulence strength were created, to represent what is seen by the detector at a snapshot in time, Fig. 1(d), (e), and (f). However, for these images there were some implementation imperfections that made the beam to have a peak of power in the center. Nevertheless, the beams seen at the receiver were far enough from a single mode beam to prove the system is able to handle turbulence patterns.
Fig. 1. Set of turbulence patterns from the simulations that were applied to the spatial light modulator. The average turbulence for low (a), medium (b) and strong (c) turbulence. The spread in the power can be appreciated for stronger turbulence due to peak of the beam oscillating around the center. A single snapshot of a typical low (d), medium (e) and strong (f) turbulence pattern. The three bottom row images have been normalized to demonstrate the system with the same intensity. Therefore, giving a fair comparison with the top row results.
The turbulence images from the simulations were uploaded to the SLM. Optical and image pre-processing was used to ensure the image produced was representative of the simulation. The beam profiles after the SLM are shown in Fig. 2, which was measured using a complementary metal oxide semiconductor (CMOS) camera.
Fig. 2. Beam shape at the transmitter output after being modulated by a spatial light modulator. The first row represents the average turbulence for low (a), medium (b) and strong (c) conditions. The second row shows a single snapshot of a turbulence pattern for low (d), medium (e) and strong (f) conditions. The SLM modulation is not perfect, and an important amount of the power is always at the center of the beam. Nevertheless, the non-average patterns are far from single modes and close to turbulence patterns so they can be used to prove the system.
For optical optimization the collimated optical input propagated through a polarizer and a half-wave plate, which rotated the polarization incident on the SLM to horizontal, a requirement for optimal operation of the SLM.
For pre-processing, a grating pattern (changing phase information) was applied to the simulation image in order to spatially separate the higher order diffraction patterns generated from the SLM, ensuring that only the first order was aligned with the optical system. The position of the pattern on the SLM screen was also optimized (spatial alignment) to ensure the correct pattern was generated. Finally, an optimization following the Ref. [48] is necessary to optimize the modulation of the images. This modulation consists in encoding the image as:
where $A$ is the amplitude of the image and $\varphi $ its phase. It can be seen in Fig. 2 that as the average turbulence increases, the maximum intensity falls. The measured optical loss difference between the average low (a) and average medium (b), and average low and average strong (c) was 3.2 dB and 9.3 dB respectively. The values match well measured atmospheric turbulences losses reported in the literature for both theoretical and experimental scenarios [18,33]. In practice, there are several reasons for the additional optical loss from turbulence. The most significant are beam wandering (beam tilting that decrease the efficiency of light collection), beam scintillation (which causes random fading in the beam intensity), and an increase in the beam divergence [49]. Here, the simulation increases the attenuation by increasing the coupling to higher orders of diffraction and it was modelled with the variance of the random log-amplitude [42].3. Experimental setup and implementation
Figure 3 shows the setup of both transmitter and receiver for the COW protocol used for this paper. The transmitter was constructed of fiber-based and coupled components while the receiver was constructed of free-space optics. The light source was a long coherence length, fiber coupled, continuous wave (CW), distributed feedback laser, which emitted at a wavelength of 852 nm. An isolator was coupled after the source to reduce back reflections, which would detrimentally affect the coherence properties of the laser source. The CW light was coupled into an intensity modulator (IM), driven by an electrical signal previously amplified, that modulated the light. This electrical signal was a pseudo-random pulse pattern that codified the logical ‘0s’ and ‘1s’ of the COW protocol at a 500 MHz, leading to an actual clock frequency of 1 GHz. The extinction ratio of the on-to-off was 15 dB, which limited the achievable quantum bit error rate (QBER). To maintain a stability over time, the intensity modulator was connected to a continuously tunable modulator bias controller with a feedback loop. Before coupling into free space, the light was attenuated, with a variable electrical attenuator and two 90:10 couplers, to a mean photon number per pulse of 0.07, the optimal value for the COW protocol [49].
Fig. 3. Schematic of the system to implement the coherent one-way (COW) protocol over free space. At the transmitter the light was generated by a distributed feedback (DFB) laser at 850 nm with a long coherence length. An optical isolator prevented optical back reflections that could degrade coherence. The light was modulated into pulses with an intensity modulator, which was connected to a feedback loop at the bias driver controller to maintain stability over time. An electrical variable attenuator and two 90:10 couplers introduce the attenuation required to reach single-photon level. Before sending to the free-space receiver a spatial light modulator (SLM) modulated the beam shape to obtain a turbulence pattern. At the receiver a 50:50 beamsplitter divided the light between the key stage and the interferometer. The interferometer was an unbalanced Michelson with a path difference match for pulse rate of 1 GHz.
Contained on the same breadboard in Alice was the turbulence simulator, which consisted of the SLM, optimization optics, and two beam steering mirrors for aligning the free-space channel. Light from Alice’s transmitter was coupled into free-space through a beam collimator. The collimated light was then polarization aligned with the SLM to optimize the SLM output (process described previously). After going through the turbulence simulator, the beam was sent to a beam expander with a factor of 3x and propagated to the receiver across a free space channel of approximately one meter.
The receiver, Bob, was a composed of three stages; a beam contractor and beam steering, a key detection stage, and an asymmetric interferometer with relay lens optical elements [30,32]. It was constructed of commercial-off-the-shelf (COTS) inch (2.54 cm) and half inch (1.27 cm) diameter free-space optics. In the beam contractor stage, a beam expander was used in reverse with a multiplication factor of 10x. The larger beam contracting factor was used at the receiver to reduce the beam size to avoid aberrations, particularly in the asymmetric interferometer. Beam steering mirrors before the time-bin receiver allowed initial optical alignment. After the beam steering stage, the beam was then split between the two elements of the COW time-bin receiver, the key detection stage and the interferometer stage, via a 50:50 beam-splitter. The key detection stage, used to measure raw key information, was composed of a focus lens followed by a silicon single-photon avalanche diode (SPAD) detector with a quantum efficiency of 45% at 852 nm and an average dark count of 200 c/s. The interferometer stage was an unbalanced Michelson interferometer with an optical path different match for a pulse operational frequency of 1 GHz. The long interferometer path length consisted of a delay made up by relay optical elements with focal lengths of 37.5 mm. The use of these as the path length delay was chosen over other designs [31] because of the operational frequency, which caused an important difference in size with a similar performance on the thermal stability [33]. These designs have been extensively proved to be robust against high multimode beam profiles both in the original paper [30] and in following works [31,32]. The short arm had a mirror mounted on z-stage with a piezo that allowed the interferometer to be manually tuned to monitor the maximum and the minimum of the interference measurement. The SPAD at the output of the interferometer stage was the same model as the key detection stage, but with a dark count of 400 c/s. Both the key detection and interferometer stage had 10 nm full-width at half-maximum spectral filters to reduce background noise from the laboratory.
To acquire data a repetitive pattern sequence of 32 modulated pulses was applied, there were four cases where two consecutives non-empty pulses were sent, giving four interfering peaks on the measurement histogram. The synchronization signal was sent every 128 ns, enabling the pattern to be visually interpreted during post-processing. The accumulative counts at all the interfering pulses were used to calculate the interferometric visibility. To measure the system at different optical losses, extra attenuation was applied to the variable optical attenuator in Alice. The time-of-arrival of the incident photons was measured using a 4 picosecond resolution time-correlated single-photon counting hardware. Each SPAD’s TTL electrical output was measured on a separate channel. During the experiment, the visibility of the interferometer stage was monitored in real time, tuning the short arm of the interferometer with the piezo stage to create the maxima and minima of the interference. The visibility was calculated using a time-gate width of 200 ps around the expected time of arrival. The raw key was acquired for 15 minutes per measurement.
Post-processing stages were applied offline, using customized code. First, a sifting process was performed, using a time-gate width of 500 ps on the key stage to reduce contribution of the dark counts. Then an estimation of the QBER was calculated using 5% of the sifted key, other works have used between 10-20% [50–52] of the bits, however, we found that sacrificing more than a 5% in our system did not have any benefits, since only an error of ${\pm} $0.01% with respect to the real QBER was found. Error correction codes were then used to further process the sifted key and the QBER estimation. In this work, the error correction was performed using the Cascade protocol [53]. However, in recent years protocols such as Low-Density Parity Check has raised in popularity because of being less interactive [54], this could be an advantage for a satellite scenario where the communication window is limited. Finally, privacy amplification was applied to give the final secure key rate. It was performed theoretically, due to a lack of efficiency in the implementation, a software estimated the key reduction size based on the experimental parameters (Eq. (9)), and then the key was reduced by that amount.
4. Results and discussion
The full QKD system was measured with the three average turbulence patterns for several fixed channel losses. The fixed channel loss considers all the losses associated with other sources aside from turbulence, for instance; beam divergence, atmospheric absorption, pointing and tracking, and transmitter and receiver optical loss. Medium and strong turbulence scenarios have an extra loss (coming from stronger beam wandering, beam scintillation effects and a bigger divergence) not included in the fixed loss used in the model, but applied in the pattern images, so they can be compared with the low turbulence case for the same physical channel. For the non-average turbulence, we demonstrated that the visibility remains the same, at a constant number of counts per pattern, to discard the losses as a reason for downgrades in the system.
The QBER model considered three factors: the noisy counts (dark counts and background noise), the encoding errors and the detector jitter [11].
$\textrm{QBE}{\textrm{R}_{\textrm{noise}}}$ considers the dark counts and the background noise at the detectors, as is common in the literature [37]:
$\textrm{QBE}{\textrm{R}_{\textrm{encoding}}}$ considers the extinction ratio of the intensity modulated states with imperfections from the device and state setting by the pulse pattern generator (PPG). Although the intensity modulator itself has an extinction ration of 15 dB, the resultant QBER contribution was 2.3%, due to the on-off switching time of the PPG, and the limit on the bandwidth of the electrical amplifier.
Finally, we also considered the effects on using SPADs when working in the gigahertz region [11]. The contribution from the $\textrm{QBE}{\textrm{R}_{\textrm{jitter}}}$ is 0.7% for the pulsing frequency of 1 GHz and a time-gate width of 500 ps. Narrower time-gate widths could be applied at the expenses of extras losses, but the protocol will lose performance either way.
The results of the QBER versus the fixed channel loss for the three levels of turbulence can be seen in Fig. 4. The black line is the model for the system for low turbulence, while the dotted lines are experimentally measured data points for the low, medium, and strong turbulence cases. We see as turbulence is applied with the SLM, the trend of data points rises away from the model. This is due to the increase in channel loss experienced with increased turbulence in the channel, 3.2 dB and 9.3 dB for medium and strong respectively.
Fig. 4. QBER of the system for a fixed channel loss from 1 to 35 dB. The model is estimated for a low turbulence. Higher turbulences introduce extra losses that deteriorates the performance. The QBER matches the model well, for higher losses than 32 dB, there is a small discrepancy that we attribute to not have perfectly modelled the average dark counts per second at the detectors. To generate the error bars, the QBER was monitored every second during the 15 minutes of the measurement, then an average and standard deviation was calculated.
To obtain the SKR we followed the analysis presented in [55] and [49], where more realistic assumptions than in other works [56] are made, in order to apply a correct post-processing. A final secret key of size $l$ is extracted from a sifted and corrected one:
The results of the SKR versus fixed channel loss can be seen in Fig. 5(a), where the black line represent the theoretical model for a 92% visibility and a low turbulence scenario, although in a real scenario it is likely to have a varying turbulence (varying loss) with time hence the SKR will not be constant, but it could take any value between the low and the strong condition. For a modest SKR of 1 kbits, the fixed channel loss is 25 dB for the model and low turbulent conditions, while for medium and strong, the fixed channel loss can only be 18.75 dB or 13.5 dB. Considering the additional loss (and extra post-processing associated with it due to higher QBER) for each of the turbulences brings the total loss in line. With this system, the maximum fixed loss where a positive SKR was obtained was at 32 dB with a value of 5 bps. At high losses, the SKR disagrees with the model due to the extra QBER measured. Medium and strong turbulences introduce large additional attenuation, the maximum fixed channel loss where a positive SKR was obtained was 16 dB due to the extra post-processing needed. The additional losses could be compensated, to a degree, with an AO system that would focus again the beam into the active area of the detector. Due to the passive interferometer design, which is capable of working with multimode beams, the requirements for the AO system would not be as high and other solutions, therefore lower specifications of AO could be used. In addition, if a larger active area detector was used, the losses would decrease, but this creates an issue with background noise due to large field-of-view. However, pixelated single-photon detector arrays could be used to with post-processing to increase the field-of-view while decreasing background noise through signal selection [32]. Figure 5(a) also shows with a dotted line the raw key of the experiment, which demonstrates the key is always proportional to the channel loss.
Fig. 5. Secret key rate and visibility for the COW system. The black line in (a) is the theoretical model for a 92% visibility and a low turbulence scenario. The solid points in (a) shows the secret key rate for a fixed channel loss (see above), for three different turbulence regimes. In dotted line, the sifted key is shown following the same color legend. Figure (b) plots the visibilities recorded for every point in (a). We attribute the fluctuations on the measurement due to following a manual process that not always was able to obtain the maximum visibility of the interferometer.
Like previous demonstrations of the asymmetric interferometer that have simulated turbulence with multimode optical fiber [30,32,34], we showed that our method of simulating turbulence is also valid and maintains high visibility, see Fig. 5(b). For both the average and “snap-shot” turbulence patterns (Fig. 2), we see that the single-photon-level visibility remains high before the background noise dominates the measurement; 35 dB for low, ∼35 dB for medium, 28 dB for high. The visibility drop off point is correlated with the signal to noise of the measurement, and due to the additional loss for higher orders of turbulence, the medium and strong drop off at lower fixed channel losses than the low. All this verify that the optical design is robust against atmospheric aberration in the beam, making it suitable for free-space QKD.
The SKR can be improved in future demonstrations by focusing on critical QBER contributors, and optimization of the receiver splitting ratios and optical design. In our experiment, the main QBER contribution was primarily the on-off time of the PPG, which gave the largest QBER contribution of 2.3%. By increasing the bandwidth of the PPG and of the electrical amplifier, a lower QBER can be obtained. In fiber-based implementations of the COW protocol, the splitting at the receiver is more asymmetric, more light to the key detection stage, due to the low background counts of the detectors used in the implementations [26,55]. As the application here is free-space, where there will be more noise, a study on optimization of the beamsplitter ratio would likely improve the SKR. Finally, the half inch optics used in the receiver reduces the field of the view of the optical system and results in aberrations at low beam angles. Increasing the diameter of critical elements, such as the relay optics, would improve the visibility and alignment of the system.
5. Conclusion
In conclusion, in this work we have demonstrated an in-lab free space turbulence simulator that can be used to simulate atmospheric turbulence and, to the best of our knowledge, the first demonstration of the COW protocol using a free-space optical receiver.
Our COW demonstration utilized an asymmetric interferometer free-space optical receiver with passive relay-lenses, which was designed to operate at a pattern frequency rate of 500 MHz, clock frequency of 1 GHz. Atmospheric turbulence was modelled for low, medium, and high values. The resultant turbulent images were applied to a SLM to simulate atmospheric turbulence of a transmitted QKD signal. Over all values for turbulence, our interferometer maintained an interferometric visibility of 92%. For a fixed channel loss of 16 dB, discounting losses associated with turbulence, we show secure key generation rates of 6.4 kbps, 3.4 kbps, and 270 bps for low, medium, and high turbulence levels, respectively. The results highlight that atmospheric turbulence is a non-negligible parameter for free-space optical channel modelling.
The COW protocol, and time-bin protocols in general, is a promising candidate for free-space QKD, and could be an alternative to the dominant polarization protocol implementations. The COW protocol is less complex to implement when operated at higher bandwidth frequencies, and it only requires two detectors and one light source, instead of multiple lasers and single-photon detectors commonly used in polarization protocols. Alternative compact designs for free-space receiver interferometers also make the COW, and other time or phase based protocols, attractive for free-space implementation [31,33]. The COW protocol is also polarization insensitive, meaning polarization compensation before the receiver is not required, which would be beneficial in satellite implementations.
Funding
Innovate UK (TS/S009353/1); Royal Academy of Engineering (RF\201718\1746); Engineering and Physical Sciences Research Council (EP/M013472/1, EP/T001011/1).
Acknowledgements
We would like to acknowledge contributions from Umberto Nasti (HWU) during experimental set up, Catarina Novo (HWU) during experimental design, and Robert Starkwood, Philip Dolan, and Christopher Chunnilall (NPL) for characterizing the photon number of the source reported in this paper.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are available in Ref. [60]
References
1. R. J. Collins, R. Amiri, M. Fujiwara, T. Honjo, K. Shimizu, K. Tamaki, M. Takeoka, M. Sasaki, E. Andersson, and G. S. Buller, “Experimental demonstration of quantum digital signatures over 43 dB channel loss using differential phase shift quantum key distribution,” Sci. Rep. 7(1), 3235 (2017). [CrossRef]
2. R. J. Donaldson, R. J. Collins, K. Kleczkowska, R. Amiri, P. Wallden, V. Dunjko, J. Jeffers, E. Andersson, and G. S. Buller, “Experimental demonstration of kilometer-range quantum digital signatures,” Phys. Rev. A 93(1), 012329 (2016). [CrossRef]
3. J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of quantum fingerprinting beating the classical limit,” Phys. Rev. Lett. 116, 240502 (2016). [CrossRef]
4. Y. Liu, Y. Cao, M. Curty, S.-K. Liao, J. Wang, K. Cui, Y.-H. Li, Z.-H. Lin, Q.-C. Sun, D.-D. Li, H.-F. Zhang, Y. Zhao, T.-Y. Chen, C.-Z. Peng, Q. Zhang, A. Cabello, and J.-W. Pan, “Experimental Unconditionally Secure Bit Commitment,” Phys. Rev. Lett. 112(1), 010504 (2014). [CrossRef]
5. R. Amiri, R. Stárek, M. Mičuda, L. Mišta, M. Dušek, P. Wallden, and E. Andersson, “Imperfect 1-out-of-2 quantum oblivious transfer: bounds, a protocol, and its experimental implementation,” PRX Quantum 2010335 (2021). [CrossRef]
6. W. Tittel, H. Zbinden, and N. Gisin, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
7. S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” Adv. Opt. Photonics 12(4), 1012 (2020). [CrossRef]
8. Y.-A. Chen, Q. Zhang, T.-Y. Chen, W. Cai, S. Liao, J. Zhang, K. Chen, J. Yin, J.-G. Ren, Z. Chen, S.-L. Han, Q. Yu, K. Liang, F. Zhou, X. Yuan, M.-S. Zhao, T.-Y. Wang, X. Jiang, L. Zhang, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, C.-Y. Lu, R. Shu, J.-Y. Wang, L. Li, N.-L. Liu, F. Xu, X.-B. Wang, C.-Z. Peng, and J.-W. Pan, “An integrated space-to-ground quantum communication network over 4,600 kilometres,” Nature 589(7841), 214–219 (2021). [CrossRef]
9. J. F. Dynes, A. Wonfor, W. W.-S. Tam, A. W. Sharpe, R. Takahashi, M. Lucamarini, A. Plews, Z. L. Yuan, A. R. Dixon, J. Cho, Y. Tanizawa, J.-P. Elbers, H. Greißer, I. H. White, R. V. Penty, and A. J. Shields, “Cambridge quantum network,” npj Quantum Inf. 5(1), 101 (2019). [CrossRef]
10. M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S. Miki, T. Yamashita, Z. Wang, A. Tanaka, K. Yoshino, Y. Nambu, S. Takahashi, A. Tajima, A. Tomita, T. Domeki, T. Hasegawa, Y. Sakai, H. Kobayashi, T. Asai, K. Shimizu, T. Tokura, T. Tsurumaru, M. Matsui, T. Honjo, K. Tamaki, H. Takesue, Y. Tokura, J. F. Dynes, A. R. Dixon, A. W. Sharpe, Z. L. Yuan, A. J. Shields, S. Uchikoga, M. Legré, S. Robyr, P. Trinkler, L. Monat, J.-B. Page, G. Ribordy, A. Poppe, A. Allacher, O. Maurhart, T. Länger, M. Peev, and A. Zeilinger, “Field test of quantum key distribution in the Tokyo QKD Network,” Opt. Express 19(11), 10387 (2011). [CrossRef]
11. P. J. Clarke, R. J. Collins, P. A. Hiskett, M.-J. García-Martínez, N. J. Krichel, A. McCarthy, M. G. Tanner, J. A. O’Connor, C. M. Natarajan, S. Miki, M. Sasaki, Z. Wang, M. Fujiwara, I. Rech, M. Ghioni, A. Gulinatti, R. H. Hadfield, P. D. Townsend, and G. S. Buller, “Analysis of detector performance in a gigahertz clock rate quantum key distribution system,” New J. Phys. 13(7), 075008 (2011). [CrossRef]
12. J.-P. Chen, C. Zhang, Y. Liu, C. Jiang, W.-J. Zhang, Z.-Y. Han, S.-Z. Ma, X.-L. Hu, Y.-H. Li, H. Liu, F. Zhou, H.-F. Jiang, T.-Y. Chen, H. Li, L.-X. You, Z. Wang, X.-B. Wang, Q. Zhang, and J.-W. Pan, “Twin-Field Quantum Key Distribution over 511 km Optical Fiber Linking two Distant Metropolitans,” Nat. Photon. 15, 570–575 (2021). [CrossRef]
13. M. Pittaluga, M. Minder, M. Lucamarini, M. Sanzaro, R. I. Woodward, M.-J. Li, Z. Yuan, and A. J. Shields, “600 Km Repeater-Like Quantum Communications With Dual-Band Stabilisation,” Nat. Photon. 15, 530–535 (2020). [CrossRef]
14. P. Shi, S.-C. Zhao, W.-D. Li, and Y.-J. Gu, “Feasibility of underwater free space quantum key distribution,” J. Opt. Soc. Am. A 32, 349–356 (2015). [CrossRef]
15. L. Ji, J. Gao, A.-L. Yang, Z. Feng, X.-F. Lin, Z.-G. Li, and X.-M. Jin, “Towards quantum communications in free-space seawater,” Opt. Express 25(17), 19795 (2017). [CrossRef]
16. Z. Feng, S. Li, and Z. Xu, “Experimental underwater quantum key distribution,” Opt. Express 29(6), 8725 (2021). [CrossRef]
17. F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. w. Boyd, and E. Karimi, “Quantum cryptography with twisted photons through an outdoor underwater channel,” Opt. Express 26(17), 22563 (2018). [CrossRef]
18. S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549(7670), 43–47 (2017). [CrossRef]
19. D. Cozzolino, B. Da Lio, D. Bacco, and L. K. Oxenløwe, “High-Dimensional Quantum Communication: Benefits, Progress, and Future Challenges,” Adv. Quantum Technol. 2(12), 1900038 (2019). [CrossRef]
20. X. Wang, X. Wang, T. Wu, C. Dong, H. Zhu, Z. Zhu, and S. Zhao, “Integrating deep learning to achieve phase compensation for free-space orbital-angular-momentum-encoded quantum key distribution under atmospheric turbulence,” Photonics Res. 9(2), B9–B17 (2021). [CrossRef]
21. D. M. Benton, P. M. Gorman, P. R. Tapster, and D. M. Taylor, “A compact free space quantum key distribution system capable of daylight operation,” Opt. Commun. 283(11), 2465–2471 (2010). [CrossRef]
22. H. Chun, I. Choi, G. Faulkner, L. Clarke, B. Barber, G. George, C. Capon, A. Niskanen, J. Wabnig, D. O’Brien, and D. Bitauld, “Handheld free space quantum key distribution with dynamic motion compensation,” Opt. Express 25(6), 6784 (2017). [CrossRef]
23. D. Lowndes, S. Frick, A. Hart, and J. Rarity, “A low cost, short range quantum key distribution system,” EPJ Quantum Technol. 8(1), 15 (2021). [CrossRef]
24. D. Stucki, N. Brunner, N. Gisin, V. Scarani, and H. Zbinden, “Fast and simple one-way quantum key distribution,” Appl. Phys. Lett. 87(19), 194108 (2005). [CrossRef]
25. N. Namekata, H. Takesue, T. Honjo, Y. Tokura, and S. Inoue, “High-rate quantum key distribution over 100 km using ultra-low-noise, 2-GHz sinusoidally gated InGaAs/InP avalanche photodiodes,” Opt. Express 19(11), 10632 (2011). [CrossRef]
26. P. Sibson, C. Erven, M. Godfrey, S. Miki, T. Yamashita, M. Fujiwara, M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. L. O’Brien, and M. G. Thompson, “Chip-based quantum key distribution,” Nat. Commun. 8(1), 13984 (2017). [CrossRef]
27. A. Ocampos-Guillén, J. Gómez-García, N. Denisenko, and V. Fernandez, “Double-loop wavefront tilt correction for free-space quantum key distribution,” IEEE Access 7, 114033–114041 (2019). [CrossRef]
28. Y. Cao, Y. H. Li, K. X. Yang, Y. F. Jiang, S. L. Li, X. L. Hu, M. Abulizi, C. L. Li, W. Zhang, Q. C. Sun, W. Y. Liu, X. Jiang, S. K. Liao, J. G. Ren, H. Li, L. You, Z. Wang, J. Yin, C. Y. Lu, X. Bin Wang, Q. Zhang, C. Z. Peng, and J. W. Pan, “Long-Distance Free-Space Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 125, 260503 (2020). [CrossRef]
29. G. Vallone, D. Dequal, M. Tomasin, F. Vedovato, M. Schiavon, V. Luceri, G. Bianco, and P. Villoresi, “Interference at the Single Photon Level Along Satellite-Ground Channels,” Phys. Rev. Lett. 116(25), 253601 (2016). [CrossRef]
30. J. Jin, S. Agne, J. P. Bourgoin, Y. Zhang, N. Lütkenhaus, and T. Jennewein, “Demonstration of analyzers for multimode photonic time-bin qubits,” Phys. Rev. A 97(4), 1–10 (2018). [CrossRef]
31. C. Cahall, N. T. Islam, D. J. Gauthier, and J. Kim, “Multimode Time-Delay Interferometer for Free-Space Quantum Communication,” Phys. Rev. Appl. 13(2), 024047 (2020). [CrossRef]
32. R. Donaldson, D. Kundys, A. Maccarone, R. Henderson, G. S. Buller, and A. Fedrizzi, “Towards combined quantum bit detection and spatial tracking using an arrayed single-photon sensor,” Opt. Express 29(6), 8181 (2021). [CrossRef]
33. A. Tello, C. Novo, and R. Donaldson, “Prospects of time-bin quantum key distribution in turbulent free-space channels,” in Emerging Imaging and Sensing Technologies for Security and Defence V; and Advanced Manufacturing Technologies for Micro- and Nanosystems in Security and Defence III, M. Farsari, J. G. Rarity, F. Kajzar, A. Szep, R. C. Hollins, G. S. Buller, R. A. Lamb, M. Laurenzis, A. Camposeo, L. Persano, L. E. Busse, M. Dušek, P. M. Alsing, M. L. Fanto, and R. Zamboni, eds. (SPIE, 2020), 1154006(September), p. 2.
34. J. Jin, J.-P. Bourgoin, R. Tannous, S. Agne, C. J. Pugh, K. B. Kuntz, B. L. Higgins, and T. Jennewein, “Genuine time-bin-encoded quantum key distribution over a turbulent depolarizing free-space channel,” Opt. Express 27(26), 37214 (2019). [CrossRef]
35. H. Chen, J. Wang, B. Tang, Z. Li, B. Liu, and S. Sun, “Field demonstration of time-bin reference-frame-independent quantum key distribution via an intracity free-space link,” Opt. Lett. 45(11), 3022 (2020). [CrossRef]
36. B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Corrigendum: Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication (2014 New J. Phys. 16 033020),” New J. Phys. 16(8), 089501 (2014). [CrossRef]
37. N. Gisin, G. Ribordy, H. Zbinden, D. Stucki, N. Brunner, and V. Scarani, “Towards practical and fast Quantum Cryptography,” arXiv:quant-ph/0411022 (2004).
38. Z. Ghassemlooy, W. Popoola, and S. Rajbhandari, Optical Wireless Communications : System and Channel Modelling with MATLAB® (CRC Press, 2019).
39. A. Arockia Bazil Raj, J. A. V. Selvi, and S. Raghavan, “Real-time measurement of meteorological parameters for estimating low-altitude atmospheric turbulence strength (C2n),” IET Sci. Meas. Technol. 8(6), 459–469 (2014). [CrossRef]
40. Y. Wang, H. Xu, D. Li, R. Wang, C. Jin, X. Yin, S. Gao, Q. Mu, L. Xuan, and Z. Cao, “Performance analysis of an adaptive optics system for free-space optics communication through atmospheric turbulence,” Sci. Rep. 8(1), 1124 (2018). [CrossRef]
41. L. Burger, I. A. Litvin, and A. Forbes, “Simulating Atmospheric Turbulence Using a Phase-Only Spatial Light Modulator,” S. Afr. J. Sci. 104, 3 (2008).
42. B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16(3), 033020 (2014). [CrossRef]
43. N. Chimitt and S. H. Chan, “Simulating anisoplanatic turbulence by sampling intermodal and spatially correlated Zernike coefficients,” Opt. Eng. 59(08), 1 (2020). [CrossRef]
44. P. Chaiwongkhot, K. B. Kuntz, Y. Zhang, A. Huang, J. P. Bourgoin, S. Sajeed, N. Lütkenhaus, T. Jennewein, and V. Makarov, “Eavesdropper’s ability to attack a free-space quantum-key-distribution receiver in atmospheric turbulence,” Phys. Rev. A 99(6), 062315 (2019). [CrossRef]
45. C. Liorni, H. Kampermann, and D. Bruß, “Satellite-based links for quantum key distribution: Beam effects and weather dependence,” New J. Phys. 21(9), 093055 (2019). [CrossRef]
46. Y. Chahine, S. A. Tedder, B. E. Vyhnalek, and A. C. Wroblewski, “Beam propagation through atmospheric turbulence using an altitude-dependent structure profile with non-uniformly distributed phase screens,” Proc. SPIE 11272, 1127215 (2020). [CrossRef]
47. G. S. Buller and R. J. Collins, “Single-photon generation and detection,” Meas. Sci. Technol. 21(1), 012002 (2010). [CrossRef]
48. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24(11), 3500 (2007). [CrossRef]
49. C. Branciard, N. Gisin, and V. Scarani, “Upper bounds for the security of two distributed-phase reference protocols of quantum cryptography,” New J. Phys. 10(1), 013031 (2008). [CrossRef]
50. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009). [CrossRef]
51. M. Li, T. Patcharapong, C. M. Zhang, Z. Q. Yin, W. Chen, and Z. F. Han, “Efficient error estimation in quantum key distribution,” Chin. Phys. B 24(1), 010302 (2015). [CrossRef]
52. A. Poppe, A. Fedrizzi, R. Ursin, H. R. Böhm, T. Lorünser, O. Maurhardt, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, “Practical quantum key distribution with polarization entangled photons,” Opt. Express 12(16), 3865 (2004). [CrossRef]
53. G. Brassard and L. Salvail, “Secret-key reconciliation by public discussion,” in Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Springer Verlag, 1994), 765 LNCS, pp. 410–423.
54. M. Mehic, M. Niemiec, H. Siljak, and M. Voznak, “Error reconciliation in quantum key distribution protocols,” in Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Springer, 2020), 12070 LNCS, pp. 222–236.
55. B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015). [CrossRef]
56. D. Stucki, C. Barreiro, S. Fasel, J.-D. Gautier, O. Gay, N. Gisin, R. Thew, Y. Thoma, P. Trinkler, F. Vannel, and H. Zbinden, “Continuous high speed coherent one-way quantum key distribution,” Opt. Express 17(16), 13326 (2009). [CrossRef]
57. J. González-Payo, R. Trényi, W. Wang, and M. Curty, “Upper Security Bounds for Coherent-One-Way Quantum Key Distribution,” Phys. Rev. Lett. 125(26), 260510 (2020). [CrossRef]
58. R. Trényi and M. Curty, “Zero-error attack against coherent-one-way quantum key distribution,” New J. Phys. 23(9), 093005 (2021). [CrossRef]
59. M. Curty, “Foiling zero-error attacks against coherent-one-way quantum key distribution,”Phys. Rev. A 104, 062417 (2021). [CrossRef]
60. A. Tello and R. Donaldson, “In-Lab Demonstration of Coherent One-Way Protocol over Free Space with Turbulence Simulation”, Herit Watt University (2022), https://doi.org/10.17861/5d51762d-e19c-438f-8e03-e4b43c67f774
