The fringes a clock's own ticking erases

The wave a row of moving clocks leaves on an instant closed the argument about whose clock reads what with a question that joins two subjects usually kept apart. Relativity says that clocks following different paths through spacetime record different elapsed times. Quantum mechanics says that a particle can follow two paths at once. What happens when the particle is itself a clock, sent along two paths that differ in proper time? Does it come back having aged by both amounts at once — and if so, what does that do to its interference?

The answer, worked out in detail by Magdalena Zych, Fabio Costa, Igor Pikovski and Časlav Brukner in 2011, is that the interference fades, by an amount set by how far apart the two readings are. The clock becomes a record of which path the particle took, written in time rather than in any interaction, and complementarity does the rest.

Gravity shifts fringes #

Begin with a particle that is not a clock. In an interferometer whose two arms are at different heights in the Earth’s gravity, a particle travelling the upper arm has more potential energy than one on the lower, and its quantum phase advances at a different rate. When the two halves recombine, the difference in phase shifts the interference pattern. The fall that leaves the mass in the phase followed that effect from the experiment of Colella, Overhauser and Werner in 1975, who saw it with neutrons, to the atom interferometers that now measure gravity with it to parts in a billion.

The phase difference is $m g \Delta h T/\hbar$, for a particle of mass $m$, arms separated by height $\Delta h$, for a time $T$. It can be written another way. The upper arm’s proper time runs faster than the lower’s by the gravitational redshift, $\Delta\tau = g\Delta h T/c^2$, and the phase difference is $mc^2\Delta\tau/\hbar$: the particle’s rest energy acting as the frequency of a clock, ticking at different rates on the two arms. On that reading every massive particle is already a clock, its rest energy setting a frequency of $mc^2/h$ — about $10^{25}$ hertz for an atom — and the gravitational phase shift is a proper-time difference read by that clock. This reading has been argued about, but the arithmetic is the same either way, and it leaves the fringes at full strength: they move, and that is all.

A clock that can be read #

Now give the particle a clock that can be read separately from its path. An atom can be put into a superposition of two internal energy levels, separated by $\Delta E$; that superposition evolves, the relative phase between the two levels advancing at $\Delta E/\hbar$, and that advancing phase is a clock. Atomic clocks work by reading it.

Send such an atom through the interferometer. On the upper arm its internal clock runs slightly fast; on the lower, slightly slow. When the two halves come back together, they carry clock states that differ by a phase $\Delta E\,\Delta\tau/\hbar$. If that phase is small, the two clock states are nearly the same and the halves interfere as before. If it is half a cycle, the two clock states are orthogonal — one could, in principle, read the clock and learn with certainty which path the atom took — and there is no interference at all.

The visibility of the fringes is the overlap of the two returning clock states,

$$V = \left|\cos\frac{\Delta E\,\Delta\tau}{2\hbar}\right| = |\cos(\pi\nu\,\Delta\tau)|,$$

where $\nu$ is the clock’s frequency. It falls to zero when the two readings differ by half a tick. For a microwave clock, the hyperfine transition of rubidium at 6.8 gigahertz, half a tick is 73 picoseconds of proper-time difference, which needs a height difference times time of about 670,000 metre-seconds — a kilometre of height held for eleven minutes. For an optical clock, strontium’s transition at 429 terahertz, it needs 10.7 metre-seconds. For the nuclear transition of thorium-229, whose frequency is higher still, about 2.3.

Moved, and faded #

The two effects — the shift that gravity always produces, and the fading that a clock adds — are different and separable.

The shift is a phase, common to both internal levels, and moves the whole pattern. The fading is a loss of contrast, which appears only when the internal state is a superposition. An atom prepared in a single energy level, not ticking, would show the shift and no fading. That makes the effect a clean test of something specific: it is the ticking, the time evolution of the internal state, that is sensitive to proper time, and the experiment checks that a quantum superposition of paths carries a superposition of elapsed times.

The clock is a which-path detector #

The fading is complementarity, in a new form.

Where the interference goes followed the rule that interference survives only as long as no record of the path exists anywhere, and that a partial record degrades it by a definite amount: the visibility $V$ and the distinguishability $D$ of the paths satisfy $V^2 + D^2 \le 1$, with equality for a pure state. The clock is a perfect example. Nothing collides with the atom or scatters from it; no detector is placed on either arm. The record is made by the passage of proper time itself, which differs on the two arms, and the atom carries the record in its own internal state. The measurement that never touched it found a related subtlety from the other side: information about a path can be gained without disturbing the particle. Here information about a path is created without anything happening to it at all, except that time passed at different rates.

That the record exists whether or not anyone reads it is the essential point. The fringes fade because the information is present in the atom, not because an observer looks at it. Remove the information — make the clock states identical again — and the fringes return.

A whole tick brings them back #

They do return, periodically.

A two-level clock is a dial with no hour hand. If the readings differ by exactly one tick, the two clock states are identical and contain no which-path information, so the fringes are fully restored. A real clock with a counter, or a clock made of many levels with a spread of energies, records elapsed time without that ambiguity and would not revive. The revival is a test of the clock’s structure: a two-level system and a multi-level one, given the same proper-time difference, would fade the fringes in different patterns.

A clock is a superposition of masses #

There is a second way to see the same result, and it removes any impression that time is doing something mysterious. Mass is a form of energy: an atom in its upper level is heavier than in its lower by $\Delta E/c^2$. An atom in a superposition of the two levels is therefore in a superposition of two masses, and gravity’s phase shift, $m g\Delta h T/\hbar$, is different for the two. The heavier component’s fringes are shifted slightly more than the lighter one’s. Added together, two patterns shifted by different amounts blur, and when the difference reaches half a fringe they cancel completely.

The difference in shifts is $(\Delta E/c^2)\,g\Delta h T/\hbar = \Delta E\,\Delta\tau/\hbar$, exactly the clock-phase difference. The two descriptions — a clock reading different proper times on the two arms, and a superposition of masses each shifted by gravity in proportion to its weight — are the same calculation. One reads the fading as relativity’s proper time recorded by a quantum clock; the other as the equivalence of inertial and gravitational mass applied to the energy of an internal state. Both rely on the same fact: that the energy of the clock’s ticking gravitates, which is what the equivalence principle requires and what the mass weighed without one of its pieces checked for the binding energy of nuclei.

So the experiment, if done, tests the equivalence principle in a new regime as well: for a body in a superposition of internal energies. A violation, in which internal energy did not gravitate as rest mass does, would show up as a fading different from the prediction. Some proposals frame the experiment exactly this way, as a quantum test of the universality of free fall.

The twin paradox in superposition #

The clock interferometer is also the twin paradox with the twins made of one atom. The longest way round is the shortest clock explained that two clocks taking different paths between the same two events age differently, and that the comparison is unambiguous when they meet again. Here the “twins” are the two halves of one atom’s wave, taking two paths between the beam splitter and the final recombination. The upper twin ages more, by $g\Delta h T/c^2$. In the classical paradox the twins compare their clocks when they meet and find them different. In the quantum version they meet and do not compare anything: they recombine, and the difference in their ages appears as a loss of interference.

That is the sense in which the atom has aged by both amounts at once. Its final state is not one age or the other, but a superposition of the two ages entangled with the two paths, and no measurement of the clock alone, after recombination, gives a single answer. The now a travelling twin can measure was careful about what each twin can know about the other’s time while they are apart; in the quantum case that question has no classical answer, because there are not two twins, only one atom in two places with two ages.

What it would take #

The required product $\Delta h \cdot T$ falls as the clock’s frequency rises, since a faster clock reaches half a tick of difference sooner.

The largest atom interferometers on the ground, in ten-metre vacuum towers, have separated the two halves of an atom’s wave by more than half a metre for about a second: half a metre-second. That falls short of the ten metre-seconds a strontium clock needs for full fading, but a tenth of a cycle’s difference, enough to see a measurable drop in visibility, needs only about two metre-seconds, and towers of a hundred metres and interferometers in space have been proposed partly to reach it. The thorium-229 nucleus, whose low-lying excited state was excited with a laser for the first time in 2024 and is now being developed as a clock, would bring the target lower still.

Height and time are not the only demands. The clock superposition itself has to survive for the whole of the interferometer’s run, a second or more, without losing its phase to anything other than proper time: stray magnetic fields that shift the two levels differently, collisions between atoms, and the finite lifetime of the upper level all destroy it, and each looks, in the data, exactly like the fading being sought. Strontium’s clock transition has an upper-level lifetime of over a hundred seconds, long enough, and optical lattice clocks routinely hold such superpositions for seconds. The lasers that split, steer and recombine the atom also act on its internal state, and their phase noise has to be the same on both arms or it adds a spurious difference. An experiment would therefore look for the fading’s specific signature — its dependence on height, its revival at a whole tick, its absence for atoms prepared in a single level — rather than for a loss of contrast alone. The idea has already been tested in a form that does not need gravity. In 2015, a group led by Ron Folman in Israel split a Bose–Einstein condensate of rubidium atoms into two wavepackets, prepared each in a superposition of internal states, and used a magnetic field gradient to make the internal clock run at different rates in the two packets — a synthetic proper-time difference. The interference faded as the clocks’ readings diverged and revived when they came back into step, exactly as the formula predicts. What remains is to do it with gravity, so that the difference in rates is a real difference in proper time rather than a magnetic stand-in.

What the experiment would establish #

A space version has particular appeal. In orbit an interferometer’s arms can stay separated for many seconds while the atoms float freely, and the height difference can be replaced by a difference in distance from the Earth along the orbit, so the product of separation and time can be made far larger than in any tower; several of the proposed space missions for atom interferometry list a clock-interference test among their aims. That last step matters because it would test something no experiment has yet touched: how general relativity’s proper time behaves in a quantum superposition. Every test of gravitational time dilation so far — clocks on towers, in aircraft, in orbit — has used clocks that were in one place at a time. The experiments described in one arrival at a time showed that single particles interfere, and the molecule that interferes until it glows found how a particle’s own emission destroys its interference by leaking path information to the outside. The clock interferometer is the case in which the leak is internal, and the record is written by gravity. If the fringes fade as predicted, proper time behaves in superposition the way quantum mechanics treats every other quantity: the clock is in a superposition of having aged by two amounts. If they did not, the meeting point of the two theories would need repair.

What the picture cannot show #

The visibility formula is for an ideal two-level clock, prepared in an equal superposition, in a perfectly symmetric interferometer whose arms differ only in height. Real atom interferometers have the two wavepackets move along different trajectories at different speeds, which adds special-relativistic time-dilation differences to the gravitational ones, and the full calculation includes both; in the usual symmetric geometries the kinematic parts largely cancel, but not entirely. The atoms’ finite temperature, the laser pulses’ imperfections and stray fields all reduce the visibility on their own, and an experiment must separate the clock-induced fading from these — which is why the revival at a whole tick, which no ordinary loss of contrast would produce, is valuable as a signature.

The figures treat $\Delta h \cdot T$ as if the height difference were held for the whole time; in a real fountain the arms separate and rejoin, and the effective product is an integral over the trajectory, somewhat smaller than the peak separation times the duration. The 0.5 metre-seconds shaded for current ground fountains is an order-of-magnitude marker, not a specification.

Still open: whether proper time is quantum #

The theory predicts the fading unambiguously, and the magnetic-gradient experiment confirmed the quantum-mechanical part. The open question is whether an experiment with real gravitational time dilation can be done, and some physicists have argued that it is the first experiment that would probe the interplay of quantum mechanics and general relativity in a regime where both matter, short of the Planck scale where quantum gravity is expected to rule. Others have argued that the effect follows so directly from known physics that its observation would confirm nothing new. Either way the experiment has not yet been done, and the instruments that could do it — long-baseline atom interferometers on the ground and in space, carrying optical or nuclear clocks — are being designed and built.

An atom carrying a ticking clock, sent through an interferometer with arms at different heights, returns with readings differing by $g\Delta h T/c^2$, so its fringes keep gravity’s shift and lose visibility as |cos(πνΔτ)| — gone at Δh·T = 10.7 m·s for strontium’s optical clock, back at a whole tick — because the clock is a which-path record written by proper time, trading visibility against distinguishability as $V^2 + D^2 = 1$. The habit worth carrying is to ask what a particle carries along each path, because anything that could tell the paths apart, even its own age, takes the interference away.