Optical Clock Frontiers

Status: optical lattice clocks, trapped-ion optical clocks, optical frequency combs, and phase-stabilized fibre comparison are established. International optical-clock networks, entanglement-assisted clocks, transportable relativistic geodesy, and nuclear-clock systematics are active. Projected nuclear-clock superiority, global optical time distribution at full clock accuracy, and discovery reach for future clock networks are conjectural until demonstrated. No clock experiment has confirmed dark matter or a variation of a fundamental constant.

Last reviewed: 26 July 2026. Record values, publication status, and governance milestones below are date-stamped. They are not permanent rankings.

Core Question

How can an optical transition become a reproducible, continuously useful, and internationally comparable reference whose uncertainty, instability, uptime, transfer, and interpretation all remain controlled at or below the $10^{-18}$ level?

The question is wider than finding the narrowest transition. A clock result passes through a chain:

$$\text{reference transition} \longrightarrow \text{interrogated ensemble} \longrightarrow \text{disciplined oscillator} \longrightarrow \text{comparison link} \longrightarrow \text{inference}.$$

Each arrow can add noise, bias, dead time, covariance, or model dependence. The clock frontier is therefore a systems problem. It includes the atomic or nuclear reference, but also the laser, servo, synthesis chain, link, relativistic correction, network adjustment, and statistical decision rule.

Why It Matters

Redefining and disseminating the second

Several optical standards outperform the best caesium-fountain realizations of the present SI second. That is not by itself enough to redefine the unit. The new realization must be reproducible across laboratories, accessible internationally, compatible with continuity of time scales, and disseminated without throwing away its advantage.

The Bureau International des Poids et Mesures considers both a definition based on one optical transition and definitions based on an ensemble of transitions. As of this review, the SI second is still defined by the caesium-133 ground-state hyperfine frequency. Ratification of a new definition is possible no earlier than 2030, and no final transition or ensemble has been selected.

Turning relativity into a measurement channel

Clocks at different gravitational potentials should tick at different rates. Once clock and link uncertainty reach the low-$10^{-18}$ range, frequency comparison becomes a direct probe of geopotential differences at centimetre-equivalent scales near Earth’s surface. The measurement is of potential, not geometric height alone. Tides, loading, reference frames, survey ties, and link endpoints become part of the experiment.

Testing physics beyond the calibration task

Ratios between dissimilar transitions respond differently to changes in dimensionless constants. Networks can search for drifts, oscillations, transients, spatial gradients, and symmetry-violating modulations. The same data can constrain ultralight-field models, but only after sensitivity coefficients, field coherence, sampling response, environmental witnesses, and statistical trials are specified.

Extending precision spectroscopy to a nucleus

The unusually low-energy isomeric transition in $^{229}\mathrm{Th}$ is laser accessible near $148\ \mathrm{nm}$. Laser excitation, coherent frequency linkage to an atomic clock, cross-crystal reproducibility, and feedback operation have moved the nuclear-clock programme from a spectroscopic search to an experimental clock platform. Its present performance is not yet competitive with the best atomic optical clocks. Its long-term potential remains an active empirical question.

Canonical Boundary

This page owns the dated frontier assessment: present evidence, open bottlenecks, contested interpretations, and changes since the previous review. It does not repeat the mature clock derivations.

• Optical Clocks owns the clock chain, ion and lattice architectures, comb readout, systematic shifts, clock comparisons, relativistic geodesy, and frequency-ratio response in detail.
• Atomic Clocks owns passive clock operation, Allan deviation, quantum projection noise, dead time, the Dick effect, and accumulated time error.
• Frequency Standards owns SI traceability, primary and secondary standards, TAI, UTC, dissemination, and metrological governance.
• Frequency Combs owns comb self-referencing, optical ratios, transfer oscillators, and optical frequency division.
• Ion Traps and Optical Lattices own the confinement physics.
• Variation of Constants Searches owns drift, oscillation, transient, transfer-function, and sensitivity-coefficient models.
• Tests of Fundamental Symmetries owns the broader effective-theory and symmetry-test interpretation.
• Precision AMO Frontiers compares clocks with magnetometers, atom interferometers, constants measurements, and new-force searches.

The nuclear-clock material here is an overview because no separate canonical nuclear-clock chapter is yet planned in this volume. It develops only the concepts needed to assess the frontier and links to the primary literature for platform-specific details.

Minimal Background

A clock has several performance axes

For a clock output $\nu_{\mathrm{out}}$ referred to an unperturbed frequency $\nu_0$, write the fractional result as

$$y = \frac{\nu_{\mathrm{out}}-\nu_0}{\nu_0} = y_{\mathrm{stat}} + \sum_k c_k + y_{\mathrm{link}} + y_{\mathrm{rel}}.$$

Here $c_k$ are systematic corrections, $y_{\mathrm{link}}$ is the transfer contribution, and $y_{\mathrm{rel}}$ is the correction to a declared reference potential. A complete report distinguishes:

• systematic uncertainty, attached to estimated corrections and model parameters;
• instability, describing how fluctuations average with time;
• uptime and dead time, determining whether the clock can steer a time scale or only make intermittent comparisons;
• reproducibility, tested by independent devices, laboratories, epochs, species, or host materials;
• transfer uncertainty, including fibre, free-space, satellite, comb, counter, and endpoint effects; and
• application uncertainty, such as geopotential, sensitivity coefficients, or a dark-matter likelihood.

No scalar ranking combines these without a declared task and weighting.

For projection-noise-limited Ramsey operation, an order-of-magnitude instability is

$$\sigma_y(\tau) \simeq \frac{1} {2\pi\nu_0TC\sqrt{N}} \sqrt{\frac{T_c}{\tau}},$$

where $T$ is coherent interrogation time, $C$ is contrast, $N$ is detected atom number, and $T_c$ is cycle time. This expression explains why neutral ensembles can be statistically powerful and why single-ion clocks value long coherence and high duty factor. It does not include local-oscillator noise, Dick aliasing, state-preparation overhead, technical detection noise, or systematic floors.

Ratios form a consistency graph

Let

$$x_{ij} = \ln R_{ij} = \ln\left(\frac{\nu_i}{\nu_j}\right).$$

An ideal triangle obeys

$$x_{AB} + x_{BC} - x_{AC} = 0.$$

A nonzero closure residual can arise from a clock bias, transfer bias, reference-potential mismatch, asynchronous sampling, or an incorrect covariance model. It is not automatically evidence that one transition varied. If $\boldsymbol{x}$ has covariance matrix $\boldsymbol{\Sigma}$ and $\boldsymbol{a}=(1,1,-1)^{\mathsf T}$, then

$$u_c^2 = \boldsymbol{a}^{\mathsf T} \boldsymbol{\Sigma} \boldsymbol{a}.$$

Shared lasers, combs, links, environmental corrections, and evaluation models make ratio data correlated. Network adjustment must preserve those correlations rather than average every published number as if independent.

Relativistic comparison measures potential

For stationary clocks in a weak field,

$$\frac{\Delta\nu}{\nu} \simeq \frac{\Delta U}{c^2},$$

where the sign depends on the ordering of the clocks and the definition of $\Delta U$. Near Earth’s surface one may estimate

$$\Delta U \simeq g\,\Delta h, \qquad \Delta h \simeq \frac{c^2}{g} \frac{\Delta\nu}{\nu}.$$

The local conversion is useful for intuition: $10^{-18}$ corresponds to roughly one centimetre. A rigorous geodetic result uses a relativistic reference system and an uncertainty budget for potential, not just a height conversion with a nominal $g$.

Clock ratios are model-dependent field sensors

For a ratio $R_{AB}=\nu_A/\nu_B$, a linearized variation model is

$$\delta\ln R_{AB} = \sum_X \Delta K_X^{AB} \delta\ln X, \qquad \Delta K_X^{AB} = K_X^{(A)}-K_X^{(B)}.$$

The $X$ are dimensionless parameters or effective couplings, and the $K_X$ are structure-dependent sensitivity coefficients. A sampled clock record also has a frequency-domain response:

$$\widetilde y_{\mathrm{obs}}(f) = H_{\mathrm{clock}}(f) H_{\mathrm{link}}(f) \widetilde y_{\mathrm{physical}}(f) + \widetilde n(f).$$

Dead time, servo filtering, counter gates, asynchronous nodes, and data cuts all enter $H(f)$. A bound on $\delta R/R$ becomes a dark-matter bound only after a field model, local density convention, coherence model, coupling basis, and trials correction are supplied.

What makes a nuclear clock a clock

A laser-observed resonance is not yet a running clock. Clock operation requires the transition to generate an error signal that steers an oscillator. A simplified loop is

$$e_n \simeq D\left( \nu_{\mathrm{LO},n}-\nu_{\mathrm{nuc}} \right) + \epsilon_n, \qquad \nu_{\mathrm{LO},n+1} = \nu_{\mathrm{LO},n} - G e_n,$$

where $D$ is the discriminator slope, $G$ is the loop gain, and $\epsilon_n$ is measurement noise. A nuclear-clock claim should therefore state whether the nuclear resonance merely calibrates a laser, is periodically sampled, or continuously disciplines it.

For solid-state $^{229}\mathrm{Th}$, the host crystal supplies a large number of confined nuclei but also introduces quadrupole splitting, inhomogeneous broadening, temperature shifts, defects, radiation damage, and crystal-to-crystal variation. Large signal and weak electronic coupling do not remove solid-state metrology.

What Is Known

Optical lattice clocks are established precision references

Neutral-atom lattice clocks combine thousands of atoms with confinement near a magic wavelength, where the leading differential electric-dipole light shift vanishes. Strontium and ytterbium systems have demonstrated low-$10^{-18}$ systematic evaluations and excellent short-term stability. Independent lattice clocks can compare rapidly enough to resolve gravitational redshift within a millimetre-scale atomic sample.

The mature result is not that the lattice shift is absent. Residual electric-dipole detuning, hyperpolarizability, multipolar response, motional-state dependence, density shifts, line pulling, blackbody radiation, Zeeman shifts, servo errors, and optical-path phase remain in the ledger. The platform succeeds because those terms can be measured, reversed, bounded, and independently compared.

Large atom number improves projection noise but can increase collisions, inhomogeneity, many-body interactions, and readout demands. Three- dimensional lattices and optical tweezers can isolate atoms and provide site-resolved control, but their additional light fields and control sequences must enter the clock model.

Trapped-ion clocks are established accuracy references

A single trapped ion removes ensemble density shifts and allows highly controlled state preparation and detection. Ion clocks based on Al$^+$, Yb$^+$, Sr$^+$, Ca$^+$, and other species use direct fluorescence or quantum-logic readout. Their principal burdens include secular and micromotion time dilation, electric-quadrupole shifts, blackbody radiation, magnetic fields, probe shifts, collisions, trap-induced fields, and one-ion projection noise.

In 2025, a quantum-logic Al$^+$ clock reported $5.5\times10^{-19}$ fractional systematic uncertainty and instability $3.5\times10^{-16}/\sqrt{\tau/\mathrm{s}}$. The result combined a one- second probe, a remote cryogenic-silicon optical reference delivered over $3.6\ \mathrm{km}$ of stabilized fibre, improved micromotion control, and reduced collision uncertainty. It is an established device result, not a universal statement that every ion architecture is more accurate than every lattice clock.

Independent ratios now probe hidden inconsistency

The 2021 Boulder Atomic Clock Optical Network compared Al$^+$, Yb, and Sr with total ratio uncertainties at or below $8\times10^{-18}$. In July 2026, a peer-reviewed successor reported Al$^+$/Yb, Al$^+$/Sr, and Yb/Sr ratios with total fractional uncertainties no larger than $3.2\times10^{-18}$. A common ultrastable reference and stabilized fibre improved comparison speed.

The later campaign also found differences from earlier ratio results: approximately $10^{-16}$ in ratios involving Sr and $1.6\times10^{-17}$ for Al$^+$/Yb. These observations are not a failure of comparison science. They show why redefinition criteria require repeated, high-accuracy measurements rather than a single record. Resolving which clock, correction, epoch, or covariance contribution explains a residual is now part of the frontier.

A 2025 coordinated international campaign compared ten optical clocks in six countries using fibre and satellite links and reported 38 ratios. It demonstrated that multi-institute network adjustment is operationally possible while also exposing the different uncertainty and uptime properties of fibre and GNSS transfer.

Fibre transfer can outperform the clocks it connects

Phase-stabilized optical fibre links have reached continental distances with transfer contributions below the uncertainty of current clocks. In 2026, a field-deployed $2067\ \mathrm{km}$ optical-frequency dissemination experiment reported modified Allan deviation $2.9\times10^{-21}$ at one day while maintaining lock for more than four days. This was a link demonstration, not an end-to-end comparison of two independently evaluated clocks.

Current GNSS time-transfer links remain adequate for the best microwave standards at practical averaging times but do not expose the full accuracy of the best optical clocks. Fibre cannot connect every ocean, mobile platform, or satellite. Free-space optical transfer, advanced two-way methods, transportable clocks, and space links remain necessary.

Relativistic redshift has been resolved in controlled experiments

Transportable clocks have compared separated sites and tested the gravitational redshift. Lattice clocks have resolved the redshift across a millimetre-scale sample. These are established demonstrations of clock response to potential.

An operational geodetic service is a stronger claim. It requires field uptime, transport reproducibility, independent clocks, stable transfer, survey ties to the atomic reference points, temporal gravity corrections, and comparison against conventional levelling and gravimetry. The clock measures a potential difference during a stated epoch; an equivalent height is a model-derived representation.

Thorium-229 is now a laser-addressed clock candidate

Several milestones are established by peer-reviewed work:

1. Tunable laser excitation of the $^{229}\mathrm{Th}$ isomer was demonstrated in 2024.
2. Independent solid-state spectroscopy measured the transition near $148\ \mathrm{nm}$ and its long lifetime.
3. A vacuum-ultraviolet frequency comb connected the nuclear transition coherently to a $^{87}\mathrm{Sr}$ optical clock and measured their frequency ratio.
4. A 2026 study compared differently doped Th:CaF$_2$ crystals, identified an operating temperature near $196\ \mathrm{K}$ where the first-order temperature sensitivity vanishes, and reported cross-crystal frequency agreement at the $1.1\times10^{-13}$ fractional level over seven months.

Two independent June 2026 preprints then reported feedback operation in which a continuous-wave VUV laser was locked to a resolved nuclear transition in Th:CaF$_2$. One reported instability $2\times10^{-12}/\sqrt{\tau/\mathrm{s}}$ and agreement between two crystals at the $10^{-13}$ level. The other continuously compared a subharmonic of the locked VUV laser with a Yb$^+$ clock and reported approaching $10^{-15}$ instability over one day.

Those feedback demonstrations satisfy an important operational meaning of “clock,” but they remain preprints at this review date. Their instability and reproducibility are several orders of magnitude away from the best atomic optical clocks. Claims that the nuclear clock has already surpassed atomic clocks are false.

Clock searches have set limits, not found dark matter

Optical ratios have constrained oscillations and drifts of dimensionless constants. A 2023 Yb$^+$ comparison improved limits on ultralight bosonic dark matter coupled to photons. A 2025 analysis used spatially separated clocks and cavities, including a $2220\ \mathrm{km}$ fibre comparison and GPS clock data, to constrain scalar coupling to the electron mass over a specified mass interval.

These are model-dependent null results. They do not establish that the searched field composes the Galactic dark matter, that its local density equals a standard halo value, or that untested couplings vanish. No statistically and systematically validated clock signal has been confirmed as dark matter.

What Is Actively Developing

Redefinition options and optical time scales

The metrology community is deciding not merely which transition is excellent, but how the definition should remain robust as clocks improve. A single-transition definition is conceptually simple. A weighted ensemble can distribute risk and include several mature species, but it requires a governed adjustment procedure and makes isolated realization less direct.

The mandatory work includes:

• repeated absolute frequencies and optical ratios with full covariance;
• agreement among independent clocks of the same and different species;
• reliable contributions to TAI and national UTC realizations;
• transport and transfer that do not erase optical accuracy;
• continuity with the caesium-defined second;
• international accessibility beyond a few leading laboratories; and
• procedures for a newly discovered systematic shift after redefinition.

The earliest possible date is not a prediction that redefinition will occur then. It is a governance boundary conditional on evidence.

Better lattice clocks without hidden interaction costs

Lattice-clock research is improving laser coherence, atom number, interrogation duty factor, cryogenic control, blackbody thermometry, operational magic conditions, three-dimensional confinement, and many-body-shift evaluation. Synchronous comparison can reject common laser noise and expose atomic noise more rapidly.

Tweezer clocks add deterministic loading, rearrangement, single-site readout, local control, and Rydberg-mediated gates. In 2024, tweezer platforms demonstrated universal operations and repeated ancilla-based readout on optical clock qubits. Another experiment generated GHZ states of up to nine clock qubits and observed sub-standard-quantum-limit instability for up to four qubits at short dark time. The same work showed the central caveat: a fixed GHZ size lost its advantage at the optimal dark time because of reduced phase range.

The active target is an end-to-end wall-clock gain under realistic laser noise, state-preparation time, contrast loss, readout, phase ambiguity, and systematic evaluation. State entanglement alone is not that demonstration.

Faster and more continuously operating ion clocks

Ion-clock accuracy is mature, while stability and uptime remain active engineering targets. Better local oscillators, remote coherence transfer, continuous or interleaved cooling, nondestructive readout, multi-ion architectures, improved traps, and automated recovery can reduce the one-particle statistical cost without compromising systematic control.

Highly charged ions and new electronic transitions offer different polarizabilities and sensitivity coefficients. Their promise depends on state preparation, cooling, spectroscopy, structure calculations, and reproducible systematics, not only on a theoretically favourable transition.

Networks across laboratories, media, and continents

The frontier is shifting from point-to-point frequency transfer to heterogeneous networks:

• fibre links with relay stations and active noise cancellation;
• free-space links to mobile, airborne, mountain, and space nodes;
• optical-to-microwave bridging for existing time infrastructure;
• asynchronous comparison with flywheel oscillators;
• common-view and two-way observables;
• network-wide covariance and closure diagnostics; and
• automated cycle-slip, relock, and anomaly records.

An extremely stable link over one path does not solve endpoint motion, relativistic reference transfer, intercontinental access, or clock uptime. Networks must be validated as integrated measurement systems.

Relativistic geodesy outside controlled demonstrations

Clock geodesy is moving toward transportable lattice and ion clocks, hybrid fibre/free-space comparison, mountain-to-plain campaigns, height- datum unification, and monitoring of time-dependent mass redistribution. The active bottleneck is often no longer the redshift formula. It is the joint clock–link–geodesy uncertainty model.

Useful campaigns need:

1. atomic reference-point coordinates and local survey ties;
2. clock transport and re-evaluation before and after deployment;
3. fibre, free-space, or satellite endpoint definitions;
4. tidal, ocean, atmosphere, hydrology, and loading corrections;
5. a relativistic reference system and potential convention;
6. covariance with gravimetric and levelling observations; and
7. enough uptime to separate drift from geophysical variation.

The frontier claim is not “a clock can see a centimetre.” It is that a complete campaign estimates geopotential with traceable uncertainty and adds information beyond existing methods.

Nuclear-clock accuracy and reproducibility

The immediate $^{229}\mathrm{Th}$ research programme includes:

• reproducible VUV laser generation and frequency synthesis;
• discriminator slope, servo bias, and long-term lock characterization;
• host-dependent line centres and inhomogeneous broadening;
• temperature-insensitive operating points and in situ thermometry;
• crystal growth, isotope concentration, defects, radiation damage, and radioluminescent background;
• quadrupole-resolved transition choice and polarization control;
• lifetime, quenching, and photo-induced effects;
• independent comparison between crystals and laboratories;
• trapped-ion alternatives with fewer nuclei but cleaner environments; and
• atomic-to-nuclear ratio comparisons with complete systematic budgets.

The theoretical attraction of a nuclear transition does not determine the realized susceptibility to the host. Conversely, present solid-state broadening does not prove that future nuclear clocks cannot improve.

Dark-matter searches with heterogeneous clocks

Searches are expanding from co-located ratios to regional and global networks. Different species provide different sensitivity vectors; separated nodes probe spatial phase and transient propagation; cavities and molecular references can break otherwise degenerate coupling directions. Nuclear transitions may offer enhanced sensitivity to strong-interaction parameters, but the enhancement and its uncertainty are themselves a theory problem.

The analysis frontier includes irregular sampling, nonstationary clock noise, network covariance, uncertain field coherence, stochastic rather than monochromatic fields, transient templates, environmental vetoes, blind injection, and global significance. Better raw clock uncertainty does not automatically improve every mass range because cadence, baseline, servo response, and campaign duration set the accepted band.

What Is Debated or Uncertain

Should the second use one transition or an ensemble?

A single optical transition gives a simple invariant definition and a clear primary realization. It concentrates vulnerability to an unknown species-specific shift and makes consensus over the chosen species harder. An ensemble distributes that risk and can evolve with performance, but it requires weights, covariance, governance, and access to a network of realizations. The choice is metrological policy informed by physics, not a contest settled by the smallest published number.

Are ion or lattice clocks fundamentally better?

Ion clocks offer isolation and clean one-particle control; lattice clocks offer atom number and rapid stability. Both inherit architecture-specific systematics. The answer changes with the task: lowest systematic uncertainty, one-day ratio precision, continuous time-scale steering, transportability, network simplicity, or new-physics sensitivity. “Fundamentally better” is not a well-defined platform metric.

Will entanglement improve the best operating clocks?

Sub-standard-quantum-limit phase estimation and clock-qubit entanglement are established in controlled regimes. Whether entanglement lowers uncertainty per wall-clock time in a fully optimized clock depends on local- oscillator noise, dynamic range, loss, state preparation, readout, servo design, and systematic shifts introduced by the entangling interaction. The useful comparison is against the best classical protocol with the same resources and accepted operating conditions.

Will a nuclear clock surpass atomic clocks?

The $^{229}\mathrm{Th}$ transition has a high intrinsic quality factor, weak direct coupling of the nucleus to some external perturbations, and potentially enhanced sensitivity to fundamental couplings. Real devices must also control VUV synthesis, solid-state host shifts or trapped-ion systematics, nuclear-structure coefficients, and reproducibility. Superiority is a plausible research programme, not an established outcome.

Is relativistic geodesy ready as routine infrastructure?

The physical principle and laboratory demonstrations are established. Routine centimetre-level or better potential transfer across arbitrary sites is not. Clock availability, transport, fibre access, free-space weather, reference frames, geophysical variability, and survey ties all matter. Readiness should be assessed for a specified baseline, averaging time, environment, and required traceability.

How much does a null clock search say about dark matter?

A null ratio constrains a response model over a finite band. Translating it into a coupling bound can depend on local density, field composition, coherence, screening, multiple couplings, nuclear and electronic sensitivity coefficients, and the chosen confidence construction. Different papers may give different limits from the same data because they ask different model questions. That is not automatically inconsistency.

Key Experimental Platforms

Platform	Established strength	Leading frontier	Principal caution
neutral-atom optical lattice	large atom number, high stability, low-$10^{-18}$ evaluations	longer coherence, cryogenic control, high duty factor, international reproduction	lattice, density, blackbody, and many-body shifts
single trapped ion	isolation, state control, very low systematic uncertainty	faster cycles, higher uptime, improved coherence transfer	one-particle projection noise, motion, trap fields
quantum-logic ion clock	access to species without convenient cooling or readout	simpler logic, continuous cooling, new ions	mapping, shared motion, heating, added control chain
tweezer clock	deterministic loading, site resolution, rearrangement, gates	end-to-end quantum-enhanced operation	light shifts, gate overhead, loss, limited phase range
local multispecies network	rapid ratios and closure with shared oscillator	identifying campaign-to-campaign discrepancies	shared-reference covariance and common bias
continental fibre network	phase-coherent transfer below clock uncertainty	uptime, relay automation, international scale	path asymmetry, cycle slips, endpoint definitions
free-space or satellite link	access to mobile, remote, and intercontinental nodes	optical time transfer through atmosphere and space	weather, pointing, reciprocity, relativistic motion
transportable optical clock	direct field comparison and geodesy	routine deployment with retained accuracy	transport shift, environmental control, low uptime
solid-state thorium-229	many nuclei, room-temperature-compatible host, VUV absorption	reproducible feedback clock and host-shift control	broadening, temperature, defects, radioactivity
trapped thorium ion	cleaner and more controllable environment in principle	nuclear-state preparation and readout	low signal, complex charge-state and nuclear spectroscopy

Key Theoretical and Statistical Tools

Clock sensitivity functions

Sensitivity functions propagate local-oscillator phase and frequency noise through finite pulses, dead time, and servo updates. They determine Dick aliasing and the response to a searched signal. For network searches, every node and link needs its own transfer function and time convention.

Covariance-aware uncertainty propagation

Uncertainty budgets are not just lists of diagonal entries. Shared thermometry, polarizability calculations, gravitational models, reference lasers, combs, links, and data selections create covariance. The relevant rule is

$$u_f^2 = \boldsymbol{J} \boldsymbol{\Sigma} \boldsymbol{J}^{\mathsf T},$$

where $\boldsymbol{J}$ is the Jacobian of the reported quantity with respect to the inputs. Variance and Covariance develops the mathematical background.

Graph adjustment and closure

A clock network is naturally a graph whose nodes are transitions and whose edges are measured ratios. Weighted least squares or Bayesian hierarchical models estimate consistent node frequencies, between-campaign scatter, and latent biases. Independent cycles in the graph provide closure tests. Leave-one-edge-out and leave-one-laboratory-out analyses reveal excessive influence.

Atomic, molecular, and nuclear structure

Electronic-structure calculations provide polarizabilities, blackbody shifts, quadrupole moments, Zeeman coefficients, isotope shifts, and fundamental-constant sensitivities. Nuclear-structure and solid-state models enter $^{229}\mathrm{Th}$ sensitivity, quadrupole splitting, host fields, and line shifts. Theory uncertainty and convention must accompany any inferred coupling.

Relativistic reference systems

Chronometric levelling requires more than $\Delta\nu/\nu=\Delta U/c^2$. The endpoint worldlines, coordinate-time convention, geopotential model, Earth rotation, tides, and moving-platform effects must be defined. Geometric height, orthometric height, and geopotential number are not interchangeable observables.

Search likelihoods and detection efficiency

Dark-matter and transient searches use matched filters, periodograms, cross-spectra, Gaussian-process or state-space noise models, and network coincidence tests. The analysis must include the spectral window, frequency trials, nuisance lines, data-quality cuts, and injected-signal recovery. Fisher Information helps design local sensitivity, but global discovery significance requires the full likelihood and search procedure.

Evidence Required for a Frontier Claim

A clock-accuracy record

Report the correction convention, complete budget, uncertainty coverage, evaluation duration, environmental range, sensitivity coefficients, reversals, servo behaviour, raw-to-final audit trail, and independent comparison. A record systematic budget without a comparison remains a device evaluation, not independent proof of the unperturbed frequency.

A stability or quantum-enhancement record

State the Allan statistic, averaging-time range, estimator, duty factor, dead-time treatment, atom number, contrast, laser noise, postselection, state-preparation and readout overhead, and equal-resource classical baseline. Show absolute performance as well as a normalized gain.

A network comparison

Publish link and clock uptimes, synchronization, cycle-slip handling, flywheel extrapolation, covariance, gravitational reference potentials, closure residuals, and robustness to alternate data cuts. A link floor measured by loopback is not the uncertainty of the full network.

A geodetic result

Identify both atomic reference points, the comparison epoch, clock and link budgets, local ties, relativistic convention, gravity and tide model, survey covariance, and the distinction between measured potential and reported height. Compare against an independent geodetic method where possible.

A nuclear-clock milestone

Distinguish laser excitation, absolute frequency measurement, frequency ratio, repeated reference measurement, closed-loop operation, and systematic clock evaluation. Report host, transition component, temperature, line model, discriminator, servo, instability statistic, and independent material reproduction.

A dark-matter limit or candidate

Specify the effective interaction, coupling normalization, assumed field density and coherence, sensitivity coefficients, transfer functions, search band, trials, nuisance treatment, environmental witnesses, injection efficiency, and confidence construction. A candidate additionally requires persistence under analysis changes and predictive confirmation in an independent instrument with the expected phase and sensitivity pattern.

Common Misconceptions

The smallest uncertainty number identifies the best clock

Systematic uncertainty, instability, uptime, reproducibility, linkability, and task sensitivity are different axes. A clock optimized for one need not dominate the others.

An optical clock is already the SI definition

Optical standards contribute as secondary representations and outperform caesium in laboratory performance, but the SI second remains defined by the caesium-133 hyperfine transition as of this review.

A shared laser makes two clocks independent

Common interrogation can reject local-oscillator noise and accelerate a comparison. It also creates covariance and can hide a shared optical-path or synthesis bias. Independence must be established at the level relevant to the claim.

A fibre link at the $10^{-21}$ level creates a $10^{-21}$ clock network

The reported statistic may characterize one stabilized link under a specific estimator and averaging time. Endpoint clocks, local fibres, combs, counters, gravitational potentials, and uptime still bound the end-to-end comparison.

A clock measures geometric height directly

It measures a frequency ratio that responds to a relativistic potential difference. Converting that result to a height requires a gravity field, datum, reference system, and local survey.

A nuclear transition is immune to the environment

The nucleus can be less sensitive to some electronic perturbations, but a solid host produces quadrupole structure, temperature shifts, inhomogeneous broadening, defects, and radiation-related effects. These are measurable systematics, not absent interactions.

The first feedback loop means the nuclear clock has won

Feedback operation is a major platform milestone. The June 2026 demonstrations remain preprints and their present instability is much larger than that of mature optical atomic clocks.

A periodic ratio residual is dark matter

Environmental cycles, sampling aliases, servo response, link noise, analysis choices, and ordinary oscillator behaviour can all create periodicity. Dark-matter attribution requires the predicted coupling, frequency, coherence, spatial phase, and cross-species pattern.

Connections to Other Volumes

• Density Operators supply the state language for dephasing, mixtures, and collective clock probes.
• Composite Systems and Entanglement supplies entangled-probe, reduced-state, and correlation concepts used by quantum-enhanced clocks.
• Quantum Channels and Noise supplies noisy evolution, loss, and information-processing language.
• Continuous Quantum Measurement supplies filtering and feedback concepts for nondestructive readout and disciplined oscillators.
• Approximation and Error Estimates supplies controlled-expansion and model-validity discipline.
• Precision AMO Frontiers places clocks beside other quantum sensors and new-force searches.
• Computational Reproducibility Benchmarks supplies versioning, convergence, uncertainty, and independent- reproduction practices for structure calculations.
• Frontiers and Open Problems supplies the claim taxonomy and living-review policy.

What Changed This Year

This review records five changes that were current on 26 July 2026.

1. Multispecies ratios became more precise and more diagnostic. Aeppli et al. published Al$^+$/Yb, Al$^+$/Sr, and Yb/Sr ratios with total uncertainties at or below $3.2\times10^{-18}$ in Physical Review Letters on 14 July 2026. Differences from the earlier BACON campaign sharpen the reproducibility problem; they do not establish new physics.
2. Solid-state nuclear references gained a peer-reviewed reproducibility study. Ooi et al. reported a temperature-insensitive operating region and $1.1\times10^{-13}$ cross-crystal fractional reproducibility over seven months.
3. Two groups reported nuclear-clock feedback operation. The June 2026 Th:CaF$_2$ demonstrations are independent preprints. Both used a resolved nuclear transition to steer a continuous-wave VUV oscillator. Peer review and complete systematic evaluations remain pending.
4. Long-haul optical dissemination advanced. A $2067\ \mathrm{km}$ field link reported $2.9\times10^{-21}$ modified Allan deviation at one day with more than four days of continuous lock. This is a transfer result, not a global-clock accuracy demonstration.
5. The SI decision remains open. The BIPM roadmap still treats 2030 as the earliest possible ratification. Single-transition and ensemble-transition definitions remain under consideration, and present GNSS transfer does not fully expose the best optical-clock performance.

No result reviewed here constitutes a confirmed detection of dark matter, variation of a constant, or violation of general relativity.

Representative Papers and Reviews

Optical-clock foundations and standards

• A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Reviews of Modern Physics 87, 637–701 (2015). The standard comprehensive review of ion and lattice clock physics.
• N. Dimarcq et al., “Roadmap towards the redefinition of the second,” Metrologia 61, 012001 (2024), together with the current BIPM FAQ. These are the authoritative sources for criteria, options, and timeline.
• T. M. Fortier, A. N. Luiten, and H. S. Margolis, “Optical atomic clocks: defining the future of time and frequency metrology,” Optica 13, 143–163 (2026). A current review of clock progress and redefinition.

Lattice, ion, and quantum-enhanced clocks

• M. C. Marshall et al., “High-stability single-ion clock with $5.5\times10^{-19}$ systematic uncertainty,” Physical Review Letters 135, 033201 (2025).
• E. Oelker et al., “Demonstration of $4.8\times10^{-17}$ stability at $1\ \mathrm{s}$ for two independent optical clocks,” Nature Photonics 13, 714–719 (2019).
• E. Pedrozo-Peñafiel et al., “Entanglement on an optical atomic-clock transition,” Nature 588, 414–418 (2020).
• R. Finkelstein et al., “Universal quantum operations and ancilla-based read-out for tweezer clocks,” Nature 634, 321–327 (2024).
• A. Cao et al., “Multi-qubit gates and Schrödinger cat states in an optical clock,” Nature 634, 315–320 (2024). This work explicitly demonstrates both sub-standard-quantum-limit operation and the dynamic-range limitation.

Ratios and networks

• Boulder Atomic Clock Optical Network Collaboration, “Frequency ratio measurements at 18-digit accuracy using an optical clock network,” Nature 591, 564–569 (2021).
• A. Aeppli et al., “Atomic clock frequency ratios with fractional uncertainty $\leq3.2\times10^{-18}$,” Physical Review Letters 137, 033201 (2026).
• T. Lindvall et al., “Coordinated international comparisons between optical clocks connected via fiber and satellite links,” Optica 12, 843–852 (2025).
• E. Caldwell, T. Triano, and L. Sinclair, “High-precision optical time and frequency transfer,” Advances in Optics and Photonics 17 (2025). A review separating time transfer, frequency transfer, fibre, and free-space methods.
• F.-X. Chen et al., “$10^{-21}$-level optical frequency dissemination over $2067\ \mathrm{km}$ of noise-loaded field-deployed fiber network,” Light: Science & Applications 15, 276 (2026).

Relativistic geodesy

• W. F. McGrew et al., “Atomic clock performance enabling geodesy below the centimetre level,” Nature 564, 87–90 (2018).
• J. Grotti et al., “Geodesy and metrology with a transportable optical clock,” Nature Physics 14, 437–441 (2018).
• M. Takamoto et al., “Test of general relativity by a pair of transportable optical lattice clocks,” Nature Photonics 14, 411–415 (2020).
• T. Bothwell et al., “Resolving the gravitational redshift across a millimetre-scale atomic sample,” Nature 602, 420–424 (2022).

Thorium-229 nuclear clocks

• J. Tiedau et al., “Laser excitation of the Th-229 nucleus,” Physical Review Letters 132, 182501 (2024).
• R. Elwell et al., “Laser excitation of the $^{229}\mathrm{Th}$ nuclear isomeric transition in a solid-state host,” Physical Review Letters 133, 013201 (2024).
• C. Zhang et al., “Frequency ratio of the $^{229\mathrm m}\mathrm{Th}$ nuclear isomeric transition and the $^{87}\mathrm{Sr}$ atomic clock,” Nature 633, 63–70 (2024).
• T. Ooi et al., “Frequency reproducibility of solid-state thorium-229 nuclear clocks,” Nature 650, 72–78 (2026).
• L. Toscani De Col et al., “A thorium-229 optical nuclear clock with feedback loop,” arXiv preprint (2026). Publication status: not peer reviewed at this review date.
• B. Huang et al., “A nuclear clock based on $^{229}\mathrm{Th}$,” arXiv preprint (2026). Publication status: not peer reviewed at this review date.

Clocks and dark-matter searches

• M. Filzinger et al., “Improved limits on the coupling of ultralight bosonic dark matter to photons from optical atomic clock comparisons,” Physical Review Letters 130, 253001 (2023).
• M. Filzinger et al., “Ultralight dark matter search with space-time separated atomic clocks and cavities,” Physical Review Letters 134, 031001 (2025).
• E. Fuchs et al., “Searching for dark matter with the $^{229}\mathrm{Th}$ nuclear lineshape from laser spectroscopy,” Physical Review X 15, 021055 (2025).

Exercises

Exercise 1: Projection-noise clock estimate

A lattice clock interrogates $N=1000$ atoms on a transition with $\nu_0=4.29\times10^{14}\ \mathrm{Hz}$. Take interrogation time $T=1.0\ \mathrm{s}$, contrast $C=0.80$, and cycle time $T_c=1.5\ \mathrm{s}$.

1. Estimate the projection-noise-limited Allan deviation at $\tau=1\ \mathrm{s}$.
2. Estimate the value at one hour under white-frequency-noise averaging.
3. Name two reasons the measured one-second instability could be worse.

Solution

Using

$$\sigma_y(\tau) \simeq \frac{1} {2\pi\nu_0TC\sqrt{N}} \sqrt{\frac{T_c}{\tau}},$$

the one-second estimate is

$$\begin{aligned} \sigma_y(1\ \mathrm{s}) &\simeq \frac{\sqrt{1.5}} {2\pi(4.29\times10^{14})(1.0)(0.80)\sqrt{1000}} \\ &\simeq 1.80\times10^{-17}. \end{aligned}$$

For white frequency noise,

$$\sigma_y(3600\ \mathrm{s}) \simeq \frac{1.80\times10^{-17}}{\sqrt{3600}} = 3.00\times10^{-19}.$$

Real performance can be worse because of local-oscillator phase noise, Dick aliasing during dead time, atom-number fluctuations, imperfect state preparation, detection noise, collisions, contrast drift, or nonwhite systematics. The one-hour extrapolation is valid only while the white-noise law persists.

Exercise 2: Covariant ratio closure

Three logarithmic ratio measurements $\boldsymbol{x}=(x_{AB},x_{BC},x_{AC})$ each have standard uncertainty $u=3.0\times10^{-18}$. Their correlation coefficients are

$$\rho_{AB,BC}=0.40, \qquad \rho_{AB,AC}=0.20, \qquad \rho_{BC,AC}=0.10.$$

Find the standard uncertainty of the closure $c=x_{AB}+x_{BC}-x_{AC}$. Compare it with the result obtained by falsely treating all three ratios as independent.

Solution

The variance is

$$\begin{aligned} u_c^2 &= u^2+u^2+u^2 +2(0.40u^2) -2(0.20u^2) -2(0.10u^2) \\ &= 3.2u^2. \end{aligned}$$

Therefore

$$u_c = \sqrt{3.2}(3.0\times10^{-18}) \simeq 5.37\times10^{-18}.$$

Ignoring covariance would give

$$u_{c,\mathrm{diag}} = \sqrt{3}\,u \simeq 5.20\times10^{-18}.$$

The numerical difference is modest in this example, but its sign and size depend on the covariance structure. A shared reference can either increase or decrease a particular closure uncertainty because the closure has both positive and negative coefficients.

Exercise 3: From clock ratio to geopotential

Two corrected clocks show a fractional difference

$$y = (2.50\pm0.60)\times10^{-18}$$

at one standard uncertainty. Using $c=299\,792\,458\ \mathrm{m\,s^{-1}}$ and $g=9.81\ \mathrm{m\,s^{-2}}$, estimate:

1. the potential difference $\Delta U$;
2. the local equivalent height difference $\Delta h$; and
3. its standard uncertainty.

State one reason this is not yet a geodetic height result.

Solution

The potential difference is

$$\Delta U = c^2y \simeq (8.98755\times10^{16}) (2.50\times10^{-18}) = 0.2247\ \mathrm{m^2\,s^{-2}}.$$

The local height estimate is

$$\Delta h = \frac{\Delta U}{g} \simeq \frac{0.2247}{9.81}\ \mathrm{m} = 2.29\times10^{-2}\ \mathrm{m} = 2.29\ \mathrm{cm}.$$

Its standard uncertainty is

$$u(\Delta h) = \frac{c^2}{g} (0.60\times10^{-18}) \simeq 5.50\ \mathrm{mm}.$$

A geodetic result still needs the reference system, clock endpoint coordinates, local survey ties, gravity and tide model, temporal epoch, link correction, and covariance with conventional observations.

Exercise 4: A nuclear-clock stability milestone

A preprint reports a solid-state nuclear-clock instability

$$\sigma_y(\tau) = \frac{2.0\times10^{-12}} {\sqrt{\tau/\mathrm{s}}}.$$

1. What value does this model predict after one day?
2. How long would it take to reach $1.0\times10^{-15}$ if the white-noise law continued without a floor?
3. Why does neither answer establish clock accuracy?

Solution

One day is $86\,400\ \mathrm{s}$, so

$$\sigma_y(1\ \mathrm{day}) = \frac{2.0\times10^{-12}} {\sqrt{86\,400}} \simeq 6.80\times10^{-15}.$$

To reach $10^{-15}$,

$$10^{-15} = \frac{2.0\times10^{-12}} {\sqrt{\tau/\mathrm{s}}},$$

which gives

$$\tau = \left( \frac{2.0\times10^{-12}}{10^{-15}} \right)^2 \mathrm{s} = 4.0\times10^6\ \mathrm{s} \simeq 46.3\ \mathrm{days}.$$

Instability measures fluctuations under a stated statistic. Accuracy requires corrections and uncertainties for temperature, crystal shifts, line model, servo, frequency synthesis, reference comparison, and other systematics. The white-noise extrapolation may also fail before 46 days.

Exercise 5: Signal response and an ultralight-field limit

A clock ratio has differential fine-structure sensitivity $\Delta K_\alpha=6.0$. At one searched frequency, the analysis limits the observed fractional ratio amplitude to $8.0\times10^{-19}$. The magnitude of the combined clock-and-link transfer function is $|H|=0.40$.

Assuming only $\alpha$ varies, find the corresponding amplitude limit on $|\delta\alpha/\alpha|$. Then list three ingredients still required before presenting a dark-matter coupling limit.

Solution

The observed amplitude is

$$|y_{\mathrm{obs}}| = |H|\,|\Delta K_\alpha| \left| \frac{\delta\alpha}{\alpha} \right|.$$

Therefore

$$\left| \frac{\delta\alpha}{\alpha} \right| < \frac{8.0\times10^{-19}} {(0.40)(6.0)} = 3.33\times10^{-19}.$$

A dark-matter coupling limit additionally needs a field normalization and local-density assumption, a relation between the field and $\alpha$, a coherence and spectral model, global trials treatment, sensitivity- coefficient uncertainty, environmental vetoes, and detection-efficiency validation. It must also state whether other couplings are fixed to zero or profiled.

Exercise 6: Does an entangled protocol win in wall-clock time?

Protocol A uses an unentangled ensemble. Its phase variance is $V_A=1.0\times10^{-3}\ \mathrm{rad^2}$ per accepted shot, its cycle time is $1.0\ \mathrm{s}$, and every shot is accepted.

Protocol B uses an entangled state. Its accepted-shot phase variance is $V_B=2.5\times10^{-4}\ \mathrm{rad^2}$, its cycle time is $2.5\ \mathrm{s}$, and its acceptance probability is $0.70$.

For independent shots, compare their Fisher-information rates using $F=1/V$ for an ideal local phase estimator. Which protocol has the lower statistical variance after equal wall-clock time?

Solution

For A,

$$\dot F_A = \frac{1/V_A}{1.0\ \mathrm{s}} = 1.0\times10^3\ \mathrm{s^{-1}}.$$

For B, the accepted information per attempted cycle is $0.70/V_B$, so

$$\dot F_B = \frac{0.70/V_B}{2.5\ \mathrm{s}} = \frac{0.70} {(2.5)(2.5\times10^{-4})} \mathrm{s^{-1}} = 1.12\times10^3\ \mathrm{s^{-1}}.$$

Protocol B has $12\%$ larger ideal Fisher-information rate, so its asymptotic variance is smaller by the factor

$$\frac{\operatorname{Var}_B} {\operatorname{Var}_A} \simeq \frac{\dot F_A}{\dot F_B} \simeq 0.893.$$

This modest wall-clock gain is much smaller than the fourfold accepted-shot variance improvement. A real comparison must also include phase range, laser noise, failed-shot correlations, systematics, and estimator bias.

Exercise 7: Classify a nuclear-clock claim

A group reports a continuous-wave VUV laser locked to a resolved $^{229}\mathrm{Th}$ transition in one crystal. The lock operates for a day, the Allan deviation averages approximately as white frequency noise, and a subharmonic is compared with an atomic clock. No complete systematic budget or independent peer-reviewed reproduction is available.

State:

1. one established conclusion supported by the described experiment;
2. three active questions; and
3. one overclaim.

Solution

An established conclusion is that, under the reported apparatus and analysis conditions, the resolved nuclear transition supplied a usable frequency discriminator that steered the VUV-related oscillator for the demonstrated interval. This is stronger than observing laser excitation alone.

Active questions include whether the line centre is reproducible across crystals, laboratories, and epochs; whether temperature, quadrupole, defect, radiation, and servo shifts support a complete accuracy budget; whether the observed averaging persists for longer campaigns; whether the frequency synthesis is independently validated; and whether peer review and reproduction preserve the reported performance.

It would be an overclaim to say that the nuclear clock is already more accurate or stable than the best atomic optical clocks, or that it has detected dark matter.