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

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.