Quantum random number generation by coherent detection of phase noise

TL;DR

This paper presents a quantum random number generator that uses phase noise from two independent lasers, measured with telecom equipment, to produce intrinsically uniform random bits without the need for post-processing.

Contribution

It introduces a robust quantum random number generation method based on phase difference measurement using standard telecom equipment, enabling high-speed and intrinsic uniformity.

Findings

Generates random bits at a speed determined by laser linewidth and detector bandwidth

Produces a probability distribution of quantum origin that is inherently uniform

Uses readily available telecom equipment for phase measurement

Abstract

We demonstrate a quantum random number generator based on the random nature of the phase difference between two independent laser sources. The speed of random bit generation is determined by the photodetector bandwidth and the linewidth of the lasers used. The system implemented is robust and generates a probability distribution of quantum origin which is intrinsically uniform and thus in principle needs no randomness extraction. The phase is measured with telecom equipment routinely used for high capacity coherent optical communications, which allows to keep track of the phase drift of the lasers and is readily available in the telecommunication industry.

Equations

$$⟨x\left(\varphi \right)⟩\propto \sqrt{I_1{I}_2}\cos \left({\Delta }_{\omega }t+\xi \left(t\right)+\varphi \right),$$

$$\text{var}\left(\xi \left(t\right)\right)=2t\left(\frac{1}{{\tau }_{c,1}}+\frac{1}{{\tau }_{c,2}}\right)=\frac{t}{\overline{{\tau }_c}},$$

$$P\left(\xi \left(t\right)=\theta \right)=\frac{e^{\overline{{\tau }_c}\cos (\theta )/t}}{2\pi I_0(t/\overline{{\tau }_c})}.$$

$$\Theta \left(t\right):=\text{unwrap}\left(\text{arg}\left(I+iQ\right)\right)={\Delta }_{\omega }t+\xi \left(t\right).$$

$$\xi \left(t\right)=\int \left(\frac{d\Theta }{dt}-{\Delta }_{\omega }\right)dt.$$

Full text

I Introduction

Random numbers are routinely needed in many branches of science and technology. They are a key element in the development of secure communications channels, since random keys can provide unbreakable encryption systems Shannon1949 . They are used in banking, which uses the RSA algorithm that relies on the generation of random numbers RSA . They are important in gambling, where excellent random number generators are needed to guarantee the fairness of used machines. Indeed, governments are implementing technical standards on the usage of random number generators Turan2018 . In scientific applications, Random Number Generators are behind powerful simulation methods such as Monte Carlo Gentle . Random numbers are generated in many different ways. There are algorithms that generate streams of numbers (Pseudo-Random Number generators, PRNGs) that, in spite of not being truly random, can faithfully simulate true random sequences. There also exist methods which are based on deterministic processes, but for which our ignorance of the many variables involved make the possible outcomes random. This is the case, for instance, of Random Number Generators (RNGs) based on the behavior of chaotic systems Bonilla2016 , certain geological events such as earthquakes, astronomical events, motion of computer mice, and interactions in social media Leung ; Chen ; ZhouMiceMotion .

Randomness sources based on the principles of quantum mechanics can provide true randomness which stems from fundamental physical phenomena Herrero . In principle, the generated random keys are intrinsically random, as opposed to other random number generators based on different physical principles which might contain biases due to phenomena which are deterministic, yet unknown to an experimenter. Due to this, there is a lot of interest in the generation of random numbers based on the intrinsic and fundamental random character of quantum phenomena.

One particular phenomena that can be considered for Quantum Random Number Generation (QRNG) is the temporal drift of the phase of any laser source. Spontaneous emission of radiation, a quantum phenomena whose presence is inevitable in any lasing process, is responsible for such a phase drift. Phase noise based QRNGs, in combination with the use of pulsed lasers, have achieved speeds of the order of 43 Gbit/s Abellan . Lately, these phase noise-based QRNGs have been a key element to close certain loopholes existing in Bell’s inequalities tests Giustina .

In 2010, Qi et al. Qi demonstrated a phase noise based random number generator by passing the signal of a laser by an unbalanced Mach-Zehnder interferometer. The idea behind this experiment is interfering the laser signal at two different times so that the phase difference between these two signals is known to evolve randomly. The experimental setup for this situation is shown in Figure 1 (a). If $\tau_{\text{coh}}$ is the coherence time of the laser, $T_{\text{det}}$ is the response time of the photodetectors, and $T_{\text{sample}}$ is the time an Analog to Digital Converter (ADC) takes between successive measurements, three conditions should be fulfilled for Random Number Generation:

$T_{\text{delay}}\gg\tau_{\text{coh}}$ to guarantee that the phases in both arms of the interferometer are uncorrelated. 2. 2.

$T_{\text{sample}}>\tau_{\text{ coh }}$ to take samples after the phase difference has effectively changed. 3. 3.

$T_{\text{det}}<\tau_{\text{coh}}$ to detect a fixed phase and make sure we are not integrating over a phase change.

These conditions pose limitations on experimental realizations of this QRNG, as time delays of the order of $\tau_{c}$ imply long path delays. Such delays are usually implemented using optical fiber cables. As an example, a coherence time of 0.01 milliseconds (a linewidth of 100 kHz), would require 3 additional kilometers of fiber in one of the interferometer arms.

The presence of long delays pose challenges due to the losses that fibres present, especially in non-telecom wavelengths, as well as their difficulty for possible integration in chips. One way to circumvent this drawback is using a slightly unbalanced interferometer that would generate a random phase with a non-uniform Gaussian-like probability distributionQi . However, this calls for randomness extraction and the active stabilization of the interferometer that increases the technical difficulty of the implementation of the system.

Here we put forward and demonstrate a new scheme where the quantum origin of the phase noise can be exploited for the generation of random number sequences without the need of using a highly unbalanced interferometer with long delays or phase stabilization in a slightly unbalanced interferometer. We use a second laser and measure the interference of the two laser sources with a coherent detector (homodyne detection) that measure the phase variation directly. This method, which readily exposes the random behavior of the phase, requires a minimal amount of post-processing from the measured signal, since the probability distribution of the phase is uniform. Homodyne detection of optical signals is routinely done in optical communications laboratories with the help of optical hybrids. In optical communications, one is interested in extracting the real and imaginary part of a signal, which carries the information, with the help of a local oscillator that acts as reference signal. For random number generation, we are interested in extracting the phase difference between two signals.

In particular, we perform balanced coherent detection of two narrowband laser signals in the telecom band generated with external cavity lasers. By using a fast sampling of the order of 100 MSamples/s, we are able to characterize the random walk performed by their phase difference, and determine their coherence times, i.e., the time over which the phases of the two lasers can be considered constant. However, by using slow sampling, of the order of 100 kSamples/s, we are able to generate random numbers which do not require any sort of randomness extraction and thus are ready to be used after being measured. We will show that we do not require randomness extraction to pass almost all of the standard randomness tests designed for RNGs. This is a feature that can be used for future, faster QRNGs, as we are able to provide true randomness with keys that are fast enough to feed randomness extractors with nonuniform distributions of quantum origin.

II Methods

The source of randomness comes from the random phase difference between the signals coming from two different lasers. To measure such phases we make both signals to interfere with the help of balanced coherent detectors as shown in Figure 1 (b) and (c). These are optical arrangements which involve the use of phase delays and beam splitters. The signal of interest comes from the subtraction of the photocurrents measured by optical detectors at the two output ports of the beam splitter. The use of two balanced detectors with different phase delays makes it possible to recover the complex information of the interfering waves, in a configuration that is called a coherent detector. QRNGs using coherent detectors and the interference of the vacuum field with coherent fields have been described theoretically Zhou and demonstrated experimentally Avesani .

Using a balanced detector with a phase delay $\phi$, it is possible to measure the mean value of the quadrature $\hat{x}\left(\phi\right)$ of the interference between two lasers with frequencies $\omega_{1/2}$ and intensities $I_{1/2}$. This gives rise to a Skellam probability distribution for the measurement of the quadrature $\hat{x}\left(\phi\right)$ Zhou , which can be approximated as a normal probability distribution with mean $\sqrt{I_{1}I_{2}}\cos\left(\Delta_{\omega}t+\phi\right)$ and variance $2\left(I_{1}+I_{2}\right)$, where $\Delta_{\omega}=\omega_{1}-\omega_{2}$.

As the phase changes in time due to spontaneous emission events occurring in the laser cavities, the signal measured by a balanced detector with a phase delay $\phi$ can be written as

[TABLE]

where $\xi\left(t\right)$ is a normally distributed phase with zero mean and a variance that becomes broader as time progresses Sun2017 :

[TABLE]

where $\tau_{c,1/2}$ are the coherence times of the input lasers, defined as the inverse of their linewidths. Since $\xi\left(t\right)$ can only take values on the range $\left]-\pi,\pi\right]$, phases are wrapped around this range and are distributed according to a von Mises distributionMardia :

[TABLE]

where $I_{0}\left(x\right)$ is the zeroth-order modified Bessel function of the first kind AbramowitzStegun .

If the values of the phase are digitalized, the probability distribution of the phase can be made arbitrarily similar to a uniform distribution by measuring at times which are much longer than the coherence time. In fact, for a digitalization of $8$ bits in $\theta$ and $t>2\overline{\tau_{c}}$, the maximum probability difference between the von Mises distribution and a uniform distribution is smaller than $10^{-5}$, as shown in the Supplementary Information, section A.

III Results

We performed experiments with two external cavity lasers (HP 8168A and Agilent 8164A), with wavelengths of around $1550\text{nm}$, i.e., central frequencies $\nu_{0}\approx 193.4\text{THz}$. Typical central frequency differences measured between the two lasers are of the order of $1\text{GHz},$ resulting in wavelength differences of the order of $8\text{pm}$. The linewidths of the lasers are $\Delta\nu\approx 100\text{kHz}$, thus providing coherence times of \tau_{c}\approx 20\text{\mus}. The laser powers are of $P=0.1\text{mW}$ at the photodetector end, corresponding to mean photon flux numbers of $\Phi=\left(P/h\nu_{0}\right)\approx 7.8\times 10^{14}\text{ photons/s}$.

We use balanced detectors (Thorlabs PDB480C-AC) with 1.6 GHz Bandwidth ($T_{\text{det}}=625\text{ps}$). With this response time, detectors are able to measure $\approx 4.9\times 10^{5}\text{photons}$ per sample. The readout is then digitalized with an oscilloscope (Tektronix MSO 70804C) at sample rates ranging from 156.25kSamples/s to 25GSamples/s.

With such values of photon numbers it is possible to measure clean sinusoidal signals for the quadratures $I$ and $Q$ when they are sampled at sample rates on the order of GHz, as illustrated in Figure 2(a). This is in contrast to experiments in which the vacuum field is one of the signals considered, in which case the shot noise is the determinant origin of randomness Avesani ; Zhou ; Gabriel .

With measurements in two complementary quadratures, i.e. by measuring $I\equiv\left\langle x\left(0\right)\right\rangle$ and $Q\equiv\left\langle x\left(\pi/2\right)\right\rangle$, it is possible have access to the complete phase dynamics of $\xi\left(t\right)$ directly. $I$ and $Q$ follow probability distributions which are correlated and complementary. Having an additional quadrature provides additional information on the value of the phase in the unit circle instead of just one projection onto the axis, which has been the customary technique for developing phase noise QRNGs which utilize randomness extractionMitchell ; Abellan ; Sun2017 . With this additional information, it is possible to retreive a distribution which can be made arbitrarily similar to a Uniform distribution stemming from quantum randomness. With the information provided by the signals $I$ and $Q$ and the unwrapping of the phase, we can reconstruct the phase behavior of the laser interference:

[TABLE]

It is possible to eliminate the main trend, $\Delta_{\omega}t$, to obtain a phase variation which describes a random walk whose origin are the spontaneous emission kicks inside of the laser cavities. This is done by differentiating, subtracting the offset, and integrating again, i.e.:

[TABLE]

The random values are obtained by further differentiating this phase ($d\xi/dt$), wrapping it to the range $\left]-\pi,\pi\right]$ and discretizing the possible outcomes to 8 bits. With this processing technique it is possible to recover the kicks in the phase distribution, following equation 3. The measurement of such phase distributions is shown in Figure 3. With this technique in the measured experimental data, we are also able to recover directly the particular and different random trajectories of the phase.

As predicted by Equation 3 and shown in Figure 3, phases start becoming uniformly distributed as time progresses. This can be seen on the histograms, which widen and flatten as the time between successive measurements becomes larger and thus converge to uniform distributions of uncorrelated data sets. The autocorrelation coefficient (as defined in ParkBook ) of different data points never exceeds $10^{-2}$, and a figure of the autocorrelation for up to $2.5\times 10^{6}$ samples is shown in the Supplementary information.

For random number generation, the measurements of $I$ and $Q$ are performed at 156.25 kSamples/s and digitalized at an 8-bit depth. Values of $\theta$ are saved as 8-bit values to reach a random number generator speed of 1.25MBits/s. We have tested the generation of uniform random numbers in a binary file of 21.21 Gbytes of data $(\approx 2^{37.3}\text{ bits}$), corresponding to $2.12\times 10^{10}$ measurements (as every measurement provides one byte) gathered in the experimental setup shown in Figure 1 d). We have used three statistical suites generated for the verification of Random Number Generators: Robert G. Brown’s dieharder Dieharder , United States’ National Institute of Standards and Technology (NIST)’s Statistical Testing Suite NIST , and Pierre L’Ecuyer’s TestU01’s Alphabit testing battery, specifically designed for the testing of hardware Random Number Generators TestU01 . The random data sequence passes 304 out of the 311 random tests applied to it. The results of the dieharder tests are presented in the Supplementary Information C. A possible application for such a generator is the provision of true, quantum random data to feed the extraction of faster generators. Faster sources could be achieved by larger laser bandwidths, but this is a problem to be addressed in future work.

IV Conclusions

We have demonstrated a quantum random number generator based on the extraction of the random phase difference between two laser beams. At slow speeds with high photon numbers, the phase noise is the dominant contribution to the random character of the interference of two lasers signals. The value of the phase is obtained by measuring two quadratures of the electric field of the two interfering lasers with a coherent detector. This randomness can be exploited to produce random numbers which do not require randomness extraction to pass almost all of the standard randomness tests designed for Random Number Generators, thus providing a reliable source of true randomness that can be used for further calibration of faster randomness extractors.

The simplicity of the processing techniques used and the availability and sturdiness of the components in telecom laboratories makes it easy to produce trusted random numbers for cryptographic purposes. The technique presented can be further improved by measuring the randomness of the phase with broader sources, such as Erbium Doped Fiber Amplifiers (EDFAs).

The usage of narrowband sources such as the ones considered in this paper are a drawback to the speed of the key generation, as the coherence times of these lasers are considerably slow. However, it is important to remark that even though certain applications might require high-speed random number generation, in excess of a few Gbit/s, this is not a restrictive requisite in all possible applications. In important cases, the use of mature and easily accessible technology, the robustness or even the availability of the components needed for random number generation can be far more important considerations to take into account than the RNG speedChen .

Acknowledgements.

JRA acknowledges V. Vicuña, M. Mitchell, M. Avesani, S. Zubieta, C. Leung and E. Kassa for useful discussions, as well as Fundació Catalunya-La Pedrera’s ICFO Summer Fellows and the Europhotonics POESII Master. SS acknowledges MINECO FPI-BES-2015-074302. JAL acknowledges ALLIANCE (TEC2017-90034-C2-2-R), project co-funded by FEDER. JPT acknowledges financial support from Fundació Cellex, from the Government of Spain through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV- 2015-0522), and from Generalitat de Catalunya under the programs ICREA Academia and CERCA.