--- title: Optical Atomic Clocks and Precision Measurement module: Modern Atomic Physics moduleNumber: 9 lessonNumber: 4 order: 904 summary: > An atomic clock counts the oscillations of a field locked to an atomic transition. The cesium microwave standard defines the second through the 9.19 GHz ground-state hyperfine transition, interrogated by Ramsey's separated-oscillatory-field method whose fringe width is set by the free-precession time. Optical clocks replace the microwave transition with an optical one five orders of magnitude higher in frequency, raising the quality factor and the fractional stability in proportion. Lattice and single-ion clocks reach fractional uncertainties near ten-to-the-minus- eighteen by trapping the atoms at a magic wavelength that cancels the light shift, and at that level they measure the gravitational redshift over centimetres of height. topics: [Modern Atomic Physics] sources: - book: Foot ref: "Ch. 6 §6.4 (hyperfine standard), Ch. 8 §8.4 (Ramsey fringes), Ch. 9 (cold-atom clocks)" - book: Demtröder ref: "Ch. 9 — Doppler-Free Spectroscopy; Ch. 11 — Optical Frequency Standards (concepts)" - book: Foot ref: "Ch. 8 — Doppler-Free Laser Spectroscopy; the frequency-comb link" draft: false --- 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: > **Definition (The second).** The second is the duration of $9\,192\,631\,770$ > periods of the radiation from the ground-state hyperfine transition of the > cesium-133 atom (at rest, at $0\ \mathrm{K}$). The frequency > $\nu_{\mathrm{Cs}} = 9\,192\,631\,770\ \mathrm{Hz}$ is exact by definition. The transition is between the $F = 3$ and $F = 4$ hyperfine levels of the $6s\,{}^2S_{1/2}$ ground state, split by the magnetic-dipole coupling of the nuclear spin to the valence electron, treated for hydrogen in [hyperfine structure](/atomic-physics/qed-corrections-and-hyperfine-structure/hyperfine-structure-21cm). 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.[^ramsey] The excitation probability then oscillates with detuning, $$ P_e(\delta) = \tfrac12\,\bigl[1 + \cos(\delta\, T)\bigr] \quad(\text{on resonance in each pulse}), \qquad \delta = 2\pi(\nu - \nu_0), $$ 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. $$ % caption: Ramsey fringes: two separated pulses with free precession time T between % them give an excitation probability that oscillates in detuning with fringe spacing % one over two T; the central fringe is the lock point. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-4.0,0) -- (4.2,0) node[right] {detuning}; \draw[->, black] (0,-0.2) -- (0,2.9) node[above] {excitation}; % slowly-varying envelope (single-pulse Rabi) \draw[black, thick, domain=-3.8:3.8, samples=140, smooth] plot (\x, {2.3*exp(-0.18*\x*\x)}); % fast Ramsey fringes under the envelope \draw[acc, very thick, domain=-3.8:3.8, samples=320, smooth] plot (\x, {2.3*exp(-0.18*\x*\x)*0.5*(1+cos(deg(6*\x)))}); \node[black, anchor=west] at (1.7,1.7) {envelope}; \draw[->, black, thick] (0.9,2.75) -- (0.1,2.55); \node[black, anchor=west] at (0.9,2.75) {central fringe}; \end{tikzpicture} $$ 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** $$ Q = \frac{\nu_0}{\Delta\nu}, $$ 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 $\nu_0 \sim 5\times10^{14}\ \mathrm{Hz}$, 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. | Standard | Transition | $\nu_0$ | Interrogated line | Order of $Q$ | | --- | --- | --- | --- | --- | | Cesium fountain | ground hyperfine, microwave | $9.19\ \mathrm{GHz}$ | $\sim 1\ \mathrm{Hz}$ | $10^{10}$ | | Strontium lattice | ${}^1S_0 \to {}^3P_0$, 698 nm | $429\ \mathrm{THz}$ | $\sim 1\ \mathrm{Hz}$ | $10^{15}$–$10^{17}$ | | Aluminium ion | ${}^1S_0 \to {}^3P_0$, 267 nm | $1.12\ \mathrm{PHz}$ | sub-Hz | $10^{17}$ | $$ % caption: A microwave and an optical resonance of the same linewidth: the optical % line sits at a frequency higher by a factor of order ten-to-the-fifth, so its % quality factor and the precision of the lock are larger in the same proportion. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % microwave: peak near origin \draw[->, black] (0,0) -- (8.6,0) node[right] {frequency}; \draw[->, black] (0,0) -- (0,2.7) node[above] {response}; \draw[black, very thick, dashed, domain=0.4:2.0, samples=100, smooth] plot (\x, {2.2*exp(-9*(\x-1.2)*(\x-1.2))}); \node[black, anchor=south] at (1.2,2.2) {microwave}; % optical: peak far right, same width \draw[acc, very thick, domain=6.0:7.6, samples=100, smooth] plot (\x, {2.2*exp(-9*(\x-6.8)*(\x-6.8))}); \node[acc, anchor=south] at (6.8,2.2) {optical}; \node[black, anchor=north] at (1.2,-0.05) {$9\ \mathrm{GHz}$}; \node[black, anchor=north] at (6.8,-0.05) {$400\ \mathrm{THz}$}; \end{tikzpicture} $$ 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.[^comb] 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. > **Definition (Magic wavelength).** The trapping-laser wavelength $\lambda_m$ at which > the dynamic polarizabilities of the two clock states are equal, so the optical > lattice shifts both levels identically and leaves the transition frequency > unperturbed. For the strontium ${}^1S_0 \to {}^3P_0$ clock, > $\lambda_m \approx 813\ \mathrm{nm}$. $$ % caption: The two clock levels are light-shifted by the trapping laser through their % polarizabilities; at the magic wavelength the two shifts cross and become equal, so % the transition frequency does not move with trap intensity. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right] {trap wavelength}; \draw[->, black] (0,-0.2) -- (0,2.9) node[above] {light shift}; % ground and excited light shifts cross once at the magic wavelength \draw[acc, very thick] (0.4,0.7) -- (6.0,2.4); \node[acc, anchor=south] at (1.9,1.15) {ground}; \draw[black, very thick, dashed] (0.4,0.3) -- (6.0,2.7); \node[black, anchor=north] at (5.2,2.25) {excited}; % crossing point at x = 3.6 \fill[black] (3.6,1.67) circle (2.6pt); \draw[black, dashed] (3.6,0) -- (3.6,1.67); \node[black, anchor=north] at (3.6,-0.05) {magic point}; \end{tikzpicture} $$ 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 $1\times10^{-18}$. ## 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 $$ \sigma_y(\tau) \simeq \frac{1}{2\pi\,\nu_0\,T}\,\frac{1}{\sqrt{N}}\, \sqrt{\frac{T_c}{\tau}}, $$ 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. $$ % caption: Allan deviation versus averaging time on a log-log plot falls as tau to % the minus one-half until it reaches the systematic floor set by uncorrected shifts; % the optical clock has both a lower starting noise and a lower floor. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.6,0) node[right] {averaging time (log)}; \draw[->, black] (0,0) -- (0,3.4) node[above] {Allan deviation (log)}; % microwave: higher, sloping down then floor \draw[black, very thick, dashed] (0.5,3.0) -- (4.3,0.9); \draw[black, very thick, dashed] (4.3,0.9) -- (6.2,0.9); \node[black, anchor=south west] at (2.9,2.35) {microwave}; % optical: lower, steeper reach, lower floor \draw[acc, very thick] (0.5,2.1) -- (5.0,0.25); \draw[acc, very thick] (5.0,0.25) -- (6.2,0.25); \node[acc, anchor=north east] at (1.9,1.35) {optical}; \node[black, anchor=south] at (5.3,0.95) {systematic limit}; \end{tikzpicture} $$ 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 $\Delta\nu/\nu = g\,\Delta h/c^2 \approx 1.1\times10^{-16}$ 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.[^redshift] - **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](/atomic-physics/modern-atomic-physics/laser-cooling-doppler): 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. [^ramsey]: **Foot**, _Atomic Physics_, §8.4, derives the two-pulse fringe pattern $P_e \propto 1 + \cos(\delta T)$ and the fringe width $1/2T$; the method is due to N. F. Ramsey, _Phys. Rev._ **78**, 695 (1950), for which he received the 1989 Nobel Prize. The cesium definition of the second is the SI standard, https://www.nist.gov/pml/time-and-frequency-division. [^comb]: **Demtröder**, _Atoms, Molecules and Photons_, Ch. 11, describes the optical frequency comb linking optical and microwave frequencies (T. W. Hänsch and J. L. Hall, 2005 Nobel Prize). Strontium clock transition and magic wavelength: NIST, https://www.nist.gov/pml/time-and-frequency-division; Sr ${}^1S_0 \to {}^3P_0$ at 698 nm ($429\ \mathrm{THz}$), magic wavelength $\approx 813\ \mathrm{nm}$. [^redshift]: The gradient $g/c^2 = 1.09\times10^{-16}\ \mathrm{m^{-1}}$ follows from $g = 9.81\ \mathrm{m\,s^{-2}}$ and $c = 2.998\times10^8\ \mathrm{m\,s^{-1}}$; centimetre-scale redshifts have been resolved between optical lattice clocks (e.g. T. Bothwell et al., _Nature_ **602**, 420 (2022)). CODATA constants: https://physics.nist.gov/cuu/Constants/.