Optical atomic clocks

Optical atomic clocks represent the state of the art in the frontier of modern measurement science. In this article a detailed review on the development of optical atomic clocks that are based on trapped single ions and many neutral atoms is provided. Important technical ingredients for optical clocks are discussed and measurement precision and systematic uncertainty associated with some of the best clocks to date are presented. An outlook on the exciting prospect for clock applications is given in conclusion.

Figure 1

(a) In the time domain, the laser output generates femtosecond pulse-width envelopes separated in time by $1/f_{\text{rep}}$. Another important degree of freedom is the phase difference between the envelope maximum and the underlying electric field oscillating at the carrier optical frequency. (b) By Fourier transformation to the frequency domain, the corresponding frequency comb spectrum is revealed. Each tooth in the comb, a particular single-frequency mode, is separated from its neighbor by $f_{\text{rep}}$. The relative carrier-envelope phase in the time domain is related to the offset frequency $f_{\mathrm{CEO}}$ in the frequency domain. $f_{\mathrm{CEO}}$ is given by the frequency of one mode of the comb (e.g., ${\nu }_n$) modulo $f_{\text{rep}}$, and can be measured and stabilized with a $f\text{−}2f$ interferometer. In this interferometer, one comb mode ${\nu }_n$ is frequency doubled and heterodyne beat with the comb mode at twice the frequency ${\nu }_{2n}$. Thus, by stabilizing $f_{\mathrm{CEO}}$ and $f_{\text{rep}}$ to a well-known frequency reference, each comb mode frequency is well known. Measurement of the frequency of a poorly known optical frequency source (e.g., previously measured at the resolution of a wave meter) can be determined by measuring the heterodyne beat between the frequency source and the frequency comb.

Figure 2

Electrode configurations for spherical Paul traps. (a) Cut through the cylindrically symmetric electrode geometry used by Paul. The oscillating potential $U_{\mathrm{rf}}=V_{\mathrm{rf}}\cos {\mathrm{\Omega }}_{\mathrm{rf}}t$ is applied between the ring and the end cap electrodes. (b) Paul trap used for the ${\mathrm{Yb}}^{+}$ frequency standard at PTB. From [506]. (c) Paul trap used for the ${\mathrm{Hg}}^{+}$ frequency standard at NIST. From [394]. (d) End cap trap. From [475]. The inner end caps are 0.5 mm in diameter and are separated by 0.56 mm. The oscillating potential $U_{\mathrm{rf}}$ is applied to the inner end cap electrodes. The outer electrodes (outer diameter 2 mm) are normally grounded; however, if required small potentials can be applied to compensate micromotion in the axial direction.

Figure 3

Linear ion-trap electrode geometries. (a) The Innsbruck trap geometry [upper panel ([209])] is implemented using elongated blades for the rf and two conical tips for the dc electrodes (lower panel). (b) The NIST trap geometry [upper panel ([454])] is implemented using microstructured segmented electrodes (lower panel). This allows splitting the tip electrodes and moving them away from the axial symmetry line, enabling improved laser access. Unsegmented electrodes are connected to rf potential $V_{\mathrm{rf}}\cos {\mathrm{\Omega }}_{\mathrm{rf}}t$, outer-segment electrodes are at a positive dc potential $V_{\mathrm{d}\mathrm{c}}$, and inner-segment electrodes are at ground.

Figure 4

Schematic energy level diagram of clock ions with a single valence electron and ${\mathrm{Yb}}^{+}$, together with a table of the most relevant transition wavelengths and linewidths. Data are from the NIST database ([429]). Energy levels are not to scale and the term notation follows [352].

Figure 5

Schematic level structure of ion clocks based on group 13 (formerly group IIIA) singly charged ions, together with a table of the most relevant transition wavelengths and linewidths. Data are from the NIST database ([429]). Energy levels are not to scale.

Figure 6

Quantum logic spectroscopy sequence. Shown are the clock ground (${{}^{1}S}_0$) and excited (${{}^{3}P}_0$) clock states and an auxiliary metastable state (${{}^{3}P}_1$) together with the logic ion (qubit states $|\downarrow {⟩}_L$, $|\uparrow {⟩}_L$). In addition, two vibrational levels ($|0{⟩}_n$, $|1{⟩}_n$) of a common motional mode of the ions in the trap are shown. (a) Initially, both ions are prepared in the electronic and motional ground states. (b) After clock interrogation, the spectroscopy ion is in an equal superposition of the two clock states. (c) Resolved-sideband pulse on the blue sideband of the clock ion to the auxiliary state, mapping the ground-state amplitude onto the first excited motional state. (d) Resolved-sideband pulse on the red sideband of the logic ion, mapping the first excited motional state amplitude to the electronically excited state of the logic ion. (e) Detection of the logic ion’s internal state via the electron-shelving technique ([117]). Energy levels are not to scale.

Figure 7

Resonance of the ${\mathrm{Al}}^{+}$ clock transition using quantum logic spectroscopy. From [97].

Figure 8

Quantum logic clock interrogation cycle. The following sequence describes interrogation of the ${{}^{1}S}_0{m}_F=5/2\to {{}^{3}P}_0{m}_F=5/2$ state. A similar protocol is used for the other ($m_F=-5/2$) stretched state interrogation. (a) Sympathetic Doppler cooling, reaching $\overline{n}\approx 3$ in all modes with a tilted ion crystal. (b) Probing of the ${\mathrm{Al}}^{+}$ clock transition ${{}^{1}S}_0\leftrightarrow {{}^{3}P}_0$ with simultaneous application of cooling light to maintain steady-state motional occupation. (c) Optical pumping on the ${{}^{1}S}_0\leftrightarrow {{}^{3}P}_1$ transition to ${{}^{1}S}_0$, $m_F=5/2$ state. (d) Sympathetic Doppler cooling followed by ground-state cooling of a selected axial mode to a mean motional excitation of $\overline{n}\approx 0.05$ (e). (f) Quantum logic state detection (see Fig. 6 for details). Steps (c)–(f) are repeated 10 times for improved readout fidelity. From [451].

Figure 9

(a) A simplified energy level diagram representative of many group 2 (-like) atoms. Some of the most relevant transitions are indicated with arrows, and their wavelengths (in nanometers) and natural linewidths are specified in the adjacent table for the case of Sr, Yb, and Hg. (b) The importance of gravity on narrow-line cooling dynamics is clearly seen from the in situ MOT images as the laser detuning $\delta$ is varied in a narrow line MOT for Sr. The dashed ovals represent the spatial position where the photon scattering rate is the highest as the laser frequency detuning matches the Zeeman level shift induced by the MOT magnetic field. In the absence of gravity, the dashed ovals define the MOT region.

Figure 10

(a) Sketch of a generic absorption spectrum in the well-resolved-sideband limit. (b) Absorption spectrum of the ${{}^{1}S}_0-{{}^{3}P}_0$ clock transition in ultracold ${}^{87}\mathrm{Sr}$ under various levels of confinement. Tighter confinement corresponds to higher trap frequency and smaller $\eta$. The spectral lines are well resolved, and the strong confinement curve falls in the Lamb-Dicke regime.

Figure 11

Calculations of the wavelength-dependent ac-Stark shift for the ${{}^{1}S}_0$, ${{}^{3}P}_0$, and ${{}^{3}P}_1(m_J=0,\pm 1)$ states in ${}^{88}\mathrm{Sr}$. Values are given for linear ($\pi$) and circular (${\sigma }^{\pm }$) polarizations. The ${{}^{1}S}_0$ (solid line) and ${{}^{3}P}_0$ (long-dashed line) states exhibit no polarization dependence and cross at a wavelength of 815 nm in good agreement with experimental results. The ${{}^{3}P}_1$ state reveals a significant polarization and $m_J$ dependence due to the tensor and vector nature of the light shifts.

Figure 12

Spectroscopy of the clock transition in the optical lattice. When the clock transition is strongly driven into saturation, the motional sidebands can be more easily observed. From these spectra trap parameters such as motional frequency, trap depth, Lamb-Dicke parameter, as well as the atomic temperature can be extracted. Examples of sideband spectra for (a) Yb and (b) Sr are shown under trapping conditions described in the text.

Figure 13

(a) High resolution spectroscopy of ${}^{87}\mathrm{Sr}$ yielding a FWHM linewidth of 0.5 Hz. (b) High resolution spectroscopy of ${}^{171}\mathrm{Yb}$, yielding a FWHM linewidth of 1 Hz.

Figure 14

(a) Typical experimental field orientation for lattice spectroscopy. The lattice laser propagates along the $z$ axis and is linearly polarized along the $x$ axis, parallel to the bias magnetic field such that $\phi \approx \pi /2$. The probe laser propagates colinearly with the lattice beam and the linear probe polarization can be rotated relative to the quantization ($x$) axis by an angle $\theta$. [(b) inset] The large nuclear spin ($I=9/2$ for ${}^{87}\mathrm{Sr}$) results in 28 total transitions, and the labels $\pi$, ${\sigma }^{+}$, and ${\sigma }^{-}$ represent transitions where $m_F$ changes by 0, $+1$, and $-1$, respectively. The hyperfine interaction state mixing modifies the ${{}^{3}P}_0$ $g$ factor, making the magnitude about 60% larger than that of ${{}^{1}S}_0$. (b) Observation of the ${{}^{1}S}_0-{{}^{3}P}_0$ $\pi$ transitions when $\theta =0$ in the presence of a $58\mu \mathrm{T}$ magnetic field. (c) Observation of the $18\sigma$ transitions when the probe laser polarization is orthogonal to that of the lattice ($\theta =\pi /2$) when a field of $69\mu \mathrm{T}$ is used. (b), (c) Data are shown in gray and fits are shown as solid lines. The peaks are labeled by the ground-state sublevel of the transition [and the relevant polarization in (c)]. The relative transition amplitudes for the different sublevels are strongly influenced by the Clebsch-Gordan coefficients. Here Fourier-limited transition linewidths of 10 Hz are used. (d), (e) Calculations of the $18\sigma$-transition frequencies in the presence of a $69\mu \mathrm{T}$ bias field, including the influence of Clebsch-Gordan coefficients. The solid and dash-dotted curves show the ${\sigma }^{+}$ and ${\sigma }^{-}$ transitions, respectively. (d) Spectral pattern for $g$ factors $g_I{\mu }_B/h=-1.85\mathrm{Hz}/\mu \mathrm{T}$ and $\delta g{\mu }_B/h=-1.09\mathrm{Hz}/\mu \mathrm{T}$. (e) Same pattern as in (d) but with $\delta g{\mu }_B/h=+1.09\mathrm{Hz}/\mu \mathrm{T}$. The qualitative difference in the relative positions of the transitions allows an absolute determination of the sign of $\delta g$ compared to that of $g_I$.

Figure 15

(a) Frequency stability of the JILA Sr vs NIST Yb lattice clocks as measured in 2009. The total deviation confirms the predicted stability limit given by the clock laser. (b) Measurements of the improved Yb clock laser instability at one to a few clock cycles. Circles are measured using atomic excitation as an (out-of-loop) frequency discriminator of the optical LO frequency and include a contribution to the instability from the Dick effect, atomic detection noise, and the local oscillator (LO) free-running instability. The Dick-effect-limited instability (black dashed line) is $1.5\times {10}^{-16}/\sqrt{\tau }$. (c) Recently improved measurements of the fractional frequency instability for both the Yb and Sr lattice clocks. From [484].

Figure 16

Measurement of key systematic frequency shifts in a ${}^{87}\mathrm{Sr}$ optical lattice clock. (a) The lattice Stark shifts as well as (b) second-order and [(b), inset] first-order magnetic-field shifts are measured by direct optical comparison to a stable calcium optical standard. On the other hand, (c) shows a measurement of the density shift using interleaved conditions of a single standard. In this case, a large number of measurements facilitate precise determination of the shift. All histograms are accompanied by Gaussian fits to the data set.

Figure 17

Absolute frequency measurement of the ${{}^{1}S}_0-{{}^{3}P}_0$ transition in ${}^{87}\mathrm{Sr}$. (a) Histogram of 880 measurements (without nuclear spin polarization) taken over a 24 h period. The corresponding Gaussian fit (solid curve) and data have a mean value of 71.8(6) Hz. (b) Histogram of the 50 h absolute frequency measurement (with nuclear spin polarization) using a hydrogen maser calibrated in real time to a caesium fountain. The resulting frequency is 70.88(35) Hz and the distribution is Gaussian as shown by the fit (solid curve). In (a) and (b) the offset frequency $f_0$ is 429 228 004 229 800 Hz and the data sets are corrected only for the maser offset. When the Sr systematics are included the frequencies are in excellent agreement. (c) Total deviation of the Sr-maser comparison for the data set in (b). The line is a fit to the data yielding a stability of $2.64(8)\times {10}^{-13}{\tau }^{-0.48(1)}$ and extends out to the entire measurement time.

Figure 18

Absolute frequency measurements of the ${{}^{1}S}_0-{{}^{3}P}_0$ transition in ${}^{87}\mathrm{Sr}$ and ${}^{171}\mathrm{Yb}$ (see also Table 6).

Figure 19

Measurements between atomic clocks of different species can constrain possible variation of fundamental constants. A number of comparisons between distinct atomic-clock species are used here to constrain time variation of $\alpha$ and $\mu$. From [243].