Optical Atomic Clocks and Precision Measurement

A clock is an oscillator and a counter. Its accuracy is the reproducibility of the oscillator's frequency; its stability is how little that frequency wanders while it is averaged. An atomic clock uses an atomic transition as the oscillator, because the transition frequency is fixed by atomic structure — the same for every atom of a species, independent of when or where the clock runs. The engineering problem is to lock a laboratory field to the transition and count its cycles without perturbing the atom. This lesson develops the interrogation method that makes the lock precise (Ramsey's separated fields), the reason optical transitions outperform microwave ones (the quality factor), the traps that hold the atoms without shifting the line (the magic wavelength), and the fractional-frequency stability that follows.

The cesium definition of the second

Since 1967 the SI second has been defined as a fixed number of oscillations of a specific atomic transition:

The transition is between the $F=3$ and $F=4$ hyperfine levels of the $6s{}^2{S}_{1/2}$ ground state, split by the magnetic-dipole coupling of the nuclear spin to the valence electron, treated for hydrogen in hyperfine structure. A microwave field at $9.19\mathrm{GHz}$ drives it. A cesium clock counts these oscillations; every other frequency and time interval is referred to this count.

Ramsey's separated oscillatory fields

Interrogating the transition means measuring how close the applied microwave frequency $\nu$ is to ${\nu }_{\mathrm{Cs}}$. Rabi's method drives the atom with a single field pulse and reads the excitation probability, whose linewidth is set by the pulse duration. Ramsey's improvement is to split the interaction into two short pulses separated by a long dark interval $T$, during which the atom precesses freely.¹ The excitation probability then oscillates with detuning,

a fringe pattern whose central maximum is at $\delta =0$ and whose fringe spacing is $\Delta \nu =1/(2T)$. The narrow central fringe, not the broad single-pulse envelope, is what the servo locks to. Making the fringe narrow requires a long free-precession time $T$: a long microwave cavity for a thermal beam, or — far better — cold atoms.

A cesium fountain launches laser-cooled atoms upward through a single microwave cavity; they pass through it once going up and once coming down under gravity, realizing the two Ramsey pulses with a dark time $T$ of order one second. The fringe width shrinks to about $1\mathrm{Hz}$, so the line is resolved to a part in $10^{10}$ of its frequency in a single interrogation. Fountains are the primary realizations of the SI second.

The quality factor and why optical wins

The sharpness of a resonance is its quality factor

the transition frequency divided by the linewidth. A servo locks to the line centre with a precision that improves with $Q$: a higher $Q$ means the discriminator slope is steeper, so the same signal-to-noise resolves a smaller frequency error. For a fixed achievable linewidth $\Delta \nu$ (set by the interrogation time), $Q$ grows in direct proportion to ${\nu }_0$. This is the entire case for optical clocks.

An optical transition oscillates at ${\nu }_0\sim 5\times 10^{14}\mathrm{Hz}$, roughly $10^5$ times the cesium microwave frequency. With a comparably narrow line — a forbidden transition of sub-hertz natural width, interrogated for a second — the quality factor reaches $Q\sim 10^{17}$, five orders of magnitude beyond the cesium fountain. The precision to which the line centre can be found scales with $Q$, so the optical clock is intrinsically the more precise oscillator.

Counting an optical oscillation directly is impossible — no electronics runs at $10^{14}\mathrm{Hz}$. The optical frequency comb bridges the gap: a mode-locked laser emits a spectrum of evenly spaced sharp lines, ${\nu }_m=m{f}_{\mathrm{rep}}+f_0$, a ruler in frequency space whose two radio-frequency parameters $f_{\mathrm{rep}}$ and $f_0$ are measured and controlled. The comb phase-coherently divides the optical frequency down to a countable microwave rate, making the optical clock a usable time standard.²

The offset $f_0$ is fixed by self-referencing. If the comb spans a full octave, so that a mode $m$ near the low-frequency end and a mode $2m$ near the high end both exist, then frequency-doubling the low tooth and beating it against the high tooth gives $2(m{f}_{\mathrm{rep}}+f_0)-(2m{f}_{\mathrm{rep}}+f_0)=f_0$ directly. With $f_0$ and $f_{\mathrm{rep}}$ both measured and locked, every optical tooth position is known to the accuracy of the radio-frequency reference, and the beat note between a clock laser and the nearest tooth reads out the optical frequency in countable hertz. The comb is what turned optical transitions from spectroscopic curiosities into clocks.

Systematic shifts and the magic wavelength

Accuracy, distinct from stability, is limited by every effect that shifts the transition frequency from its unperturbed value. The largest in a trapped-atom clock would be the AC Stark shift of the trapping light itself: confining the atoms in an optical-lattice or dipole trap light-shifts the two clock levels, and if they shift by different amounts the transition frequency moves with the trap intensity.

The resolution is the magic wavelength. The light shift of a level depends on the wavelength of the trapping light through the atom's dynamic polarizability $\alpha (\lambda )$. At a specially chosen wavelength the polarizabilities of the two clock states are equal, ${\alpha }_g({\lambda }_m)={\alpha }_e({\lambda }_m)$, so both levels shift by the same amount and the transition frequency is unshifted to first order, independent of the trap intensity.

Other systematics are controlled to comparable precision: the blackbody radiation shift from room-temperature thermal photons (evaluated from the polarizability and the environment temperature, or suppressed by cryogenic shields), the second-order Zeeman shift (measured by interleaving field directions), collisional shifts (small in a lattice with one atom per site or in a single trapped ion), and the first-order Doppler shift (eliminated by tight confinement in the Lamb-Dicke regime, where the atom is localized to less than a wavelength). The best optical clocks report total fractional systematic uncertainties near $1\times 10^{-18}$.

Stability and the tests it enables

Stability is how quickly the clock averages down to its accuracy. It is quantified by the Allan deviation ${\sigma }_y(\tau )$, the root-mean-square fractional frequency fluctuation between measurements averaged over an interval $\tau$. For a clock limited by the quantum projection noise of $N$ independent atoms interrogated for time $T$, the standard-quantum-limit stability is

with $T_c$ the cycle time. Three features improve it: a high ${\nu }_0$ (optical), a long free-precession time $T$, and many atoms $N$ (a lattice clock with $10^3$ atoms averages down faster than a single ion). The ${\tau }^{-1/2}$ scaling is the ordinary averaging of white frequency noise: run longer and the statistical uncertainty falls as the square root of the number of samples.

At fractional uncertainties of $10^{-18}$ the clocks resolve physics that lower precision hides:

• Gravitational redshift. General relativity predicts a clock runs faster higher in a gravitational potential by $\Delta \nu /\nu =g\Delta h/c^2\approx 1.1\times 10^{-16}$ per metre of elevation. A $10^{-18}$ clock resolves a height difference of about a centimetre, and such shifts have been measured between two clocks in the same laboratory.³
• Drift of fundamental constants. Because different transitions depend differently on the fine-structure constant $\alpha$, comparing two optical clocks over years bounds any time variation $\dot{\alpha }/\alpha$, currently below about $10^{-18}$ per year.
• Relativistic geodesy. A network of clocks measures differences in gravitational potential — hence height — through the redshift, giving a geodetic tool tied to a frequency rather than to a tide gauge.

The line of development runs unbroken from the scattering force of the first lesson: laser cooling supplies the slow atoms, sub-Doppler cooling and traps hold them still at a magic wavelength that hides the trap from the transition, and the narrow optical line interrogated by Ramsey's method for a full second, read out through a frequency comb, gives a clock precise enough to see gravity bend time across the height of a table.