The outcomes identical photons refuse

Send two identical photons into a half-silvered mirror from opposite sides and they never leave by opposite sides. The force with no force in it traced that to the sign of an exchange: the two ways of parting — both photons transmitted, or both reflected — lead to the same final state with amplitudes that are equal and opposite, and cancel. Hong, Ou and Mandel measured it in 1987 by sweeping one photon’s arrival time and watching coincidences between the two outputs fall towards zero when the photons overlapped.

That is where the effect is usually left, as a demonstration of bosonic symmetry. It carries more than that. The depth of the dip is a test that distinguishes quantum light from every classical field; it is a measurement of how identical two photons are, in every respect at once; and the cancellation, extended from two photons to three, forbids whole classes of outcome and turns into a sum that becomes intractable to compute as the photons multiply.

The two ways of parting #

The beam splitter multiplies an amplitude by $1/\sqrt2$ for transmission and by $i/\sqrt2$ for reflection. Both transmitted gives $\tfrac12$; both reflected gives $i^2/2 = -\tfrac12$. For particles that can be told apart the two routes end in different final states, their probabilities add, and the particles part half the time. For identical photons the two routes end in the same state — the symmetric combination that bosons are required to occupy — their amplitudes add, and the sum is zero. For identical electrons the exchange brings a minus sign that turns the cancellation into reinforcement, and they part every time, which is the exclusion principle acting at a mirror.

Nothing in that calculation depends on the photons’ phases, which is why the dip in coincidences has no optical fringes in it. The cancellation is between two routes that differ only by which photon went where, and a phase acquired on the way affects both equally.

A test that no classical wave passes #

Classical light at the same beam splitter also changes its coincidences with delay, and that has to be dealt with before the photons’ zero means anything. A detector responds to intensity. Two classical fields meeting at a 50:50 splitter produce intensities $|a + b|^2/2$ and $|a - b|^2/2$ in the two outputs, and coincidences follow the average product of those two intensities. When the fields overlap in time the cross terms can make one output bright while the other is dim, so the product falls; when they do not overlap, it does not.

For two laser beams of equal strength with a random relative phase, sampled over forty thousand trials, the coincidences at zero delay fall to 0.499 of their far value, against an analytic one half. For two thermal sources they fall only to 0.669, against two thirds. The reason there is a floor is that intensities are never negative, and for classical fields of any kind meeting at a balanced splitter that constraint alone is enough to prove the coincidences cannot fall below half their uncorrelated value.

Single photons go to zero. A dip deeper than one half is evidence of light that no classical field describes, which makes it a companion to the experiment a wave cannot pass, and a sharper cousin of the reason two lamps never interfere: that one tests a single source for photons arriving alone, this one tests two sources for photons that cannot be told apart. Neither implies the other. A source can emit one photon at a time and still emit photons that differ from one another in colour or timing, and then the dip will be shallow; two sources can be individually imperfect and still produce a dip below one half if their photons match.

How deep the dip goes is how identical the photons are #

“Identical” here means identical in every degree of freedom that a detector could in principle resolve: arrival time, polarisation, colour, transverse shape. Any difference that would let a hypothetical instrument say which photon went where makes the two routes end in different final states, and the cancellation weakens in proportion.

A difference of polarisation by an angle $\theta$ leaves an overlap of $\cos^2\theta$ between the photons’ polarisation states, and the dip’s depth is exactly that: at 30° the photons part 12.5 per cent of the time at zero delay, at 60° 37.5 per cent, and at 90° — orthogonal polarisations — half the time, which is no interference at all. A difference of colour works the same way through the overlap of the two spectra: shifted by half their width, the dip keeps 94 per cent of its depth; shifted by twice their width, 37 per cent.

The striking thing is what the apparatus does not contain. There is no polariser in the left-hand case and no spectrometer in the right. Nobody checks which photon went where. The possibility that it could be checked is enough, because the final states differ whether anyone looks or not. That is the same statement as the fading of fringes when a path can be told, made with two particles instead of one.

The depth has therefore become a standard measure. A source of single photons for quantum technologies is specified by how deep a dip two of its successive photons make at a beam splitter, because that one number summarises every way the emission could vary from one photon to the next — jitter in time, drift in wavelength, a stray polarisation component — without having to measure any of them separately.

A clock made of a dip #

The width of the dip is set by how long each photon’s wavepacket lasts — its coherence time, the reciprocal of its spectral width, which sets how far any wave can remember its own phase. For photons from the down-conversion sources used in the 1987 experiment it is of order a hundred femtoseconds, which is thirty micrometres of path. Moving one photon’s route by a few micrometres moves the coincidence rate measurably.

That turned the effect into an instrument almost at once. The detectors in the experiment responded in nanoseconds, ten million times too slowly to resolve a femtosecond directly, and they did not have to: the only thing they recorded was whether both fired, and the timing information lives entirely in whether the two routes cancel. Relative delays between two photons were measured to a fraction of a femtosecond with electronics that could not have seen a picosecond. It is the same move that intensity interferometry makes with starlight: put the fine structure into a correlation, and let a slow detector record the correlation.

The dip also has a property that ordinary interference lacks. When the two photons are born together in down-conversion, their frequencies are anticorrelated — if one is a little bluer, the other is a little redder — and a slab of glass in one photon’s path, which would spread an ordinary pulse by dispersion, leaves the dip’s width almost unchanged, because the spreading suffered by each frequency component is matched by an opposite spreading in its partner’s. That cancellation of dispersion was demonstrated in 1992 and used to measure the group velocity of single photons in glass. It depends on the two photons being born together, and it has no counterpart in the fringes of a single beam, which measure the spectrum rather than cancel it.

Three photons, and outcomes that cannot happen #

A 50:50 beam splitter is a network with two inputs and two outputs. The same cancellation happens in any network, and with more photons it acquires a structure that two photons do not show.

Take the most symmetric splitter with three inputs and three outputs, in which each input sends its photon to each output with probability one third, and the phases are arranged so that the network treats the three ports as positions around a circle. Put one photon into each input. There are ten ways for three photons to leave — all three by one output, two by one and one by another, or one by each — and for photons that can be told apart all ten occur.

For identical photons six of the ten never happen. Every probability in the figure is a sum of amplitudes over the distinct ways of sending the three photons to that outcome, and for those six the sum is zero. The rule that picks them out is arithmetic: number the outputs 0, 1 and 2, add up the output numbers of the three photons, and if the total is not a multiple of three the outcome is forbidden. It is a consequence of the network’s circular symmetry combined with the photons’ exchange symmetry, and it holds for any number of photons in the analogous network with that many ports — a suppression law, which is what it is called.

The allowed outcomes absorb the probability. One photon per output happens a third of the time; all three together in any one output, two ninths each. Identical fermions do the opposite of piling up: they can only leave one per output, and they do so every time.

A sum without its minus signs #

Written out, the amplitude for identical photons to reach an outcome is a sum over every way of pairing the photons with the outputs they arrive in, each term a product of the network’s amplitudes, all added with a plus sign. That sum is what mathematics calls the permanent of a matrix. For fermions the same sum carries the sign of each pairing, and it is a determinant.

The difference between the two is not cosmetic. A determinant of a matrix with a hundred rows can be computed in a fraction of a second, because the alternating signs allow the elimination that ordinary linear algebra uses. No comparable method is known for the permanent: the best general algorithms take a time that grows as two to the power of the matrix’s size, and computing permanents exactly is believed to be intractable in general. A network of identical photons does not compute that sum. It simply produces outcomes whose probabilities are those sums, and that is the basis of the proposal, made in 2011, that sampling outcomes from a large enough network of identical photons is a task no ordinary computer can imitate efficiently.

Experiments with tens of photons in networks with many modes have been reported as reaching beyond what classical simulation can match. Each claim has been followed by classical algorithms that exploit what the experiments could not avoid — lost photons and photons that were not perfectly identical — and narrowed the margin. Where the boundary really lies is being settled experiment by experiment.

The largest of those experiments do not start with single photons at all, because making dozens of identical single photons on demand, at the same instant, is harder than the interference itself. They start instead with light that contains photons in pairs — squeezed light — fed into every input of a large network, and they record how many photons arrive at each output. The probabilities are then not permanents but a closely related quantity with the same believed difficulty, and the scheme trades the problem of synchronising single photons for the problem of making every pair-source identical.

Either way, the physics doing the work is the one drawn in the first figure: routes that end in the same final state add as amplitudes, and a network of many ports and many photons simply has an astronomical number of such routes to each outcome. The question of whether a machine built this way can do something no ordinary computer can is a question about how faithfully the network preserves that addition — how few photons it loses and how identical it keeps them — rather than about any new principle.

The forbidden outcomes return #

The last figure shows why imperfection matters so much for that question. If each pair of photons has an indistinguishability below one — an overlap of internal states that is imperfect — the forbidden outcomes start to happen, and the probability that they do can be computed exactly as a mixture over which photons happen to share a common internal state.

For two photons the dependence is a straight line: they part with probability one half times one minus the indistinguishability, so a quarter of the time when it is one half. For three photons the six forbidden outcomes together have probability two thirds when the photons are completely distinguishable, 0.431 when each pair is half-identical, and still 0.097 when each pair is 90 per cent identical. The curve approaches zero steeply only near perfection, because the cancellation for three photons needs every pair to be alike at once, and the requirement compounds.

That compounding is the general lesson. The larger the network and the more photons in it, the more nearly identical the photons must be for the interference to survive intact, and a small imperfection per photon becomes a large departure for many. Partially distinguishable photons behave increasingly like classical particles, and classical particles are easy to simulate — which is the precise sense in which the difficulty of the sum depends on the quality of the photons producing it.

What the ideal network assumes #

No photon is lost. Every probability in the figures sums to one over the outcomes. A real network absorbs and scatters light, and with losses the missing photons carry away exactly the information that would have been needed to distinguish a genuine suppression from a failure to arrive.

The detectors count photons. Outcomes like “three in one output” can only be told from “one in that output” by a detector that resolves how many photons arrived, and most single-photon detectors register only that at least one did.

The sources emit exactly one photon. Common sources of photon pairs sometimes emit two pairs at once, and an extra photon produces coincidences that imitate distinguishability. Correcting for it is part of every real measurement of a dip.

The network is exactly symmetric. The suppression law holds for the perfectly balanced splitter drawn. A real three-way splitter with slightly unequal splitting ratios or phases leaves the forbidden outcomes small rather than zero, and how small is itself a test of how well the network was built.

No path for a photon that has none #

The bar chart gives each outcome a probability and says nothing about which photon went where, because in the interference that question has no answer. Drawing one photon’s path through the network would be drawing a distinguishable particle, and the whole content of the figures is that identical photons do not have separable paths. The amplitudes themselves, which do all the work, are complex numbers that no detector records; only their summed and squared totals appear, and the chart shows those.

The arrows in the first figure are the one place the amplitudes are drawn, and they are drawn for two photons because two is the only case with a picture that small. For three photons there are six pairings to add for each outcome, and for sixty photons there are more pairings than atoms in the observable universe — which is the permanent’s difficulty, stated as a problem of drawing.

What else is not classical about light #

The dip and the refusal of a single photon to divide are both statements about counting photons. Light can be non-classical in a way that does not involve counting at all: its fluctuations can be pushed below the level that even a perfectly steady classical wave shows, in one property at the expense of another. The noise pushed below the floor is that other kind, and it is the one now used inside gravitational-wave detectors — and it turns out to be made of photons in pairs, the same pairs whose interference this essay has been drawing.

The habit worth carrying away is to ask of any interference effect what is being added. When the things that add are amplitudes for different particles to have gone different ways, the result depends on whether the particles can be told apart — and on whether the sum carries signs — which decides between photons that bunch, electrons that separate, and a computation that is easy for one and believed intractable for the other.