48 History: Poisson photon statistics and vision
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The story of the discovery of the photoelectric effect, and the role of Hertz, Einstein, Planck, and Lenard, is fascinating. The stochastic nature of photon absorption was recognized from the earliest days of quantum mechanics. Yet when I spent some time trying to understand who developed that specific insight—not just that absorptions are random, but specifically that the count of photo-electrons follows a Poisson distribution—it was a genuine surprise to discover that a rigorous mathematical proof had to wait until about 1958–1960, more than fifty years after Einstein’s 1905 paper.
48.1 The 50-year puzzle: From Einstein to Poisson photo-electrons
Why did this derivation take so long? Part of the answer seems to be technological: as a friend pointed out to me, practical devices capable of measuring and counting individual electrons only began to mature in the late 1950s, 1960s, and 1970s. Once counting electrons became common practice, a precise statistical theory of photo-electron arrivals had a clear practical purpose, and formal theory followed.
Another part of the answer is that the idea was already “in the air” decades earlier, but it developed along two separate, largely independent tracks:
- Sensory physiology and vision science, where researchers studying the absolute limits of human vision recognized that dark-adapted photoreceptors can respond to single quanta, leading them to grapple with Poisson absorption statistics as early as the 1940s and 1950s.
- Quantum optics and physics, where theorists in the late 1950s formalized semiclassical and quantum photon-counting distributions for optical fields.
Here is my understanding of how those threads unfolded.
48.2 Einstein (1909): Poisson-like, but more than Poisson
Einstein’s 1909 analysis of energy fluctuations in blackbody radiation examined the variance of the energy contained within a small subvolume of a cavity assumed to be in thermal equilibrium. He found it consisted of two distinct terms: one linear and one quadratic in the mean photon number:
\[ \langle (\Delta E)^2 \rangle = h \nu \langle n \rangle + \left( h \nu \langle n \rangle \right)^2 \]
Here, \(\langle n \rangle\) is the average number of photons at frequency \(\nu\), and \(h\) is Planck’s constant.
Einstein interpreted this two-part structure as evidence that light cannot be fully described either as a particle or as a wave alone—it exhibits both aspects, depending on the observable under consideration: - The linear term corresponds to fluctuations from discrete, independently arriving particles—what we now recognize as shot noise, characteristic of Poisson statistics. - The quadratic term, however, arises from wave interference effects and reflects wave-like correlations, which do not occur in a purely Poisson process.
This insight was a remarkable early hint of wave-particle duality, predating the formal development of quantum electrodynamics by decades.
While this analysis includes a term that resembles Poisson fluctuations, it goes beyond the framework of simple photon counting. By considering energy fluctuations in an entire cavity volume—rather than discrete detection events at an absorbing surface—Einstein’s approach was more general and conceptually distinct from later treatments of photon counting in the photoelectric effect.
In summary, Einstein’s analysis of the energy in the cavity has a Poisson-like term, but wave-induced correlations make it more complex than a purely Poisson model. The full realization that photon arrivals at a detector follow a Poisson distribution—under suitable conditions—emerged later.
48.3 The vision science path: Thresholds and retinal noise
Vision scientists in the 1940s and 1950s were at the forefront of describing the Poisson characteristics of absorption, driven by the extraordinary sensitivity of the dark-adapted human eye.
48.3.1 Hecht, Shlaer, and Pirenne (1942)
In their landmark paper, “Energy, Quanta and Vision” (Hecht et al. 1942), Selig Hecht, Simon Shlaer, and Maurice Pirenne demonstrated that the threshold for human vision corresponds to the absorption of just a handful of photons—roughly 5 to 7 at the cornea, with only a small subset absorbed in the retina. While they did not explicitly name a statistical distribution or write down a formal proof, their analysis implied that trial-to-trial variability in perception arose from random fluctuations in photon absorption rather than biological instability. In modern terms, these fluctuations are described by the Poisson distribution.
48.3.2 Pirenne (1951) and Barlow (1956)
In 1951, Maurice Pirenne took the next step in a theoretical review (Pirenne 1951). Pirenne included a whole section—see Equation (2)—deriving the Poisson distribution for retinal photon absorptions from first principles. Because the probability of any given visual pigment molecule absorbing a quantum from a weak flash is independent and very small, the number of absorbed quanta in a given retinal area and time interval follows a Poisson distribution.
Pirenne’s theoretical insights directly influenced Horace Barlow’s classic 1956 paper, “Retinal noise and absolute threshold” (Barlow 1956). Barlow proposed that visual sensitivity is limited not by retinal inefficiency, but by the quantum nature of light itself, alongside spontaneous thermal isomerizations in the photoreceptors (“retinal noise” or “dark light”). Barlow explicitly cites Pirenne for the Poisson distribution:
“This probability is quite independent of the number of similar events in the past or the neighboring retina, and therefore the number of events in a given retinal area and time follows the Poisson distribution as shown in detail by Pirenne.” (Barlow 1956, p. 634)
Barlow thus extended Pirenne’s ideas, crystallizing a central principle of sensory neuroscience: perception at threshold is governed by quantum shot noise.
48.3.3 Sakitt and Barlow (1972)
I should also mention that Barbara Sakitt and Horace Barlow continued working on this problem together at Cambridge and Berkeley. In her 1972 paper, “Counting Every Quantum” (Sakitt 1972), Sakitt showed through a rating-scale experiment that human observers can reliably distinguish between flashes that result in 1, 2, 3, 4, 5, or 6 quantum absorptions in the rod retina—a direct psychophysical demonstration of Poisson photon counting in vision.
48.4 The quantum optics path: Mandel, Wolf, and photo-electron counting
While vision scientists were modeling retinal absorptions, physicists were approaching the problem from electromagnetic field theory and quantum optics.
Leonard Mandel, collaborating with Emil Wolf (with whom he later wrote the definitive treatise Optical Coherence and Quantum Optics), provided the rigorous theoretical foundations for the Poisson statistics of coherent light and photo-electron counting (Mandel 1958, 1959).
Mandel’s 1958 and 1959 papers derived what is now known as Mandel’s formula: the probability \(P(n, T)\) of emitting \(n\) photo-electrons during a time interval \(T\) given an integrated light intensity \(W\):
\[ P(n, T) = \int_0^\infty \frac{(\alpha W)^n}{n!} e^{-\alpha W} P(W) \, dW \]
where \(\alpha\) is the quantum efficiency of the detector. When the light source is stable and coherent (so that intensity fluctuations are negligible and \(P(W)\) is a delta function), this expression reduces directly to a pure Poisson distribution:
\[ P(n) = \frac{\langle n \rangle^n}{n!} e^{-\langle n \rangle} \]
For fluctuating thermal sources, Mandel’s formula reproduces Bose-Einstein statistics, elegantly capturing both the discrete particle shot noise and the wave interference correlations that Einstein had identified in 1909.
Mandel and Wolf were focused on quantum electrodynamics and statistical optics rather than physiological studies of vision. In their early papers, they did not cite Pirenne or Barlow. Their mathematical formalism emerged independently to support the physics of lasers and photoelectric detectors.
48.5 Reflections: Two parallel traditions
Looking back, Barlow and Pirenne were conceptually ahead of Mandel and Wolf in applying Poisson reasoning to photon absorption in the retina. But Mandel and Wolf provided the formal, quantum-theoretic justification for using Poisson statistics in photon counting, which wasn’t fully developed until the late 1950s and early 1960s.
It makes good sense why each community arrived when they did:
- The vision scientists were studying an extraordinary biological detector: the dark-adapted human rod, which can respond to single photons. Because human threshold vision operates right at the quantum limit, sensory physiologists had to confront Poisson statistics decades before electronic image sensors existed.
- The physicists were developing the formal theory of quantum optics just as laboratory technology (photomultiplier tubes, coincidence counters, and lasers) made counting individual photo-electrons a practical reality in the late 1950s and 1960s.
Although Pirenne clearly articulated the Poisson distribution for light absorption in 1951, his work was published in a biophysics journal and remained outside the citation scope of the physics community. The convergence of these two traditions into modern biophotonics and quantum optics took several decades, but both paths arrived at the same fundamental truth: at low light levels, the world is governed by the Poisson statistics of quanta.