The molecule that interferes until it glows

One arrival at a time sent particles through a double slit so slowly that only one was in the apparatus at once, and found the interference pattern assembling all the same, dot by dot. The particles in that experiment were electrons, and later neutrons and atoms. From 1999 Anton Zeilinger’s group in Vienna sent fullerenes — football-shaped molecules of sixty and seventy carbon atoms — through gratings and found them interfering too, and the grating made of light followed the technique as it was extended to molecules of thousands of atoms, diffracted by a standing wave of laser light.

A fullerene is different from an electron in a way that matters here: it has an inside. Its hundreds of internal vibrations hold heat, and a fullerene leaving an oven at 900 K carries several electronvolts of vibrational energy. A body that hot glows, and every photon it emits during its flight through the interferometer is a small encounter with the outside world. Where the interference goes found that interference is lost not by being destroyed but by being moved into correlations with an environment. A glowing molecule supplies its own environment, and in 2004 Lucia Hackermüller and colleagues measured what that does, by heating fullerenes with a laser before they entered the interferometer and watching the fringes fade.

What one photon can say #

The first question is what a single emitted photon does to the fringes. A molecule in an interferometer is in a superposition of two positions, one in each arm, a distance $d$ apart. If it emits a photon, the photon’s state depends on where it came from: a photon emitted from one position carries a phase relative to one emitted from the other, $\mathbf k\cdot\mathbf d$, where $\mathbf k$ is its wavevector. The molecule and the photon are now entangled, and the molecule’s fringes are multiplied by the overlap between the two photon states. For a photon emitted in a random direction, that overlap averages to

$$\frac{\sin kd}{kd}, \qquad k = \frac{2\pi}{\lambda}.$$

The second figure draws that factor. A photon whose wavelength is ten times the separation costs the fringes six per cent; one whose wavelength is twice the separation leaves nothing. The intermediate values mean what they look like: partial information. A photon of intermediate wavelength, caught by a perfect detector and traced back to its point of emission, would locate the molecule only to within about half a wavelength, and so would favour one arm over the other without deciding between them, and the fringes lose exactly as much contrast as the photon gains certainty.

This is Heisenberg’s microscope read backwards, and its trade is the Fourier one that sharpness has to be paid for found in every wave: a probe that resolves a distance must carry wavenumbers of about the inverse of that distance. Heisenberg argued that observing an electron with light locates it only to about the wavelength of the light, and disturbs its momentum by about Planck’s constant over that wavelength. Here the molecule is the source rather than the object and nobody is looking, but the arithmetic is the same: the light is able to locate its source to half a wavelength, and a superposition across a smaller distance than that is not resolved by it, whether or not any detector is placed to catch the photon. The photon need not be detected. Its existence, somewhere in the room, carrying the phase, is enough.

The oscillation beyond $\lambda/2$ in the figure is not a mathematical nicety. It says that a photon emitted with a definite wavelength by a molecule on two paths about seven-tenths of a wavelength apart leaves the fringes not erased but reduced to a fifth of their contrast and shifted by half a fringe, because the phase it imprints, though random in direction, is correlated with the separation in a way that partly survives the averaging. Michael Chapman, David Pritchard and their colleagues at MIT tested exactly this in 1995 with sodium atoms in an interferometer whose arms could be pulled apart by a controlled amount. Each atom was made to scatter a single photon of laser light, and as the arms’ separation was increased the fringe contrast fell, passed through zero near half the photon’s wavelength, and came back with reversed sign and reduced size, tracing the oscillating curve this figure draws. A year earlier Tilman Pfau and colleagues in Konstanz had seen the same loss of coherence from a single spontaneous emission. Coherence lost to one photon could be regained in part by moving the paths, which is the clearest demonstration that what the photon carries is information about position, measured at the resolution of its wavelength.

What a hot molecule emits #

The third figure supplies the photons. A molecule at temperature $T$ emits the spectrum of a small hot body: by detailed balance, it emits at each frequency at the rate at which it would absorb from blackbody radiation at its own temperature, so its spectrum is Planck’s law multiplied by its absorption cross-section. The curve that would not come down traced the history of that law; what matters here is its scaling with temperature. The number of photons emitted grows as the cube of the temperature, and the typical wavelength shrinks as its inverse. The figures take the absorption cross-section to be the same at all wavelengths in the thermal band, and they set its size so that a molecule at 3000 K emits three photons during its passage — an assumption, of the order the Vienna experiments inferred for heated fullerenes, rather than a computed property of the molecule. The shape of the result does not depend on it; where the visibility falls does.

At 1000 K a molecule emits a tenth of a photon during its flight, and nearly all its photons are longer than three micrometres. At 3000 K it emits three, peaking at 1.2 micrometres. The interferometers in which fullerenes have been studied separate the paths by distances of the order of a micrometre — the grating period of the Vienna Talbot–Lau interferometer was 991 nanometres — and the dashed line marks where a photon’s wavelength is twice that: shorter photons can locate the molecule well enough to tell the paths apart. Heating does two things at once. It raises the number of photons, and it moves them towards wavelengths short enough to matter.

The fringes fading with heat #

The fourth figure combines the two. Each photon emitted multiplies the visibility by its own factor, so the visibility after a whole transit is the cold value multiplied by the exponential of minus the total decoherence, an integral over the spectrum weighted by $1 - \sin(kd)/kd$. Up to about 1500 K the fringes are untouched; the molecules emit too few photons, and those too long. Then the visibility falls steeply: to half its cold value at a little over 2100 K, and to a few per cent at 3000 K.

That is the behaviour the 2004 experiment saw. As the heating laser’s power was raised and the molecules’ mean temperature climbed past about 2000 K, the fringes lost contrast, and by about 3000 K they had all but gone; the measured fall was matched by a calculation of this kind, made with the measured absorption properties of $\mathrm{C}_{70}$ and the known heating rather than fitted to the fringes. A heated molecule is not deflected, ionised or broken by emitting — the mean count rate in the fringes stayed the same — and the photons themselves were never detected. The interference was lost because the light could, in principle, have said which way the molecule went.

The dashed curve is the comparison that makes the wavelength’s role visible. If every emitted photon were a complete which-path record, three photons would leave $e^{-3}$ of the fringes, 2.3 per cent. The computed value at 3000 K is twice that, because about half the photons are too long to locate the molecule and so cost little. A photon’s mere emission is not a measurement. Its wavelength decides how much of one it is.

A filter on scale #

The last figure turns the comparison around and fixes the temperature, varying the separation instead. For molecules at 3000 K the fringes between paths ten nanometres apart are untouched, those between paths a tenth of a micrometre apart lose a few per cent, those a micrometre apart lose nine-tenths, and beyond a few micrometres every photon is a complete record and the survival levels off at $e^{-N}$, the probability that the molecule emitted none at all.

This is the most useful lesson of the experiment, because it describes decoherence everywhere, not only for hot fullerenes. An environment does not destroy superpositions as such. It destroys superpositions of states it can tell apart, and it tells apart states separated by more than the wavelength of whatever it uses to look — thermal photons, scattered air molecules, the vibrations of a support. Superpositions over shorter distances survive the same environment intact. A dust grain in air is localised to the wavelength of the air molecules that bounce off it, a fraction of a nanometre, within a tiny fraction of a second; the electrons in an atom, in the same air, keep their coherent superpositions across the atom’s size because nothing in the room has a wavelength short enough to resolve them. The probability that goes below zero found the sign of a superposition in the negative regions of the Wigner function; decoherence washes those regions out on the scales the environment resolves, and leaves them on finer scales.

Why bigger molecules are harder #

The interferometers that have since diffracted molecules of tens of thousands of atomic mass units have had to manage exactly this. A larger molecule has more internal vibrations, holds more heat at a given temperature, and has a larger absorption cross-section, so it emits more photons; it also has a shorter de Broglie wavelength, as everything has a wavelength put it, so its fringes are finer and the gratings closer together. The paths are then separated by less, which helps. The experiments cool the molecules before they enter, use beams that are slow enough to diffract but fast enough to cross before emitting much, and work in vacuums good enough that collisions with residual gas, which decohere in exactly the same way, are rare.

A molecule in flight is also not at a temperature in the ordinary sense. It is isolated, with a fixed amount of vibrational energy shared among its modes, and it emits at random: one molecule may emit early and cool, another emit nothing and arrive as hot as it left. The temperature on the figures’ axes is the average over such molecules, and the visibility is an average too, over molecules that emitted none, one or several photons at different points of their flight. The fringes of the molecules that happened to emit nothing are perfect, and those of the ones that emitted a short photon are gone; the measured visibility is their weighted mean. A detector that could pick out the molecules that had not emitted would see full fringes at any temperature.

Each of those is a budget against one decoherence mechanism, and the budgets are well understood. What they show, so far, is that every loss of interference observed with molecules has been accounted for by known interactions with the environment — emission, collisions, stray fields — and none has required anything else. Some proposals would add something else. The fall that leaves the mass in the phase found gravity entering a neutron’s interference pattern as an ordinary phase, and some theorists have argued that gravity must also, at large enough masses, entangle the paths with the gravitational field itself and decohere them: that, or a spontaneous collapse of the wavefunction at a rate growing with mass, would explain why large objects are never seen in superposition. Molecular interferometry sets limits on them by finding the fringes still there. The best known of these, proposed by GianCarlo Ghirardi, Alberto Rimini and Tullio Weber in 1986 and refined since, has the same structure as the figures. It supposes that every nucleon is spontaneously localised at a tiny rate, about once in $10^{16}$ seconds, to within a fixed length of about a tenth of a micrometre. Like a thermal photon, the localisation erases superpositions over separations larger than its length and spares those over smaller ones, and like the glow of a hot molecule its effect grows with the number of particles involved. The length was chosen so that atoms remain coherent and dust grains do not, and molecular interferometers, with their micrometre separations and growing masses, close in on the rate from below.

One gas molecule is enough #

The emitted photons decohere gradually because they are poor microscopes. A photon several micrometres long, scattered or emitted by a molecule whose two paths are a fraction of a micrometre apart, carries away only a little information about which path the molecule took, and many such photons are needed to destroy the fringes. That was the filter on scale of the fourth figure. A collision with a molecule of the residual gas in the vacuum chamber is a very different probe. A nitrogen molecule at room temperature has a de Broglie wavelength of a few hundredths of a nanometre, thousands of times shorter than the separation of the paths, and a single collision with it records the position of the fullerene well enough to tell the paths apart completely. There is no gradual fading per collision: one is decisive.

So the prediction for collisions is simpler than for emission, and it was tested by letting gas into the interferometer on purpose. If each collision destroys the fringes for the molecule that suffers it, the visibility is the visibility of the molecules that suffered none, and the chance of suffering none falls exponentially with the number of gas molecules along the path. The fringes of fullerenes in a Talbot–Lau interferometer were found in 2003 to fade exponentially with the pressure of the added gas, at the rate the expected number of collisions predicted, and differently for different gases in proportion to their collision cross-sections. A pressure of around a millionth of a millibar, a vacuum that would be excellent for most purposes, was enough to remove a large share of the contrast.

The two experiments together give the theory of decoherence its clearest test. The same molecules, in the same interferometer, lose their fringes to two environments — heat radiation and gas — at two different rates, with two different dependences on the path separation, and both rates are calculated from ordinary scattering theory with no free parameter.

Where the grey-body model stops #

The figures treat the molecule as a small grey body at a fixed temperature, and a real fullerene is neither grey nor at a fixed temperature. Its absorption has structure, strong in the ultraviolet and weak in the infrared where most thermal photons would be, and the 2004 analysis used the measured spectrum. It cools as it emits, so its temperature falls along its flight, and a molecule that emits an energetic photon early is cooler afterwards. And it emits at the first grating as well as the last, where the path separation is different: in a Talbot–Lau interferometer the separation between the paths that interfere grows and shrinks between the gratings, and a photon emitted in one place matters more than one emitted in another. The single separation of a micrometre drawn here is a summary of that geometry, and the calibration to three photons at 3000 K is an assumption made to show the shape. The experiment’s comparison with theory was made with the full calculation; the figures reproduce its form, not its numbers.

What a fringe pattern cannot record #

None of the figures shows a photon being detected, because none was. The decoherence they compute happens whether or not anyone collects the light, and the information it describes is information in principle — the photon’s state is correlated with the molecule’s path, and that correlation is enough. Nor do the figures show what would happen if the photons could be collected and the correlation undone, which in principle restores the fringes in the subset of events selected by the photon’s measured state. For thermal photons radiated in every direction into a vacuum chamber that is impossible in practice. The measurement that never touched it found information extracted without any interaction; here interaction happens and nothing is extracted, and the fringes are lost all the same.

Still open: how heavy an object can keep its superposition #

The heaviest objects put into superposition so far are molecules of about 25,000 atomic mass units, and their fringes are accounted for by known decoherence alone. Pushing to masses a thousand or a million times larger, with nanoparticles held in optical traps and released, is the next step, and the decoherence budgets for those experiments are dominated by exactly the mechanisms drawn here: thermal emission and scattering of gas and light. Whether, once those are controlled, some further loss of coherence appears that no known interaction explains — which several proposed modifications of quantum mechanics predict — is not known, and experiments now under construction are designed to look.

The habit worth carrying away is to ask what an environment can resolve. A hot molecule’s own thermal photons erase its interference only in proportion to how well each can locate it: one photon multiplies the fringe visibility by $\sin(kd)/kd$, nothing when its wavelength is much longer than the separation $d$ of the paths and everything when it is shorter than $2d$ — so fringes a micrometre wide survive molecules at 1500 K and are gone by 3000 K, while fringes a tenth as wide survive the same glow. Decoherence is a measurement made by the world, and like any measurement it has a resolution.