A lab-based test of the gravitational redshift with a miniature clock network

The study tests a central prediction of general relativity—the gravitational redshift—at millimeter to centimeter length scales using quantum systems (optical atomic clocks). While general relativity has been confirmed across many regimes, there is no unified theory with quantum mechanics, motivating precision tests at new length scales and with quantum sensors. Optical clocks now achieve instabilities and inaccuracies at or below 10^-18, enabling sensitivity to geopotential differences at sub-centimeter scales. The research question is whether the measured fractional frequency gradient across a miniature, vertically separated network of atomic ensembles matches the general relativity prediction (≈ -1.1 × 10^-18 per cm near Earth’s surface), and how precisely such a miniature clock network can measure gravitational potential differences while being independent of local oscillator coherence limits. The purpose is to perform a blinded, laboratory-based test using local-oscillator-independent synchronous differential comparisons and to quantify and correct spatially varying systematics at the 10^-19 level.

Atomic clocks have enabled gravitational redshift tests over varied length scales: comparisons between single-ion clocks separated by 30 cm, satellite-based tests using microwave clocks on eccentric orbits yielding the strongest constraints on deviations (~10^-5), and portable Sr optical lattice clocks compared over 450 m (Tokyo Skytree) achieving the most precise terrestrial constraint on deviations (~10^-5). Recent in-situ measurements resolved redshift across a millimeter-scale ensemble with differential precision of 7.6 × 10^-21. Transportable clocks essential for relativistic geodesy currently underperform laboratory systems that use ultra-stable cavities with second-scale coherence. Differential measurement techniques between ions and neutral atom ensembles, and phase-coherently linked clocks, have enabled interrogation beyond local oscillator limits, opening applications for transportable or space-based clocks. Prior multiplexed lattice clock work demonstrated sub-10^-19 differential imprecision but without full systematic attribution. This study builds on these advances to perform a blinded, fully systematic-evaluated redshift test using synchronous Ramsey spectroscopy.

Experimental platform: a spatially multiplexed 87Sr optical lattice clock with five vertically separated atomic ensembles in a one-dimensional optical lattice, equally spaced by 2.5 mm over a total of 1.0 cm. The lattice operates near the magic wavelength (zero differential polarizability between 1S0 and 3P0). Improvements enabling this work include deeper initial loading (130 E_rec) with in-lattice cooling to achieve >2000 atoms per ensemble, >99% axial ground-band occupancy, and <200 nK radial temperatures; operational Ramsey interrogation at 15 E_rec; suppression of residual magnetic field gradient along the lattice axis by >10×; and long atom-atom coherence times of 32 s using synchronous Ramsey spectroscopy (beyond ~100 ms atom-laser coherence). Interrogation and readout: Synchronous Ramsey spectroscopy with a shared clock laser interrogates magnetically insensitive transitions |g, m_f = ±5/2⟩ ↔ |e, m_f = ±3/2⟩ in an interleaved sequence. Averaging opposite nuclear spin states cancels first-order Zeeman and vector AC Stark shifts. Typical Ramsey free evolution time T_R ≈ 10 s with ~2 s dead time yields ~83% duty cycle. Spatially resolved fluorescence imaging provides parallel readout of all ensembles. Phase and frequency extraction: For each ensemble pair (10 total per run), excitation fractions P_i and P_j form ellipses due to random laser phase; least-squares ellipse fitting yields the differential phase Φ = 2π δf T_R and hence differential frequency δf. Statistical uncertainties are obtained by jackknife-based Allan deviation analysis fitted to white-frequency-noise 1/√τ scaling, consistent with QPN, giving differential instabilities < 1 × 10^-17/√τ per pair. Blinding: A pseudo-random constant offset gradient drawn uniformly from ±5 × 10^-18/cm is automatically added post-ellipse fitting to all measurements (including systematics runs). The blind was removed only after finalizing corrections and uncertainties. The revealed blinded offset was +3.7 × 10^-18/cm. Systematic evaluation: Spatially varying systematics evaluated and corrected include: - Density shifts from p-wave collisions: measured by varying atom-number differences; for a representative symmetric pair (2,4) at 15 E_rec the differential shift is -0.7(1) × 10^-19 per 100 atom difference; per-pair corrections applied accounting for trap-volume variation. - Second-order Zeeman gradient: background gradient ≈ -1.5 mG/cm with bias field ≈ -5.5 G; differential field inferred from splitting of ±m_f transitions and independent field calibration; gradient -95.3(1.0) × 10^-20/cm with ≤1 × 10^-19/cm uncertainty limited by coefficient knowledge. - Black-body radiation (BBR) gradient: induced viewport temperature differences up to ±1 K to map sensitivity; measured -4.2(1) × 10^-18/cm per 1 K difference; under normal conditions weighted BBR gradient -15.7(1.5) × 10^-19/cm from continuously monitored calibrated sensors. - Lattice light shifts: differential shift from relative trap depth difference δμ due to beam profile (linear in δμ). A residual spatial light shift gradient of -8.0(1.1) × 10^-20/E_rec/cm observed, likely due to differential tensor Stark shift from slight magnetic field orientation variations, supported by a measured vector Stark gradient of -2.5(2) × 10^-18/E_rec/cm. Varying lattice depth allows separation of δμ-dependent and residual spatial gradients and extraction of operational detuning from the effective magic wavelength. - Probe Stark, DC Stark, and ellipse fitting biases assessed and included. Data analysis: 14 blinded runs over 3 weeks (1–4 h each). Approach 1: For each run, linear fit of ten pairwise differential frequencies versus their height differences, including covariance between pairs sharing an ensemble; apply spatially varying systematic corrections (BBR, lattice light, Zeeman) and pair-specific corrections (density, ellipse fitting). Weighted average of run slopes forms the final result; statistical error inflated by √χ^2_red with χ^2_red = 1.16. Approach 2: Weighted average of each pair’s differential frequency across runs with per-pair systematic corrections, then a final linear fit vs height difference. Height reconstruction: assuming Eq. (1), extract relative heights from measured redshifts and independently measured g = -9.803 m/s^2; verify ordering and compare to known separations.

- Measured fractional frequency gradient: [-12.4 ± 0.7(stat) ± 2.5(sys)] × 10^-19/cm, consistent with the expected general relativity redshift gradient of -10.9 × 10^-19/cm within 1σ total uncertainty. - Hypothesis of zero gravitational redshift for mm–cm separations is rejected at 4.9σ. - Deviation parameter a in f = (1 + a) g h / c^2 constrained to a = 0.13 ± 0.23 at mm–cm scales. - Alternative analysis yields (-11.9 ± 2.6) × 10^-19/cm, again consistent with theory. - Gravitational potential measurement resolution equivalent to 1.3 mm (extrapolated to 0 cm separation), with systematic uncertainty dominated by differential density shifts. - Height inference: using measured redshifts and local g, the relative height ordering is recovered, and extracted pairwise height differences are within 2 mm of known values. - Systematics characterization (Table 1 representative gradients ×10^-19/cm and uncertainties): BBR: -15.7 ± 1.5; Lattice light: -11.8 ± 1.2; Density: ±1.0 (pair dependent, non-linear with separation); Second-order Zeeman: -95.3 ± 1.0; Probe Stark: 0 ± 0.5; DC Stark: 0 ± 0.1; Ellipse fitting bias: ±0.5. Total systematic correction +122.8 ± 2.5; statistical gradient -135.2 ± 0.7; corrected gradient -12.4 ± 2.6. BBR sensitivity measured at -4.2(1) × 10^-18/cm per 1 K viewport temperature difference. Residual spatial lattice light shift gradient -8.0(1.1) × 10^-20/E_rec/cm.

The experiment directly addresses whether a miniature, laboratory-based network of spatially separated optical clock ensembles measures the gravitational redshift predicted by general relativity at sub-centimeter scales, independent of local oscillator limitations. By employing synchronous Ramsey spectroscopy, interleaved spin-state interrogation to cancel first-order Zeeman and vector Stark shifts, and a blinded protocol with comprehensive systematics evaluation, the measured frequency gradient matches the theoretical expectation within combined uncertainties. The 4.9σ rejection of no-redshift confirms sensitivity at the mm–cm regime. The ability to reconstruct relative heights and resolve ~1.3 mm differences demonstrates practical utility for relativistic geodesy in controlled lab conditions. Moreover, the methodology highlights the strengths of local-oscillator-independent differential comparisons for precision sensing, characterizing spatially varying shifts (e.g., BBR gradients, tensor lattice shifts) at the 10^-19 level. These results bolster the case for using multiplexed optical lattice clocks in geodesy, searches for beyond-Standard-Model physics, and gravitational wave detection schemes where long interrogation times and low differential instabilities are crucial.

This work presents a blinded, laboratory-based test of the gravitational redshift using a miniature network of five spatially multiplexed 87Sr ensembles over 1 cm separation. The measured fractional frequency gradient agrees with general relativity and rules out zero redshift at high confidence, achieving mm-scale gravitational potential resolution. The experiment introduces a complete 10^-19-level systematic evaluation with synchronous Ramsey spectroscopy, independent of ultra-stable local oscillators, and characterizes key spatially varying systematics (BBR and tensor lattice light shifts). Future directions include scaling the apparatus to ~1 m separations to enhance signal by ~100× and reducing systematic uncertainties by over an order of magnitude via density shift cancellation (e.g., magic excitation fraction), improved thermal control to suppress BBR gradients, and elimination of magnetic field gradients to reduce tensor Stark effects. Such improvements could enable constraints on the deviation parameter at the 10^-5 level, making lab-based tests competitive with satellite and transportable-clock experiments and broadening applications in geodesy and fundamental physics.

- Common-mode rejection benefits from shared lattice and chamber; this advantage will not apply to geographically separated clocks, where uncontrolled environmental differences and frequency transfer phase noise over long baselines (>1 km) limit coherence and accuracy. - Systematic uncertainty is currently dominated by differential density shifts, which depend on atom-number balance and trap volume; although mitigated, residuals set the mm-scale resolution. - BBR gradients from viewport temperature differences contribute significantly to uncertainty; improved thermal control is required for sub-10^-19/cm accuracy. - Residual spatial lattice light shift (likely tensor Stark from field orientation variations) requires careful control of magnetic field gradients and lattice parameters. - The study focuses on vertical separations within 1 cm in a controlled lab; external validity to field-deployed, transportable systems with lower stability and accuracy is not directly established.