Optical, Microwave, and Nuclear Clocks

The best clock for a gravity experiment is the one whose transition sensitivity, instability, uptime, transfer link, and systematic covariance match the signal’s amplitude and timescale. Optical clocks offer high carrier frequency and low evaluated uncertainty; microwave clocks provide mature, robust, often continuous operation; thorium-229 nuclear clocks promise different sensitivity to nuclear parameters but remain a developing platform. “Most precise” is not a complete design criterion.

Required background. Equivalence Principles and Gravitational Measurement supplies proper-time and clock-comparison observables. Probability and Statistical Inference supplies covariance and likelihood concepts; the variance and residual models below give the needed local bridge.

Helpful background. Weak-Field Gravity supplies the potential perturbation that a clock must resolve.

Match a platform to the signal

A clock realizes a transition frequency $\nu_0$ by steering a local oscillator from repeated interrogations. For a gravitational signal $s(t)=\Delta\Phi(t)/c^2$ or an anomalous coupling, compare three quantities on the signal timescale $\tau_s$:

$$u_{\mathrm{total}}^2(\tau_s) =\sigma_y^2(\tau_s) +\mathbf k^{\mathsf T}C_{\rm sys}\mathbf k +u_{\mathrm{link}}^2+u_{\Phi}^2.$$

Here $\sigma_y$ is fractional-frequency instability, $C_{\rm sys}$ is the systematic covariance, $\mathbf k$ maps systematic shifts into the chosen comparison, $u_{\mathrm{link}}$ is fractional transfer uncertainty, and $u_\Phi$ is potential uncertainty divided by $c^2$. The sum assumes these four blocks are independent and that instability has been converted to uncertainty for the actual estimator; otherwise cross-covariance terms are required. Allan deviation at $\tau_s$ is not automatically the standard error of a signal amplitude. A clock with spectacular long-term accuracy can be unsuitable for a transient if dead time and short-term local-oscillator noise dominate.

Instability and systematic uncertainty behave differently under averaging. Independent white fractional-frequency samples of standard deviation $\sigma$ give a mean with uncertainty $\sigma/\sqrt N$. An unknown common offset $b$ does not average down: if it has calibration uncertainty $u_b$, the mean has variance $\sigma^2/N+u_b^2$. A modulated signal can separate itself from a constant offset, but only if enough modulation phases are observed. This is the practical reason to compare a platform’s noise and uptime with the signal’s timescale.

Platform	Principal strength	Leading systematic structure	Gravity-test niche
Caesium fountain or microwave maser	Mature SI traceability, long operation, established satellite use	Distributed-cavity phase, collisions, blackbody and Zeeman shifts, microwave electronics	Long satellite arcs, time transfer, durable reference
Optical ion clock	Strong control of one ion and small collisional shifts	Micromotion, quadrupole, blackbody, Zeeman and probe shifts	High-accuracy local comparison and potential changes
Optical lattice clock	Many atoms and rapid averaging	Lattice Stark, density, blackbody, Zeeman, tunnelling and wavefront effects	Networks, geodesy, transient and spatial-gradient searches
Thorium-229 nuclear clock	Nuclear transition with potentially enhanced sensitivity to hadronic and electromagnetic variation	Crystal or ion environment, isomer population, VUV laser, nuclear and electronic bridge shifts	Future varying-constant and complementary coupling tests

The BIPM list of secondary representations records evaluated standard frequencies, not a ranking for gravity experiments BIPM 2025, official standard-frequency table. The SI second remains defined through the unperturbed caesium-133 hyperfine transition at this evidence date BIPM 2026, §2.3.1, p. 126, PDF. Optical standards can outperform a particular caesium realization without silently redefining the unit.

Data, calibration, and selection

The minimum data product includes clock-ratio samples, interrogation and servo metadata, instability estimates, evaluated systematic budgets, dead time, environmental monitors, transfer-link records, and the gravitational template. Pin the transition realization, hardware and software configuration, frequency-comb and link calibration, and reference timescale.

Calibration is platform-specific. Microwave fountains require microwave-field and collision characterization. Optical ion clocks require secular motion and excess micromotion, electric-field gradients, and blackbody response. Lattice clocks require trap depth, density, polarization, magic-frequency detuning, and spatial gradients. Nuclear-clock work must separate the intrinsic nuclear frequency from host-crystal or electronic environment and VUV-laser response. Ludlow et al. organize optical-ion and lattice operation and uncertainty evaluation in Ludlow et al. 2015, §§II–V.

Selection includes lock loss, low atom number, laser excursions, frequency-comb cycle slips, environmental vetoes, clock uptime, and link outages. Report dead-time distribution because periodic gaps alias daily, sidereal, annual, or transient templates. Do not select only intervals in which two clocks agree.

For ratio samples $y_i$, a useful likelihood is built from

$$r_i=y_i-s(t_i;\boldsymbol\theta)-\sum_k k_kX_k(t_i),$$

with colored covariance and priors on calibrated response coefficients. Cross-clock correlations matter when clocks share a laser, frequency comb, environment, reference, or transfer path. Allan deviation diagnoses averaging behavior but does not replace a likelihood for a deterministic gravity signal.

Current platform status

Evidence assessed through 2026-08-19 shows optical clocks resolving redshift gradients within a single sample: Bothwell et al. use spatial density and Zeeman corrections, then independent region comparisons Bothwell et al. 2022, Figs. 3–4, Methods “Analysis” and “Systematics,” and Table 1. This is a complete clock-plus-systematics example, not a general performance guarantee for every optical platform.

Thorium-229 progressed from observation of radiative decay in crystals to a frequency ratio against a strontium atomic clock. The latter establishes a direct metrological connection Zhang et al. 2024, pp. 63–70 and Methods. A June 2026 preprint reports a feedback-stabilized nuclear-clock oscillator Toscani De Col et al. 2026, §§2–4, PDF, but at the cutoff it is not yet a mature gravity instrument with the independently compared systematic record of leading atomic clocks.

Reproduction means propagating each platform’s full covariance to the target signal and verifying that its measured instability reaches the required uncertainty at $\tau_s$. A useful control injects the same blind gravitational template into optical, microwave, and candidate nuclear time streams with their actual sampling. A platform fails the design test if dead time, an environmental regressor, or link noise absorbs the injection.

Exercise

Clock A has lower systematic uncertainty than clock B after one day, but its lock duty cycle is 30% and the target is a coherent two-hour transient. Which clock is automatically preferable?

Solution

Neither is automatically preferable. One must compare instability and effective transfer function on the two-hour timescale, including the probability that clock A is unlocked during the event. Its lower one-day systematic uncertainty does not compensate for missing or aliased transient data. Therefore propagate each clock’s sampling, link covariance, and systematic response through the transient likelihood, inject a blind signal, and choose the platform with better calibrated recovery rather than the smaller headline uncertainty.

References

• Bothwell, Tobias, et al. “Resolving the Gravitational Redshift across a Millimetre-Scale Atomic Sample.” Nature 602 (2022): 420–424. DOI.
• Bureau International des Poids et Mesures. “Recommended Values of Standard Frequencies.” Official table, updated 2025. Standard-frequency table.
• Bureau International des Poids et Mesures. The International System of Units (SI), 9th ed., version 4.01. Sèvres: BIPM, 2019, updated June 2026. DOI; Open PDF.
• Ludlow, Andrew D., et al. “Optical Atomic Clocks.” Reviews of Modern Physics 87 (2015): 637–701. DOI.
• Toscani De Col, Lorenzo, et al. “A Thorium-229 Optical Nuclear Clock with Feedback Loop.” Preprint (2026). arXiv.
• Zhang, Chuankun, et al. “Frequency Ratio of the $^{229\mathrm m}$Th Nuclear Isomeric Transition and the $^{87}$Sr Atomic Clock.” Nature 633 (2024): 63–70. DOI.