A neutron floating a hundredth of a millimetre above a mirror is the only bound gravitational eigenstate anyone has ever measured
Drop almost anything and gravity accelerates it smoothly, continuously, all the way to the floor. Drop an ultracold neutron onto a flat mirror and something else happens: below a few tens of micrometres, the neutron cannot occupy just any height above the mirror. It can only sit at a handful of discrete levels, the lowest one centred barely more than a hundredth of a millimetre — about 13.7 micrometres — above the glass, with nothing continuous in between. Nesvizhevsky and collaborators first resolved this in 2002, scanning the height of an absorbing edge above a neutron mirror and watching the transmitted flux drop to nothing once the gap closed below the size of the lowest bound state, then climb again only once the gap opened enough to admit it [1]. It was the first time anyone had measured a bound quantum state of a genuinely gravitational potential rather than an electromagnetic or nuclear one.
A new paper, “The Oldest Quantum-Gravity Experiment Is a Neutron Falling,” argues that this system has quietly become something larger than a striking demonstration: it is the most quantum-gravitational experiment ever actually run, in the narrow but real sense that its energy levels are set by a combination of the gravitational and inertial mass no other device measures, its transition frequencies already bound dark-energy theories in a range no torsion balance can reach, and its own honest scope — Newtonian potential, not curved spacetime — can be stated as an exact number rather than a hand-wave. This article opens the second entry in The Crossed Fields Papers, following the series’ first piece on a nuclear clock’s redshift sensitivity, and walks through the argument in the paper’s own order: the ladder of levels and its real numbers, the mass combination that makes the ladder a distinct equivalence-principle axis, the wedge of interaction range where the bouncing neutron already leads every mechanical force search, what another decade of spectroscopy would retire, and the honest limit on what any of it can currently say about curvature itself.
The ladder has exact numbers: a ground state at 1.407 picoelectronvolts, a length scale of 5.87 micrometres, and transitions in the hundreds of hertz
Above a perfectly reflecting mirror, a neutron’s Schrödinger equation is about as simple as a bound-state problem gets: an infinite wall at the mirror’s surface and a linear potential rising above it, gravity playing the same role a laser trap or a magnetic bottle plays in every other cold-atom experiment. The exact solutions are Airy functions, and the allowed energies are
where
A static ladder of levels is a spectrum; it becomes spectroscopy once something can drive transitions between the rungs. Jenke and collaborators supplied that piece in 2011, mechanically vibrating a bottom mirror at a tunable frequency to resonantly drive the neutron between adjacent gravitational states — the technique the field now calls gravity-resonance spectroscopy, and the direct analogue of Rabi’s decades-older method for driving transitions in a molecular beam [3]. Run on a three-region state-select, drive, state-select apparatus modelled explicitly on that Rabi geometry, the qBounce collaboration measured the transition between the first and third states at 464.1 plus or minus 1.2 hertz and the transition between the first and fourth states, observed for the first time, at 648.8 plus or minus 1.6 hertz, extracting a local gravitational acceleration of 9.84 plus or minus 0.04 metres per second squared directly from the quantum spectrum along the way [4]. A two-mirror variant of the same technique measured the first-to-third transition at 539.1 hertz and the second-to-fourth transition at 679.5 hertz, with a resonance linewidth of 53 hertz set purely by how long a neutron spends crossing the drive region [5]. Every one of those numbers — hundreds of hertz, not the kilohertz-to-gigahertz range of an atomic clock transition — is a direct, published readout of the peV-scale ladder the 2002 discovery paper first sketched in outline.
Which mass combination a device measures is itself a fact, and this one is a new column
Write the gravitational mass and the inertial mass as if they were obviously the same number and ordinary mechanics never notices the difference — that is Galileo’s and then Einstein’s equivalence principle, and it is why a feather and a hammer fall together. But a measurement does not test “the equivalence principle” as an undifferentiated whole; it tests whatever specific combination of the two masses its own dynamics happen to depend on, and different devices depend on different combinations. Kajari and collaborators worked this out explicitly in 2010, cataloguing three cases side by side: a free-falling wave packet’s classical-looking trajectory depends only on the ratio of gravitational to inertial mass; an energy eigenfunction’s spatial oscillation depends on their product to the one-third power; and a particle bouncing off a hard floor in a linear gravitational potential — exactly the neutron-on-a-mirror problem — has energy levels that scale as the gravitational mass squared over the inertial mass, all to the one-third power, a third, logically distinct combination [6]. The gravitationally bound neutron states are not a slightly different way of measuring the same thing an atom interferometer measures. They are reading a different row of the same underlying matrix.
That distinction has real history behind it. The Colella–Overhauser–Werner neutron-interferometer experiment, the founding demonstration that a gravitational potential produces a quantum-mechanical phase shift at all, measures a phase proportional to the product of the two masses [7] — Kajari’s second row, not the third. Refining that same measurement over two more decades produced one of neutron physics’ longest-running open questions: Littrell, Allman, and Werner’s 1997 two-wavelength measurement pushed the statistical precision to one part in a thousand and still reported “a discrepancy between the theoretically predicted and experimentally measured values of the phase shift due to gravity… observed at the 1% level” [8], a gap only partly narrowed by later work identifying nanoradian-scale crystal misalignments in the interferometers themselves as a likely contributor. Atom interferometry occupies Kajari’s first row instead: Asenbaum and collaborators’ 2020 dual-species rubidium interferometer, comparing free-fall trajectories rather than bound-state energies or diffraction phases, reached an Eötvös-parameter precision of 1.6 plus or minus 1.8 statistical plus or minus 3.4 systematic, in units of ten to the minus twelve, with no violation found [15] — a spectacular number, but one that, by Kajari’s own accounting, is simply not sensitive to the same mass combination the bouncing neutron’s ladder is. None of this makes the gravitationally bound states more fundamental than COW-type phases or free-fall interferometry. It makes them a third, independent axis in the same coverage matrix, exactly the packaging this paper’s authors say is their own contribution rather than an inherited result — the mass-combination theorem itself traces to Kajari and collaborators; its placement as a distinct row worth stating explicitly is the paper’s.
In a wedge from roughly one to a few dozen micrometres, the falling neutron already outperforms every torsion balance
The reason any of this connects to dark energy is a specific proposed mechanism, not a vague gesture at new physics. Khoury and Weltman’s 2004 chameleon field solves an old problem with light scalar fields — that they should already have shown up in solar-system tests of gravity — by giving the field a mass that depends on the local matter density: heavy and short-ranged in the dense laboratory environment, essentially massless and free to act over astronomical distances in the near-vacuum of space, with an interaction range near Earth’s surface density that the original paper puts at around a millimetre [9]. Hinterbichler and Khoury’s 2010 symmetron achieves a similar screening through a different route, a spontaneous-symmetry-breaking transition that switches the field’s coupling to matter on or off depending on the local density, with an unscreened force otherwise comparable in strength to gravity itself over megaparsec scales [10]. Both mechanisms are built precisely to hide from torsion balances and planetary orbits alike, which is exactly why a bouncing neutron, small enough and light enough to sit inside the screening radius of an ordinary laboratory object, is not a redundant way of looking for them.
Brax and Pignol supplied the theoretical bridge in 2011, showing that a chameleon field sitting over the neutron mirror shifts the gravitational energy levels directly, with the shift growing sharply once the chameleon-matter coupling exceeds about one hundred million; existing Grenoble-era bouncing-neutron data already excluded couplings above roughly one hundred billion, three orders of magnitude tighter than the atomic-spectroscopy bound of one hundred trillion that had stood before [12]. Jenke and collaborators’ 2014 resonance-spectroscopy run turned that theoretical bound into a sharper measured one, excluding a coupling above 5.8 times ten to the eighth at 95 percent confidence for a range of chameleon self-interaction exponents — five orders of magnitude below the prior atomic-spectra limit — while also excluding an axion-like spin-mass coupling at a 20-micrometre Yukawa range roughly thirty times tighter than the group’s own earlier bound [5]. The three-region qBounce run pushed a complementary chameleon exclusion down to 6.9 times ten to the sixth and bounded a Yukawa-type force at a 5.6-micrometre range as well [4].
Set those numbers beside the best mechanical force search at the same scale. Lee and collaborators’ 2020 Eöt-Wash torsion balance, a stationary detector paired with a rotating attractor carrying two different azimuthal symmetries at once, tested Newtonian gravity down to plate separations of 52 micrometres and excluded gravitational-strength Yukawa interactions at ranges below 38.6 micrometres [11] — an extraordinary result, and very close to the physical floor for a torsion pendulum, whose two plates simply cannot be pushed much closer without touching. A bouncing neutron’s wavefunction does not have that problem: its lowest states already sample the field within a few micrometres of the mirror surface without any mechanical contact at all. In the resulting wedge, roughly one to several dozen micrometres of interaction range, the falling neutron is not merely competitive with the torsion-balance frontier — it is currently the more sensitive probe, for the structural reason that it can get closer to the source without breaking.
What ten times narrower resonances would retire, and what they would not
None of the bounds above are the ceiling of the method — they are where a still-young technique happens to sit today. Abele, Jenke, Leeb, and Schmiedmayer proposed in 2010 adapting Ramsey’s method of separated oscillating fields to gravity-resonance spectroscopy, projecting that the approach could in principle reach a sensitivity to short-range interactions roughly 21 orders of magnitude below the electromagnetic energy scale [14] — a headline number that is the proposal’s own stated ceiling for the technique, not yet a measured result, and the qBounce collaboration’s own subsequent proof-of-principle run treated it exactly that way, as a milestone toward the method rather than a finished measurement. What has moved since 2010 is the supply side: the TUCAN collaboration’s newly completed superthermal ultracold-neutron source at TRIUMF reported in 2026 a sustained production rate of 6.75 times ten to the fifth neutrons per second, described in the paper’s own abstract as higher than any other ultracold-neutron source operating anywhere in the world [13]. More neutrons per second means more counts per resonance scan in the same running time, which is the most direct route from today’s few-hertz statistical uncertainties — 1.2 hertz on the 464.1-hertz transition, 1.6 hertz on the 648.8-hertz transition [4] — toward the sub-hertz precision a Ramsey-type protocol is built to exploit.
Follow that arithmetic forward, structurally rather than exactly: a source an order of magnitude brighter, feeding a Ramsey-type interrogation with a correspondingly longer coherence time, is the obvious next target for pushing the transition frequencies another factor of ten past today’s precision. That improvement would retire specific, named pieces of parameter space rather than physics in general — it would press the chameleon bound another notch past Jenke and collaborators’ exclusion, narrow the symmetron and Yukawa islands the qBounce collaboration has already carved out, and extend the wedge where the neutron beats Eöt-Wash-class torsion balances further below Lee and collaborators’ 38.6-micrometre floor. It would not, on its own, do anything at all to the question the next section exists to answer.
The honest sentence: this is Newton’s potential, not curvature, and here is the number that says so
It is tempting, and wrong, to call gravitationally bound neutron states a test of general relativity’s curved spacetime. They are not, at least not yet, and the paper is explicit about exactly how far that is from changing. Jenke and collaborators’ own 2014 systematics budget states the sizes plainly: tidal effects, the general-relativistic curvature correction to a flat, uniform gravitational field, shift a level by roughly ten to the minus nineteen electronvolts; Coriolis-force effects from Earth’s rotation shift it by less than ten to the minus seventeen electronvolts; van der Waals and Casimir forces from the mirror surface itself shift it by less than ten to the minus twenty-eighth electronvolts — and all three sit far below the roughly ten to the minus fourteenth electronvolt energy sensitivity the same measurement actually achieved [5]. A gap of five orders of magnitude between what curvature would contribute and what the experiment can currently see is not a rounding error; it is the entire reason the honest description of this system is quantum mechanics in a uniform Newtonian field, not a quantum probe of spacetime curvature. Getting from here to there is not a matter of one clever upgrade — it needs several more orders of magnitude of energy resolution than even the ten-times improvement the previous section describes, and Coriolis- and Casimir-type systematics, smaller than tidal effects but still far above the eventual curvature signal, would have to be understood and subtracted first regardless.
That is also, in a specific sense, the whole point of calling this the oldest quantum-gravity experiment. It earns the label operationally — it is the oldest running experiment in which a genuinely quantum system sits in a genuinely gravitational potential and its bound states have been measured, going back to 2002 — not because it currently touches the Planck-scale unification physicists usually mean by quantum gravity. Conflating the two claims would overstate what a peV-scale, curvature-blind measurement can say; keeping them separate is what lets the chameleon and symmetron bounds above stand on their own without needing a bigger claim to justify them.
Read from a freely falling frame, the mirror does the accelerating
There is one more reading worth stating plainly, because it clarifies what the mass-combination axis is actually testing rather than adding a new claim. Step into the frame that falls freely with the neutron, and the gravitational potential the textbook problem starts from disappears; what replaces it is a mirror accelerating upward at one Earth gravity and imposing a boundary condition on an otherwise free particle. The bound states are still there — Airy functions do not care which frame derived them — but they are now a statement about how a quantum wavefunction responds to an accelerating wall, which is precisely the equivalence-principle question in its most literal form. Zych and Brukner’s 2018 quantum formulation of the equivalence principle makes the needed distinction rigorous: they define a quantum Einstein equivalence principle in terms of equality between an object’s rest, inertial, and gravitational internal-energy operators, and show explicitly that satisfying the ordinary classical equivalence principle does not automatically guarantee this quantum version holds, so each new class of system has to be checked on its own terms rather than assumed [16]. Read that way, the gravitationally bound neutron is not a curiosity sitting beside the equivalence principle; every one of its levels is a boundary-condition statement about which of the principle’s several components a falling quantum system is actually respecting, at the specific mass combination Kajari’s table assigns to exactly this geometry.
What belongs to fifty years of neutron physics, and what this paper adds
None of the physics above is new, and the paper does not present it as new. The 2002 discovery of the bound gravitational states themselves belongs to Nesvizhevsky and collaborators, as does the 2005 position-sensitive refinement that pinned down the 5.87-micrometre length scale and the full Airy-zero ladder [1, 2]. The gravity-resonance-spectroscopy technique that turned a static ladder into a measurable spectrum belongs to Jenke, Geltenbort, Lemmel, and Abele, with the chameleon and axion-coupling exclusions built on it belonging to the wider qBounce collaboration across its 2014 and three-region 2016 runs [3, 5, 4]. The chameleon and symmetron mechanisms themselves belong to Khoury and Weltman and to Hinterbichler and Khoury, with the theoretical bridge connecting them specifically to the bouncing-neutron geometry belonging to Brax and Pignol [9, 10, 12]. The COW phase-shift lineage and its still-debated one-percent discrepancy belong to Colella, Overhauser, and Werner and to Littrell, Allman, and Werner [7, 8]; the mass-combination theorem that separates a bouncer’s energy levels from a COW phase or a free-fall trajectory belongs to Kajari and collaborators [6]; the competing torsion-balance frontier belongs to Lee, Adelberger, Cook, Fleischer, and Heckel [11]; the atom-interferometric equivalence-principle record belongs to Asenbaum and collaborators [15]; the Ramsey-spectroscopy proposal and the new source that could realize it belong, respectively, to Abele, Jenke, Leeb, and Schmiedmayer and to the TUCAN collaboration [14, 13]; and the quantum formulation of the equivalence principle that makes the freely-falling-frame reading rigorous belongs to Zych and Brukner [16].
What the paper adds, and states plainly is all it adds, is the packaging: placing the gravitationally bound states’ mass combination as an explicit, independent row in the equivalence principle’s coverage matrix rather than an unremarked fact buried in a 2010 table; compiling the chameleon, symmetron, and Yukawa exclusions from separate qBounce runs onto one range-coupling map next to the torsion-balance frontier so the micrometre wedge is visible as a single claim rather than scattered footnotes; structuring what a funded, in-progress order-of-magnitude improvement in precision would and would not retire; and stating, as an exact number rather than a qualitative aside, exactly how many orders of magnitude separate today’s sensitivity from a measurement that would actually see spacetime curvature rather than a uniform field. A reader can take fifty years of neutron physics as excellent, independently established results, and take this paper’s contribution as the argument for reading them together as one coverage axis, without either claim weakening the other.
