Effect of photon statistics on vacuum fluctuations based QRNG

The significance of random number generators (RNGs) is growing constantly, alongside the advancement of technology and science. Nowadays, there is a wide range of applications for RNGs such as games, simulations, statistics, financial transactions, and communication security [1–4]. For most of these applications, random numbers are required to be generated at a high rate [5]. Pseudo-RNGs are based on complex mathematical algorithms and can satisfy the high bit rate need. However, their randomness relies on the seed bits and they fail to be truly random as there exist algorithms that can examine the pattern of the pseudo-RNGs and predict the upcoming bits.

On the other hand, quantum random number generators (QRNGs) are proven to be the most secure RNGs because their entropy source is completely random and unpredictable [6]. Over the last few years, many QRNG schemes have been implemented in the literature based on different quantum phenomena, such as Raman scattering [7], quantum vacuum fluctuations [8–12], spontaneous emission noise [13, 14], uncertainty principle [15], photon polarization state [16], and using the LED of a mobile phone [17].

Photon statistics of a coherent light source, such as a continuous wave (CW) laser, follow the Poissonian distribution. The scattering of a coherent light by scattering centres leads to a greater variance in distribution due to the fact that each scattering centre acts as a pseudo-thermal light source. Consequently, the distribution becomes super-Poissonian, in which the fluctuations in the measured optical intensity have a broader distribution compared to the Poissonian distribution with the same mean value. The scattering measurement must not be considered as a separate process from quantum random number generation. Including an optical scattering mechanism only to one arm of the homodyne detection works on increasing the randomness of the system. An experimental QRNG scheme based on the balanced homodyne detection using a chaotic light source was proposed previously [18]. Unlike this implementation, in our setup the light distribution in one arm of the optical propagation path is Poissonian; the intensity distribution on the other arm is super-Poissonian due to the scattering process. The autocorrelation function (ACF) of the raw signal for a coherent light and scattered light show different lag values for minima (ideally zero) of the ACFs. The introduction of scattering into this setup allows for a sampling rate faster than coherent light sources without compromising the randomness characteristics which are enhanced by the high frequency of this system.

We classify light sources based on the variance of their photon distribution. They are broadly classified into; super-Poissonian, Poissonian, and sub-Poissonian. The super-Poissonian is either chaotic or thermal light. A perfectly coherent light has a Poissonian distribution. A non-classical light source can be represented by sub-Poissonian distribution. Although they share the same mean, the super-Poissonian (sub-Poissonian) distribution has a larger (smaller) variance than the Poissonian [19]. The variance of the Poissonian statistics determines the noise level, which is the shot noise.

To describe the statistics of the photons emitted from CW lasers, we illustrate a method of detecting them. A strong light beam with a high amount of photon flux cannot be detected using single-photon counters or photomultiplier tubes. Rather it needs a photodetector that generates free electrons by excitation from incident photons. These electrons flow in the form of photocurrents and can be represented by

$$\begin{equation}I = \Re \left( \lambda \right)P,\end{equation} \tag{ 1 }$$

where $\Re \left( \lambda \right)$ is the responsivity of the photodetector at a specific wavelength, and P is the power of the source. Inside the photodetector, the photocurrent flows through a load resistor R_L which creates a time varying voltage. Also, the photocurrent passes through a capacitor which blocks the DC signal in order to produce a time dependent voltage which fluctuates with incident photons. The photocurrent that fluctuates with time $I\left( t \right)$ and passes through the load resistor R_L generates a time-varying noise power P_n(t) which is defined by

$$\begin{equation}{P_{\text{n}}}\left( t \right) = {R_{\text{L}}}{\left( {\Delta I\left( t \right)} \right)^2}.\end{equation} \tag{ 2 }$$

Since the photocurrent variance $\Delta I\left( t \right)$ is proportional to the average photocurrent $\langle I \rangle$, taking the Fourier transform of the photocurrent gives

$$\begin{equation}{\left( {\Delta I } \right)^2} = 2e \langle I \rangle \Delta f,\end{equation} \tag{ 3 }$$

where e corresponds to the charge of the electron, $\langle I \rangle$ to the average time-independent photocurrent, and $\Delta f$ to the bandwidth of the detector. Thus, the quantum noise P_Q can be found by

$$\begin{equation}{P_{\text{Q}}} = {\left( {\Delta I} \right)^2}{R_{\text{L}}}.\end{equation} \tag{ 4 }$$

Equations (3) and (4) are used to find the theoretical shot noise in figure 3.

When applying a coherent laser source directly to the photodetector, the detection probability of the photons will follow the statistics of the Poisson distribution indicated by (P_Po) where their average number is equal to the variance in the same beam segment and is given by

$$\begin{equation}{{\text{P}}_{{\text{Po}}}}\left( k \right) = \frac{{{{\text{e}}^{ - \mu }}{\mu ^k}}}{{k!}},\end{equation} \tag{ 5 }$$

where k is the photon number and $\mu$ is the mean. For a scattered light source, the photons follow the super-Poissonian probability distribution indicated by (P_s-Po) as the variance is larger than the mean number of photons and is described as follows:

$$\begin{equation}{{\text{P}}_{{\text{s-Po}}}}\left( k \right) = \frac{1}{{\mu + 1}}{\left( {\frac{\mu }{{\mu + 1}}} \right)^k}.\end{equation} \tag{ 6 }$$

This probability is used to quantify the min-entropy of the QRNG as will be described in the latter section.

In order to choose the most suitable scattering sources (SSs) we focus on the practical scattering mechanisms to be applied in the experimental setup. For that, we have used unpolished aluminium and half-polished silver coated objects. This preference is made due to the high reflectivity of aluminium and silver for infrared, and even near-infrared/visible wavelengths [20]. The common feature of both aluminium and silver 5 mm radii objects, is their rough surface that allows light beams to diffract in different directions and behave as pseudo-thermal light sources.

The elliptical 900 silver object with a relatively high purity has a rough surface with root mean square roughness of 1.10 ± 0.05 mm. It is known that mainly the macroscopic deficiencies like deeper scratches and digs are the major cause of scattering for infrared light sources [21]. The aluminium object has a significantly low thickness of 0.02 mm and it is due to its highly uneven surface [22]. It is out of the scope of this paper to investigate the characteristics of the scattering materials further.

As seen in the experimental scheme in figure 1, we use the homodyne detection method for eliminating the laser power fluctuations and environmental temperature related noise. An NKT Koheras BASIK CW laser (E15), with central wavelength of 1550 nm and linewidth of 100 Hz, is passed through a 50/50 beam splitter (BS) which divides the signal into two beams, one of which is directed into a 5 mm diameter elliptical object (aluminium or silver) as a static scattering object [23]. The scattered beam is detected by a photodetector of 5 GHz bandwidth (DET08CFC), alongside the other signal which is detected at the same time with another detector of the same properties. Both signals are measured using a high-speed oscilloscope (OS) (WaveRunner 640Zi).

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In the experiment, unpolished aluminium and half-polished silver are used as the SSs. While different SSs have different scattering parameters, all result in a broadened distribution of the photon statistics. Figure 2 shows the distributions of the detected signal in homodyne configuration from a coherent light (blue), scattered light from half-polished silver (green), and scattered light from unpolished aluminium (red). The light from the laser exhibits Poissonian distribution. The random nature of scattering from half-polished silver enhances the chaotic (thermal) characteristics of the light and broadens the distribution. Unpolished aluminium has a higher degree of scattering centres and the scattering is proportional to the surface roughness. Therefore, the distribution is further broadened compared to the silver's case. This shows that the chaotic behaviour becomes more dominant when the scattering centres are more irregularly distributed.

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One crucial aspect of proving the randomness of a stream of bits is the entropy it can provide per output bit. The perfect entropy in a system exists only if the number of 0's and 1's is exactly equal in a bit stream. That is, each bit can be used as an independent randomness source where it resembles a fair coin toss. Min-entropy is the most moderate bound of a functional entropy for a randomness source, where it is not feasible to quantify the randomness by other entropies such as Shannon Entropy [24]. In order to ensure that the randomness we are extracting is predominantly quantum, and not just electronic noise, we need to understand each noise source separately. This is necessary to guarantee the validity of the extracted quantum noise and predict its entropy. To show the dominance of the quantum signal over the classical noise, the power spectra of two different signals are shown in figure 3. The blue one shows the laser signal from the balanced homodyne detection configuration. The red one is the classical noise which is measured by turning the laser off.

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Min-entropy is used to quantify the amount of randomness in the system. Whereby using min-entropy we can decide on the number of bits to be received from each set of samples. In the process of calculating min-entropy, a few assumptions are made; the signal detected by the detector is the sum of two different noises; quantum noise (shot noise) and electronic noise (classical noise) given by ${X_{\text{t}}} = {X_{\text{q}}} + {X_{\text{e}}}$. The lower bound is determined by the min-entropy as it is explained above. Another assumption is that they are both independent of each other and that they will be digitized [8] as voltage values by an OS, as can be seen in figure 3. The third assumption is that an eavesdropper has full knowledge of ${X_{\text{e}}}$ which is the worst case-scenario. The last assumption is that the photon statistics of the signal will follow Gaussian distribution in the output of the homodyne detector [25]. Similar to the total noise, the total variance $\sigma _{\text{t}}^2$ equals the sum of the electronic noise variance $\sigma _{\text{e}}^2$ and the quantum variance $\sigma _{\text{q}}^2$ [26]. In our setup, we found $\sigma _{\text{q}}^2$ to be equal to 4159.6². After finding the variance, min-entropy is calculated by [8]:

$$\begin{equation}{H_\infty }\left( {{X_{\text{q}}}} \right) = - {\log _2}\left\{ {\max \left[ {{\text{P}}\left( k \right)} \right]} \right\},\end{equation} \tag{ 7 }$$

where ${H_\infty }\left( {{X_{\text{q}}}} \right)$ is the min-entropy used to quantify the quantum noise ${X_{\text{q}}}$, and ${\text{P}}\left( k \right) = \sqrt {2\pi } {\sigma _{\text{q}}}$ is defined to be the probability distribution function depending on the type of light as can be seen in equations (5) and (6). Out of 16 bits we get a min-entropy of 13.4 bits per sample, which is also the maximum entropy an eavesdropper can guess from a random bit sample.

There are some limitations in the experimental setups in the literature such as the frequency bandwidth for continuous measurement and the dead time of the detector for single photon experiments. Using 5 GHz frequency bandwidth detectors limits the bandwidth of the analog signal, which is reasonably large compared to the ones in the previous works [27]. However, the biggest limitation on the bit rate of the RNGs originate from the post processing (also known as randomness extraction) [6].

Randomness extraction is a method used to elicit a bunch of bits in the shape of a distribution that is almost uniform coming from biased and correlated bits [8]. The main purpose of randomness extraction methods are to improve the randomness of the raw data of a QRNG. This purpose is still attractive if used for the same reason, but recently this method has gained notable importance especially in the field of cryptography and computer science. It is also important because when we smooth the entropy source, Shannon Entropy function is maximized [28]. There are different methods in the literature that are used to extract randomness of QRNGS, some use Toeplitz-matrix hashing [6, 28] which needs a long seed to be used. Another extractor is the Trevisan's extractor which needs a relatively shorter seed than the Toeplitz-matrix method. Although it is tested to have high security, the price to be paid for this high security is the limitation on its speed [6]. In this work, a randomness extractor is not deployed in order to compare the raw data of the coherent and thermal signals. Instead, the collected raw data of both signals is digitized by a fast comparator logic where below and above the mean values are labelled as 0 and 1, respectively.

5.1. Autocorrelation

There are a few methods to determine the randomness quality of any QRNG depending on the sampling rate of the raw signal. One of them is the ACF of the bit stream as it shows the correlation of a signal with a delayed version of it. When the sampling rate is too high, the correlation with the next sampling point is also higher. Hence, to determine the optimal sampling rate, the autocorrelation coefficient of the output signal needs to be calculated at assorted sampling frequencies. The sampling rate of the OS is 40 GSps, which is much faster than the bandwidth of our detector (5 GHz). Therefore, the signal requires undersampling. The ideal undersampling value can be extracted from the ACF, whose samples shape the primary approximation for the bit sequence quality. The lag value corresponding to minimum autocorrelation indicates better randomness. If the minimum autocorrelation occurs at small lag values, the sampling rate can be faster and therefore a higher final bit rate can be achieved. As shown in figure 4, the ACFs of the scattered signals come around zero at an earlier lag value than the laser signal. This lag point is used for undersampling. If the data is sampled with double of this rate, it is more likely that no pattern can be extracted within the bit string. Therefore, the bit string is more likely to pass the randomness tests.

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For the scattered source, the minimum autocorrelation value is much smaller than the signal from the laser fluctuations. Combining figures 3 and 4, we can conclude that the optical scattering works as a bit extraction process. The lower value of the ACF coefficient in the scattered data compared to the laser data implies that the higher quality randomness can be obtained with scattering. The autocorrelation of the laser intersects with zero at a lag value of 44, half-polished silver at 22, unpolished aluminium at lag 10. Dividing the OS's maximum sampling rate (40 GSps) by this 10 lag value gives 4 GSps. According to the Nyquist–Shannon sampling theorem we can sample our signal with twice the value found at the first lag which comes to zero [29], which means that 8 Gbps can be achieved maximally with the laser signal that is scattered from an unpolished aluminium SS. However, for the coherent source we only get a rate of 3.2 Gbps. This is no surprise as the bandwidth of the detectors are 5 GHz and the subtraction of the signals is performed in the homodyne configuration.

5.2. Randomness statistical tests

It has been shown that the sources with super-Poissonian distribution have better randomness characteristics compared to the ones with Poissonian statistics. Optical scattering is proposed to convert the Poissonian distribution into super-Poissonian distribution to have better randomness characteristics and also to allow a faster bit generation rate. This conclusion is achieved by analysing the minimum lag value that corresponds to the minimum autocorrelation, supported by the randomness test suit results.

In summary, the proposed method of using a scattering source (SS) in one arm of the homodyne detection scheme allows faster random bit rate without compromising the quality of the randomness. By just including a SS to the homodyne detection, the random bit rate can be increased from 3.2 Gbps to 8 Gbps. The limitation of this value is due to the bandwidth of the detectors. Therefore, faster detectors can further increase the random bit rate. The method is used to perform a fast-optical post processing compared to the raw data of a coherent source. Our method of using a SS, proposes a solution to slow digital bit extraction process. Nowadays, commercial QRNGs are mostly using shot noise for the random bit generation. The proposed method has the potential to improve the speed and randomness quality of the optics based QRNGs.

This research was supported by the Silicon Photonics for Quantum Fibre Networks (SQUARE) project under the QuantERA ERA-NET Cofund in Quantum Technologies program and by the National Funding Authority TUBITAK-ARDEB with the project number: 117F289.