Experiment: Precision Spectroscopy and Atomic Clocks

Contents

Historical context and the prediction under test
Apparatus
Procedure
Observations
Interpretation
Modern repetitions and precision

Frequency is the quantity physics measures best. Nothing else is known to eighteen significant figures, and the reason is structural: a frequency is a count of cycles against a clock, so its uncertainty falls with averaging time rather than being set by the resolution of an instrument. This chapter is the experimental record of that advantage. It begins with the microwave measurement by Lamb and Retherford of a splitting the Dirac equation of The Dirac Equation forbids [Lamb:1947], whose explanation by Bethe [Bethe:1947] started renormalized quantum electrodynamics (Quantum Electrodynamics and Renormalization). It continues through the hydrogen \(1S\)–\(2S\) two-photon transition, whose natural linewidth of about \(1.3\,\mathrm{Hz}\) on a carrier of \(2.5\times 10^{15}\,\mathrm{Hz}\) makes it the sharpest transition in the simplest atom [Parthey:2011] [Matveev:2013], and through the femtosecond frequency comb [Diddams:2000] [Jones:2000] [Udem:2002], which replaced building-sized frequency chains with a single laser and made optical clocks possible. It ends with clocks whose systematic uncertainty is below \(10^{-18}\) [Brewer:2019] [McGrew:2018] [Bothwell:2022].

When written the chapter will use the structured experiment environment, with apparatus, procedure, observations, interpretation and references as separate fields, and the measured frequencies with their uncertainty budgets lodged as a dataset in the evidence store. Its place in the book is as the quantitative test bench for The Hydrogen Atom and Atoms and Molecules: the atomic theory of those chapters is where the numbers come from, and the disagreements recorded here — above all the proton-radius puzzle [Pohl:2010] [Antognini:2013], which moved the accepted proton charge radius by seven standard deviations — are what a \(10^{-15}\) measurement can do to a theory. The chapter also supplies evidence used elsewhere: the constancy of the fundamental constants (Measurement, SI Units, and the Theory of Errors) [Rosenband:2008], the gravitational redshift of The Equivalence Principle and Classical Tests measured across a millimetre [Bothwell:2022], and the CPT test of Discrete Symmetries and CPT in antihydrogen [Ahmadi:2018]. The review is [Ludlow:2015].

Derivation pending.

Experiment: Precision Spectroscopy and Atomic Clocks: all derivations of this chapter are pending.

Historical context and the prediction under test

The degeneracy the Dirac equation predicts

[Reserved: the Dirac spectrum of hydrogen (The Dirac Equation), in which the energy depends only on \(n\) and \(j\), so that \(2S_{1/2}\) and \(2P_{1/2}\) are exactly degenerate; the pre-war optical evidence, which was consistent with the degeneracy to within its resolution; why an optical measurement could not settle it and a radio-frequency measurement between the two levels could.]

The Lamb shift

[Reserved: the Lamb–Retherford experiment, in which a beam of metastable \(2S_{1/2}\) hydrogen atoms was driven to \(2P\) by a microwave field and the quenching detected as a drop in the beam current, giving a splitting of about \(1000\,\mathrm{MHz}\) where theory predicted zero [Lamb:1947]; Bethe's nonrelativistic estimate, made within weeks, which located the effect in the electron's interaction with its own radiation field and got the number essentially right by a mass subtraction [Bethe:1947]; the modern value near \(1057.8\,\mathrm{MHz}\), and the pointer to Quantum Electrodynamics and Renormalization and Experiment: The Electron Anomalous Magnetic Moment.]

What a frequency measures

[Reserved: the chain from a measured transition frequency to a constant of nature — the Rydberg constant \(R_{\infty}=10973731.568157(12)\,/\mathrm{m}\) as the most precisely known of the CODATA inputs [Mohr:2025] [Tiesinga:2021], the proton charge radius entering through the finite-size shift of the \(S\) levels, and the fine-structure constant entering through the QED terms; that a single transition determines a combination and not a constant, so at least two independent measurements are always needed; the definition of the second in the SI [BIPM:2019] as the fixed point of the whole construction.]

Phenomenon 90.1 (Frequency uncertainty falls as the inverse square root of the averaging time).

The fractional frequency instability of an atomic clock — the deviation between successive averages of its output — falls as \(\tau^{-1/2}\) with the averaging time \(\tau\), over many decades, before flattening on to a floor set by systematic effects [McGrew:2018] [Brewer:2019]. Nothing else in physics is measured this way. The resolution of the instrument is not the limit; a longer measurement buys precision at a known and verifiable rate, and the rate itself is checked by plotting the deviation against \(\tau\) and reading off the slope. This is why frequency is the best-measured quantity there is, and why a precision spectroscopy paper is mostly about its systematic budget rather than about its statistics.

Derivation. Interrogate \(N\) uncorrelated atoms with a Ramsey sequence of free-evolution time \(T\) inside a cycle of total duration \(T_{\mathrm{c}}\geq T\), and steer the oscillator to the centre of the resulting fringe. Each atom yields one of two outcomes, so the transition probability is estimated with the binomial uncertainty \(1/(2\sqrt{N})\) at the half-height point of the fringe, where its slope is steepest. By Phenomenon 90.2 that slope is \(\pi T\) per unit frequency, so one cycle determines the frequency to

\begin{equation}\tag{90.1} \delta\nu\sim\frac{1}{2\pi T\sqrt{N}}\ep \end{equation}

Successive cycles are statistically independent, so after a time \(\tau\) containing \(M=\tau/T_{\mathrm{c}}\) of them the error falls as \(M^{-1/2}\) and the fractional instability is

\begin{equation}\tag{90.2} \sigma_{y}(\tau)\sim\frac{1}{2\pi\nu_{0}T\sqrt{N}} \sqrt{\frac{T_{\mathrm{c}}}{\tau}}\ec \end{equation}

with \(\nu_{0}\) the transition frequency. Three readings of Equation (90.2) between them account for the shape of the whole field. The \(\tau^{-1/2}\) law is nothing but the independence of the cycles, so it persists exactly until some drift correlates them, and the \(\tau\) at which the measured curve departs from the slope is therefore a diagnostic. The instability is inversely proportional to \(\nu_{0}\), so replacing a microwave transition near \(10^{10}\,\mathrm{Hz}\) by an optical one near \(10^{15}\,\mathrm{Hz}\) buys five orders of magnitude at equal \(T\) and \(N\) — which is the entire case for optical clocks, and the reason the counting problem of Section 90.2.5 had to be solved before they could be used at all. And \(T\) is in practice limited by the coherence of the interrogating laser rather than of the atom, which is why Section 90.2.4 is a section about lasers in a chapter about atoms.

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Apparatus

Resonance methods

[Reserved: Rabi's molecular-beam magnetic resonance, which turned a frequency into a deflection and a null [Rabi:1938]; Ramsey's method of separated oscillatory fields, whose interference fringe is narrower than the transit-time limit of a single interaction region and which is the operating principle of every atomic clock since [Ramsey:1950]; the hydrogen maser as an active oscillator on the ground-state hyperfine transition [Goldenberg:1960]; the microwave method of [Lamb:1947] as the same idea applied to an excited state.]

Phenomenon 90.2 (Separated fields give a fringe narrower than the interaction).

Splitting the interrogation into two short pulses separated by a free interval \(T\), instead of applying one continuous pulse, produces a transition probability that oscillates with the detuning with period \(1/T\). The central fringe of that pattern is narrower than the resonance obtained from a single interaction of the same total duration, and narrower than the transit-time width of either pulse taken alone: the width is fixed by the time the atom spends undisturbed between the pulses, not by the time it spends interacting. Every atomic clock built since 1950 rests on this, and the SI second [BIPM:2019] is realized with it.

Derivation. Treat the two levels as a spin one-half in the frame rotating at the drive frequency, where a detuning \(\delta\) in angular measure appears as a field along the polar axis and the drive as a field in the equatorial plane. A pulse of area \(\pi/2\), short enough that \(\delta\) may be neglected during it, carries the initial state from the pole to the equator, \(\ket{0}\mapsto(\ket{0}+\ket{1})/\sqrt{2}\). During the free interval the two components acquire a relative phase \(\delta T\),

\begin{equation}\tag{90.3} \tfrac{1}{\sqrt{2}}\bigl(\ket{0}+\ket{1}\bigr) \longmapsto \tfrac{1}{\sqrt{2}}\bigl(\ket{0} +\ee^{\ii\delta T}\ket{1}\bigr)\ec \end{equation}

and a second identical pulse converts that phase back into a population,

\begin{equation}\tag{90.4} P_{1}(\delta)=\tfrac{1}{2}\bigl[1+\cos(\delta T)\bigr]\ep \end{equation}

The fringes are periodic in \(\delta\) with period \(2\pi/T\), that is \(1/T\) in ordinary frequency; the central fringe has full width at half maximum \(1/2T\); and the steepest slope, at the half-height point, is \(\pi T\) per unit frequency, which is the number Phenomenon 90.1 needs. Nothing in Equation (90.4) refers to the pulse duration at all.

Two consequences constrain the apparatus. The pattern is periodic, so the resonance must first be located to better than \(1/T\) before the central fringe can be identified, which is why an interrogation is stepped down in \(T\) from a broad line. And any phase acquired between the pulses that is not \(\delta T\) — from a stray field, a residual motion, or an imperfect drive — is indistinguishable from a frequency offset. That is the reason the systematic budget of Section 90.3 is, item for item, a list of phases.

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Doppler-free laser spectroscopy

[Reserved: the first-order Doppler width, which at room temperature exceeds the natural width of an optical transition by four orders of magnitude and is the obstacle every technique here addresses; saturated-absorption spectroscopy; Doppler-free two-photon excitation, in which counter-propagating beams cancel the linear shift for every velocity class at once — applied to hydrogen \(1S\)–\(2S\) by Hänsch and collaborators [Hansch:1975b]; the residual second-order (time-dilation) shift, which is not cancelled and which is why the atoms must also be slowed.]

Cooling and trapping

[Reserved: the proposal to cool a gas by resonant radiation pressure [Hansch:1975a]; deceleration of an atomic beam by frequency-chirped light [Phillips:1982]; three-dimensional optical molasses [Chu:1985]; the magneto-optical trap; the optical lattice at the magic wavelength, at which the light shifts of the two clock levels are equal so that trapping does not move the clock transition [Katori:2003]; cryogenic hydrogen beams and the nozzle temperature that sets the residual second-order Doppler shift in [Parthey:2011].]

Phenomenon 90.3 (A trap that does not move the clock).

Atoms held in an optical lattice have both clock levels displaced by the trapping light, and at a general trap wavelength the two displacements differ, so a trapped atom does not have the transition frequency of a free one. At one particular wavelength, different for each species and found by measurement, the two displacements are equal [Katori:2003] [Takamoto:2005]: the transition frequency of the trapped sample then agrees with the free-atom value, and what residual dependence on the trap depth remains is small enough to be extrapolated to zero. This is what permits some \(10^{4}\) atoms to be interrogated simultaneously while confined tightly enough that the recoil and the first-order Doppler shift both vanish, without the confinement itself displacing the clock.

Derivation. A non-resonant field of amplitude \(E_{0}\) and angular frequency \(\omega\) displaces a level \(i\) by the second-order energy

\begin{equation}\tag{90.5} U_{i}(\omega)=-\tfrac{1}{4}\alpha_{i}(\omega)\,E_{0}^{2}\ec \end{equation}

where \(\alpha_{i}\) is the dynamic polarizability, a sum over the dipole transitions out of level \(i\) weighted by \((\omega_{ki}^{2}-\omega^{2})^{-1}\). The displacement of the clock transition is the difference,

\begin{equation}\tag{90.6} \Delta U(\omega) =-\tfrac{1}{4}\bigl[\alpha_{e}(\omega) -\alpha_{g}(\omega)\bigr]E_{0}^{2}\ec \end{equation}

which vanishes wherever the two polarizability curves cross, and does so at every intensity: the condition \(\alpha_{e}=\alpha_{g}\) fixes a wavelength and says nothing whatever about the trap depth. Such a crossing exists because the two levels are connected by dipole transitions to different sets of states, so their polarizabilities have poles at different frequencies and their difference must change sign between consecutive poles; a usable magic wavelength is a zero of that difference lying far enough from every pole that the induced scattering loss is tolerable.

The distinction that makes this an instrument rather than a trick is worth stating. A cancellation arranged by tuning an intensity would have to be maintained, and every fluctuation of the intensity would reappear as a frequency shift. Here the cancellation is a property of the frequency of the light alone — a quantity that can itself be stabilized to the precision of the clock — so the residual sensitivity to depth is second order and the leading systematic disappears from Section 90.3 rather than being corrected for.

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Stable lasers and optical cavities

[Reserved: the Pound–Drever–Hall scheme for locking a laser to a reference cavity [Drever:1983], the same technique that holds the interferometers of Experiment: Gravitational Waves on resonance; the thermal noise floor of the mirror coatings, and the cryogenic single-crystal silicon cavity that reached a linewidth below \(40\,\mathrm{mHz}\) [Kessler:2012]; why the interrogation laser, not the atom, has been the limiting element of optical clocks.]

The frequency comb

[Reserved: the spectrum of a mode-locked femtosecond laser as a comb of lines at \(f_{n}=nf_{\text{rep}}+f_{\text{ceo}}\), so that an optical frequency is measured by counting two radio frequencies; self-referencing by octave-spanning supercontinuum, which measures \(f_{\text{ceo}}\) directly [Jones:2000]; the first direct microwave-to-optical link [Diddams:2000]; the review that established the method [Udem:2002]; comb transfer over a \(920\,\mathrm{km}\) fibre link between Garching and Braunschweig, which let the hydrogen transition be compared with a primary caesium standard without moving either [Matveev:2013].]

Phenomenon 90.4 (A pulse train is a comb of exactly equidistant optical lines).

The spectrum of a mode-locked femtosecond laser consists of hundreds of thousands of sharp lines spanning hundreds of terahertz, and their spacing is uniform across the whole span to the precision with which it can be compared. Each line sits at

\begin{equation}\tag{90.7} f_{n}=n\,f_{\text{rep}}+f_{\text{ceo}}\ec \end{equation}

with \(n\) an integer of order \(10^{6}\) and \(f_{\text{rep}}\) and \(f_{\text{ceo}}\) both radio frequencies that an ordinary counter measures directly. An optical frequency is therefore determined by counting two radio frequencies and one integer [Diddams:2000] [Jones:2000]. Before this, the gap between a caesium standard and an optical transition had to be bridged by a chain of phase-locked oscillators that filled a building and was operated by a team; the comb replaced it with one laser, and that is what made optical clocks usable rather than merely conceivable.

Derivation. Let \(A(t)\) be the complex field of a single pulse — envelope times carrier — and let the train repeat with period \(T=1/f_{\text{rep}}\). The carrier is not locked to the envelope: group and phase velocities in the laser cavity differ, so the carrier slips under the envelope by a fixed phase \(\varphi\) from one pulse to the next, and

\begin{equation}\tag{90.8} E(t)=\sum_{m}A(t-mT)\,\ee^{\ii m\varphi}\ep \end{equation}

Taking the Fourier transform of Fourier Analysis and Integral Transforms,

\begin{equation}\tag{90.9} \tilde{E}(\omega)=\tilde{A}(\omega) \sum_{m}\ee^{\ii m(\varphi-\omega T)}\ec \end{equation}

and the sum is a periodic comb of delta functions: it vanishes unless \(\omega T-\varphi\) is an integer multiple of \(2\pi\), that is unless \(\omega_{n}=(2\pi n+\varphi)/T\). In ordinary frequency this is Equation (90.7), with

\begin{equation}\tag{90.10} f_{\text{rep}}=\frac{1}{T}\ec\qquad f_{\text{ceo}}=\frac{\varphi}{2\pi}\,f_{\text{rep}}\ep \end{equation}

The pulse shape enters only through \(\tilde{A}(\omega)\), which decides which lines carry appreciable amplitude and never where a line sits. The spacing is therefore exactly \(f_{\text{rep}}\) everywhere in the spectrum, with no error accumulating across the span, because it descends from a repetition period rather than from a dispersion relation.

Measuring \(f_{\text{rep}}\) is trivial. Measuring the offset is not, and the trick that makes it possible requires the comb to span an octave: frequency-double the line \(n\) and beat it against the line \(2n\). Their difference is \(2(nf_{\text{rep}}+f_{\text{ceo}})-(2nf_{\text{rep}}+f_{\text{ceo}}) =f_{\text{ceo}}\), in which the unknown integer has cancelled identically. That self-referencing is why an octave-spanning supercontinuum is not a convenience but the enabling step.

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Procedure

[Reserved: the measurement cycle — prepare, interrogate with a Ramsey sequence, read out, steer the laser to the line centre, repeat; line-centre determination and the statistical uncertainty falling as the inverse square root of averaging time; the systematic budget, which is the real content of a precision measurement and is to be tabulated in full: second-order Doppler, recoil, DC and AC Stark, Zeeman, blackbody radiation, collisional, servo and gravitational shifts, each with its measured or bounded size [Ludlow:2015] [Parthey:2011]; extrapolation to zero density and to zero light intensity; the practice of splitting the analysis between groups and of publishing the budget before the value.]

Observations

[Reserved: the quantities to be tabulated in SI with uncertainties per [JCGM:2008]. The hydrogen \(1S\)–\(2S\) centroid frequency \(2466061413187035(10)\,\mathrm{Hz}\), a fractional uncertainty of \(4.2\times10^{-15}\) [Parthey:2011], and its confirmation through the fibre link [Matveev:2013]; the \(2S\)–\(4P\) and \(1S\)–\(3S\) transitions as the second input needed to separate \(R_{\infty}\) from the proton radius [Beyer:2017] [Fleurbaey:2018]; the \(2S\)–\(2P\) Lamb shift measured in electronic hydrogen [Bezginov:2019] and in muonic hydrogen [Pohl:2010] [Antognini:2013]; the derived root-mean-square charge radius, \(0.84184(67)\,\mathrm{fm}\) from muonic hydrogen against the then-accepted \(0.8768(69)\,\mathrm{fm}\) from electronic hydrogen and electron scattering; clock fractional uncertainties — caesium fountains at a few parts in \(10^{16}\) [Weyers:2018], \(\mathrm{Al}^{+}\) at \(9.4\times10^{-19}\) [Brewer:2019], ytterbium at \(1.4\times10^{-18}\) [McGrew:2018].]

Interpretation

Tests of quantum electrodynamics

[Reserved: hydrogen as the system in which a bound-state QED calculation and a measurement meet with no adjustable parameters beyond \(R_{\infty}\) and the proton radius; the hierarchy of contributions — self-energy, vacuum polarization, recoil, two-loop terms — and the order at which theory rather than experiment currently limits the comparison; the complementary tests in the electron anomalous moment [Ludlow:2015] and in Experiment: The Electron Anomalous Magnetic Moment, and in helium (Section 90.6.3).]

The proton-radius puzzle

[Reserved: the discrepancy as it arose — muonic hydrogen giving a radius smaller by about \(4\,\mathrm{\%}\), some seven standard deviations from the CODATA value, with the muon's larger overlap with the nucleus amplifying the finite-size shift by roughly \((m_{\mu}/m_{e})^{3}\) [Pohl:2010] [Antognini:2013]; the subsequent electronic measurements that moved toward the small radius [Beyer:2017] [Bezginov:2019] [Grinin:2020], the low-momentum-transfer scattering experiment that agreed with them [Xiong:2019], and the \(1S\)–\(3S\) result that did not [Fleurbaey:2018]; the CODATA adjustment that followed [Mohr:2025]; the honest verdict, that the puzzle is largely resolved in favour of the small radius and that the remaining tension is a datum, not an embarrassment to be smoothed.]

Constants, and whether they are constant

[Reserved: the comparison of two optical clocks with different sensitivity to the fine-structure constant as a null test of its variation, and the aluminium–mercury ratio measurement that bounded \(\dot{\alpha}/\alpha\) at the \(10^{-17}\) level per year [Rosenband:2008]; the improved bounds from clock networks and the associated limits on violations of local position invariance [Lange:2021]; the relation to the astrophysical claims and to the Oklo bound already assessed in [Uzan:2011]; why a laboratory null result at this precision is the strongest available statement.]

Phenomenon 90.5 (The fine-structure constant does not measurably change).

Two optical clocks whose transition frequencies depend differently on the fine-structure constant have been compared over long periods, and their ratio is found not to drift. The comparison of an aluminium-ion clock with a mercury-ion clock bounds the fractional rate of change of \(\alpha\) at the level of parts in \(10^{17}\) per year [Rosenband:2008], and it is consistent with the geological and astrophysical bounds assessed in [Uzan:2011]. What is measured is a ratio of two frequencies in one laboratory, so no assumption about the constancy of any unit enters anywhere — which is the point, since a dimensional constant has no observer-independent rate of change at all and only a dimensionless one can be said to drift.

Derivation. Any atomic transition frequency may be written

\begin{equation}\tag{90.11} \nu_{i}=c\,R_{\infty}\,A_{i}\,F_{i}(\alpha)\ec \end{equation}

with \(A_{i}\) a pure number fixed by the level structure and \(F_{i}\) collecting the relativistic and radiative corrections, which are the only place \(\alpha\) enters once \(cR_{\infty}\) has been factored out. Define the sensitivity \(K_{i}=\pp\ln F_{i}/\pp\ln\alpha\), computed once per species and large for heavy atoms, where the relativistic corrections grow steeply with nuclear charge. In the ratio of two such frequencies the factor \(cR_{\infty}\) cancels, and with it every dimensional quantity in the problem, leaving

\begin{equation}\tag{90.12} \frac{\dd}{\dd t}\ln\frac{\nu_{1}}{\nu_{2}} =(K_{1}-K_{2})\,\frac{\dot{\alpha}}{\alpha}\ep \end{equation}

A bound on the drift of the measured ratio therefore converts directly into a bound on \(\dot{\alpha}/\alpha\), divided by the difference of the two sensitivities — which is why the experiment pairs species chosen to make \(K_{1}-K_{2}\) as large as possible rather than to make either clock individually best.

Two features of Equation (90.12) carry the argument. Because it is a ratio it is a dimensionless statement, immune to the objection that the second might be drifting along with everything else. And because it is tested against zero rather than against a predicted value, its reach improves twice over: through the statistical uncertainty of each comparison, which falls by Phenomenon 90.1, and through the span of time over which the comparisons are separated, which enters the slope directly.

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Relativity measured with clocks

[Reserved: the Gravity Probe A hydrogen-maser rocket flight, confirming the gravitational redshift to about \(0.007\,\mathrm{\%}\) [Vessot:1980]; the demonstration of both the velocity and the height effect with optical clocks at everyday scales — speeds of \(10\,\mathrm{m}/\mathrm{s}\) and height differences of \(0.33\,\mathrm{m}\) [Chou:2010]; the resolution of the redshift across a single millimetre of a strontium sample [Bothwell:2022]; the consequence that relativistic geodesy is now a measurement technique (The Equivalence Principle and Classical Tests and Experiment: Time Dilation and Relativistic Kinematics), and that a clock comparison at \(10^{-18}\) requires the geoid height of each clock to a centimetre.]

Phenomenon 90.6 (Two clocks one metre apart run at different rates).

Two clocks of identical construction, at rest in the laboratory and separated vertically, do not keep the same time: the higher runs faster, by the fractional amount

\begin{equation}\tag{90.13} \frac{\Delta\nu}{\nu}=\frac{g\,\Delta h}{c^{2}}\ec\qquad \frac{g_{0}}{c^{2}}=1.09\times 10^{-16}\,/\mathrm{m}\ec \end{equation}

and the effect has been resolved at height differences of a fraction of a metre and, most recently, across a single millimetre of one trapped atomic sample [Chou:2010] [Bothwell:2022]. It is not a correction to be applied at the end of an analysis. At the \(10^{-18}\) level of the best clocks the height of each clock above the geoid must be known to about a centimetre before two of them can be compared at all, so that a clock comparison has become a way of measuring that height (The Equivalence Principle and Classical Tests).

Derivation. Compare a clock at height \(h\) with one at the floor, in a field treated as uniform over that height, and use the equivalence principle: the laboratory is locally indistinguishable from a frame accelerating upwards with acceleration \(g\). A signal of frequency \(\nu\) emitted downwards takes a time \(h/c\) to arrive, and during that time the receiver has acquired an upward speed \(v=gh/c\) relative to the frame in which the emitter was at rest at the moment of emission. The first-order Doppler shift then makes the received frequency higher by

\begin{equation}\tag{90.14} \frac{\Delta\nu}{\nu}=\frac{v}{c}=\frac{gh}{c^{2}}\ec \end{equation}

which is Equation (90.13). Inserting the conventional standard value \(g_{0}=9.80665\,\mathrm{m}/\mathrm{s}^{2}\) and the exactly defined \(c=299792458\,\mathrm{m}/\mathrm{s}\) [BIPM:2019] gives \(g_{0}/c^{2}=1.09\times 10^{-16}\,/\mathrm{m}\), the number quoted — a shift that the best clocks resolve in seconds and that was, within living memory, at the edge of what a Mössbauer experiment could reach over the height of a tower.

Two limits on the reading. The derivation is first order in \(gh/c^{2}\) and uses only the equivalence principle together with special relativity, so what it tests is the universality of free fall and not any field equation; the general-relativistic treatment, and what a measured violation would mean, belong to The Equivalence Principle and Classical Tests. And the comparison is one of rates, not of readings, so it is unaffected by the arbitrary epoch at which either clock was started — which is exactly why an atomic transition, whose frequency is fixed by physics rather than by a convention, is the right instrument for the measurement.

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Modern repetitions and precision

Optical clocks

[Reserved: the two families — single trapped ions, with essentially no collisional shift but a small signal [Rosenband:2008] [Brewer:2019], and neutral atoms in a magic wavelength lattice, with \(10^{4}\) atoms interrogated at once [Katori:2003] [Takamoto:2005] [McGrew:2018] [Bothwell:2022]; quantum logic spectroscopy, which reads out a clock ion through a co-trapped logic ion [Brewer:2019]; the blackbody shift as the dominant systematic for most species and the cryogenic environments that suppress it; the stability–accuracy distinction, which is routinely conflated and must not be here [Ludlow:2015].]

Redefining the second

[Reserved: Essen and Parry's caesium standard [Essen:1955] and the 1967 definition it enabled [BIPM:2019]; the accuracy of present primary fountains [Weyers:2018], now two orders of magnitude behind the best optical clocks; the recommended frequency values for secondary representations of the second [Riehle:2018] and the criteria a redefinition must satisfy; the practical obstacle, which is not clock performance but the comparison links between national laboratories.]

Two-electron and exotic atoms

[Reserved: helium as the two-electron benchmark, with the \(2\,^{3}S_{1}\to2\,^{1}S_{0}\) transition measured in a quantum degenerate gas [vanRooij:2011] and the \(2\,^{3}S\)–\(2\,^{3}P\) interval as a route to the fine-structure constant [Zheng:2017]; muonic helium, giving the alpha-particle charge radius [Krauth:2021]; positronium, a purely leptonic atom with no nuclear structure at all, in the \(1\,^{3}S\)–\(2\,^{3}S\) interval [Fee:1993] and in the \(n=2\) fine structure [Gurung:2020]; muonium, in the ground-state hyperfine interval [Liu:1999] and in the \(1S\)–\(2S\) transition [Meyer:2000]. Each removes one source of theoretical uncertainty that hydrogen carries.]

Antihydrogen

[Reserved: trapped antihydrogen and the measurement of its \(1S\)–\(2S\) transition, consistent with hydrogen at the level of \(2\times10^{-12}\) [Ahmadi:2018] — the sharpest existing test of CPT symmetry in a bound system (Discrete Symmetries and CPT); the production, trapping and laser-cooling chain that makes a spectroscopy experiment out of a few thousand atoms per run; the gravitational free-fall measurement as the complementary programme.]

Toward a nuclear clock

[Reserved: the anomalously low-lying isomeric state of \(^{229}\mathrm{Th}\), at about \(8.4\,\mathrm{eV}\), the only nuclear transition within reach of a laser; its direct laser excitation [Tiedau:2024] and the measurement of its frequency ratio against a strontium optical clock [Zhang:2024]; why a nuclear transition is expected to be far less sensitive to external fields and far more sensitive to a variation of the constants of Section 90.5.3 than any electronic one; the status, which is a demonstrated transition and not yet a clock.]