  Some 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 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
qualitatively change 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.
