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# Harvesting entropy and quantifying the transition from noise to chaos in a photon-counting feedback loop

Edited by Katepalli R. Sreenivasan, New York University, New York, NY, and approved June 23, 2015 (received for review April 3, 2015)

## Significance

The unpredictability of physical systems can depend on the scale at which they are observed. For example, single photons incident on a detector arrive at random times, but slow intensity variations can be observed by counting many photons over large time windows. We describe an experiment in which we modulate a weak optical signal using feedback from a single-photon detector. We quantitatively demonstrate a transition from single-photon shot noise to deterministic chaos. Furthermore, we show that measurements of the entropy rate of a system with small-scale noise and large-scale deterministic fluctuations can resolve both behaviors. We describe how quantifying entropy production can be used to evaluate physical random number generators.

## Abstract

Many physical processes, including the intensity fluctuations of a chaotic laser, the detection of single photons, and the Brownian motion of a microscopic particle in a fluid are unpredictable, at least on long timescales. This unpredictability can be due to a variety of physical mechanisms, but it is quantified by an entropy rate. This rate, which describes how quickly a system produces new and random information, is fundamentally important in statistical mechanics and practically important for random number generation. We experimentally study entropy generation and the emergence of deterministic chaotic dynamics from discrete noise in a system that applies feedback to a weak optical signal at the single-photon level. We show that the dynamics transition from shot noise to chaos as the photon rate increases and that the entropy rate can reflect either the deterministic or noisy aspects of the system depending on the sampling rate and resolution.

Continuous variables and dynamical equations are often used to model systems whose time evolution is composed of discrete events occurring at random times. Examples include the flow of ions across cell membranes (1), the dynamics of large populations of neurons (2), the birth and death of individuals in a population (3), traffic flow on roads (4), the trading of securities in financial markets (5, 6), infection and transmission of disease (7), and the emission and detection of photons (8). We can identify two sources of unpredictability in these systems: the noise associated with the underlying random occurrences that comprise these signals, which are often described by a Poisson process, and the macroscopic dynamics of the system, which may be chaotic. When both effects are present, the macroscopic dynamics can alter the statistics of the noise, and the small-scale noise can in turn feed the large-scale dynamics. This can lead to subtle and nontrivial effects including stochastic resonance and coherence resonance (9, 10). Dynamical unpredictability and complexity are quantified by Lyapunov exponents and dimensionality, whereas shot noise is characterized by statistical metrics like average rate, variance, and signal-to-noise ratio. Characterizing the unpredictability of a system with both large-scale dynamics and small-scale shot noise remains an important challenge in many disciplines including statistical mechanics and information security.

Many cryptographic applications, including public key encryption (11), use random numbers. Because the unpredictability of these numbers is essential, physical processes are sometimes used as a source of random numbers (12⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–25). Physical random number generators are usually tested using the National Institute of Standards and Technology (26) and Diehard (27) test suites, which assess their ability to produce bits that are free of bias and correlation. These tests are an excellent assessment of the performance of a physical random number generator in practical situations but leave an important and fundamental problem unaddressed. Deterministic postprocessing procedures, such as hash functions (25), are often used to remove bias and correlation. Because these procedures are algorithmic and reproducible, they cannot in principle increase the entropy rate of a bit stream. Thus, the reliability of a physical random number generator depends on an accurate assessment of the entropy rate of physical process that generated the numbers (28). It remains difficult to assess the unpredictability of a system based on physical principles.

Evaluation of entropy rates from an information-theoretic perspective is also centrally important in statistical mechanics (29⇓⇓⇓⇓⇓⇓–36). One might expect that the unpredictability of a system with both small-scale shot noise and large-scale chaotic dynamics would depend on the scale at which it is observed. In many systems, the dependence of the entropy rate on the resolution, *ε*, and the sampling interval, *τ*, can reflect the physical origin of unpredictability (37⇓⇓–40). This dependence has been studied experimentally in Brownian motion, RC circuits, and Rayleigh–Bénard convection (34, 35, 37, 41, 42).

Here, we present an experimental exploration and numerical model of entropy production in a photon-counting optoelectronic feedback oscillator. Optoelectronic feedback loops that use analog detectors and macroscopic optical signals produce rich dynamics whose timescales and dimensionality are highly tunable (43⇓⇓⇓–47). Our system applies optoelectronic feedback to a weak optical signal that is measured by a photon-counting detector. The dynamic range of this system (eight orders of magnitude in timescale and a factor of 256 in photon rate) allows us to directly observe the transition from shot noise-dominated behavior to a low-dimensional chaotic attractor with increasing optical power—a transition that, to our knowledge, has never been observed experimentally. We show that the entropy rate can reflect either the deterministic or stochastic aspects of the system, depending on the sampling rate and measurement resolution, and describe the importance of this observation for physical random number generation.

## Experiment and Results

Fig. 1 shows a schematic of our experimental configuration. Our system has a similar architecture to earlier experiments involving optoelectronic feedback loops but differs in that we use a photon-counting detector, whereas previous experiments used an analog photodetector. In either case, the signal from the detector is time-delayed and filtered, and the output of the filter drives the Mach–Zehnder electrooptic modulator (MZM), which in turn controls the light incident on the detector, forming a feedback loop. When an analog photodiode is used, the feedback loop is modeled by a time-delayed nonlinear differential equation:**2** can be replaced with its expectation value, **1**.

In our implementation, the time delay is *β* is kept constant at 8.87.

Fig. 2 shows several time series recorded with this system with increasing photon rate, showing a transition from Poisson noise to deterministic chaos. We plot *t*]. In Fig. 2, all of the plots were generated with **1**. This time series was smoothed with a moving average over a time window of width *w* to be directly comparable with *δ*-like peak, characteristic of a Poisson process, to an oscillatory function that shows correlations at long timescales (tens of milliseconds). The autocorrelation function of the deterministic simulation time series is in close agreement with the autocorrelation function of the photon arrivals with

To visualize the development of chaos with increasing photon rate, we show Poincaré surfaces of section in Fig. 3. We perform a time delay embedding of the experimental time series *w* is used so that the smoothed intensity time series, *D* can be regarded as the infinite photon rate limit of the photon-counting system.

Fig. 4 shows the dependence of the variance of *w* and offers another indication of the transition from shot noise to deterministic chaos. The time integral of an uncorrelated random signal executes a random walk in which the variance grows linearly with the integration time. For this reason, we plot *w*. When *w* is small, the variance reflects the Poissonian nature of the photon arrivals, and the growth rate of the variance has roughly constant value of *w*. As we increase the photon rate from **4** is found in section 14.9.2 of ref. 8):**4** accounts for the difference between the observed variance and the variance of a Poisson process with the same rate. The quantity *w* increases, accounting for the shape of the curves shown in Fig. 4. In deterministic simulations, we find

We characterize the entropy production using the *ε*, and sampling time interval, *τ*, which are natural parameters for most experiments because measurement devices record data to finite resolution at discrete times. In addition to being experimentally relevant, the dependence of

In chaotic systems, unpredictability is due to the sensitive dependence on initial conditions. Because small perturbations grow exponentially in time, chaotic systems generate information. The growth of uncertainty is quantified by the Lyapunov exponents, **1** (52, 53). There is only one positive Lyapunov exponent with a value of

The *τ* (37, 39).

Another advantage of the

The first step to computing the entropy rate of an experimental signal is to generate a list of points in *d*-dimensional space using time delay embedding with a delay of *τ*. These vectors can be regarded as samples of a *d*-dimensional probability distribution over phase space. The entropy of this probability distribution, *d* (41). In principle *ε* and applying Shannon’s formula, *M* of reference points from the time series. In our case, *i*, one computes *ε* centered on the reference point. The only difference between a direct calculation of the Shannon entropy and the Cohen–Procaccia procedure is that, in a direct calculation, a rectangular array of bins is used, rather than a set of bins centered on random points chosen from the dataset. In searching for neighbors for the *i*th reference point, we exclude points within a time window of *τ* of that point, as suggested by Theiler (57). The pattern entropy is then estimated by the following:*d* in the limit that *d* is large, and the entropy rate is the slope of this linear increase:*A* shows that, in the deterministic simulation, the entropy rate remains flat as *ε* decreases. As *d* increases, this plateau approaches **5**. In Fig. 5*B*, we see that, as the photon rate increases, the dependence of the entropy on *ε* becomes progressively flatter at high *ε*. Furthermore, in the region that this flattening is present, the value of *ε* decreases, which is due to the shot noise inherent in the system. It is natural to compare the entropy rates we observe to a constant-rate Poisson process with the same average rate. The Poisson curves in Fig. 5 were calculated by approximating the Shannon entropy of a Poisson distribution by an integral over a Gaussian distribution with a mean and variance of *B*. For small ε, *ε*, and parallels this curve. This logarithmic dependence is more pronounced at lower

## Discussion

We show in this paper that the choice of the resolution with which we observe our system allows us to see either noisy or deterministic dynamics. By counting photon arrivals over timescales on the order of the delay time and filter time constants, we see deterministic dynamics in the time series, Poincaré sections, and the autocorrelation functions. Furthermore, when we observe the dynamics on large scales of both value (*ε*) and time (*w* and *τ*), we find that the entropy rate is close in value to the metric entropy calculated from the positive Lyapunov exponents of the deterministic model, which shows that the entropy generation is dominated by the deterministic exponential amplification of small perturbations in this regime.

In contrast, by using high resolution in photon counts and timescales, we see that both the entropy rate and variance reflect the stochastic nature of the photon arrivals. For small values of *w*, the variance of the number of photon counts is equal to the average number of counts, characteristic of a Poisson process. The logarithmic dependence of the entropy on *ε* shown in Fig. 5 offers another indication of the noisy nature of the dynamics at small scales. In addition to showing both shot noise and chaos at different scales, our experiment also shows a transition from shot noise to chaos with increasing photon rate. The precise control over the rate of photon arrivals and dynamical timescales afforded by our experiment allows for experimental observation of the interplay of noise and dynamics. Our results can be seen to bridge two widely used methods of physical random number generation.

Two prevalent methods have attracted attention for optical random number generation: those based on single-photon detection from strongly attenuated light sources (58, 59), and those based on digitized high-speed fluctuations from chaotic lasers (13). In the former case, the entropy is claimed to originate entirely from quantum mechanical uncertainty, yet in practice these methods are also subject to unpredictable drift and environmental variations. In the latter case, the entropy is attributed to the dynamical unpredictability of chaos, but the unavoidable presence of spontaneous emission is thought to play a role in seeding these macroscopic fluctuations (20, 21). The system presented here is unprecedented in that it can approach macroscopic chaos from the single-photon limit, thereby revealing the transition from noise to chaos. Moreover, the analysis offers a unified measure of entropy that captures both behaviors and clarifies the relationship between sampling frequency, measurement resolution, and entropy rate.

The designer of a physical random number generator must choose the sampling rate and resolution that they will use to collect numbers from a physical system. These decisions will impact the entropy rate. Heuristically, finer discretization (smaller *ε*) and more frequent sampling (smaller *τ*) lead to higher entropy rates, but without the methods presented here it is difficult to assess the dependence of the entropy rate on these parameters in any given system. The statistical tests that are usually used to evaluate physical random number generation (26, 27) were not designed to answer these questions, but rather to certify that a stream of bits is free of bias and correlation. If a random number generator employs postprocessing (as most do), existing statistical tests applied to the output binary sequence provide no insight into whether the entropy originates from the physical process or the postprocessing algorithm used. The (ε, τ) entropy clarifies the origin and nature of uncertainty and informs the choice of sampling rate and measurement resolution.

## Acknowledgments

We acknowledge the University of Maryland supercomputing resources (it.umd.edu/hpcc) made available in conducting the research reported in this paper. We gratefully acknowledge Grant N000141410443 from the Office of Naval Research.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: aaron.hagerstrom{at}gmail.com.

Author contributions: A.M.H., T.E.M., and R.R. designed research; A.M.H. performed research; A.M.H. analyzed data; and A.M.H., T.E.M., and R.R. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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