The experiment a wave cannot pass

Every introduction to quantum mechanics offers the photoelectric effect as the demonstration that light is made of particles. Below a threshold frequency nothing happens whatever the brightness; above it, electrons come out immediately with an energy set by the frequency alone. That is exactly what a stream of lumps would do and exactly what a wave, it is said, could not.

The argument is wrong, and it has been known to be wrong since 1926. What the photoelectric effect proves is that matter is quantised, which is a different and equally important statement. The experiment that light cannot get past without being quantised is elsewhere, and it is the one this essay is about.

What a wave can do that it is not supposed to #

Take a classical electromagnetic field — a continuous wave, no photons anywhere in it — and let it fall on an atom described by quantum mechanics, with discrete levels. Compute the rate at which the atom is ionised by the field, using time-dependent perturbation theory.

A wave can do more than it is usually given credit for, and the photoelectric effect is the standard example of the confusion. Its stopping voltage against frequency has a slope of $h/e$ and an intercept at the work function — and every feature of that plot can be reproduced by a semiclassical model in which the light is a wave and only the atom is quantised. So the photoelectric effect does not establish the photon, which is why an experiment that does is worth having.

The answer has a threshold, because the atom cannot absorb less energy than its binding energy and the field can only supply $\hbar\omega$ per absorption event — the discreteness coming from the atom’s levels rather than from the field. The rate is proportional to the intensity, so brightness controls how many electrons and not how fast each one comes out. And the emission is immediate, because the perturbation acts on the atom from the moment the field arrives.

That is the whole of what the photoelectric effect measures, and the semiclassical calculation gets all of it. Einstein’s 1905 argument was correct and was ahead of the evidence; the evidence caught up much later, and not here.

It is worth being clear about what Einstein’s argument actually rested on, since it was not the photoelectric effect. His 1905 paper derived the entropy of high-frequency blackbody radiation in the Wien limit and observed that it has the form an ideal gas of independent particles would have — a statistical argument about the spectrum that would not come down rather than a mechanical one about electron emission. The photoelectric prediction was offered as a testable consequence, and it was tested and confirmed by Millikan over the following decade. The confirmation established the relation between the stopping voltage and the frequency; it did not establish the premise, and Millikan himself said so.

The same is true of the Compton effect and of the shot noise in a photocurrent, both of which are also routinely offered as proof and both of which have semiclassical accounts. The pattern is worth naming: an experiment in which light is absorbed is an experiment about the absorber, and the absorber is quantised whatever the field is.

The measurement that cannot be faked #

Send light at a beam splitter and put a detector in each output. Count the singles at each, and count the coincidences — the events where both fire inside a short window.

Divide the coincidence rate by what it would be if the two detectors fired independently. Call that $\alpha$. For a classical field of any kind it is at least one; for a source emitting one photon at a time it is zero, because a photon that went one way did not go the other.

The classical bound is not an empirical observation but a theorem, and the theorem is about squares rather than about light. Both detectors respond to the same instantaneous intensity, so the coincidence rate is proportional to $\langle I^2\rangle$ while the product of the singles rates is proportional to $\langle I\rangle^2$. And the mean of a square is at least the square of the mean, for any real quantity whatever.

That figure is the bound attacked rather than asserted. Two hundred different intensity distributions, chosen to be as awkward as possible, and the smallest value found is 1.05. Approaching the bound requires an intensity that never varies, which is a laser; going below it requires something a classical intensity cannot be.

Reading the three curves #

The correlation function in the opening figure separates three kinds of light by one number, and each of the three values has a mechanism behind it.

Thermal light gives two. Its intensity fluctuates, and a photon is more likely to be found where the intensity happened to be high, so photons arrive with company for as long as a fluctuation lasts — which is the coherence time. That is bunching, and it was the first thing measured this way: Hanbury Brown and Twiss used it in 1956 to measure the angular size of Sirius, and were widely disbelieved.

A laser gives one. A coherent state has no intensity fluctuation to correlate with, so the arrivals are a Poisson process and knowing about one says nothing about the next. That is the boundary case, and it is the best a classical field can do.

There is a detail in the thermal curve that repays attention. Its value at zero delay is two, not something larger, and the two is not a coincidence: for a field whose real and imaginary parts are independent Gaussians, the mean square intensity is exactly twice the square of the mean. That is the same circular-Gaussian statistics that make a speckle pattern have a contrast of exactly one, and the two numbers are the same fact stated for time and for space. A measurement of $g^{(2)}(0) = 2$ is therefore a certificate that the source is thermal, and a measurement of anything between one and two says the source is partly coherent and by how much.

A single emitter gives zero. After emitting, the atom is in its ground state and cannot emit again until it has been re-excited, which takes about a radiative lifetime. There is no intensity distribution, however peculiar, that produces this, and that is the point.

The recovery of a single emitter’s correlation from zero is an excited state’s survival curve read forwards. The atom cannot emit again until it has been re-excited, so the probability of a second detection climbs from nothing on the timescale of the excited state’s lifetime — and that timescale is measurable independently, which turns the shape of the dip into a quantitative check rather than a qualitative one.

The width of the dip is therefore a direct measurement of the excited-state lifetime, made without ever timing an individual decay. That is one of the more useful by-products of the technique.

Why one atom and not two #

The experiment is delicate for a reason that the arithmetic makes precise, and the reason is not detector noise.

One emitter gives zero. Two side by side give a half, because a coincidence can now come from two different atoms. Ten give nine tenths. A lamp with $10^{15}$ of them is indistinguishable from a classical field, and this is exactly what it means for the quantum signature to disappear in bulk: it is not lost gradually into noise, it is divided by the number of emitters.

The same arithmetic explains a fact about the world that would otherwise be puzzling: why nothing in ordinary experience looks quantum. Every everyday light source contains an astronomical number of independent emitters, so its correlation is one minus something like $10^{-15}$, which no measurement will ever distinguish from a classical field. The quantum character of light is not hidden by being small; it is divided by the number of things emitting, and there is no way to see it without arranging for that number to be one.

That arithmetic is why the experiment waited until 1977, when Kimble, Dagenais and Mandel used an atomic beam thin enough that two atoms were rarely in the observation region at once, and why the dip they measured went to about 0.4 rather than to zero. Grangier, Roger and Aspect improved it in 1986 by using a heralding photon — a cascade in calcium emits two photons, and detecting the first announces that a single second one is on its way — so the window could be short without the run taking a month.

A detector’s dead time and a finite timing resolution fill the dip in as well, which means the residual has at least three causes and telling them apart is most of the experimental work. That is a fair description of nearly every measurement whose headline is a number close to zero.

What the same apparatus says about interference #

The Grangier experiment did something else in the same session that is worth as much as the anticorrelation.

The same source that refuses to trigger two detectors at a beam splitter produces a full-visibility fringe pattern when the two paths are recombined. Both facts are true of the same light on the same apparatus, and neither a wave picture nor a bullet picture accounts for both — which is the honest statement of what these experiments establish, and it is a statement about detection statistics rather than about what light is.

Having established that the source refuses to send anything to both detectors, they replaced the beam splitter’s outputs with an interferometer and recombined them. The fringes came back with a visibility of 98 per cent, built up one detection at a time over minutes.

So the same light that will not divide at a beam splitter interferes with itself when the two paths are brought back together. That is the two-slit result with the arrival statistics under control, and it is the sharpest statement of the situation available: the pattern accumulates from individual arrivals, and each arrival is indivisible.

Nothing in that pair of results is compatible with either a wave or a particle in the classical sense. It is compatible with the quantum-mechanical account and, as far as anybody has been able to arrange, with nothing else.

What the argument does not say #

It is easy to over-read the conclusion, and two over-readings are worth heading off, because each replaces a precise statement with a vaguer and more exciting one.

Nothing in the anticorrelation experiment says light is a particle in the sense a bullet is. What it says is that a detection is a discrete event and that the source can be made to supply them one at a time — and de Broglie’s relation runs the other way for matter, giving a wavelength to things that are unarguably lumps. The two results together make “wave or particle” the wrong question rather than a question with an answer.

It does not show that light is made of little balls. A photon has no trajectory in this experiment, no position between emission and detection, and no size. What the measurement establishes is that a detection event cannot be halved — that the apparatus never records half an arrival at each detector — and the very next measurement in the same paper shows that whatever crossed the apparatus went both ways at once.

And it does not make the wave description wrong. Everything about where the light goes, how it interferes and how strongly it is absorbed is computed from a field. What the field does not do is arrive; the quantisation is in the coupling between the field and a detector, and the correlation measurement is precisely a measurement of that coupling. The interference pattern built one arrival at a time is the same statement seen from the other side.

The honest summary is narrower and more useful than either slogan. There exist states of the electromagnetic field with no classical description at all, they can be made, and one number measured on a beam splitter distinguishes them from every state that does have one.

What this is used for now #

Antibunching stopped being a demonstration and became a specification.

A source that emits one photon at a time needs exactly one emitter in the beam, and the whole engineering problem is isolating one and keeping it still. That is what the modern applications turn on: a quantum key distribution system whose source occasionally emits two photons has a security hole, so the dip measured in these experiments is now a specification rather than a demonstration.

Single-photon sources are components now, made from quantum dots, colour centres in diamond, and single molecules, and the number quoted on the datasheet is $g^{(2)}(0)$. A source with $g^{(2)}(0) = 0.02$ emits a second photon two per cent of the time, and for quantum key distribution that fraction is exactly the fraction of pulses an eavesdropper can split off undetected.

Quantum key distribution depends on it directly, and the dependence is quantitative rather than decorative: the security proof bounds the information an eavesdropper can obtain by the multi-photon fraction, so $g^{(2)}(0)$ is not a figure of merit but an input to the key rate.

Super-resolution microscopy uses it the other way. Counting how far below one the correlation falls counts the emitters in a diffraction-limited spot: two emitters give 0.5, three give 0.67, so the dip is a photon-counting measurement of a number that no amount of optical resolution could supply.

And astronomy uses the bunching, which is where the technique began. Intensity interferometry correlates the fluctuations at two widely separated telescopes rather than the fields, which makes the baseline insensitive to atmospheric phase and to path length errors of many wavelengths — a robustness that has brought the technique back with modern detectors after fifty years of disuse.

The correlation that was declared impossible #

The bunching curve was mentioned as something Hanbury Brown and Twiss measured and were disbelieved about. The disbelief was more specific and more interesting than that, and the resolution is the boson half of this essay’s subject.

Hanbury Brown was a radio astronomer with a practical problem. Measuring the angular size of a source by conventional interferometry requires the phases at two receivers to be compared, which means the two signal paths must be matched to a fraction of a wavelength — feasible over hundreds of metres of cable and hopeless over kilometres. His idea, worked out with Twiss, was to correlate the intensities at the two receivers instead of the fields. The correlation still depends on the source’s angular size, and it is completely insensitive to path length and atmospheric phase.

It worked in the radio, on Cygnus A and Cassiopeia A. Then they proposed doing it with light, and the objection was immediate: at optical frequencies a receiver counts photons, and photons arriving at two separate detectors are supposed to be independent events. A correlation between them appeared to require that a photon going to one detector somehow knew about a photon going to the other.

Brannen and Ferguson published a null result in Nature in 1956 and stated flatly that a positive result would call for a major revision of some fundamental concepts in quantum mechanics.

Their experiment was not sensitive enough by about three orders of magnitude, and the effect is real. Purcell settled the matter in the same journal within months, and his resolution is exactly the statement this essay is about read with the opposite sign: photons are bosons, they are more likely to be detected together than independent particles would be, and the factor of two at zero delay is what Bose statistics predicts for a thermal source. There was no revision required; there was a piece of quantum statistics that nobody had thought to apply to a photocurrent.

So the same apparatus, with the same theorem behind it, gives two above one for a thermal source and zero for a single emitter — and both values are impossible for a classical field, in opposite directions. The first was measured twenty years before the second and was regarded as the anomaly.

The instrument that came out of it measured the angular diameters of thirty-two stars from Narrabri in New South Wales through the 1960s, using mirrors of poor optical quality on a railway track — poor quality being acceptable precisely because the phase does not matter. The technique fell out of use when conventional interferometry caught up, and it has come back: arrays of Cherenkov telescopes built for gamma-ray astronomy have exactly the right properties — enormous, cheap, widely separated light buckets — and are now being used to correlate intensities and resolve stars at fractions of a milliarcsecond.

The number that is a security parameter #

The claim that the multi-photon fraction is an input to a key rate rather than a figure of merit deserves the arithmetic, because the attack it enables was thought for some years to be fatal.

Most quantum key distribution does not use a true single-photon source. It uses a laser pulse attenuated until its mean photon number is around a tenth, which is cheap and reliable and has Poisson statistics — so about five per cent of the pulses that contain anything contain two or more photons.

That fraction is exploitable. An eavesdropper who can measure how many photons a pulse contains, without disturbing their polarisation, can block every single-photon pulse, split one photon out of every multi-photon pulse, and let the rest through. She keeps her copy until the sender and receiver announce which basis each used, then measures hers in the right basis and learns the bit with no error at all. She has replaced a lossy channel with a lossless one and helped herself to the pulses that could be divided.

The attack is not blocked by anything in the protocol, and its effect on the key rate is severe: the secure rate falls as the square of the channel transmittance rather than in proportion to it, which puts long distances out of reach.

The answer, found around 2003, is to make the eavesdropper’s ignorance work against her. Randomly vary the intensity of the pulses between the working value and one or two lower decoy values, chosen and recorded by the sender and announced only afterwards. An eavesdropper cannot tell a decoy from a signal at the time she has to act, so whatever she does she does to both — and any attack that treats multi-photon pulses differently from single-photon ones changes how the detection rate depends on the intensity, in a way the two legitimate parties can see by comparing their statistics at each level.

Comparing the yields at two or three intensities bounds the contribution from single-photon pulses separately from the rest, which is exactly the quantity the security proof needs. The linear scaling is restored, and decoy-state protocols are now the standard.

The point for this essay is what happened to the number. A fraction that is a property of the light’s photon statistics — the same quantity a correlation measurement reports — went from being an unavoidable flaw to a parameter that a protocol measures around, and the measurement is made by varying it deliberately and watching what changes.

What the pictures cannot show #

The correlation functions drawn are models. The single emitter’s exponential recovery assumes a two-level atom with no coherent driving; a strongly driven atom shows oscillations in the dip — Rabi oscillations in the correlation function — which is a further quantum signature and a different measurement.

The simulated run has ideal detectors. Real ones have dead times, dark counts and finite efficiency, and the last of those is unusually benign: a low efficiency reduces both the coincidences and the singles product, so $\alpha$ is unbiased by it — which is why the experiment is possible at all with detectors that miss most of the light.

The classical bound is tested numerically. That is a demonstration rather than a proof; the proof is the Cauchy–Schwarz inequality and takes one line. What the two hundred attempts add is the assurance that nothing exotic in the choice of distribution evades it.

And nothing here is about a photon’s shape or position. The word “photon” in this essay means only an indivisible detection event, which is all the experiment establishes and all it needs.

The ladder from here #

Later rungs on this anchor: the quantum theory of photodetection and why the correlation function is normally ordered; squeezed light, where the fluctuations are pushed below the coherent-state level in one quadrature and above it in the other; the Hong–Ou–Mandel dip, where two identical photons entering a beam splitter from opposite sides always leave together; and photon-number states, which have a correlation of $1 - 1/n$ and are what a heralded source approximates.

The neighbouring ladders are light arrives in lumps, which is the effect this essay declines to accept as proof, one arrival at a time, where the same indivisibility builds an interference pattern, and why two lamps never interfere, which is the coherence property the thermal curve above measures.