Open Access
Add to BibSonomy
Abstract
We performed the first direct measurement of the frequency ratio between a mercury (199Hg) and an ytterbium (171Yb) optical lattice clock to find νHg/νYb = 2.177 473 194 134 565 07(19) with the fractional uncertainty of 8.8 × 10−17. The ratio is in excellent agreement with expectations from the ratios νHg/νSr and νYb/νSr obtained previously in comparisons against a strontium (87Sr) optical lattice clock. The completed closure (νHg/νYb)(νYb/νSr)(νSr/νHg) − 1 = 0.4(1.3) × 10−16 tests the frequency reproducibility of the optical lattice clocks beyond what is achievable in comparison against the current realization of the second in the International System of Units (SI).
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The desire for better measurement accuracy of time and frequency is driving the rapid progress of atomic clocks based on optical transitions, which achieve fractional uncertainties smaller than the current primary frequency standards [1] in the International System of Units (SI), based on the microwave transition of 133Cs atoms at 9.2 GHz. Several groups have realized optical clocks with a fractional frequency uncertainty close to ${10^{ - 18}}$ using single-ion clocks [2–4] or optical lattice clocks [5–7]. The uncertainty in measuring the absolute frequency of such optical atomic clocks is dominated by the inaccuracy of the realization of the SI second [8]. Thus, measurements of the frequency ratios between clocks with an uncertainty of 10−16 or less [9–15] play a crucial role. Taken together, three individually measured ratios should satisfy the relationship of the loop closure,
2. Experimental setup
Figures 1(a) and 1(b) show the energy diagrams of 199Hg and 171Yb and the schematic of the experiment. The detailed experimental setups of the Hg and Yb clocks are described in Refs. [10,11], respectively. The 1S0—3P0 transitions of 199Hg and 171Yb, both with nuclear spin 1/2, are used as their clock transitions. We alternately interrogate the two π transitions ${m_F} ={\pm} 1/2 \to {m_F} ={\pm} 1/2$ to cancel out the 1st-order Zeeman shift and the vector light shift [16]. Yb clock spectroscopy is performed in a cryogenic environment to suppress the blackbody radiation shift, while the Hg spectroscopy is performed at room temperature since its susceptibility to blackbody radiation is small compared to Yb and Sr.
Fig. 1. (a) Relevant energy levels for 199Hg and 171Yb, both with nuclear spin 1/2. Blue and red text gives the transition wavelengths for Hg and Yb, respectively. (b) Experimental configuration of the clock frequency comparison. Fundamental lasers for both clocks at 1063 nm and 1157 nm are phase-locked to an Er-fiber-based optical frequency comb stabilized to an optical reference at 1397 nm. The optical lattice for Hg is formed inside an optical enhancement cavity at 363 nm. The optical lattice for Yb is formed by two counter-propagating beams at 759 nm, and the trapped atoms are transported into a cryogenic environment at T = 160 K by a moving lattice technique. Lasers for cooling, detection, and repumping are not shown. ECDL, external-cavity diode laser; SHG, second-harmonic generator; TA, tapered semiconductor laser amplifier; YbFL, Yb-doped fiber laser; AOM, acousto-optic modulator; CCD, charge-coupled device camera.
Hg atoms are laser-cooled and magneto-optically trapped on the 1S0—3P1 transition at a wavelength of 254 nm. Atoms are loaded into a one-dimensional (1D) optical lattice formed inside a vertically-oriented power-enhancement cavity. The optical lattice is operated at 363 nm, the magic wavelength for the Hg clock, and an optical lattice potential depth during clock interrogation of ${V_0} = 43\; {E_\textrm{R}}$, where ${E_\textrm{R}}$ is the lattice photon recoil energy. After optical-pumping the atomic population in the 1S0 state to ${m_F} ={+} 1/2\;\textrm{or}$ $- 1/2$ by exciting the 1S0—3P1 transition, the clock spectroscopy is carried out on the 1S0—3P0 transition at 266 nm.
Yb atoms are laser-cooled by a two-stage magneto-optical trap first on the 1S0—1P1 transition at 399 nm and then on the 1S0—3P1 transition at 556 nm. Atoms are loaded into a 1D optical lattice at the magic wavelength of 759 nm. An optical power of 0.55 W per lattice beam enables us to trap and transport the Yb atoms into a cryogenic chamber by using a moving lattice technique [6]. The cryogenic chamber is made of copper with small holes to admit the lattice beams, and is cooled to T=160.0(5) K by a Stirling refrigerator. The inner surface of the cryogenic chamber is black-coated with high emissivity. A sequence of interleaved pulses of axial sideband cooling and optical-pumping to the ${m_F} ={+} 1/2$ or $- 1/2$ Zeeman substate leaves more than 96% of atoms in the desired 1S0 ground state with a mean longitudinal vibrational quantum number of typically ${n_\textrm{Z}} = 0.1$. This state preparation and the clock spectroscopy on the 1S0—3P0 transition at 578 nm are both conducted in the cryogenic chamber. During the clock interrogation, the lattice intensity is set to provide a potential depth of ${V_0} = 77\; {E_\textrm{R}}$.
The clock laser for Hg consists of an Yb-doped fiber laser (YbFL) at 1063 nm, a tapered semiconductor laser amplifier (TA) and two second-harmonic generators (SHGs). The first SHG is a single-pass periodically-poled lithium niobate (PPLN) waveguide for 532 nm generation, and the second is a beta-barium borate (β-BBO) crystal inside an optical enhancement cavity for 266 nm generation. The Yb clock laser utilizes an external-cavity diode laser (ECDL) at 1157 nm and a single-pass PPLN waveguide for 578 nm generation. Both laser frequencies at 1063 nm and 1157 nm are heterodyne-locked to an Er-fiber-based optical frequency comb. The repetition rate of the comb is stabilized through a heterodyne lock of the optical beat note with a Sr clock laser at 1397 nm, which is referenced to a 40-cm-long optical cavity with fractional frequency instability of $3 \times {10^{ - 16}}$ at an averaging time of 1 s. The carrier envelope offset frequency ${f_{\textrm{CEO}}}$ of the frequency comb is obtained by a self-referencing f-2f interferometer, and is stabilized with reference to the same GPS-conditioned RF standard used in all the locks. In this way, both the Hg and the Yb clock laser frequencies rely on the common optical reference. The phase noise from the spectral purity transfer between the clock frequencies is much smaller than the cavity instability [17]. After frequency doubling, the clock lasers are superimposed with the lattice lasers to interrogate Hg and Yb atoms.
Both clocks have an approximately equal cycle time of 1.5 s. The Yb clock laser at 518 THz applies Rabi π-pulses of 250 ms. At the Hg clock frequency of 1129 THz, a shorter pulse duration is needed to match the Fourier-limited interrogation linewidth to the clock laser instability. Since the reduced duty cycle additionally increases the sensitivity to high-frequency laser noise, the Hg clock typically operates with a pulse duration of 50 ms.
Frequency deviations of the interrogation lasers from the atomic resonances are corrected by acousto-optic modulators (AOMs). Hg and Yb clock frequencies are described as
3. Experimental results
In the following discussion, ${\Delta }{R_{\textrm{Hg}/\textrm{Yb}}} = {R_{\textrm{Hg}/\textrm{Yb}}} - {R_0}$ stands for the deviation of the measured frequency ratio ${R_{\textrm{Hg}/\textrm{Yb}}}$ from the averaged value ${R_0}$. The fractional deviation $\Delta {R_{\textrm{Hg}/\textrm{Yb}}}/{R_0}$ provides a convenient link to the individual fractional frequency measurements $\varDelta {\nu _i}/{\nu _i}$ through the equation,
Table 1. Typical correction and uncertainty budget for Hg and Yb clock frequencies (upper) and for frequency ratio ${{\boldsymbol\nu }_{{\rm Hg}}}/{{\boldsymbol\nu }_{{\rm Yb}}}$ (lower). All values are shown in fractional units of 10−17.
Figure 2(a) shows the time series data of the frequency ratio measurement after adding the frequency corrections for systematic effects. We acquired ${n_\textrm{d}} \approx $ 100 000 data points where both clocks simultaneously recorded data, representing 79% of the total operating time. No interpolation for the lost data points is applied. The standard error of the mean for the whole dataset is $4 \times {10^{ - 18}}$. The average over all valid data points is ${R_0}$ = 2.177 473 194 134 565 07. Figure 2(b) gives averages per measurement to summarize the frequency ratio campaign. The uncertainties vary with the number of contributing data points, ranging from ${n_\textrm{d}} = $7 000–17 000, while systematic contributions remain largely unchanged. Figure 2(c) shows the overlapping Allan deviation of the fractional frequency ratio $\varDelta {R_{\textrm{Hg}/\textrm{Yb}}}/{R_0}$ for the last measurement run in Fig. 2(b). The instability of the fractional frequency ratio falls as quickly as $2.0 \times {10^{ - 15}}/\sqrt {\tau /\textrm{s}} $ with averaging time $\tau $ as displayed by the dashed line. This is close to the limit imposed by the frequency instability of the Hg clock laser when considering the Dick effect due to the small duty cycle (50 ms/1500 ms) of interrogation to cycle time. The stability improves on the previous measurement of ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Sr}}}$ [10] owing to an upgrade of the frequency comb [17]. We consider the Allan deviation a reliable confirmation of stability up to a quarter of the measurement time, and thus assign a statistical uncertainty of $2.4 \times {10^{ - 17}}$ (representing $\tau $ = 6 000 s) for this measurement run. Following the same procedure for all runs [as shown in Fig. 2(b)], the combined expected uncertainty of the mean is then $1.4 \times {10^{ - 17}}$ with a reduced chi-squared statistic due to day-to-day variations of $\sqrt {\chi _{\textrm{red}}^2} = 2.5 > 1$, which is consistent with fluctuation of systematic effects within the stated uncertainty. We expand the statistical uncertainty to a final value of $3.5 \times {10^{ - 17}}$ by inflating with $\sqrt {\chi _{\textrm{red}}^2} $. As a result, we determine the frequency ratio between Hg and Yb to be ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Yb}}}$ = 2.177 473 194 134 565 07(19) with the total fractional uncertainty of $8.8 \times {10^{ - 17}}$.
Fig. 2. (a) Time series data of the frequency ratio measurement of a Hg and an Yb optical lattice clock after correcting the systematic frequency shifts. Graph shows fractional values with ${R_0}$ = 2.177 473 194 134 565 07. (b) Summary of the frequency ratio measurements with uncertainty bars representing the statistical uncertainty obtained from the Allan deviation. The dashed lines show the total uncertainty from statistical and systematic contributions. (c) Overlapping Allan deviation of the fractional frequency ratio for the last measurement run. The dashed line shows a $2.0 \times {10^{ - 15}}/\sqrt {\tau /\textrm{s}} $ trend.
Figure 3 compares frequency ratio values determined from the measurements compiled in Table 2. The black triangle indicates a frequency ratio ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Yb}}}$ = 2.177 473 194 134 565 2(15) calculated from the absolute frequencies ${\nu _{\textrm{Hg}}}$ and ${\nu _{\textrm{Yb}}}$ recommended by the CIPM in 2018 [1]. The uncertainty bar represents the quadrature sum of the individual uncertainties. The blue filled square is $\frac{{{\nu _{\textrm{Hg}}}}}{{{\nu _{\textrm{Yb}}}}} = \frac{{{\nu _{\textrm{Hg}}}/{\nu _{\textrm{Sr}}}}}{{{\nu _{\textrm{Yb}}}/{\nu _{\textrm{Sr}}}}} = $ 2.177 473 194 134 565 09(21) derived from the frequency ratios ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Sr}}}$ [10] and ${\nu _{\textrm{Yb}}}/{\nu _{\textrm{Sr}}}$ [11] previously obtained in our group. The green open square is a value calculated from weighted averages for ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Sr}}}$ and ${\nu _{\textrm{Yb}}}/{\nu _{\textrm{Sr}}}$, taking into account all published values. In both cases, the combined uncertainties assume no correlation between the two ratio measurements. The red circle is the frequency ratio obtained in this work. All frequency ratios ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Yb}}}$ show agreement within the uncertainties.
Fig. 3. Summary for the ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Yb}}}$ ratio. The black triangle is the calculated value from absolute frequencies recommended by CIPM in 2018 with the uncertainty bar representing uncertainties summed in quadrature. The squares are the calculated value from the frequency ratio measurements of ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Sr}}}$ and ${\nu _{\textrm{Yb}}}/{\nu _{\textrm{Sr}}}$ obtained from measurements in our group (blue filled) and by averaging all reported results (green open). The combined uncertainties assume no correlation between the measurements. The red circle is the result obtained in the direct measurements presented in this work with bar indicating the total uncertainty. The values of all the points are listed in Table 2.
Table 2. Comparison of the values of the Hg and Yb frequency ratio ${{\boldsymbol\nu }_{{\rm Hg}}}/{{\boldsymbol\nu }_{{\rm Yb}}}$.
Combined with our previously reported ratios ${\nu _{\textrm{Yb}}}/{\nu _{\textrm{Sr}}}$ and ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Sr}}}$, the results reported here complete the loop closure over the frequencies of Sr, Yb and Hg optical lattice clocks. For the three separately measured frequency ratios Hg-Yb-Sr-Hg, we find a loop misclosure $({{\nu_{\textrm{Hg}}}/{\nu_{\textrm{Yb}}}} )({{\nu_{\textrm{Yb}}}/{\nu_{\textrm{Sr}}}} )({{\nu_{\textrm{Sr}}}/{\nu_{\textrm{Hg}}}} )- 1 ={-} 8.7 \times {10^{ - 18}}$. This falls within the quadrature sum of the statistical uncertainties $\sqrt {{{3.5}^2} + {{2.9}^2} + {{3.7}^2}} \times {10^{ - 17}} = 5.9 \times {10^{ - 17}}$, where $2.9 \times {10^{ - 17}}$ is for the Yb/Sr ratio [11] and $3.7 \times {10^{ - 17}}$ for the Hg/Sr ratio [10]. The small magnitude of the misclosure supports consistency according to Eq. (1) for the contributing clocks with their uncertainty evaluations, which now stand at $7.5 \times {10^{ - 17}}$ for Hg, $1.7 \times {10^{ - 17}}$ for Yb and $5.8 \times {10^{ - 18}}$ for Sr. If we make an assumption of no correlations among these three frequency ratios, combining the statistical and systematic uncertainties in quadrature yields $\sqrt {{{8.8}^2} + {{4.6}^2} + {{8.4}^2}} \times {10^{ - 17}} = 1.3 \times {10^{ - 16}}$.
Applying the weighted averages of the reported frequency ratios in Table 2, the loop misclosure is $({{\nu_{\textrm{Hg}}}/{\nu_{\textrm{Yb}}}} )({{\nu_{\textrm{Yb}}}/{\nu_{\textrm{Sr}}}} )({{\nu_{\textrm{Sr}}}/{\nu_{\textrm{Hg}}}} )- 1 = 0.4({1.3} )\times {10^{ - 16}}$, once again assuming no correlations among the measurements. Here we adjust the combined fractional uncertainty of the Yb/Sr ratio by $\sqrt {\chi _{\textrm{red}}^2} = 1.19$, while the Hg/Sr ratio shows no need for adjustment with $\sqrt {\chi _{\textrm{red}}^2} = 0.88$. A similar loop closure was recently reported over the frequencies of Cs, Yb and Sr clocks with a misclosure of $0.8({2.4} )\times {10^{ - 16}}$. As calculated, the uncertainties of each clock enter the loop closure once for each of their ratio measurements, and this result thus approaches what is achievable within the limits of the latest Cs fountain clocks, and effectively the limits of the SI limit [8] itself. Closing the loop with only optical clocks thus probes their consistency with uncertainties beyond the SI limit.
4. Conclusion
We performed the first direct measurement of the frequency ratio between a 199Hg and a 171Yb optical lattice clock. An improved optical frequency comb and re-evaluation of systematic effects in the Yb clock improved stability and accuracy over previous measurements, and we determined the ratio to be ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Yb}}}$ = 2.177 473 194 134 565 07(19) with a fractional uncertainty of $8.8 \times {10^{ - 17}}$. Such data collected for various frequency ratios supports a future implementation of the unit of time as discussed in Ref. [21]. Combining the result with the previously reported frequency ratios ${\nu _{\textrm{Hg}}}/{\nu _{\textrm{Sr}}}$ and ${\nu _{\textrm{Yb}}}/{\nu _{\textrm{Sr}}}$, we evaluated the loop closure of the frequency ratios among 87Sr, 171Yb and 199Hg clocks to be $({{\nu_{\textrm{Hg}}}/{\nu_{\textrm{Yb}}}} )({{\nu_{\textrm{Yb}}}/{\nu_{\textrm{Sr}}}} )({{\nu_{\textrm{Sr}}}/{\nu_{\textrm{Hg}}}} )- 1 = 0.4({1.3} )\times {10^{ - 16}}$. The results confirm the consistency of Sr, Yb, and Hg optical lattice clocks beyond the SI limit, representing an important step towards the redefinition of the second.
Appendix A: Yb clock systematic frequency shifts
A.1. Second-order Zeeman shift
We experimentally determined the quadratic term of the Zeeman shift by performing a series of interleaved measurements of the clock frequency alternating between a reference magnetic field setting, corresponding to a first-order Zeeman shift of $\varDelta {\nu _{Z1}} = 120$ Hz, and a setting with high magnetic field, ranging from $\varDelta {\nu _{Zh}} = 253$ Hz to $409$ Hz. We parametrize the measured second-order Zeeman shift as $\varDelta {\nu _{Z2}} = {a_{Z2}}\varDelta \nu _Z^2 = {a_{Z2}}({\varDelta \nu_{Zh}^2 - \varDelta \nu_{Z1}^2} )$ and find a coefficient ${a_{Z2}} ={-} 1.532({15} )$ µHz/Hz2, in perfect agreement with the result reported in Ref. [8]. We set $\varDelta {\nu _Z} \sim 119$ Hz during interrogation.
A.2. Probe light shift
We evaluated the frequency shift induced by the probe laser, employing a Ramsey interrogation scheme where the free-evolution time ${T_\textrm{e}} = $ 200 ms between resonant π/2-pulses of duration ${T_\textrm{p}} = $ 15 ms is realized by detuning the probe laser (typically 200 kHz) from resonance. This avoids the excitation of the clock transition while maintaining active phase stabilization. It also allows the probe intensity $I(t )$ – continuously sampled throughout the interrogation sequence of duration ${T_\textrm{i}} = 230\; \textrm{ms}$ – to be varied through the same AOM that provides the Doppler noise cancellation. This is part of the actively controlled path and does not introduce frequency chirp. The measurement of $I(t )$ was calibrated through the intensity ${I_{{\pi }/2}}$ during the resonant pulses of average Rabi frequency ${{\Omega }_\textrm{R}} = $ 102(1) rad/s, representing a 65-fold intensity increase compared to Rabi interrogation at 250 ms.
A differential probe light shift is measured by alternating between conditions that apply 100% or 8% of probe laser intensity during the free-evolution time. In calculating the effective probe laser intensity change over the full interrogation sequence (with ${T_\textrm{i}} > {T_\textrm{e}})$ we make use of the Ramsey sensitivity function $g(t )$[22,23] $:\; {\varDelta }{I_{\textrm{prb}}} = \frac{{\mathop \smallint \nolimits_0^{{T_\textrm{i}}} {I_\textrm{H}}(t )g(t )dt - \mathop \smallint \nolimits_0^{{T_\textrm{i}}} {I_\textrm{L}}(t )g(t )dt}}{{\mathop \smallint \nolimits_0^{{T_\textrm{i}}} g(t )dt}}$, where H (L) refers to the clock cycle at high (low) intensity. Using the model relation $P = {P_\textrm{m}}{\sin ^2}({{{\Omega }_\textrm{R}}{T_\textrm{p}}/2} )$, we determine the Rabi frequency from the observed excitation probabilities P after Rabi pulses of ${T_\textrm{p}} = $ 15 and 45 ms that represent π/2 and 3π/2 conditions of maximal sensitivity to the pulse area ${{\Omega }_\textrm{R}}{T_\textrm{p}}$. These measurements are interleaved with the probe light shift cycles. With ${P_\textrm{m}}$ as a free parameter, this allows us to describe the incident intensity in terms of a resonant Rabi frequency ${{\Omega }_\textrm{R}}(t )$ with a 1% uncertainty originating from the instability of the photodetector acquisition.
We normalize the differential probe light shift $\varDelta {\nu _\textrm{p}}$ over ${{\Omega }_\textrm{R}}$ of a resonant pulse to find a coefficient of ${c_\textrm{p}} = \varDelta {\nu _\textrm{p}}/{\Omega }_\textrm{R}^2 = 0.8({2.2} )\times {10^{ - 7}}$ Hz/(rad/s)2. During standard clock operation, atoms are interrogated by a Rabi $\pi $-pulse of ${T_\pi } = 250\;\textrm{ms}$ with constant intensity and ${{\Omega }_\textrm{R}} = \pi /{T_\pi }$. The measured coefficient can then be directly applied without considering the sensitivity function. We find a shift of $3(7 )\times {10^{ - 20}}$.
A.3. Lattice light shift
We monitored the spectrum of the Ti:sapphire laser realizing the lattice at ${\nu _{\textrm{lat}}} = 394\;798\;267.5(1)$ MHz to exclude multi-mode operation, and the far-detuned broadband spectral components due to amplified spontaneous emission were filtered out by a volume Bragg grating with 40 GHz bandwidth. Sideband spectra acquired before and after each measurement characterize the trapping parameters (with typical values of ${V_0} = 77.5\; {E_\textrm{R}}$, ${n_z} = 0.09$, ${\zeta } = 0.8$, ${\delta _2} = 0.01$) describing trap depth and atomic temperature within the light shift model described in Ref. [24]. These parameters show only statistical variations over the ratio measurement campaign. Since light shift measurements acquired over two years provide different evaluations of the ${\nu _{\textrm{E}1}}$ magic frequency we assess ${\nu _{\textrm{E}1}} = 394\;798\;265.2({5.4})$ MHz as the average of the extracted values, with an uncertainty corresponding to the standard deviation. We evaluate a final shift of $0.2({1.6} )\times {10^{ - 17}}$.
A.4. Background gas shift
We evaluated the background gas shift as $- 1.9(3 )\times {10^{ - 18}}$ based on the model in Ref. [25] and the measured atom lifetime in the lattice of ${\tau _{\textrm{lft}}} = 8.6(2 )$ s, limited by collisions with mainly background H2 atoms.
A.5. Other shifts
The evaluation of the remaining frequency shifts follows the methods described in Ref. [11].
Funding
Japan Society for the Promotion of Science, Grant-in-Aid for Specially Promoted Research (JP16H06284); Japan Science and Technology Agency, Exploratory Research for Advanced Technology (JPMJER1002); Japan Science and Technology Agency, Mirai Program (JPMJMI18A1).
Acknowledgments
The authors acknowledge M. Takamoto for providing the Sr clock laser.
Disclosures
The authors declare no conflicts of interest.
References
1. F. Riehle, P. Gill, F. Arias, and L. Robertsson, “The CIPM list of recommended frequency standard values: guidelines and procedures,” Metrologia 55(2), 188–200 (2018). [CrossRef]
2. C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency Comparison of Two High-Accuracy Al+ Optical Clocks,” Phys. Rev. Lett. 104(7), 070802 (2010). [CrossRef]
3. N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm, and E. Peik, “Single-ion atomic clock with 3 × 10−18 systematic uncertainty,” Phys. Rev. Lett. 116(6), 063001 (2016). [CrossRef]
4. S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, “27Al+ Quantum-Logic Clock with a Systematic Uncertainty below 10−18,” Phys. Rev. Lett. 123(3), 033201 (2019). [CrossRef]
5. T. L. Nicholson, S. L. Campbell, R. B. Hutson, G. E. Marti, B. J. Bloom, R. L. McNally, W. Zhang, M. D. Barrett, M. S. Safronova, G. F. Strouse, W. L. Tew, and J. Ye, “Systematic evaluation of an atomic clock at 2× 10-18 total uncertainty,” Nat. Commun. 6(1), 6896 (2015). [CrossRef]
6. I. Ushijima, M. Takamoto, M. Das, T. Ohkubo, and H. Katori, “Cryogenic optical lattice clocks,” Nat. Photonics 9(3), 185–189 (2015). [CrossRef]
7. W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Schäffer, K. Beloy, D. Nicolodi, R. C. Brown, N. Hinkley, G. Milani, M. Schioppo, T. H. Yoon, and A. D. Ludlow, “Atomic clock performance enabling geodesy below the centimetre level,” Nature 564(7734), 87–90 (2018). [CrossRef]
8. W. F. McGrew, X. Zhang, H. Leopardi, R. J. Fasano, D. Nicolodi, K. Beloy, J. Yao, J. A. Sherman, S. A. Schaffer, J. Savory, R. C. Brown, S. Römisch, C. W. Oates, T. E. Parker, T. M. Fortier, and A. D. Ludlow, “Towards the optical second: verifying optical clocks at the SI limit,” Optica 6(4), 448 (2019). [CrossRef]
9. T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319(5871), 1808–1812 (2008). [CrossRef]
10. K. Yamanaka, N. Ohmae, I. Ushijima, M. Takamoto, and H. Katori, “Frequency Ratio of 199Hg and 87Sr Optical Lattice Clocks beyond the SI Limit,” Phys. Rev. Lett. 114(23), 230801 (2015). [CrossRef]
11. N. Nemitz, T. Ohkubo, M. Takamoto, I. Ushijima, M. Das, N. Ohmae, and H. Katori, “Frequency ratio of Yb and Sr clocks with 5 × 10−17 uncertainty at 150 seconds averaging time,” Nat. Photonics 10(4), 258–261 (2016). [CrossRef]
12. R. Tyumenev, M. Favier, S. Bilicki, E. Bookjans, R. Le Targat, J. Lodewyck, D. Nicolodi, Y. Le Coq, M. Abgrall, J. Guéna, L. De Sarlo, and S. Bize, “Comparing a mercury optical lattice clock with microwave and optical frequency standards,” New J. Phys. 18(11), 113002 (2016). [CrossRef]
13. R. M. Godun, P. B. R. Nisbet-Jones, J. M. Jones, S. A. King, L. A. M. Johnson, H. S. Margolis, K. Szymaniec, S. N. Lea, K. Bongs, and P. Gill, “Frequency Ratio of Two Optical Clock Transitions in 171Yb+ and Constraints on the Time Variation of Fundamental Constants,” Phys. Rev. Lett. 113(21), 210801 (2014). [CrossRef]
14. J. Grotti, S. Koller, S. Vogt, S. Häfner, U. Sterr, C. Lisdat, H. Denker, C. Voigt, L. Timmen, A. Rolland, F. N. Baynes, H. S. Margolis, M. Zampaolo, P. Thoumany, M. Pizzocaro, B. Rauf, F. Bregolin, A. Tampellini, P. Barbieri, M. Zucco, G. A. Costanzo, C. Clivati, F. Levi, and D. Calonico, “Geodesy and metrology with a transportable optical clock,” Nat. Phys. 14(5), 437–441 (2018). [CrossRef]
15. M. Fujieda, S.-H. Yang, T. Gotoh, S.-W. Hwang, H. Hachisu, H. Kim, Y. K. Lee, R. Tabuchi, T. Ido, W.-K. Lee, M.-S. Heo, C. Y. Park, D.-H. Yu, and G. Petit, “Advanced satellite-based frequency transfer at the 10-16 level,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 65(6), 973–978 (2018). [CrossRef]
16. M. Takamoto, F. L. Hong, R. Higashi, Y. Fujii, M. Imae, and H. Katori, “Improved frequency measurement of a one-dimensional optical lattice clock with a spin-polarized fermionic 87Sr isotope,” J. Phys. Soc. Jpn. 75(10), 104302 (2006). [CrossRef]
17. N. Ohmae, N. Kuse, M. E. Fermann, and H. Katori, “All-polarization-maintaining, single-port Er:fiber comb for high-stability comparison of optical lattice clocks,” Appl. Phys. Express 10(6), 062503 (2017). [CrossRef]
18. D. Akamatsu, M. Yasuda, H. Inaba, K. Hosaka, T. Tanabe, A. Onae, and F.-L. Hong, “Frequency ratio measurement of 171Yb and 87Sr optical lattice clocks,” Opt. Express 22(7), 7898–7905 (2014). [CrossRef]
19. M. Takamoto, I. Ushijima, M. Das, N. Nemitz, T. Ohkubo, K. Yamanaka, N. Ohmae, T. Takano, T. Akatsuka, A. Yamaguchi, and H. Katori, “Frequency ratios of Sr, Yb, and Hg based optical lattice clocks and their applications,” C. R. Phys. 16(5), 489–498 (2015). [CrossRef]
20. D. Akamatsu, T. Kobayashi, Y. Hisai, T. Tanabe, K. Hosaka, M. Yasuda, and F.-L. Hong, “Dual-mode operation of an optical lattice clock using strontium and ytterbium atoms,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 65(6), 1069–1075 (2018). [CrossRef]
21. J. Lodewyck, “On a definition of the SI second with a set of optical clock transitions,” Metrologia 56(5), 055009 (2019). [CrossRef]
22. S. Falke, M. Misera, U. Sterr, and C. Lisdat, “Delivering pulsed and phase stable light to atoms of an optical clock,” Appl. Phys. B 107(2), 301–311 (2012). [CrossRef]
23. G. J. Dick, “Local oscillator induced instabilities in trapped ion frequency standards,” Proceedings of the 19th Annual Precise Time and Time Interval Systems and Applications Meeting, Redondo Beach, California, December 1987, pp. 133–147
24. N. Nemitz, A. A. Jørgensen, R. Yanagimoto, F. Bregolin, and H. Katori, “Modeling light shifts in optical lattice clocks,” Phys. Rev. A 99(3), 033424 (2019). [CrossRef]
25. K. Gibble, “Scattering of Cold-Atom Coherences by Hot Atoms: Frequency Shifts from Background-Gas Collisions,” Phys. Rev. Lett. 110(18), 180802 (2013). [CrossRef]
