Quantum random number generation using an on-chip nanowire plasmonic waveguide

Quantum random number generators employ the inherent randomness of quantum mechanics to generate truly unpredictable random numbers, which are essential in cryptographic applications. While a great variety of quantum random number generators have been realized using photonics, few exploit the high-field confinement offered by plasmonics, which enables device footprints an order of magnitude smaller in size. Here we integrate an on-chip nanowire plasmonic waveguide into an optical time-of-arrival based quantum random number generation setup. Despite loss, we achieve a random number generation rate of 14.4Mbitss−1 using low light intensity, with the generated bits passing industry standard tests without post-processing. By increasing the light intensity, we were then able to increase the generation rate to 41.4Mbitss−1 , with the resulting bits only requiring a shuffle to pass all tests. This is an order of magnitude increase in the generation rate and decrease in the device size compared to previous work. Our experiment demonstrates the successful integration of an on-chip nanoscale plasmonic component into a quantum random number generation setup. This may lead to new opportunities in compact and scalable quantum random number generation.


I. INTRODUCTION
Random numbers are used extensively in cryptography [1], simulation [2] and fundamental physics tests [3], as well as in lotteries, machine learning and coordination in computer networks [4].When classical techniques are used to generate random numbers, the unpredictability relies on incomplete knowledge, which can result in the random numbers being more predictable than anticipated.In fact, it was shown in a number of recent studies that even simple machine learning algorithms can successfully predict the random numbers generated by poor quality classical pseudorandom number generators [5][6][7], rendering them useless for cryptography.While high quality cryptographically secure pseudorandom number generators have been developed, they are incredibly resource intensive [8][9][10].Furthermore, it is unclear whether these cryptographically secure classical pseudorandom number generators will be able to withstand future improved machine learning algorithms, since they are still fundamentally deterministic.
The past two decades have seen an increasing body of work aimed at understanding quantum features of plasmons and how plasmonic confinement and losses affect the transport of quantum states of light [57][58][59][60][61][62][63][64][65][66][67][68].The ability of plasmonics to confine light to subwavelength scales, well below the diffraction limit [69,70], and to simultaneously carry optical and electrical signals [71,72] suggests that quantum information processing protocols can be carried out at a much smaller scale than in the dielectric systems typically used in photonics [73,74] and that such systems can be integrated with existing electronics via electrically controlled and tuned plasmonic devices via carrier injection and electrical modulation.Studies have already resulted in an array of applications, such as quantum plasmonic sensing [75][76][77][78][79][80][81][82][83], plasmonic entanglement generation [84] and distillation [85,86], plasmonic quantum gates [87] and plasmonic quantum state engineering [88], which have revealed that, despite being inherently lossy, plasmonic components can be successfully employed in quantum information processing tasks.
Recently, the branching paths scheme for quantum random number generation was demonstrated using an on-chip plasmonic beamsplitter [89].However, other schemes exist, for example the time-of-arrival scheme is very different to the branching paths scheme and has two major advantages, namely it requires one detector instead of two, and substantially higher bit rates are possible as multiple bits of randomness can be extracted from a single photon.So far, the time-of-arrival scheme has not been demonstrated using plasmonics.More-over, most time-of-arrival generators are bulky and use a highly attenuated coherent source as a single-photon source, where photons simply propagate through free space to a single-photon detector [16][17][18][19][20][21][22][23][24].An integrated circuit time-of-arrival generator has been realised recently by adding a semi-coherent silicon LED source directly on a detector chip [41].However, an alternative and more flexible option for on-chip time-of-arrival based quantum random number generation is to embed an on-chip source [90][91][92][93] and detector [94][95][96][97] within a plasmonic waveguide.
In this paper we report a time-of-arrival based quantum random number generation scheme using a nanowire plasmonic waveguide.In particular, we integrate an on-chip nanowire plasmonic waveguide into an optical time-of-arrival based quantum random number generation setup and test its performance in the presence of loss.Despite loss, we initially managed to achieve a random number generation rate of 14.4 Mbits/s.The generated bits did not require any post-processing to pass the industry standard ENT [98] and NIST [99] Statistical Test Suites.By increasing the light intensity, we were able to increase the generation rate to 41.4 Mbits/s, however these bits required a shuffle to pass all the tests due to some correlation being introduced.Our work demonstrates the successful integration of an on-chip nanoscale plasmonic component into a time-of-arrival based quantum random number generation setup.The specific advantage offered by introducing an on-chip nanowire plasmonic waveguide into the setup is that light in the plasmonic waveguide can be confined to a length scale well below the diffraction limit, which enables the footprint of the on-chip component to be reduced to a size unattainable with dielectric hardware [69,70].This research therefore makes an important contribution to addressing the on-going challenge of miniaturising on-chip quantum random number generators, as it shows how an on-chip nanoscale plasmonic component, with a footprint well below that of equivalent state-of-the-art dielectric components, can be successfully used in quantum random number generation despite it being inherently lossy.Although our current setup employs an off-chip source and detector, future work on the integration of an on-chip source [90][91][92][93] and detector [94][95][96][97] will yield a fully integrated nanophotonic quantum random number generator chip with a footprint an order of magnitude smaller than its dielectric counterpart.This opens up new opportunities in compact and scalable quantum random number generation.

II. EXPERIMENTAL SETUP
The experimental setup used to investigate time-ofarrival based quantum random number generation using a nanowire plasmonic waveguide is shown in Fig. 1a.The plasmonic waveguide used in the experiments comprises a gold nanowire 70 nm in diameter and just over 3 µm in length with tapering and a grating with a period of 740 nm on either end (see Fig. 1b).Each 11-step grating is 2 µm in width and 70 nm in height.In the optical setup, highly attenuated coherent laser light, with a vacuum wavelength of 785 nm, is focused onto the input grating of the nanowire plasmonic waveguide using a diffraction-limited microscope (DLM) objective.At the input grating, photons are converted to surface plasmon polaritons, which propagate through the tapering regions and the nanowire waveguide to the output grating, where they are converted back to photons.The gratings provide free-space photons with the additional momentum needed to couple into the bound plasmonic waveguide mode, thereby enabling the conversion of photons to surface plasmon polaritons and vice versa [89,[100][101][102].The tapering regions adiabatically nanofocus surface plasmon polaritons into and out of the nanowire waveguide [103].The effective mode index of the characteristic mode of the gold nanowire waveguide was determined to be n eff = 1.84 + 0.0573i using a 2D finite element method simulation in COMSOL (see Appendix A).The associated electric field distribution through a crosssection of the nanowire waveguide is shown in Fig. 1c.We see that the electric field is highly confined at the corners of the nanowire waveguide, at a length scale well below the diffraction limit -something which would have been impossible with a dielectric waveguide.Photons are collected from the output grating of the nanowire plasmonic waveguide using the same DLM objective that was used to focus light onto its input grating, and are then sent to a single-photon detector.The arrival times of photons at the detector, relative to an external reference (see Fig. 1d), are then used to obtain random numbers [21], as will be explained in more detail later.
The temporal degree of freedom of photons generated during stimulated emission is a true source of randomness [23].In this work we employ a λ = 785 nm continuous-wave laser (Thorlabs LPS-785-FC) operating in the stimulated emission regime.The wavelength of our continuous-wave laser source is chosen to be well within the coupling window of the 740 nm-period gratings of our on-chip nanowire plasmonic waveguide, which ensures that the conversion between photons and surface plasmon polaritons takes place with maximal efficiency at these gratings [89,100].A polarisation-preserving singlemode optical fibre (SM) connects the continuous-wave laser to a beam expander (BE), through which polarised coherent laser light enters the optical setup in Fig. 1a.The collimated beam passes through a neutral density filter (NDF), a half-wave plate (HWP), a quarter-wave plate (QWP), a polarising beamsplitter (PBS) and a second HWP.The NDF, along with loss in the optical setup and the plasmonic waveguide, ensures that light reaching the detector is attenuated down to an appropriate level for single-photon detection.The first HWP, the QWP and the PBS are used to purify the polarisation of light from the laser.In particular, the PBS transmits only horizontally polarised photons and the preceding HWP and QWP adjust the polarisation of light incident on the PBS so as to maximise the transmission of light through the PBS.The second HWP rotates the resulting horizontally polarised light so as to maximise the conversion of photons to surface plasmon polaritons at the input grating of the nanowire plasmonic waveguide, which can only be achieved when the polarisation vector is perpendicular to the gratings [89,100].A 100x DLM objective is used to focus the beam onto the input grating of the nanowire plasmonic waveguide at a spot size of about 2 µm.
The plasmonic waveguide is fabricated on a silica glass substrate with a refractive index of 1.5255 and a thickness of 1 mm using a combination of electron beam lithography and electron beam deposition.A resist is first spin coated on the silica glass substrate and 20 nm of gold is deposited so that the surface becomes conductive.Electron beam lithography is used to define the regions for the nanowire, the taperings and the gratings.The gold is then etched and resist is developed.Next a lift-off technique is employed, where first a 5 nm thick adhesion layer of titanium is deposited and then the desired 70 nm thick gold layer on the silica glass substrate.The unexposed resist and gold is then lifted off with acetone, IPA and de-ionised water.Atomic force microscope (NT-MDT Smena) images of the fabricated on-chip nanowire plasmonic waveguide are shown in Fig. 1e.
The power transmission factor of the nanowire plasmonic waveguide was measured to be η wgd = 6.7 × 10 −5 (see Appendix B).Losses occur in the waveguide as a result of scattering during the conversion between photons and surface plasmon polaritons at the gratings, leakage from the tapering regions, and absorption during the propagation of surface plasmon polaritons along the nanowire.By using results from Appendix A, we were able to estimate the power transmission factor of the nanowire as well as the net power transmission factor of a grating and tapering region in Appendix B. Hence Appendices A and B play a very important role in the characterisation of our main component by providing us with a graded picture of the different parts of our on-chip nanowire plasmonic waveguide and a quantitative analysis of the losses occurring in each of these parts, which will be of interest to researchers looking to investigate further optimisations and extensions.Note that one can use higher intensity light to keep the photon detection rate within a certain desired range even in the presence of such losses.Unlike in a previously demonstrated plasmonic quantum random number generator [89], there is no limit on the amount by which one can increase the intensity of the coherent source, since the time-of-arrival scheme does not require the nanowire plasmonic waveguide to operate in the single-excitation regime [21].
Photons are collected from the output grating of the nanowire plasmonic waveguide by the same DLM objective that was used to focus the input beam onto the input grating.A knife-edge mirror (KM) is then used to reflect these photons into a fibre coupler (FC), which is connected to a single-photon avalanche diode (SPAD) detector (Excelitas SPCM-AQRH-15) by a multi-mode optical fibre (MM).The polarisation dependence of the photon detection rate [102,104] confirms that the collection optics is indeed capturing out-coupled photons from the output grating and not scattered photons from the input beam (see Appendix C).The SPAD detector used in the experiments has a dead time of 24 ns, a timing resolution of 350 ps and a maximum dark count rate of 50 counts/s.For data collection, the SPAD detector is connected to a channel of a Picoquant TimeHarp 260, which is connected to a PC.The TimeHarp is capable of recording the arrival time of a photon at the detector to a precision of 25 ps.
We implement a variation of the time-of-arrival scheme, first proposed by Nie et al. [21], in which random numbers are obtained from the arrival times of photons relative to an external reference (see Fig. 1d).The benefit of this variation compared to earlier versions [16,19,20], in which random numbers are obtained directly from the difference of consecutive photon arrival times, is a significant reduction in bias.In the variation proposed by Nie et al. [21], the external time reference is divided into time intervals of an arbitrary but fixed length T , each of which are in turn divided into N bins of equal width.Provided that at most one photon detection can occur in a time interval of length T , it follows that a photon detection occurring in a time interval of length T occurs in each of the N bins with probability 1 N (shown in Appendix D).Hence the numbers of the bins in which photon detections occur are uniformly distributed random log 2 (N )-bit unsigned integers.The model proposed by Nie et al. [21] for the physical system used in their experimental implementation also applies to our experimental setup and we make similar assumptions and approximations.Device imperfections which could result in deviations from uniformity and degrade the quality of the random numbers generated by the system are discussed in detail in Refs.[21,23].These include a non-unit SPAD detector efficiency, SPAD detector dark counts, SPAD detector dead time and a non-zero probability of multi-photon emission from the attenuated coherent source.
For our experiment, we set T = 12.8 ns.This ensures that T is less than the dead time of the SPAD detector, which in turn ensures that at most one photon detection can occur in a time interval of length T .Furthermore, we set N = 2 8 = 256, which enables us to extract a random 8-bit unsigned integer from each photon arrival time.We note that the bin width, T N = 50 ps, is greater than the precision of the TimeHarp, but less than the timing resolution of the SPAD detector.However, results from previous experiments [21,23] suggest that the timing resolution of the detector does not significantly affect the uniformity or the quality of the random numbers.
For data collection, we adjust the light intensity so as to give a photon detection rate of 1.8 Mcounts/s.This corresponds to an average time interval of 0.56 µs between photon detections, which is much greater than the dead time of the SPAD detector, ensuring that the majority of photons arriving at the SPAD detector are indeed detected.Random 8-bit unsigned integers extracted from recorded photon arrival times are converted to binary form, resulting in a sample of binary digits or bits.The most conservative information-theoretic measure with which to quantify the randomness or unpredictability of the bits generated by our setup is the minentropy [21,23].As part of the model of the physical system used in their experiment, Nie et al. [21] show that the min-entropy of bits generated by a non-ideal experimental implementation of their time-of-arrival scheme can be estimated using the mean number of photons in a time interval of length T , which can be calculated using the photon detection rate.In Appendix E, we show that for our chosen photon detection rate of 1.8 Mcounts/s, the mean photon number is 0.024, which gives a min-entropy of 0.998 per bit.This is very close to the informationtheoretically optimal value of 1 per bit, which shows that our experimental setup is capable of creating very highquality randomness even in the presence of the device imperfections considered in the model used by Nie et al. [21].
In their own implementation, Nie et al. [21] operated their SPAD detector at its saturation rate in order to maximise the random number generation rate.Consequently, the min-entropy of the raw bits generated by their setup was only 0.88 per bit and a randomness extractor was needed to increase the min-entropy.In contrast, the min-entropy for our experimental setup is very close to the information-theoretically optimal value of 1 per bit, even with device imperfections which can degrade the quality of the randomness taken into account, and so randomness extraction is not required here.We generated a sample of 866, 893, 768 bits in 60 s, which corresponds to a random number generation rate of about 14.4 Mbits/s.Note that a previous plasmonic quantum random number generator was only able to achieve a generation rate of 2.37 Mbits/s [89] and a previous on-chip time-of-arrival generator was only able to achieve a generation rate of 1 Mbits/s [41].Hence our work demonstrates an order of magnitude improvement in speed compared to these previous devices.In the next section, we apply a number of industry standard tests [98,99] to the first 800 Mbits generated, which we will refer to as the generated sample.

III. RESULTS
As a first test, we employ the Pearson correlation coefficient [105] to detect short-ranged correlations in the generated sample.The Pearson correlation coefficient is a real number in the interval [−1, 1], where a positive value suggests a positive correlation, a negative value suggests a negative correlation and a value close to zero suggests no correlation.As can be seen in Fig. 2, short-ranged correlations in the generated sample are negligible.Furthermore, the relative frequency of zeros and ones in the gen-  erated sample is 0.49995 and 0.50005 respectively.Hence the bias in the generated sample is also negligible.For further quality assessment, we apply the ENT Statistical Test Suite [98] to the generated sample.The ENT Statistical Test Suite comprises five simple tests in which the sample being tested is used to compute five important values.These include the entropy per byte, which quantifies the information density of the sample, the χ 2 distribution, which is known to be extremely sensitive to flaws in random number generators, the arithmetic mean of 8bit unsigned integers extracted from the sample, which aids in detecting bias, the Monte Carlo value for π, which provides a practical test of the generator's suitability for use in simulation, and the serial correlation coefficient, which quantifies correlations between adjacent bytes in the sample.The quality of the sample being tested can then be assessed by comparing the values obtained using the sample to the known values for a true random sample.The ENT Statistical Test Suite results for the generated sample are given in Table I.For all five tests, the values obtained using the generated sample show good agreement with the expected values for a true random sample.
As a final assessment of the quality, we apply the NIST Statistical Test Suite [99] to the generated sample.The NIST Statistical Test Suite consists of 15 stringent tests, which are primarily aimed at assessing a random number generator's suitability for use in cryptographic applications.To apply one of these tests to a sample of bits from a generator, the sample is first divided into sequences of a fixed length.The test is then applied to each sequence and a p-value, which can be used to assess the uniformity of the distribution of the test results obtained for the individual sequences, is determined.For the sample to pass the test, a sufficient number of sequences must pass the test and the p-value must be greater than or equal to 0.0001.The NIST Statistical Test Suite results for the generated sample are shown in Table II.
For each test, the generated sample was divided into 800 sequences 1 Mbit in length.The default values were used for the block length, except in the Block Frequency test, where the block length was adjusted from 2 7 = 128 to 2 14 = 16384.The generated sample passed all 15 NIST tests.Hence we conclude that the random numbers generated by our plasmonic system are of sufficient quality for cryptographic applications -without requiring any classical post-processing.This is a significant improvement compared to a previously reported plasmonic quantum random number generator [89] and a previously reported on-chip time-of-arrival generator [41], both of which required a randomness extractor to pass the NIST Statistical Test Suite.A more elaborate list of comparisons with previously reported work in the domain of time-of-arrival based quantum random number generation is presented in Table III.
In principle, the random number generation rate can be increased by increasing the light intensity, which increases the photon detection rate.However, with an increased photon detection rate, the detector dead time becomes non-negligible and the photon counts obtained in an experiment would typically need to be multiplied by a non-unit correction factor to compensate for the resulting underestimation of photon counts.In our experiment, the result is photon detection times, not photon counts, and so we are unable to correct for undetected photons.We therefore investigate the effect of detection with a non-unit correction factor on the quality of the random numbers generated in an experiment.To this end, we adjust the light intensity so as to give a photon detection rate of 5.2 Mcounts/s.In Appendix F, we show that despite the increased photon detection rate, higher order photon number within a time interval of length T is negligible.However, with an increased detection rate, the average time interval between photon detections decreases to 0.19 µs, which is closer to the dead time of the SPAD detector.The correction factor is about 1.143 (see Appendix F), which means that on average, one in every seven photons arriving at the SPAD detector are not detected.Furthermore, increasing the photon detection rate to 5. 0.993 per bit (see Appendix E).Nevertheless, the minentropy of the raw bits generated by our setup is still very close to the information-theoretically optimal value of 1 per bit and so we would require minimal, if any, classical post-processing.We generated 1, 242, 469, 056 bits in 30 s, which corresponds to a random number generation rate of about 41.4 Mbits/s.We then applied the same industry standard tests to the first 800 Mbits generated, which we will refer to as the raw sample.
We find that short-ranged correlations in the raw sample are mostly negligible, with non-negligible correlations present only between bits at 8-bit intervals (see Fig. 3a).To remove these correlations, we deterministically rearrange or shuffle the bits in the raw sample.In what follows, we will refer to the resulting sample of bits as the shuffled sample.As can be seen in Fig. 3b, shuffling the bits in the raw sample indeed removed the 8-bit interval correlations.The relative frequency of zeros and ones is 0.49993 and 0.50007 respectively, in both the raw sample and the shuffled sample, and so the bias is negligible in both samples.
The ENT Statistical Test Suite results for the raw sample and the shuffled sample are given in Table IV.For the raw sample, the χ 2 distribution and the serial correlation coefficient deviate significantly from the expected values for a true random sample.The large serial correlation coefficient for the raw sample seems to suggest that undetected photons result in correlations between adjacent bytes extracted from consecutive photon arrival times.In contrast, the χ 2 distribution and the serial correlation coefficient obtained using the shuffled sample show excellent agreement with the expected values for a true random sample.The negligible serial correlation coefficient for the shuffled sample shows that rearranging the bits in the raw sample, so that the eight consecutive bits which make up a given byte are extracted from eight different non-consecutive photon arrival times, removes the correlations between adjacent bytes.Finally, we note that both the raw sample and the shuffled sample passed the NIST Statistical Test Suite (see Table V).Hence both these samples are of sufficient quality for cryptographic applications.This implies that the short-ranged correlations in the raw sample, while nonnegligible, are too small to impair the quality to such an extent that the raw sample is unusable for cryptographic applications.This confirms that a deterministic shuffle, which essentially just transforms short-ranged correlations into long-ranged correlations, is sufficient to improve the quality of the raw sample and more sophisticated randomness extraction schemes are not needed.In summary, we were able to show that performing detection with a non-unit correction factor introduces small short-ranged correlations into the random bits generated by our plasmonic system, which can be removed with a simple deterministic shuffle provided that the correction factor is not too large.

IV. CONCLUSION
Our work demonstrates the successful integration of a nanowire plasmonic waveguide into an optical time-ofarrival based quantum random number generation setup.The specific advantage offered by introducing an on-chip nanowire plasmonic waveguide into the setup is that light in the plasmonic waveguide can be confined to a length scale well below the diffraction limit, which enables the footprint of the on-chip waveguide to be reduced to a size unattainable with dielectric hardware [69,70].Despite the presence of loss in the plasmonic waveguide and in the optical setup, we first managed to achieve a random number generation rate of 14.4 Mbits/s.This is an order of magnitude improvement in speed compared to both a previous plasmonic quantum random number generator [89] and a previous on-chip time-of-arrival generator [41].Furthermore, unlike these previous devices, our generator did not require any classical post-processing to pass the NIST Statistical Test Suite.We were able to increase the generation rate to 41.4 Mbits/s, with the resulting bits only requiring a shuffle to pass all the tests.Our study makes an important contribution to addressing the on-going challenge of miniaturising on-chip quantum random number generators, as it shows how an onchip nanoscale plasmonic component, with a footprint well below that of equivalent state-of-the-art dielectric components, can be successfully employed in quantum random number generation.We note that although our current setup relies on an off-chip source and the detection is also done off-chip, future work on the integration of an on-chip source [90][91][92][93] and detector [94][95][96][97] would enable a self-contained plasmonic quantum random number generator chip with a footprint an order of magnitude smaller than its dielectric counterpart.The primary challenge associated with integrating an on-chip source into our nanowire plasmonic waveguide is that the on-chip near-field single-photon sources typically used in plasmonic systems emit light of a much lower intensity than that of the off-chip source currently used in our setup [91][92][93].A reduction in the light intensity would result in a reduction in the random number generation rate.Thus, the development of a bright on-chip light source will be key in a future integrated version of our device.On the other hand, the primary challenge associated with integrating an on-chip detector into our nanowire plasmonic waveguide is that the on-chip near-field superconducting single-photon detectors typically used in plasmonic systems require cryogenic cooling to around 4 K [96,97].This would greatly increase the footprint and power consumption of the overall system.Thus, the development of compact cryogenic cooling systems will be important in a future integrated version of our device.Nevertheless, if these challenges can be overcome, then the successful demonstration of plasmonic quantum random number generation with a fully integrated on-chip near-field source and detector would lead to new opportunities in compact and scalable quantum random number generation.
the power is proportional to the light intensity, which is in turn proportional to |E| 2 , and that |E| is proportional to e −Im(k)z , and so η nwr = e −2Im(k)z = 0.06, where Im(k) = Im(n eff )k 0 = 0.459 rad/µm is the attenuation constant for a vacuum wavelength of 785 nm and z ≈ 3 µm is the length of the nanowire.It follows that η grt η tpr = 0.03.

FIG. 1 :
FIG. 1: Quantum random number generation using an on-chip nanowire plasmonic waveguide.(a) Experimental Setup shows the setup used to investigate time-of-arrival based quantum random number generation using an on-chip nanowire plasmonic waveguide.The labels used are: single-mode optical fibre (SM), beam expander (BE), neutral density filter (NDF), half-wave plate (HWP), quarter-wave plate (QWP), polarising beamsplitter (PBS), diffraction-limited microscope (DLM), knife-edge mirror (KM), fibre coupler (FC) and multi-mode optical fibre (MM).(b) Nanowire Plasmonic Waveguide shows a top view of the on-chip nanowire plasmonic waveguide used in the experiments.(c) Waveguide Mode shows the electric field distribution of the characteristic mode through a cross-section of the nanowire waveguide for a vacuum wavelength of 785 nm.The grey square shows the nanowire waveguide and the grey line shows the substrate surface.(d) Time-of-arrival Scheme illustrates the implemented variation of the time-of-arrival scheme, in which random numbers are obtained from the arrival times of photons relative to an external time reference [21].(e) Atomic Force Microscope Images shows atomic force microscope (AFM) images of the fabricated on-chip nanowire plasmonic waveguide.These include an AFM image of the entire nanowire plasmonic waveguide (left), an AFM image of the top tapering (top centre), an AFM height profile of the top tapering (bottom centre), an AFM image of the top grating (top right) and an AFM height profile of the top grating (bottom right).

FIG. 2 :
FIG.2: Pearson correlation coefficient of the generated sample with 1-bit to 15-bit delays of itself.

FIG. 3 :
FIG. 3: Pearson correlation coefficient of the (a) raw sample and (b) shuffled sample with 1-bit to 15-bit delays of itself.

TABLE II :
NIST Statistical Test Suite results for the generated sample.'Req' shows the minimum number of 800 sequences which need to pass a test for the sample to pass the test.'Prop' shows the number of sequences of the generated sample which passed each test.For tests which involve more than five subtests (marked with *) the median of the results is presented.

2
Mcounts/s increases the mean number of photons in a time interval of length T to 0.076, which slightly decreases the min-entropy to

TABLE III :
Comparison of time-of-arrival based quantum random number generators.

TABLE IV :
ENT Statistical Test Suite results for the raw sample and the shuffled sample.'Raw' shows the values obtained using the raw sample.'Shuffled' shows the values obtained using the shuffled sample.'Expected' shows the expected values for a true random sample.

TABLE V :
NIST Statistical Test Suite results for the raw sample and the shuffled sample.'Req' shows the minimum number of 800 sequences which need to pass a test for the samples to pass the test.'Prop' shows the number of sequences of the raw sample or the shuffled sample which passed each test.For tests which involve more than five subtests (marked with *) the median of the results is presented.