Absolute measurement of the 1S0 − 3P0 clock transition in neutral 88Sr over the 330 km-long stabilized fibre optic link

We report a stability below 7 × 10−17 of two independent optical lattice clocks operating with bosonic 88Sr isotope. The value (429 228 066 418 008.3(1.9)syst (0.9)stat Hz) of the absolute frequency of the 1S0 – 3P0 transition was measured with an optical frequency comb referenced to the local representation of the UTC by the 330 km-long stabilized fibre optical link. The result was verified by series of measurements on two independent optical lattice clocks and agrees with recommendation of Bureau International des Poids et Mesures.

Ultracold neutral atoms in an optical lattice 1 are seen as an alternative to single-ions 2 for development of optical frequency standards. All best present realizations of the strontium optical clocks are made with fermionic strontium isotope 87 Sr 3-7 , since the bosonic isotopes are expected to have larger collisional effects on the clock transition. Additionally, the bosonic isotopes require at least one extra field to induce the clock transition, which implies careful control of this field and its respective field shift. On the other hand, the bosonic lattice clocks have some advantages over their fermionic counterpart: no first order Zeeman shift, no vector or tensor lattice Stark shifts and much higher isotopic abundance. Lack of hyperfine structures in both 1 S 0 and 3 P 0 states and higher abundance reduce the time required for one lock cycle. Furthermore, the set-up of cooling and trapping the bosonic isotope is simpler, which is important for transportable systems.
The experimental difficulties in limiting and characterising the collisional shift in bosons are the reason why there are only two reported measurements of the 1 S 0 − 3 P 0 transition in 88 Sr so far 8,9 . To calculate the recommended frequency values for the practical realizations of the metre (MeP) and secondary representations of the second (SRS), the BIPM takes into account the weighted average of independently obtained frequencies. A limited pool of available measurements forced the BIMP to set practical relative uncertainties above the × − 1 10 14 level when the 1 S 0 − 3 P 0 transition in 88 Sr is used as MeP and restrain in recommending this transition as SRS 10 .
There are two known ways to limit the effects of the collisions: the first, the measurements in the optical lattice trap with low atomic density and high confinement to suppress tunnelling effects 11 ; the second, the use of higher dimensional optical lattice trap 9 . In our system the low value of collisional shift is ensured by a large waist of the lattice and trapping only a few atoms per lattice site in a trap. We report a system of two independent bosonic strontium optical lattice standards with 88 Sr probed with a single shared ultranarrow laser. The absolute frequency of the clock transition is measured by the use of a frequency-doubled Er:fibre polarization-mode-locked optical frequency comb referenced to the UTC(AOS) and UTC(PL) 12,13 via the 330 km-long stabilized fibre optic link of the OPTIME network 14,15 .

Methods
Optical Lattice Standards. The experimental set-up of our system has been described in detail in ref. 16, so only its most essential elements are presented below.
A simplified scheme of the system of two optical lattice clocks is depicted in Fig. 1. Two optical frequency standards (Sr1 and Sr2) are based on the 1 S 0 − 3 P 0 transition in neutral 88 Sr atoms. Two clouds of cold atoms in Sr1 and Sr2, trapped in the vertical optical lattices, are independently probed by an ultrastable laser with spectral width below 1 Hz. The laser beam is split into two optical paths. The frequencies of both beams are independently digitally locked to the narrow atomic resonances in each standard by feedback to the acousto-optic frequency shifters.
The short-time frequency reference of the optical standards, i.e. the ultrastable laser, is an Extended Cavity Diode Laser (ECDL) locked to the TEM 00 mode of the high-Q cavity. The light from the ultrastable laser is transferred to the Sr1 and Sr2 standards and to the optical frequency comb through optical fibres. Each fibre has a system of active Doppler cancellation of the fibre-link noises to assure the transfer of stable optical frequencies 17 .
In both Sr1 and Sr2 systems the Fabry-Perot diode lasers are injection-locked to the light from ultrastable laser. The master-slave system filters out any power fluctuations of the injection laser. The beam is passing the acousto-optic modulator (AOM) of the digital lock and is injected to the optical lattice such that it is exactly superimposed with the lattice. The beam waist is much bigger than the size of the sample of atoms. Stabilized fibre optic link and UTC(AOS). The frequencies of the clock transitions can be compared by the use of an optical frequency comb with the UTC(AOS) and UTC(PL) 12,13 via the OPTIME network 14 .
The 330 km-long time and frequency dissemination line between the Space Research Centre at Borowiec Astrogeodynamic Observatory (AOS) and KL FAMO in Toruń is electronically stabilized with the ELSTAB technology 18 . The underlying idea of the ELSTAB solution is to implement the compensation of the fibre delay fluctuations in the electronic domain, by using a pair of precisely matched variable delay lines. The delay lines are both placed in the forward and backward paths of the delay-locked-loop (DLL) structure (see Fig. 2, left panel).
The local module is installed at the AOS in Borowiec and the remote module is installed at the KL FAMO in Toruń. Additionally, the line contains seven specialized optical bidirectional amplifiers based on erbium-doped fibres (see Fig. 2, right panel). Thanks to bidirectional operation over the same optical path for the forward and backward directions, the propagation delay is constant for both directions. Consequently, the possible phase fluctuations compensate and the insertion of the amplifier does not destroy the symmetry of the optical path.
To estimate the quality of the link, the pre-installation tests with a 300 km-long fiber on spools and bidirectional optical amplifiers were performed. The stability of the remote 10 MHz signal was measured with respect to the local input, using the A7-MX Signal Stability Analyser. The overlapping Allan deviation is equal to × − 4 10 13 for 1 s integration period, and drops down to × − 3 10 16 within 1 h (Fig. 3). The local representation of the Coordinated Universal Time (see e.g. 19 ) at AOS in Borowiec, UTC(AOS), is realized directly in the form of a 1PPS (one-pulse-per-second) by a system of an active H-maser (CH1-75A) and an offset generator (Symmetricom Auxiliary Output Generator -AOG). The active H-maser provides good stability over measurement times of up to a few days, with an Allan deviation of × − 2 10 13 at an averaging time of τ = 1 s and decreasing as τ / 1 up to the averaging time of τ = 10 4 s. The AOG compensates the linear frequency drift of the maser on a daily basis and adds corrections in respect to the UTC, extrapolated from differences UTC-UTC(AOS) and UTCr-UTC(AOS) published monthly and weekly, respectively, in Circular-T 20 . The details of the frequency chain at the AOS are presented in Fig. 4.

Results
Statistical stability of the system. The difference between the corrections of the digital locks in the Sr1 and Sr2 standards yields the momentary frequency difference between the two clocks. The measured frequency stability in fractional units represented by the Allan standard deviation is presented in Fig. 5    The stability of the Sr1 was also compared with the stability of the UTC(AOS) maintained by the hydrogen maser in AOS in Borowiec. The comparison was made over the dedicated 330 km-long stabilized fibre optic link ( blue line in Fig. 5). This measured frequency stability provides information about the overall statistical uncertainty of the reference frequency of the hydrogen maser, the stability of the fibre link, and of optical frequency comb. For example, the plateau at around 2000 s corresponds to the ± 1 °C fluctuations of temperature in the room where part of the frequency chain at the AOS in Borowiec (micro-stepper and frequency distribution amplifiers) is placed.

Accuracy budget.
We have evaluated the main contributions to the frequency shifts in both Sr1 and Sr2 standards. The accuracy budgets are compared in Table 1 for typical experimental conditions: applied B-fields inducing clock transition equal to 2.725 and 2.383 mT, clock laser intensities equal to 207 and 488 mW/cm 2 and resultant Rabi frequencies 24 equal to 7.4 and 9.7 Hz for Sr1 and Sr2, respectively. Most of the systematic contributions presented there were evaluated by making a series of several simultaneous (interlaced) locks to the atomic line with different values of particular physical parameter in one of the standard, with the other standard serving as a stable reference. Examples of such evaluations are presented in Fig. 6. The notable exceptions were the blackbody radiation (BBR) shift, gravitational red shift and post-processed corrections between UTC(AOS), UTC and TT (the SI second on the geoid).
The frequency stability (within 1 MHz) of the lattice 813 nm light was assured by narrowing and locking the lattice laser (pre-tuned to the magic wavelength 368 554.58 (28)  The shift induced by the BBR can be described as a static shift with a small dynamic correction 26 . The static contribution is proportional to the differential static polarisability of the two clock states 27,28 and the mean square value of the electric field at temperature T. The dynamic correction is calculated similarly as in ref. 27. Two transitions to the states s p 5 5 3 P 1 and s p 5 5 1 P 1 contribute to the dynamic shift of the s 5 2 1 S 0 ground state and four transitions to the states s d 5 4 3 D 1 , s s 5 6 3 S 1 , p 5 2 3 D 1 and s d 5 5 3 D 1 contribute to the dynamic shift of the s p 5 5 3 P 0 excited state (see ref. 28,29). The relevant parameters of the transitions are taken from ref. 28,30. The temperature of crucial points of the vacuum system is monitored during the experiment cycle by calibrated thermistors. The acquired data and an accurate model of the vacuum systems and theirs surroundings are used to simulate the temperature distribution of the system (see Fig. 7). In the simulation (based on finite-element method), the atoms are represented by a small vapour ball inside the vacuum chamber. The temperature probed by this ball is used to calculate the BBR experienced by the atoms. The uncertainty of the shift is evaluated from calculations of the BBR for the maximum and minimum temperatures measured in the experiment. The UTC(AOS) signal in AOS in Borowiec is corrected with respect to the Earth's Geoid, therefore the measurements at KL FAMO also have to be corrected with respect to the Geoid with the gravitational red shift. The local height over the Geoid, 50(2) m, and the gravimetrically measured local value of the gravitational acceleration, 9.8127208(26) m/s 2 , were used for this correction.  The last evaluated uncertainty represents the finite resolution of the direct digital synthesizers (DDSs) driving the AOMs in the frequency chain of the clock lasers.

Discussion
In Fig. 9 we present comparison of the 88 Sr 1 S 0 − 3 P 0 transition frequency with the previously known values. The only direct measurement with 88 Sr we found in the literature has the uncertainty ten times bigger than the values reported in this paper 8 . The most precise value of the transition frequency was evaluated based on the measurement of the isotope shift 88 Sr-87 Sr in ref. 9 and from the frequency of the clock transition in 87 Sr 4-6 . Dashed horizontal band in Fig. 9 represents the value recommended by  (top and bottom panels, respectively). In the left panels each solid circle represent 100 s of averaged data, the light and dark-green regions represent 1 σ standard deviation and standard deviation of the mean, respectively. The offset frequency ν BIPM is the BIPM recommended frequency value 10 . The right panels show a histogram of the frequency measurements with fitted Gaussian function. the BIPM 10 . We believe that better control of the magnetic field would enable measurement of the 88 Sr 1 S 0 − 3 P 0 transition frequency with accuracy at least order of magnitude better and recommendation of this transition as SRS.

Conclusion
We have performed a series of measurements of the absolute frequency of the 1 S 0 − 3 P 0 transition in neutral 88 Sr. The measurements has been made in two independent optical lattice clocks with an optical frequency comb referenced to the UTC(AOS) by a long distance stabilized fibre optic link. Our results have comparable accuracy to those indirectly derived in ref. 9 and one order of magnitude better accuracy than value measured directly and reported in ref. 8. Presented results agree with the recommendation of Bureau International des Poids et Mesures and should improve the accuracy of future recommendation. In conclusion, 1 S 0 − 3 P 0 transition in the bosonic strontium seems to be a good candidate for practical representation of the second with stability of the order of 10 −17 , particularly for transportable systems.