Atom, Neutron, and Molecule Gravity Tests

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Atoms, neutrons, and molecules convert gravity into quantum phases, transition frequencies, or spatial distributions. The quantum observable can be exquisitely sharp, but the gravity conclusion is only as clean as the state preparation, surface interaction, field-gradient calibration, detector response, and model that connect that observable to a coupling.

Required background. Weak-Field Gravity supplies the laboratory potential and its gradients; Probability and Statistical Inference supplies likelihood, covariance, selection, and interval reasoning. Helpful background. Classical Field Theory supplies the additional-field and boundary-potential models used in short-range interpretations.

Quantum observables of a weak gravitational field

In a light-pulse atom interferometer, two momentum paths sample different spacetime trajectories. For a uniform vertical acceleration and ideal pulses, the leading Mach–Zehnder phase is

\Delta\phi=\mathbf k_{\rm eff}\!\cdot\!\mathbf g\,T^2+\phi_{\rm laser},

where $\mathbf k_{\rm eff}$ is the effective laser wave vector and $T$ is the pulse separation. The product is dimensionless. Here $\phi_{\rm laser}$ contains imposed phase offsets, chirp, and laser imperfections after the inertial contribution has been extracted; adding the full position-dependent pulse phase would count that contribution twice. Atom-Interferometric Free Fall derives this separation. Reversing $\mathbf k_{\rm eff}$ reverses the inertial phase but not every laser, magnetic, or detection offset; alternating species, launch directions, and pulse timing helps separate them. A short-range source mass must be integrated over its geometry and added to the local Earth field rather than represented by one uniform $g$ .

For an ultracold neutron above a horizontal mirror, the vertical Hamiltonian is

H_z=-\frac{\hbar^2}{2m_n}\frac{d^2}{dz^2}+m_ngz+V_{\rm surface}(z).

This null-model Hamiltonian assumes equal inertial and passive gravitational masses. To test that equality, use $m_I$ in the kinetic denominator and $m_G$ in the potential rather than silently changing both together. Ignoring $V_{\rm surface}$ away from the mirror, the characteristic height in the null model is

z_0=\left(\frac{\hbar^2}{2m_n^2g}\right)^{1/3}.

For an ideal perfectly reflecting mirror at $z=0$ , impose $\psi(0)=0$ and a normalizable wavefunction on $z\geq0$ . Setting $u=z/z_0-E/(m_ngz_0)$ reduces the Schrödinger equation to $d^2\psi/du^2=u\psi$ . Normalizability selects $\operatorname{Ai}(u)$ ; the boundary condition gives

\psi_k(z)\propto\operatorname{Ai}(z/z_0-a_k), \qquad E_k=m_ngz_0a_k, \qquad \operatorname{Ai}(-a_k)=0.

For $g=9.81\ {\rm m\,s^{-2}}$ , $z_0\simeq5.87\ \mu{\rm m}$ . The first two positive Airy zeros, $a_1\simeq2.3381$ and $a_2\simeq4.0879$ , give turning heights near $13.7$ and $24.0\ \mu{\rm m}$ and a transition frequency $(E_2-E_1)/h\simeq255$ Hz. These are ideal-model estimates, not a fit to neutron data. A weak additional potential shifts the levels by $\delta E_k=\int_0^\infty\lvert\psi_k(z)\rvert^2\delta V(z)\,dz$ for normalized states, so different transitions weight the force profile differently.

The levels are nonequidistant, allowing a driven mirror to address transitions. Jenke and collaborators measured resonance transmission and modeled damping, slit height, state populations, magnetic shielding, mirror roughness, and detector response Jenke et al. 2014, Eqs. (1)–(2) and Figs. 1–2. Neutron optical mirror potentials and magnetic effects must be modeled; electromagnetic surface forces such as Casimir–Polder effects are especially relevant to atom and molecule probes and are species dependent.

Molecule interferometers offer large masses and internal-state choices, yet polarizability, blackbody recoil, collisions, electric gradients, and velocity selection grow with complexity. Observing interference is not already a precision test of gravitational universality; the apparatus must modulate a gravitational source or compare a predicted mass and interrogation-time scaling.

Results and limitations through 2026-08-19

The following matrix separates what each platform directly establishes from a stronger gravity interpretation.

Platform	Evidence or result	Limitation and independent discriminator
Light-pulse atoms	Reproducible phases follow inertial and source-mass modulation; atom searches have also tested screened-force templates	Wavefront aberration, diffraction phase, vibration, magnetic gradients, species-dependent trajectories, and source metrology can imitate differential acceleration; reverse wave vector and source position and recover a blinded phase
Ultracold neutrons	Gravity-resonance spectroscopy resolves transitions of states supported above a mirror, with null or bounded deviations in tested potentials	Slit height, mirror roughness, state loss, magnetic gradients, surface potentials, velocity selection, and detector efficiency limit the mapping; verify quantum-number scaling and the no-extra-potential limit
Molecules and nanoparticles	Matter-wave coherence and force sensing demonstrate sensitive massive quantum probes	A gravity-specific, independently calibrated coupling is often absent or model-dependent; vary mass, charge, polarizability, field orientation, and source position before attributing a phase to gravity

The conservative position is that these platforms validate quantum evolution in controlled gravitational or inertial potentials and constrain specified added interactions. The competing, stronger claim—that a residual establishes a new gravitational coupling—depends on electromagnetic and surface models and is not settled by quantum coherence alone. Null results are evidence against only the tested range, coupling, chamber, and preparation model.

For atom-based screened-force searches, source-position modulation and control of magnetic and optical effects are central; see Hamilton et al. 2015, Figs. 2–3 and Materials and Methods. The adversarial check is a joint mass, interrogation-time, source-position, and field-reversal test whose scalings are fixed before unblinding. A phase that disappears under a harmless surface or field change is counterevidence to a universal gravitational interpretation.

Data, likelihood, and reproduction

Every empirical statement must pin the primary dataset or article, pulse or bound-state sequence, source and surface geometry, calibration version, retrieval date, and any later correction. Selection includes atomic launch and detection thresholds, neutron velocity and state filters, molecule velocity and coherence cuts, excluded transients, and detector masks. Completeness refers only to the prepared ensemble after those cuts, not to all incident particles.

A generic count likelihood is

\mathcal L(\theta,\boldsymbol\nu) = \prod_j \operatorname{Poisson} \left[n_j\mid \mu_j(\theta,\boldsymbol\nu)\right] \mathcal L_{\rm aux}(\boldsymbol\nu),

where $\mathcal L_{\rm aux}$ denotes independent calibration data, $\theta$ is a gravitational or fifth-force parameter, and $\boldsymbol\nu$ includes wave-packet populations, phase offset, contrast, slit geometry, surface potential, vibration, field gradients, background rate, and efficiency. A prior is a separate factor when forming a posterior. Poisson counts apply to a fluctuating incident flux; fixed-total, mutually exclusive output counts instead use a multinomial or binomial model. Gaussian phase summaries are acceptable only when their covariance and wrapping are handled. Priors, confidence or credible construction, trials over ranges or frequencies, and posterior-predictive residuals must be declared.

Reproduction begins with the null platform: recover $k_{\rm eff}gT^2$ for atoms or the Airy spectrum for neutrons. Then inject a known source-coupled signal, reverse electric and magnetic fields, remove or move the source, and repeat with an independent preparation or detector channel. Code or software versions, response calibration, raw-to-count pipeline, masks, and covariance must accompany a machine-readable result. No such pipeline execution is claimed here.

The conclusion remains model-dependent where preparation and surface effects overlap the proposed force. What could change the conclusion. The interpretation could change after a superseding calibrated dataset, a corrected response model, a demonstrated molecule gravity modulation, or an experiment that passes the multi-scaling blinded discriminator with independent source and field controls. A new exclusion contour without those ingredients is not enough.

Exercises

An atom interferometer phase changes sign when $\mathbf k_{\rm eff}$ is reversed, while a source mass is moved between near and far positions. Construct a four-configuration combination that rejects a phase offset independent of both reversals, and state one effect it does not automatically remove.

Solution

Let $\phi_{\sigma,\tau}$ denote the phase with wave-vector sign $\sigma=\pm1$ and source position $\tau=\pm1$ . The double difference $D=(\phi_{+,+}-\phi_{+,-}-\phi_{-,+}+\phi_{-,-})/4$ selects a component odd under both operations. Therefore a constant laser phase and an ordinary source-independent inertial phase cancel. The combination does not remove a magnetic or optical background that also changes with source position and wave-vector reversal. That possibility must be measured with field monitors, dummy masses, altered material, or an independent pulse sequence and included as a correlated nuisance parameter.

References

Hamilton, P., Jaffe, M., Haslinger, P., Simmons, Q., Müller, H., and Khoury, J., “Atom-Interferometry Constraints on Dark Energy,” Science 349 (2015) 849–851, DOI.
Jenke, T., et al., “Gravity Resonance Spectroscopy Constrains Dark Energy and Dark Matter Scenarios,” Physical Review Letters 112 (2014) 151105, DOI.