Detection and the Double Slit

We've seen that the electromagnetic field of a photon expands as a spherical shell with a structure that extends throughout space. Now we confront the deepest puzzle: how that extended field interacts with matter.

Detection is Localized and Quantized

The field is spread across space. But detection is the opposite — it happens at a single point, transferring a discrete quantum of energy:

When detection occurs, the entire spatially-extended field vanishes. Not gradually — instantly. This is the "collapse" of quantum mechanics. The field was on a sphere 16 light-minutes across, and now it's gone, its energy deposited in one atom.

The probability of detection at any point follows the Born rule:^[1]

For an electromagnetic field, the quantum probability amplitude is directly proportional to the classical field strength. This is why classical optics works: the interference patterns predicted by Maxwell's equations correctly predict the probability distribution of quantum detection events.

The Double Slit Experiment

Watch this in the double-slit experiment: the field passes through both slits, interferes with itself, then collapses to one detection event. Over many photons, the random detections build up into an interference pattern.

Fraunhofer Diffraction

The detection probability on a screen at distance $L$ from two slits of width $a$ separated by distance $d$:^[2]

where $\operatorname{sinc}(x) = \sin(x)/x$.

The $\cos^2$ term comes from the interference between the two slits — it creates the fringes. The $\operatorname{sinc}^2$ term is the diffraction envelope from each individual slit — it modulates the overall brightness. Together they predict the exact positions of every bright and dark band.

This formula is the Fraunhofer (far-field) limit, valid when $L \gg d^2/\lambda$. It assumes the path difference can be approximated as $d \sin\theta \approx dy/L$.

The Field Itself is Not Quantized

This distinction is crucial and widely misunderstood. Consider a laser beam passing through neutral density filters:

The beam starts as a continuous electromagnetic wave — a coherent field produced by stimulated emission. Each filter absorbs discrete quanta of energy from the field ($\hbar\omega$ per absorption event), but these absorption events happen randomly in time and are spread throughout the filter volume. After the filter, the remaining field is still continuous and coherent, just weaker.^[3]

Shot Noise: The Randomness of Detection

At very low light levels, individual pixels light up at seemingly random positions. This is shot noise — the inherent randomness of quantum detection events.^[4]

The spatial distribution differs between measurements even under identical conditions. Each detection is a random sample from the probability distribution $P \propto |E|^2$. Only after many detections does the pattern emerge.

Interference is never observed from a single detection. It requires the statistics of many events. What we call an "interference pattern" is really a probability distribution — the $|E|^2$ pattern made visible by accumulating thousands of individual, random, localized detection events.

The shot noise is not a limitation of the detector. It is fundamental. Even a perfect detector would show the same randomness, because the randomness is in the quantum detection process itself — the field's probability distribution determines where detections are likely, but each individual detection is irreducibly random.