One arrival at a time, and the pattern still appears

The experiment is old and the version that matters is not. Two slits, a screen, and a source turned down until the average number of particles inside the apparatus at any moment is far below one.

Both halves of that are the result, and dropping either produces a picture that is not surprising. A wave explains the fringes and predicts a continuous smear rather than dots. A particle explains the dots and predicts two bands behind two slits rather than a dozen fringes. What is actually seen is dots that land where a wave says they are likely to.

The two things the arrivals are not #

They are not a crowd effect. The obvious first explanation for interference in a beam is that the particles interact — one jostles another, and the pattern is the result of their collective behaviour. The single-particle version rules this out by construction: at the source rates used, each particle is detected before the next is emitted, and in the electron version of the experiment the mean separation between successive electrons in the apparatus is metres while the apparatus is centimetres long. There is nothing for a particle to interfere with except itself.

They are not a hidden sorting. The second explanation is that each particle goes through one slit or the other, and the pattern reflects some property the particle had on leaving the source that determined both which slit it took and where it landed. That is testable, and the test is what closing one slit does: block either slit and the pattern becomes a single broad hump, with particles arriving in places that were dark when both slits were open.

The second result is the sharper one, because it is a statement about removal. Opening a second route to the screen makes some places on the screen unreachable. Any account in which each particle takes one route and the two routes are independent predicts that opening a second route can only ever add arrivals, never subtract them.

The two things the arrivals are not are worth stating early. They are not particles that took one route and were nudged, and they are not waves that arrived spread out and condensed. The dark places in the pattern are not places where nothing was sent — they are places where two contributions arrived and destroyed each other, which requires both to have been present. A model in which each arrival went through one slit has nothing to cancel against.

Where the fringes are #

The spacing follows from the geometry and is worth deriving because the answer contains a subtlety the figure exposes.

Two slits a distance d apart, a screen a distance L away, and a point on the screen a distance y off axis. The path difference between the two routes is approximately dy/L, and the two contributions cancel when that equals a half-integer number of wavelengths. So the dark fringes sit at

$$y = \left(m+\tfrac12\right)\frac{\lambda L}{d},$$

evenly spaced by λL/d.

The bright fringes are not evenly spaced, and the reason is the second thing going on. Each slit has a width, and a slit of finite width spreads the wave by diffraction into a broad envelope that falls away from the axis. The intensity is the interference pattern multiplied by that envelope, and multiplying a symmetric hump by a falling function pulls its maximum toward the centre. With the geometry drawn above, the first two bright gaps come out at 5.98 and 5.68 millimetres against a nominal λL/d of 6.33.

The dark fringes are untouched by this, because multiplying zero by anything leaves it at zero. So the evenly spaced feature of a real double-slit pattern is the dark fringes, and the site’s figure gate measures them rather than the bright ones — a correction that was made because the check disagreed with the caption and the caption turned out to be the thing that was wrong.

What the accumulation actually shows #

The build-up is often described as the pattern “emerging”, which understates it. Something more specific is true: the arrivals are a sample from a fixed distribution, and everything about the sequence is what sampling looks like.

The figure is drawn that way rather than by scattering dots along the curve. The intensity is integrated into a cumulative distribution, inverted against a seeded generator, and each arrival’s position is drawn from it — so the twenty-arrival panel is genuinely what twenty draws look like, lopsided and gappy, and it is the same twenty on every build.

The site’s gate then tests both halves as statistics. At a thousand arrivals, the histogram’s correlation with the drawn intensity exceeds 0.9. At twenty, a chi-squared test against a flat screen fails to reject — twenty arrivals are formally consistent with there being no pattern at all. Two claims, opposite in direction, both measured off the drawing.

That second number is the one worth holding onto. It is not that the pattern is faint after twenty arrivals; it is that twenty arrivals contain no evidence for it. The information is in the accumulation, and no single arrival carries any of it.

The version that was done with electrons #

The thought experiment is Feynman’s, who called it “a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics” — and who added that it had never been done that way.

It had, twice, and the history is a small lesson in how results travel. Merli, Missiroli and Pozzi ran a single-electron interference experiment in Bologna in 1974, recording the build-up frame by frame on film, and published it in a journal that was not widely read outside Italy. Tonomura’s group at Hitachi repeated it in 1989 with a much better electron source and published the sequence that everyone has seen since.

Both experiments show what the panels above show: sparse and apparently random arrivals resolving into fringes over minutes. Tonomura’s has the added virtue that the electron biprism used to split the beam has no physical slits at all — the splitting is done by an electrostatic field, so the pattern cannot be attributed to anything scattering off an edge.

The experiment has since been done with neutrons, with atoms, with molecules and, in 2013, with an actual pair of slits and a movable mask that could close either one during the run. Nothing about the result has changed with the improvements. What the improvements have done is remove, one at a time, every classical explanation that had been proposed for the earlier versions.

What destroys it #

The pattern is fragile in a specific and informative way: it disappears exactly when it becomes possible to tell which slit was used.

Put a detector at one slit that registers a passage without absorbing the particle, and the fringes vanish, leaving the sum of the two single-slit humps. This is not because the detector is clumsy — the effect persists as the interaction is made gentler, and versions have been run in which the which-path information is carried by an internal state of the particle itself and no momentum is transferred at all.

What matters is whether the information exists anywhere, not whether anyone reads it. If the two paths leave distinguishable traces in the environment, the interference is gone; if they leave identical traces, it is not. And in a delayed-choice quantum eraser, the traces can be made distinguishable and then re-mixed after the particle has landed, and the fringes return in the correlated subset.

That is the point at which this essay hands over to what a measurement is, because “the pattern exists when the paths are indistinguishable” is not an explanation of anything. It is a very precise statement of the rule, and the rule is what the rest of the field argues about.

Why the world is not visibly like this #

Every object has a wavelength, so in principle every object could be sent through a double slit. Two things prevent it.

The first is the wavelength itself. A dust grain’s de Broglie wavelength is 10⁻²⁰ metres, and no aperture that small exists; the fringe spacing λL/d would be smaller than a nucleus for any apparatus that could be built.

The second is decoherence, and it bites long before the first does. A large object in a room is continuously struck by air molecules and thermal photons, each of which carries away a little information about where it is. Once the environment holds enough information to distinguish the two paths, the fringes are gone — and for an object of even a few thousand atomic mass units this requires a high vacuum and a cold source. The record for molecular interference stands around 25,000 atomic mass units, and every increase has come from better isolation rather than from new physics.

So the boundary is not a mass at which quantum mechanics stops. It is a threshold of isolation, and it moves whenever the vacuum improves.

The interference is between amplitudes, not between particles #

The single most useful reframing of the whole experiment is to stop saying that the particle interferes with itself and start saying that the amplitudes interfere.

There is one amplitude for arriving at a given point via the upper slit and another via the lower one. Quantum mechanics says to add them as complex numbers and square the result, and everything on this page follows from that instruction. The rule is not about particles at all; it is about how to compute a probability when there is more than one indistinguishable way for an outcome to happen.

The interference is between amplitudes and not between intensities, and the whole difference between the classical prediction and the quantum one is the order of two operations. Add the two contributions and square afterwards, and where they are out of step the result is zero. Square each first and add afterwards, and the result is twice one contribution’s intensity. Which of those is correct is settled by the dark fringes, which is exactly where the two answers disagree most — classically they should never be dark at all.

Stated that way, the surprising ingredient is easy to name: squaring at the end rather than at the beginning. Classical probabilities add; quantum amplitudes add and are then squared, and the cross term that survives the squaring is the interference. Every quantum phenomenon on this site — the fringes, tunnelling, the analyser chain — is that cross term in a different setting.

It also explains why which-path information destroys the pattern without any mechanical disturbance. If the two routes leave different traces, the outcomes “arrived via the upper slit” and “arrived via the lower slit” are distinguishable, so their probabilities are added rather than their amplitudes, and the cross term is not there to be squared.

Dim is not the same as single #

There is a version of this experiment from 1909 that is usually quoted as the first, and it does not establish what it is quoted for — which is worth a paragraph, because the gap between the two is exactly the gap between a wave picture and a quantum one.

G. I. Taylor, then a research student, photographed the diffraction fringes of a needle in light attenuated by smoked glass until the exposure required three months. The fringes came out as sharp as with a bright source. The result was taken to show that a single photon interferes with itself.

It does not, and the reason is the source. Attenuating a lamp does not produce photons one at a time; it produces a stream whose photon number in any interval is random, so at Taylor’s intensities there were intervals with two photons and intervals with none, and the whole experiment is reproduced exactly by a classical wave of very small amplitude falling on a detector that fires at random with a probability proportional to intensity. Nothing in three months of exposure distinguishes the two accounts.

The experiment that does was done in 1986 by Grangier, Roger and Aspect, and it took two measurements on one source. They used an atomic cascade in calcium, in which one photon announces the imminent arrival of its partner, so a genuine one-photon state could be prepared on demand. The first measurement sent that photon at a beamsplitter with a detector on each side and found that the two detectors essentially never fired together — an anticorrelation no classical wave can produce, since a wave divides and illuminates both. The second measurement sent the identical photons into an interferometer and recovered fringes of ninety-eight per cent visibility.

Neither result alone is remarkable. Together, from one source, they are the content of this essay stated as a laboratory fact: the thing that refuses to be divided at a beamsplitter is the thing that goes both ways through an interferometer.

The rule that could have been different #

The instruction at the heart of the previous section — add the amplitudes, then square — is Born’s rule, and it is a postulate rather than a derivation. A postulate can be tested, and there is a test of this one that the double slit almost suggests by itself: use three slits.

Born’s rule has a consequence that is easy to miss. Because the probability is the square of a sum, the cross terms it produces are all pairwise — amplitude one times amplitude two, and so on. There is no term involving all three at once. So the three-slit pattern is completely determined by the seven patterns obtainable from the slits taken singly and in pairs, with no freedom left over.

Written out, the combination

$$P_{123} - P_{12} - P_{13} - P_{23} + P_1 + P_2 + P_3$$

must vanish identically. Every genuinely three-way contribution would appear in it, and Born’s rule says there are none. A theory in which probabilities were built from some other power of the amplitude would predict a non-zero value, and the size of it would be the size of the departure.

Sorkin pointed this out in 1994 and the measurement was made in 2010, with single photons through a mask carrying three slits that could be blocked in every combination. The eight patterns were recorded, combined, and the result was consistent with zero at the level of a per cent of the ordinary two-slit interference term — since improved.

The value of the experiment is not that anybody expected it to fail. It is that the rule this essay’s conclusion rests on is stated in a form that a measurement could have contradicted, and did not.

What the picture cannot show #

Three things, and the first is the important one.

It cannot show what happens between the source and the screen. The figure has arrivals and an intensity curve and nothing in between, and that is not an omission — the theory does not supply a trajectory to draw. Asking which slit a particle went through, when no measurement distinguishes them, is asking for a quantity the state does not have. Drawings that show a wave splitting and recombining are drawing the amplitude, not the particle, and the difference is the whole content of the experiment.

It cannot show the phase. The intensity is the square of a sum of two complex amplitudes, and what makes a place dark is that the two arrive with opposite phase. Phase is invisible in every panel here, and everything interesting is a fact about it.

It cannot show the source’s coherence. Fringes appear only if the two routes stay in a fixed phase relationship over the whole run, which requires the source to have a narrow enough spread of wavelengths and a small enough angular size. A real experiment spends most of its effort on that condition, and every panel here assumes it perfectly.

It cannot show a single arrival’s history. Each dot is drawn at a position sampled from a distribution, which is a faithful representation of what the theory predicts and gives no account of why that dot went there. The theory does not offer one, and the drawing correctly declines to invent one.

The same arrangement, made into an instrument #

The experiment is not only a demonstration. Once the fringe spacing is known to depend on λL/d, an interferometer becomes a way of measuring whatever changes the phase along one arm.

For light this is old and familiar. For matter waves it produces instruments with no optical equivalent, because a matter wave’s phase responds to things a light wave’s does not: gravity, rotation, and any potential the particle feels.

An atom interferometer drops a cloud of cold atoms, splits it with laser pulses so that the two halves follow trajectories separated by millimetres, and recombines them a few hundred milliseconds later. The phase difference accumulated is proportional to g, and to the area enclosed by the two paths when the apparatus rotates. Devices of this kind measure gravitational acceleration to parts in 10⁹, and they are being built as gravimeters for geophysics and as gyroscopes that need no moving parts.

What makes them sensitive is exactly what makes the effect hard to see. The wavelength is picometres, so a picometre of path difference is a full fringe — and a fringe is easy to count. Fragility and sensitivity are the same property seen from two sides, which is a pattern this site keeps finding: the exponential that makes tunnelling invisible is the one that makes a scanning tunnelling microscope work.

Where the ladder goes next #

The rungs from here: the which-path experiments in detail, and the quantitative trade between path distinguishability and fringe visibility; the delayed-choice eraser; interference with larger and larger molecules, and what limits it; the Aharonov–Bohm effect, where the fringes shift in response to a potential in a region neither path enters; and the analyser chain, which asks the same question about a discrete two-outcome property rather than about position.

The claim to carry forward is the one that makes the accumulation the point. The pattern is a probability distribution, and probability distributions are not properties of individual events. No arrival is anomalous and no arrival is informative; what is strange is that a thousand independent arrivals, each landing in one place with no knowledge of the others, reproduce a pattern that can only be computed by supposing each of them went both ways.