Kept time in the quantum phase of free fall

About 2 × 10⁴ atoms of ⁸⁷Rb form a Bose–Einstein condensate in a magnetic trap made by current-carrying wires on an atom chip, which hangs upside down with the atoms below it. Released and collimated by a delta kick to an effective temperature of 3.3 nK, the cloud is dilute enough that each atom runs on its own. A radio-frequency π pulse moves it to |F = 2, m_F = 1⟩, |2, 1⟩ for short, and a microwave π/2 pulse splits it between that state, which the field gradient moves, and |1, 0⟩ ≡ |F = 1, m_F = 0⟩, which the gradient moves only at second order. A Stern–Gerlach kick from the chip’s wires launches the first part upward, and a microwave π pulse then swaps the two states: the launched packet, now in |1, 0⟩, flies ballistic, while the packet left behind, now in |2, 1⟩, is held about 113 µm below the chip by a gradient tuned to cancel gravity. The sequence then runs in reverse, a π pulse and the same kick, so that the packets meet with one position and one momentum. A last π/2 pulse turns their phase difference into populations, which a final gradient separates for absorption imaging.

The source scans the total duration 2τ of the ballistic flight. The population in one output port runs through thirteen oscillations, about 80 rad of phase, with an average population error of 1.8 ± 1.5% and a visibility that falls from 80% to 20% by 2τ = 2000 µs; the 633 cycles of 30 s took 5.3 hours. The phase is fitted as a cubic in 2τ, with the first and last oscillations excluded, and compared with the analytic model of the source’s Eq. 5 and with a numerical simulation that has no free parameters. The residual against the simulation is about 2 rad across the 80, about 2.5%, against a statistical phase noise of 1.3%; a change of ±0.5% in the kick current sets the systematic bands. The source reads the phase −𝓜g²τ³/(3ħ) of its Eq. 2 as the quantum phase of a freely falling packet relative to one static in the laboratory: the endpoint difference of the gauge phase of its Eq. 1, which the equivalence principle predicts in the quantum domain.

The Methods give the ballistic packet an effective acceleration g_eff = 9.91 m s⁻², which includes the acceleration due to the second-order Zeeman shift on |1, 0⟩, and apply an external bias field of 12.6 G; the supplementary text of the preprint, arXiv:2502.14535v4, splits g_eff as g + a_SOZ, with a_SOZ = 0.10 m s⁻². The holding current, found from trajectories of |2, 1⟩, is 23.07 ± 1.49 mA: a 6.5% inaccuracy in cancelling gravity, which enters the phase squared, as 0.42%. With kicks of 80 µs on average and a delay of 71 µs between each kick and the hold, the worked example 2τ = 2.4 ms takes the upper arm 7.62 µm up and the reference 0.11 µm, a maximal splitting of about 7.5 µm.

There is one substance, the omnium, and the rubidium atom is one closed pattern of it (0.1b). Its two internal states are confinement modes of that pattern: a microwave pulse changes the mode at a cost ħΔω = Δ𝓜c² and re-braids nothing, and the mass 𝓜_Rb is a confinement eigenvalue read from measurement, the operator that sets it to be derived (O-⁠9). Both arms of the interferometer are free. Each follows the geodesic of its own clock factor (0.1c): the ballistic arm in the Earth’s gravitational window and, through the moment the chip’s twist induces in it, in the twist as well; the reference in the gravitational window composed with the twist. The chip’s field is the omnium’s helical density gradient (P-⁠5), and a pattern whose rest energy depends on where it sits in a twist takes that energy as its clock factor. This is the helical window:

with E_ind the energy of the moments the twist induces, the work of inducing them included, and 𝗆 the held moments. The windows compose. A slow pattern reads the Earth’s lapse 𝒩 = (1 − α_ext)^½ times its own factor, 𝔫_j = 𝒩𝔪_j, so its weight and the twist’s push add:

The reference is held where the lapse times its own rest energy is stationary: the twist’s slope of its rest energy cancels the lapse’s, and the twist’s push on its held moment equals its weight, the pair face of I-⁠1 carried to the source’s g_lab. The ballistic state holds no moment along the field at first order, but the twist induces one, and its second-order Zeeman energy is its E_ind: it falls at g_b = g_lab + a_SOZ = 9.91 m s⁻². The helical window returns that push from the source’s bias field and the reference’s levitation, with the measured ground-state structure. The held moment of |2, 1⟩, antiparallel to the field, is 0.5037 μ_B at the 12.6 G operating field in the Breit–Rabi structure, against 0.4998 μ_B at low field, so levitation fixes the gradient at −∂_z|B_mag| = 𝓜_Rbg_lab/μ_2,1 = 0.303 T m⁻¹, or 30.3 G cm⁻¹. The induced energy of |1, 0⟩ is −K_q|B_mag|², with K_q = (g_J − g_I)²μ_B²/(4ΔE_hfs) = 1.905 × 10⁻²³ J T⁻², and in one twist the two pushes stand in the ratio of the moments along the gradient:

This is the push the preprint’s supplementary text computes, here from the 12.6 G bias, the reference’s levitation and the measured ground-state structure, with no parameter left free. The push is read from the pattern’s rest energy where it sits; the map from a circulating charge to an I-⁠1 input is to be derived (O-⁠6).

Each arm’s phase advances with the rest energy where it sits, per unit of the time it keeps (0.1a). Its action is 𝔄_j = −∫E_j dt_own, and the measured phase is the difference of the two actions over ħ. A closed pattern’s strand runs at c on a helix and keeps time by circulating, v_∥² + v_⊥² = c² with c dt_own = v_⊥ dt_lab (0.1d). In the slow limit the strand triangle supplies the kinetic term and the lapse the gravitational one, so the arms’ action rate is the laboratory Lagrangian plus the rest-energy term, with neither term added by hand. Splitting the difference into the rest-energy term and the twist’s energy, with the apex at t_lab = 0 and the endpoints at ±τ,

Gaining more time by its height in the lapse than it loses by its speed through the strand triangle, the ballistic arm keeps more time than the reference, and the interferometer counts the excess in the pattern’s own period T_Rb = h/(𝓜_Rbc²) = 5.109 × 10⁻²⁶ s. At the worked example, τ = 1.2 ms, the excess is 6.167 × 10⁻²⁵ s: 12.07 periods, 75.85 rad. The twist’s energy along the ballistic path adds 1.56 rad, for 77.41 rad in all, the source’s ideal phase with g_eff. The finite pulses add 5.7 rad more, and the record’s thirteen oscillations, 81.7 rad, lie between the two.

The second-order Zeeman push that the source books in its effective acceleration is therefore a share 2a_SOZ/g_b = 2.0% of the phase, the size of the reported 2.5% residual. The record tests the composed fall g_lab + a_SOZ, its helical part supplied by the source’s field model. The universality of free fall is the gravitational window’s, and it stands: there, at a pattern’s monopole order, 𝔪 belongs to the point, and every pattern falls alike. In the helical window 𝔪 belongs to the pattern, and patterns whose held and induced moments, summed per unit of energy, differ along the gradient fall differently where they sit. The equality of inertial and gravitational mass, which the source assumes, is derived: a stationary closed pattern, closure included, has inertia E/c² (0.1d), its gravitating mass is the same energy over c² (0.5), and the lapse multiplies that one E.

On its own clocks and rulers the ballistic arm meets nothing; the bookkeeper reads it rising, stopping and falling. The value of ln 𝔪 at a point is invisible, its gradient is the free fall of one pattern, and its second derivatives are the deviation of neighbours, removed by no choice (0.1c, I-⁠3a). The source reads the phase as that of one freely falling object; FDT reads it as the comparison of two free arms of one pattern, the gradient read between the falling arm and the held reference path (Use 7). The tide between the arms, Γ_td = 2G𝓜_⊕/R_⊕³ = 3.083 × 10⁻⁶ s⁻², changes the action difference by the fraction (2/5)Γ_tdτ² = 1.8 × 10⁻¹², or 1.4 × 10⁻¹⁰ rad.

On the A-⁠1 chart the Earth at the laboratory reads α_ext = R_s/R_⊕ = 1.392 × 10⁻⁹, deep in the general-relativistic regime (0.3a), and patterns falling from rest far away pass the laboratory at the flow speed c√α_ext = 11.19 km s⁻¹ (0.1c). The rubidium pattern’s weight is the pair face of I-⁠1, F_maxα_⊕α_Rb = G𝓜_⊕𝓜_Rb/R_⊕² = 1.417 × 10⁻²⁴ N, with α_Rb = 3.364 × 10⁻⁵⁹ a correct small reading. The atom itself is matter, a closed braid read on its own interior chart, in the matter phase, with no transfer map to the Earth’s exterior atlas (I-⁠3). The experiment reads a matter pattern’s phase, quantum mechanics’ reading, as it runs in the spacetime phase, whose reading is general relativity’s: one omnium, one chain, two readings.

The five tests use the same ⁸⁷Rb chip. Each is defined with its independent calibrations, closures and covariance. The conventional comparator is the source’s own action and finite-pulse model, and every residual 𝒟 is zero in FDT and in that model: a residual beyond the criterion falsifies the shared model it tests, and is assigned to FDT only with an additional source or response stated. A residual is resolved when |𝒟| > z_sigs_𝒟 + b_𝒟, with z_sig = 3 chosen in advance, s_𝒟 propagated from the joint covariance and b_𝒟 an independent bound on the nuisance channels; the source prints neither, so each test carries its own calibration.

1. T-⁠1
   A held-out duration contrast. Measure new phases Φ₀ to Φ₃ at four total durations spaced by h, each from populations acquired at its own duration, with kick durations, delay, field and preparation fixed and the kicks set to the measured closure at each duration. The third finite difference 𝒞₁ = −Φ₀ + 3Φ₁ − 3Φ₂ + Φ₃ cancels the constant, linear and quadratic terms of the finite-pulse polynomial exactly and equals (3C_φ/4)h³, with C_φ = 𝓜_Rbg_b²/(3ħ) = 4.480 × 10¹⁰ s⁻³ built from independent mass, gravity and push. Fitting those inputs from the same four phases would remove the test.
2. T-⁠2
   A symmetric, closed levitation scan. Set the calibrated holding acceleration to g_lab + δa, to levitation and to g_lab − δa, moving the kicks with it so that the arms still close, and read the three signed phases. The even contrast (φ₊ + φ₋)/2 − φ_centre equals 𝓜_Rbτ³δa²/(3ħ): the source’s Eq. 3 has its extremum at levitation, and the twist’s share on the ballistic arm moves the level, never the curvature.
3. T-⁠3
   Timing from the phase coefficients. Fit new phases as a cubic in τ with the pulse widths held fixed. The finite-pulse polynomial gives 𝔠₃ = C_φ, 𝔠₂ = C_φτ_k and 𝔠₁ = C_φτ_k(τ_k + τ_d), so the delay τ̂_d = 𝔠₁/𝔠₂ − 𝔠₂/𝔠₃ is free of the phase origin and of a common sign, and it meets a delay measured on the waveform with the same average of the two kicks. A timing claim needs both 𝔠₂ and 𝔠₃ resolved away from zero.
4. T-⁠4
   The current-ratio response. Interleave runs with the kick-to-hold current ratio displaced by ±ε from its nominal value, measuring trajectories, axial bias and pulse areas. The odd half-difference of the phases, divided by the total duration, meets half the response integral, measured in a separate calibration run. The source names this ratio its leading systematic; its ±0.5% protocol is an envelope, not an uncertainty distribution.
5. T-⁠5
   The m_F = 0 pair reads the two windows apart. Keep the source’s run A, reference |2, 1⟩ and ballistic |1, 0⟩, and add run B on the same chip: ballistic |2, 0⟩, with the reference held in |1, −1⟩, whose moment at 12.6 G is 0.4979 μ_B. Its holding current is recalibrated from trajectories, so the ratio of the two calibrated currents is the ratio of the held moments, r = 1.0116. The two m_F = 0 energies sum to a constant at every field, so at one gradient the twist pushes the two ballistic states equally and oppositely; each run sets its own gradient, and they fall at g_lab + a_SOZ and g_lab − ra_SOZ. From the two runs’ cubic coefficients, the weighted sum (rĝ_A + ĝ_B)/(1 + r) returns g_lab, to meet a site gravimeter, and (1 − ρ̂)/(r + ρ̂), with ρ̂ = (Ĉ_B/Ĉ_A)^½, returns a_SOZ/g_lab, to meet a field map. At the worked example the runs read 77.4 and 74.3 rad, 3.1 rad apart, with Ĉ_B/Ĉ_A = 0.960.

In the weighted sum the twist cancels and the gravitational window remains, where 𝔪 belongs to the point; in the difference gravity cancels and the helical window remains, where 𝔪 belongs to the pattern. The ratio carries that difference in units of the fall, free of every common scale of mass, duration and phase origin. This is the twist read between neighbours, between patterns whose E/E₀ and fall differ at one point (0.1c, I-⁠3a, Use 7). A residual in the sum falsifies the shared gravitational-window model at the platform’s precision, and a residual in the ratio falsifies the calibrated twist energy; neither arises from a chart label. For uncorrelated runs of equal fractional uncertainty and zero bound, resolving the ratio at z_sig = 3 takes a coefficient uncertainty under 0.97% per run. One phase sample per cycle at the record’s noise of about 1 rad over its 633 cycles gives 4.7%, so T-⁠5 takes about 23 times the record’s cycles per run at that noise, or a correspondingly lower noise, with T-⁠1’s four-point contrast as its held-out check.

X-⁠1, exploratory. On the record’s own parameters the phase splits as |Δφ_br| = 2πN_kept + φ_tw, with N_kept = 12.07 at τ = 1.2 ms and φ_tw/|Δφ_br| = 2a_SOZ/g_b = 2.0%. Three quantities of the record have one size: the twist’s share, 1.6 rad; the spread of the three timing conventions, 2.4 rad; and the reported model–data residual, about 2 rad. The reading fixes the scale at which this record tests the universality of free fall, about 2.5% of the phase, the size of the helical share. An in-sample redescription against the source’s blind simulation, kept apart from its tuned comparison and never counted with the tests.

FDT agrees with the measured record of general relativity and quantum mechanics without reservation. The convergence map is explicit: evaluate the Earth’s A-⁠1 exterior at the laboratory radius and take the source’s g_lab, or the site value; write each arm’s action 𝔄_j = −∫E_j dt_own on its composed clock factor; take the slow limit, which is the source’s Lagrangian with the twist’s energy; match the measured pulse history; fix the arm orientation and phase origin; and apply the same population readout. With the pulses shortened to zero and the hold at levitation, the finite-pulse reconstruction becomes the ideal phase −𝓜_Rbg_b²u³/(24ħ), with u = 2τ; with the push set to zero, it becomes the kept-time count −2πΔt_own/T_Rb. The laboratory Lagrangian, the equality of inertial and gravitational mass and the source’s Equations 1 to 6 follow from the core. The comparison with the source’s full dynamics keeps its reported residual, about 2 rad or 2.5% over about 80 rad, and relabelling by α adds nothing to it. No divergence is claimed: here FDT supplies no additional source, constitutive input or changed observable dictionary, and the five tests are where its readings meet new data. The published supplement, the data and the simulation code lie outside the evidence used; the push and the third timing convention come from the preprint’s supplementary text.