Open Access
Add to BibSonomyAbstract
Optical frequency comparison of the 40Ca+ clock transition νCa (2S1/2-2D5/2, 729nm) against the 87Sr optical lattice clock transition νSr (1S0-3P0, 698nm) has resulted in a frequency ratio νCa / νSr = 0.957 631 202 358 049 9(2 3). The rapid nature of optical comparison allowed the statistical uncertainty of frequency ratio νCa / νSr to reach 1 × 10−15 in 1000s and yielded a value consistent with that calculated from separate absolute frequency measurements of νCa using the International Atomic Time (TAI) link. The total uncertainty of the frequency ratio using optical comparison (free from microwave link uncertainties) is smaller than that obtained using absolute frequency measurement, demonstrating the advantage of optical frequency evaluation. We note that the absolute frequency of 40Ca+ we measure deviates from other published values by more than three times our measurement uncertainty.
©2012 Optical Society of America
1. Introduction
The most reliable means to fully evaluate the reproducibility and stability of a frequency standard is frequency comparison against independent standards. Comparison also enables the evaluation of systematic shifts by investigating the dependence of the frequency on various experimental parameters. Optical frequency standards have a significant advantage over microwave frequency standards (e.g. Cs fountain clocks) in the speed of the comparison, requiring less than 1000 seconds instead of more than six hours to evaluate a fractional frequency difference with ~10−16 uncertainty [2]. Rapid optical evaluation has enabled comparison of the frequencies of two single-ion optical clocks in same laboratory at the 10−18 level [3]. Using fiber transfer techniques [4,5] optical clocks located in distant laboratories can be compared, for example, the reproducibility of optical lattice clocks has been measured at the 10−16 level of uncertainty [6, 7]. Furthermore, an optical frequency comb can be employed as a bridge enabling the measurement of the relative instabilities of standards based on different atomic transitions. Such frequency ratio comparisons can yield information on possible temporal variations of fundamental constants [8].
In this paper, we report a frequency comparison of a 40Ca+ 2S1/2-2D5/2 single-ion clock transition against the 87Sr 1S0-3P0 lattice clock transition. The 40Ca+ clock in this work has exceptional fractional frequency instability of parts in 10−16. Neutral strontium and calcium ions are popular as quantum absorbers in the realization of optical clocks; worldwide five neutral strontium lattice clocks are already in operation [9–13], and calcium ions are ideal for portable optical clocks due to the availability of laser-diodes at all required wavelengths. 40Ca+ clocks are being developed at NICT [1], Univ. of Innsbruck [14], and WIPM [15]. With such interest and development in these two optical standards around world it is imperative that we establish a benchmark frequency ratio with high precision.
2. Single calcium ion clock
The experimental setup for the Ca+ clock has been improved since it was reported in [1]; firstly, an increase in production efficiency of 40Ca+ ions has been achieved by a photo-ionization process using lasers of wavelength 423 nm (1S0→1P1 transition) and 374 nm (1P1→ ionization). Secondly, a magnetic shield has been installed on the vacuum chamber that has reduced by more than 20 times stray ac fields that previously limited the spectral width to 300 Hz. Thirdly, mechanical shutters and acousto-optic modulator (AOM) shutters have been installed to avoid coupling of cooling-laser light during the 40 ms interrogation period. Further optimization of the clock laser [16] has reduced its spectral width to less than 5 Hz, and a noise cancellation technique [17] implemented on the 40 m of optical fiber between it and the ion trap. These improvements have resulted in an observed spectral width of the clock transition of 30 Hz, as shown in Fig. 1 .
Fig. 1 Spectra of 40Ca+ clock transition. Each point of the excitation probability resulted from 40 measurements of 40 ms duration interrogation of the clock laser. The FWHM of the clock transition is 30 Hz.
The systematic shifts and the respective uncertainties to determine the Ca+ clock-transition frequency are shown in Table 1 . Black body radiation frequency shifts were evaluated by constructing an identical ion-trap equipped with platinum resistance thermometers. We measured the temperature of the electrodes when the RF power for the ion trap is switched on and found that the temperature of the ring electrode to be at (305.4 ± 2)K, and the end cap electrodes and trap chamber to be (299 ± 2) K. The platinum resistance thermometers had no measurable effect on the trap Q as they are electrically insulated from the trap; further, their influence on the trap temperature is negligibly small as their thermal capacity is much less than that of the trap electrode. The black body radiation shift is estimated from ratios of solid angles as seen by the ion of the endcap, ring electrode and trap chamber. The total shift is calculated to be 0.39(05) Hz [18]. GPS-based elevation measurement together with a geoid map of Japan GSIGEO2000 yields an average geoid height for the Ca+ trap of 76.38 m. Although the average height is measured to within a few centimeters of uncertainty, it can vary around ± 30 cm due to tidal deformations of the Earth’s crust. When compared to this fluctuation, any possible variation of elevation due to the Tohoku 2011 earthquake is negligible. We therefore estimate the gravitational shift to be 3.4 (0.015) Hz.
Table 1. Systematic shifts and their uncertainties of the 40Ca+ clock
From knowledge of the magnetic field bias of 3 μT we estimate the quadratic Zeeman shift in the clock transition to be 0 (0.1) Hz. We estimate the ion temperature of 2 mK from the intensity of the motional sidebands, corresponding to a quadratic Doppler shift of 2 (10) mHz. We optimized micromotion compensation at the start of each measurement day using the rf-photon correlation method, and monitored the correlation throughout the measurement period. To maintain optimization we did not need to adjust compensation voltages by more than 10% of the initial daily values. We observed a gradual long-term voltage drift which may be caused by a patch effect of accumulated Ca atoms on the electrodes. To investigate possible time-dilation shifts, excess voltage was deliberately applied to the micromotion compensation electrode. This caused large correlations between fluorescence intensity and the RF phase of the trap, yet no time-dilation frequency shifts larger than the measurement uncertainty were detected when using the Sr lattice clock as a reference.
In order to evaluate the electric quadrupole shift the dependence of the resonant frequencies on the upper magnetic sublevels was investigated [19]. To do this we operate an interleaved measurement between transitions having ΔMJ = 0 (|2S1/2 MJ = ± 1/2 > - |2D5/2 MJ = ± 1/2>) and transitions having ΔMJ = 2 (|2S1/2 MJ = ± 1/2 > - |2D5/2 MJ = ± 5/2>). We measure a frequency difference due to the quadrupole shift of 0.9 Hz, from which we calculate a quadrupole shift of 0.1 Hz for the normally used transition having ΔMJ = 1 (|2S1/2 MJ = ± 1/2 > - |2D5/2 MJ = ± 3/2>). We experimentally confirmed that quadrupole shifts of six transitions varied according to the theoretical ratio when we deliberately imposed a DC electric field. When the maximum DC field was applied (limited by requiring the ion remains stably trapped) the maximal difference between the ΔMJ = 0 and the ΔMJ = 2 transitions were 4.0Hz, which corresponds to a shift of 0.44 Hz for the ΔMJ = 1 transition. Thus, the magnitude and uncertainty of the quadrupole shift for the ΔMJ = 1 transition are estimated to be 0.1(0.34) Hz so that maximum shift of the stably trapped ion is 0.44 Hz. In addition, there was no frequency difference within measurement uncertainty between the average frequency of the three transitions [20] and the |2S1/2 MJ = ± 1/2 > ― |2D5/2 MJ = ± 3/2> transition.
Potential ac Stark shift due to 397-nm light (used for cooling and detection) unintentionally coupling into the optical fiber directed to the trap was estimated as follows. Firstly, the attenuation of the mechanical shutter was measured to be more than −50 dB, limited by photodetector noise. Secondly, no definite 397 nm light intensity dependence of the clock frequency outside measurement uncertainty was observed. We estimate the ac Stark shift due to 397 nm light from linear fitting of the clock frequency to the intensity of 397 nm light which was a zero measurement of 0.27(0.54) Hz at normal operation. A further experiment when the mechanical shutter was inoperative found that the blue shift of the clock transition due to leaked 397 nm radiation from the AOM was proportional to the incident intensity to the AOM. Within measurement uncertainty there was no difference in frequency between normal operation and when the shutter is inoperative and the 397 nm intensity is extrapolated to zero. This result excludes the possibility of 397nm light scattering from the back surface of the shutter blade and coupling to the fiber. No frequency shift is observed when the AOM for the clock laser is kept on and only a mechanical shutter is used to generate clock interrogation pulses, ruling out frequency chirp due to the transient phenomena of the AOM [21].
Ac Stark shift due to the 854-nm light (used for quenching the ion from the 2D5/2 levels into the 2S1/2 levels) was estimated from the comparison between frequencies measured with and without use of 854-nm light (when we do not use 854-nm light the ion in 2D5/2 takes longer to decay via spontaneous emission.) We did not observe any frequency difference with uncertainty of 0.2 Hz. We also measured frequency shifts by changing the power of the clock laser and ac Stark shifts were evaluated to be 0 (0.3) Hz. The linear frequency drift of the clock laser is roughly compensated by a chirped AOM. The servo error was estimated from the analysis of the residual error, resulting in 0 (0.5) Hz. Summing the above, we calculate a total systematic shift of 4.2 Hz with an uncertainty of 0.89 Hz.
3. Strontium lattice clock
The optical lattice clock utilized as a stable frequency reference in this work is based on the 1S0-3P0 transition (λ = 698 nm) of optically trapped spin-polarized fermionic 87Sr atoms (87Sr lattice clock [9]). The clock is in a different room, 10m from the Ca+ clock. The absolute frequency of the 87Sr lattice clock at NICT agrees with other clocks in various institutes within the so-called Cs-limit [13]. The systematic uncertainty has been evaluated to be 5 × 10−16. Using a 60 km fiber link [7] we confirmed the agreement of the clock frequencies at the 10−16 level with another 87Sr lattice clock in The University of Tokyo. This experiment also demonstrated that the instability of the clock at NICT reached 5 × 10−16 in 1000 s [6]. Since then the instability of the clock laser has been improved by implementing optical compensation against vibration-induced phase noise, details of which are to be described elsewhere. By this technique the short term stability is <2 × 10−15 at 1 s and no longer requires the stability transfer technique employed previously [13]. Systematic shifts and their uncertainties have been recently evaluated after some minor renovations which has resulted in a systematic correction of −1.25(22) Hz. In this paper we report on the use of our lattice clock as an independent optical frequency reference for the evaluation of the Ca+ clock.
4. Frequency ratio and instability
A Ti:Sapphire-based optical frequency comb [22] bridges the two clock transitions and the frequency ratio νCa /νSr is measured as follows. The beat signal between the clock laser for the Sr lattice clock and the nearest comb component is first phase locked to a stable RF frequency fPLL which is generated by a direct digital synthesizer with reference to a hydrogen maser. In this case, the repetition frequency frep is expressed as follows,
where N1 and fceo are the mode number of the comb component and the carrier-envelope offset frequency of the comb. Measurement of the beat frequency fb between the Ca+ clock frequency and the nearest comb component (mode number N2) yields the transition frequency νCa according toThe frequency ratio νCa /νSr can be calculated asHere, the first term is the order of 1, whereas the second term is on the order of 10−7 since the numerator is a radio frequency of ~107 Hz and the denominator is an optical frequency of ~1014 Hz. Therefore, the easily attainable 10−10 fractional accuracy of all components including νSr is sufficient for evaluation of the frequency ratio at the 10−16 level.The instability of νCa / νSr is shown in Fig. 2 . The level of instability slowly decreases with increasing integration time over 1-10 s, indicating our clock laser (employing a simple cylindrical shaped high finesse optical cavity) is slightly above the thermal noise limit. The optical compensation of vibration induced noise allows the Sr clock laser to operate at the thermal noise limit, but the compensation is not applied to the Ca+ clock laser, which may prevent the short term stability from reaching the flat thermal noise limit. The long term instability of 2.4 × 10−14/τ1/2, where τ is the integration time, is limited by the Ca+ clock. The instability of the Sr lattice clock is 1 × 10−14/τ1/2 according to an interleaved stabilizing operation, where the clock laser is stabilized to a π transition from an identical stretched state with a total operation cycle of 6 s. This is shown as the thin black trace in Fig. 2.
Fig. 2 Instability of the frequency ratio νCa / νSr (red) determined by the instability of Ca+ clock. An interleaved measurement of the Sr lattice clock has resulted in lower instability as shown in black.
The Ca+ center frequency is estimated by forty quantum projection measurements in a cycle time of 17 seconds with a Fourier-limited linewidth of 30 Hz. Random initial preparation of Zeeman substates (mJ = ± 1/2) halves the number of effective measurements. The quantum projection limit estimated using these parameters is 2 × 10−14/τ1/2, and is consistent with the observed long term stability shown as a broken line in Fig. 2. Note that the bump around averaging time of 800 s is suspected to be caused by two 10 m-length optical fiber cables that do not have fiber noise cancellation. According to the thermal coefficient of the refractive index of fused-silica, peak-to-peak temperature fluctuations of 0.8 K are sufficient to induce this level of phase noise. The frequency shift due to this fiber noise is estimated to be at the 10−17 level when the signal is averaged over the campaign length of several days.
The frequency ratio νCa / νSr measured for a half year are summarized in Fig. 3 . The results indicate that both optical clocks have a day-to-day reproducibility better than 1 × 10−15, consistent with the evaluation of systematic shifts described above. Taking into account a statistical uncertainty of 1.8 × 10−16 and systematic uncertainties of Ca+ clock and Sr clock (2.2 × 10−15and 5 × 10−16 respectively), we conclude the frequency ratio νCa/ νSr is 0.957 631 202 358 049 9 with a fractional uncertainty of 2.3 × 10−15.
Fig. 3 Frequency ratio νCa / νS obtained for half a year. The reproducibility of less than 10−15 is consistent with the systematic uncertainties of two clocks. The thick and thin lines indicate the weighted average and the uncertainty, respectively.
5. Absolute frequency measurement of the Ca+ clock
The frequency of the Ca+ clock transition was also evaluated by a microwave link to International Atomic Time (TAI). We measured a beat frequency between the clock laser and the nearest comb component of a frequency comb stabilized to a hydrogen maser. The frequency of the hydrogen maser is calibrated to UTC(NICT) using dual-mixer time difference (DMTD) technique [23]. Measurements spanning one year are summarized in Fig. 4 . The link uncertainties consist of those in the UTC(NICT)-TAI link and TAI-TT link. In principle, these uncertainties are evaluated by BIPM in Circular T. However, the calibration of the UTC(NICT) to the TAI is based on time differences of two instances separated for 5 days, whereas the Ca+ clock is operated only for several hours. We therefore estimate that UTC(NICT) for this limited time could be different from the five day average by the amount of the instability of UTC(NICT) over those five days. In the duration shown in Fig. 4, this five day instability of UTC(NICT) against TAI was 3.0 × 10−15, where twelve data points based on nine calibrations of the UTC(NICT) is averaged to yield the final result. Forty-five days of signal integration (nine calibrations) resulted in an the instability of TAI-UTC(NICT) of 1.9 × 10−15. We suspect that the reduction is less than the expected factor of 3 (ie. 91/2) because of imperfect steering of UTC(NICT) to the ensemble of Cs clocks, TA(NICT). Thus, the uncertainty due to the TAI link is evaluated to be 1.9 × 10−15 including minor contributions from the calibration of TAI to the SI second (3 × 10−16). Coupled with considerations of the systematic uncertainty of the Ca+ clock shown in Table 1 and statistical uncertainty in Fig. 4, the total fractional uncertainty of the absolute frequency measurement is 3.0 × 10−15. We therefore determine that the absolute frequency for our measurement is 411 042 129 776 398.4(1.2) Hz.
Fig. 4 One year record of absolute frequency measurements of 40Ca+ clock frequency. The corrections due to systematic shifts are included. Error bars, however, include only statistical errors. The hydrogen-maser frequency used for the frequency-comb stabilization was calibrated using the TAI link. Calibrations of UTC(NICT) of two data at MJD = 55704 and 55707 as well as three data at MJD = 56009-56011 are based on same Circular T. The weighted average frequency and statistical uncertainty is 411 042 129 776 398.4 (0.2) Hz.
The instability at the 10−16 level shown in Fig. 2 makes it possible to use the Ca+ clock as a transfer oscillator, in the same way that a hydrogen maser is used in the microwave frequency regime. Combining the measured frequency ratio νCa / νSr with the absolute frequency of our Ca+ clock we evaluate the absolute frequency of the Sr lattice clock to be 429 228 004 229 871.9 (1.2) Hz. This frequency is consistent with our previous measurement [13] as well as that obtained in four other institutes [10–12, 24], indicating the validity of the TAI-link we have used.
While our Ca + clock has a day-to-day reproducibility at the 10−16 level, the frequency is disparate by more than three times the measurement uncertainty when compared to two other published results [14, 15]. Our frequency measurements using two frequency references, namely a Sr lattice clock and the UTC(NICT)-TAI microwave link, are consistent within measurement uncertainty, ruling out errors in frequency counting. Additionally, our measurements are based on atomic spectra narrower than those used in the other reported measurements. To address this discrepancy we are planning further experiments; firstly satellite-based inter-continental comparisons directly to other clocks, and secondly we will repeat the frequency measurement using an In+-Ca+ clock currently under development in our laboratory [25]
6. Summary and outlook
We have measured the frequency ratio between the 40Ca+ 2S1/2-2D5/2 transition and the 87Sr 1S0-3P0 transitions with an uncertainty of 2.3 × 10−15. The frequency ratio measurement demonstrated for the first time that the Ca+ clock is able to reach the 10−16 level of instability. All optical comparison has also enabled quick and rigorous evaluation of the systematic shifts of Ca+ ion clock. The frequency ratio νCa / νSr was consistent with the absolute frequency measurement of the Ca+ clock.
The Ca+ and Sr optical clocks investigated here are two of the most popular in ion-based and lattice-based systems. The systematic uncertainty of the Ca+ clock in this work is at the 10−15 level, and we have demonstrated it may be used as a transfer oscillator. The systematic uncertainties of both clocks are currently projected to be smaller than current state-of-the-art cesium fountain clocks and therefore further improved measurement of the ratio νCa / νSr may become an infrastructure of frequency metrology, enabling frequency comparison of two physically separated optical clocks more precisely than that of comparisons via SI second. In this case one optical clock can be locally linked to a Sr lattice clock and the remote clock linked to a Ca+ clock possibly working as a logic ion in quantum logic gate spectroscopy. Such comparison requires only 1000 seconds of frequency ratio measurement in each laboratory.
Acknowledgments
We are grateful to R. Kojima and A. Yamaguchi for considerable effort to initiate the clock operation of the Ca+ clock and the Sr lattice clock, respectively. Contributions of Y. Hanado, H. Ito, and M. Kumagai are gratefully acknowledged in discussions related to the TAI-link. The authors thank Y. Koyama and N. Shiga for fruitful discussions. Experimental support by H. Ishijima and K. Kido were indispensable for this work. This research was supported in part by the Japan Society for the Promotion of Science through its FIRST program.
References and links
1. K. Matsubara, K. Hayasaka, Y. Li, H. Ito, S. Nagano, M. Kajita, and M. Hosokawa, “Frequency measurement of the optical clock transition of 40Ca+ ions with an uncertainty of 10−14 level,” Appl. Phys. Express 1, 067011 (2008). [CrossRef]
2. J. Guéna, M. Abgrall, D. Rovera, P. Laurent, B. Chupin, M. Lours, G. Santarelli, P. Rosenbusch, M. E. Tobar, R. Li, K. Gibble, A. Clairon, and S. Bize, “Progress in atomic fountains at LNE-SYRTE,” IEEE Trans. Ultrason., Ferroelectr., Freq. Cont. 59, 391–420 (2012).
3. 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] [PubMed]
4. P. A. Williams, W. C. Swann, and N. R. Newbury, “High-stability transfer of an optical frequency over long fiber-optic links,” J. Opt. Soc. Am. B 25(8), 1284–1293 (2008). [CrossRef]
5. K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, Th. Legero, T. W. Hänsch, Th. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336(6080), 441–444 (2012). [CrossRef] [PubMed]
6. A. Yamaguchi, M. Fujieda, M. Kumagai, H. Hachisu, S. Nagano, Y. Li, T. Ido, T. Takano, M. Takamoto, and H. Katori, “Direct comparison of distant optical lattice clocks at the 10−16 uncertainty,” Appl. Phys. Express 4(8), 082203 (2011). [CrossRef]
7. M. Fujieda, M. Kumagai, S. Nagano, A. Yamaguchi, H. Hachisu, and T. Ido, “All-optical link for direct comparison of distant optical clocks,” Opt. Express 19(17), 16498–16507 (2011). [CrossRef] [PubMed]
8. 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] [PubMed]
9. M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435(7040), 321–324 (2005). [CrossRef] [PubMed]
10. G. K. Campbell, A. D. Ludlow, S. Blatt, J. W. Thomsen, M. J. Martin, M. H. G. de Miranda, T. Zelevinsky, M. M. Boyd, J. Ye, S. A. Diddams, T. P. Heavner, T. E. Parker, and S. R. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45(5), 539–548 (2008). [CrossRef]
11. X. Baillard, M. Fouche, R. Le Targat, P. G. Westergaard, A. Lecallier, F. Chapelet, M. Abgrall, G. D. Rovera, P. Laurent, P. Rosenbusch, S. Bize, G. Santarelli, A. Clairon, P. Lemonde, G. Grosche, B. Lipphardt, and H. Schnatz, “An optical lattice clock with spin-polarized 87Sr atoms,” Eur. Phys. J. D 48(1), 11–17 (2008). [CrossRef]
12. St. Falke, H. Schnatz, J. S. R. V. Winfred, T. Middelmann, S. Vogt, S. Weyers, B. Lipphardt, G. Grosche, F. Riehle, U. Sterr, and C. Lisdat, “The 87Sr optical frequency standard at PTB,” Metrologia 48(5), 399–407 (2011). [CrossRef]
13. A. Yamaguchi, N. Shiga, S. Nagano, Y. Li, H. Ishijima, H. Hachisu, M. Kumagai, and T. Ido, “Stability transfer between two clock lasers operating at different wavelengths for absolute frequency measurement of clock transition in 87Sr,” Appl. Phys. Express 5(2), 022701 (2012). [CrossRef]
14. M. Chwalla, J. Benhelm, K. Kim, G. Kirchmair, T. Monz, M. Riebe, P. Schindler, A. S. Villar, W. Hänsel, C. F. Roos, R. Blatt, M. Abgrall, G. Santarelli, G. D. Rovera, and Ph. Laurent, “Absolute frequency measurement of the 40Ca+ 4s2S1/2-3d2D5/2 clock transition,” Phys. Rev. Lett. 102(2), 023002 (2009). [CrossRef] [PubMed]
15. Y. Huang, J. Cao, P. Liu, K. Liang, B. Ou, H. Guan, X. Huang, T. Li, and K. Gao, “Hertz-level measurement of the 40Ca+ 4s2S1/2–3d2D5/2 clock transition frequency with respect to the SI second through the Global Positioning System,” Phys. Rev. A 85(3), 030503 (2012). [CrossRef]
16. Y. Li, S. Nagano, K. Matsubara, H. Ito, M. Kajita, and M. Hosokawa, “Narrow-line and frequency tunable diode laser system for S–D transition of Ca+ ions,” Jpn. J. Appl. Phys. 47(8), 6327–6332 (2008). [CrossRef]
17. L.-S. Ma, P. Jungner, J. Ye, and J. L. Hall, “Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by an optical fiber or other time-varying path,” Opt. Lett. 19(21), 1777–1779 (1994). [CrossRef] [PubMed]
18. B. Arora, M. S. Safronova, and C. W. Clark, “Blackbody-radiation shift in a 43Ca+ ion optical frequency standard,” Phys. Rev. A 76(6), 064501 (2007). [CrossRef]
19. G. P. Barwood, H. S. Margolis, G. Huang, P. Gill, and H. A. Klein, “Measurement of the electric quadrupole moment of the 4d2D5/2 level in 88Sr+.,” Phys. Rev. Lett. 93(13), 133001 (2004). [CrossRef] [PubMed]
20. H. S. Margolis, G. P. Barwood, G. Huang, H. A. Klein, S. N. Lea, K. Szymaniec, and P. Gill, “Hertz-level measurement of the optical clock frequency in a single 88Sr+ ion,” Science 306(5700), 1355–1358 (2004). [CrossRef] [PubMed]
21. C. Degenhardt, T. Nazarova, C. Lisdat, H. Stoehr, U. Sterr, and F. Riehle, “Influence of chirped excitation pulses in an optical clock with ultracold calcium atoms,” IEEE Trans. Instrum. Meas. 54(2), 771–775 (2005). [CrossRef]
22. S. Nagano, H. Ito, Y. Li, K. Matsubara, and M. Hosokawa, “Stable operation of femtosecond laser frequency combs with uncertainty at the 10−17 level toward optical frequency standards,” Jpn. J. Appl. Phys. 48(4), 042301 (2009). [CrossRef]
23. F. Nakagawa, M. Imae, Y. Hanado, and M. Aida, “Development of multichannel dual-mixer time difference system to generate UTC(NICT),” IEEE Trans. Instrum. Meas. 54(2), 829–832 (2005). [CrossRef]
24. F.-L. Hong, M. Musha, M. Takamoto, H. Inaba, S. Yanagimachi, A. Takamizawa, K. Watabe, T. Ikegami, M. Imae, Y. Fujii, M. Amemiya, K. Nakagawa, K. Ueda, and H. Katori, “Measuring the frequency of a Sr optical lattice clock using a 120 km coherent optical transfer,” Opt. Lett. 34(5), 692–694 (2009). [CrossRef] [PubMed]
25. K. Hayasaka, “Synthesis of two-species ion chains for a new optical frequency standard with an indium ion,” Appl. Phys. B 107(4), 965–970 (2012). [CrossRef]
