# Intercontinental comparison of optical atomic clocks through very long baseline interferometry

Marco Pizzocaro, Mamoru Sekido, Kazuhiro Takefuji, Hideki Ujihara, Hidekazu Hachisu, Nils Nemitz, Masanori Tsutsumi, Tetsuro Kondo, Eiji Kawai, Ryuichi Ichikawa, et al.

► To cite this version:

Marco Pizzocaro, Mamoru Sekido, Kazuhiro Takefuji, Hideki Ujihara, Hidekazu Hachisu, et al. In-tercontinental comparison of optical atomic clocks through very long baseline interferometry. Nature Physics, 2021, 17, pp.223-227. 10.1038/s41567-020-01038-6. insu-03713043

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# Intercontinental comparison of optical atomic clocks through very long baseline interferometry

Marco Pizzocaro<sup>1,2</sup>, Mamoru Sekido<sup>2,3</sup>, Kazuhiro Takefuji<sup>2,8</sup>, Hideki Ujihara<sup>1,2</sup>, Hidekazu Hachisu<sup>3</sup>, Nils Nemitz<sup>3</sup>, Masanori Tsutsumi<sup>3</sup>, Tetsuro Kondo<sup>2,4</sup>, Eiji Kawai<sup>2</sup>, Ryuichi Ichikawa<sup>3</sup>, Kunitaka Namba<sup>3</sup>, Yoshihiro Okamoto<sup>3</sup>, Rumi Takahashi<sup>3</sup>, Junichi Komuro<sup>3</sup>, Cecilia Clivati<sup>1</sup>, Filippo Bregolin<sup>1,9</sup>, Piero Barbieri<sup>1</sup>, Alberto Mura<sup>1</sup>, Elena Cantoni<sup>1</sup>, Giancarlo Cerretto<sup>1</sup>, Filippo Levi<sup>1</sup>, Giuseppe Maccaferri<sup>5</sup>, Mauro Roma<sup>5</sup>, Claudio Bortolotti<sup>5</sup>, Monia Negusini<sup>5</sup>, Roberto Ricci<sup>1,5</sup>, Giampaolo Zacchiroli<sup>5</sup>, Juri Roda<sup>5</sup>, Julia Leute<sup>6,7</sup>, Gérard Petit<sup>6</sup>, Federico Perini<sup>5</sup>, Davide Calonico<sup>1</sup> and Tetsuya Ido<sup>1,3</sup>

The comparison of distant atomic clocks is foundational to international timekeeping, global positioning and tests of fundamental physics. Optical-fibre links allow the most precise optical clocks to be compared, without degradation, over intracontinental distances up to thousands of kilometres, but intercontinental comparisons remain limited by the performance of satellite transfer techniques. Here we show that very long baseline interferometry (VLBI), although originally developed for radio astronomy and geodesy, can overcome this limit and compare remote clocks through the observation of extragalactic radio sources. We developed dedicated transportable VLBI stations that use broadband detection and demonstrate the comparison of two optical clocks located in Italy and Japan separated by 9,000 km. This system demonstrates performance beyond satellite techniques and can pave the way for future long-term stable international clock comparisons.

Atomic clocks based on optical transitions can reach fractional uncertainties at the $10^{-18}$ level$^{1-3}$, surpassing by two orders of magnitude the microwave clocks$^4$ that realize the definition of the second in the International System of Units (SI) and are the basis of international timekeeping$^5$. Given the rapid progress in optical clocks, a redefinition of the SI second as the unit of time is anticipated$^6$. Remote comparison of different clocks worldwide is fundamental to check their consistency in view of such a redefinition. It is also attractive for challenging experiments such as tests of special and general relativity$^{7-9}$, laboratory searches for the variations of fundamental constants$^{10,11}$ and measurements of Earth's gravitational potential$^{12-14}$. Future applications of clock comparisons include the search for dark matter$^{15}$, the establishment of quantum networks for secure communications and timing$^{16}$ and gravitational wave detection$^{17}$. Optical-fibre links have been shown to allow comparisons at the full accuracy of the clocks up to thousands of kilometres$^{18}$. However, they have not reached a transoceanic scale$^{19}$ and therefore intercontinental comparisons are only possible by microwave satellite transfer techniques$^{20}$. Among these are two-way time-and-frequency transfer, which makes use of geostationary television communication satellites$^{21,22}$, or precise point positioning$^{23}$, which relies on the constellation of the Global Positioning System (GPS). These techniques achieve a typical uncertainty at the level of $10^{-15}$ at 1 day of averaging time that limits the comparison of optical clocks.

Very long baseline interferometry (VLBI) is a technique developed for radio astronomy and geodesy$^{24,25}$ that is based on the

simultaneous observation of remote radio sources with distant antennas, each referenced to a local atomic clock. Since its early development, VLBI has been considered for synchronization of the reference clocks$^{26,27}$ but it has never been exploited for this task due to the convenient use of satellites. Recently, VLBI has been proposed to improve intercontinental frequency links$^{28}$ and has been suggested for the comparison of optical clocks$^{29}$. Here we demonstrate a VLBI system dedicated to frequency transfer with broadband detection that allows for better performance than satellite techniques.

VLBI cross-correlates the signals arriving from astronomical radio sources to different antenna locations. The geometrical delay between two stations, $\tau_{geom} = B \cdot S/c$, depends on the baseline between the antennas B and the unit vector to the radio source S, with c being the speed of light. The observed value $\tau_{obs}$ differs from the geometrical delay because of the timing offset between the reference clocks $\Delta\tau_{clock}$ and it is additionally affected by the station-to-station difference in excess delays introduced by the atmosphere ($\Delta\tau_{atm}$), ionosphere ($\Delta\tau_{ion}$), instruments such as antennas, cables and receivers ($\Delta\tau_{instr}$) and ultimately the radio source structure ($\Delta\tau_{source}$):

$$ \tau_{obs} = \tau_{geom} + \Delta\tau_{atm} + \Delta\tau_{ion} + \Delta\tau_{instr} + \Delta\tau_{source} + \Delta\tau_{clock} \quad (1) $$

The different components of the delay can be modelled, estimated from the observations or obtained as a result of separate measurements depending on the application. Radio astronomy is interested in studying radio source structure, while geodetic VLBI seeks to

<footnote n="1">¹Istituto Nazionale di Ricerca Metrologica (INRIM), Torino, Italy. ²National Institute of Information and Communications Technology (NICT), Kashima, Japan. ³National Institute of Information and Communications Technology (NICT), Koganei, Tokyo, Japan. ⁴Chinese Academy of Sciences, Shanghai Astronomical Observatory, Shanghai, China. ⁵Istituto Nazionale di Astrofisica (INAF), Istituto di Radioastronomia (IRA), Bologna, Italy. ⁶Bureau International des Poids et Mesures (BIPM), Sèvres, France. ⁷LNE-SYRTE, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Paris, France. ⁸Present address: Japan Aerospace Exploration Agency (JAXA), Nagano, Japan. ⁹Present address: Toptica Photonics AG, Gräfelfing, Germany. ¹⁰e-mail: m.pizzocaro@inrim.it; sekido@nict.go.jp</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 1** | Schematic of the frequency links between the optical clocks. a, In Torino, the IT-Yb1 optical lattice clock at INRIM is compared with the local hydrogen maser and the link laser using an optical frequency comb. The link laser is sent over a compensated fibre link to Medicina. b, In Medicina, at INAF Radio Observatory, the incoming link laser is compared with the local hydrogen maser using another optical frequency comb. The first transportable broadband VLBI station is set up in Medicina and referenced to this maser. c, In Koganei, the second VLBI station at NICT headquarters is referenced to a local hydrogen maser. The NICT-Sr1 optical lattice clock is compared to the local hydrogen maser using a third optical frequency comb, ending the chain. d, The 34-m-diameter radio telescope maintained by NICT in Kashima assisted in the node-hub-style observations, increasing the signal-to-noise ratio. The observational data in Medicina and Koganei were transferred to the correlation centre in Kashima over high-speed internet networks (blue arrows). The cross-correlation between the antennas and the following geodetic analysis allows measurement of the reference clock difference. e, The VLBI link is established by observing several radio sources in the ICRF. f, A satellite frequency link with GPS satellites is established between the receivers in Torino and Koganei.</figcaption></figure>

minimize its effects by selecting point-like sources in the inter-
est of determining precise station coordinates and monitoring
variations in Earth rotation. The International Celestial Reference
Frame (ICRF), as adopted by the International Astronomical Union
(IAU)³⁰, provides a list of suitable extragalactic radio sources that are
regarded as fiducial points in the sky. Calibration of excess atmo-
spheric delay is made by an atmospheric model based on weather
data³¹, which is represented by a zenith delay and a mapping func-
tion for observations at different elevation angles generated by
ray tracing techniques. The ionospheric delay has a characteris-
tic frequency dependence that allows its calibration directly from
observational data. VLBI measurements typically use local active
hydrogen masers as stable frequency references. When all effects
are considered, it is possible for VLBI measurements to provide the
frequency difference between the hydrogen masers and thus com-
pare any clocks or timescales locally linked to them.

We designed and built two transportable VLBI stations with
2.4-m-diameter Cassegrain antennas. These stations implement
the concept of broadband VLBI observation proposed as the VLBI

Global Observing System (VGOS) by the International VLBI
Service for Geodesy and Astrometry (IVS)³² to reach 1-mm posi-
tion accuracy in geodetic measurements. Whereas standard geo-
detic VLBI only makes observations in narrow frequency ranges
in the S band (2.2–2.4 GHz) and X band (8.2–8.95 GHz)²⁵, the sta-
tions in this work can access frequencies from 3 GHz to 14 GHz.
Our receivers first acquire data with a sampling rate of 16 GHz and
then digital filters extract the signals in four 1-GHz-wide channels
spread over the wide frequency range. Integration of the four chan-
nels determines precise group delays using the bandwidth synthesis
technique³³. We employ a 2,048-MHz sampling rate in each chan-
nel, whereas conventional VLBI uses 32 MHz or below, enhanc-
ing the signal-to-noise ratio by a factor of eight. The improved
signal-to-noise ratio allows for a smaller overall dish area of the
antennas. By itself, the enhanced detection bandwidth is not suf-
ficient for measurements over an intercontinental baseline with
easily transportable dish sizes. This is overcome by including the
34-m radio antenna operated by NICT in Kashima (Japan) in a
joint node-hub-style observation: the delay observable between
<table><caption>**Table 1 | Summary of the VLBI sessions used for frequency comparison of the optical clocks**</caption><thead><tr><th>Date (dd/mm/yyyy)</th><th>Duration (h)</th><th>Scans</th><th>Sources</th><th>Delay residual (ps)</th><th>Rel. freq. uncertainty</th></tr></thead><tbody><tr><td>14/10/2018</td><td>29</td><td>448</td><td>22</td><td>32</td><td>3.9 × 10<sup>-16</sup></td></tr><tr><td>04/11/2018</td><td>31</td><td>463</td><td>16</td><td>39</td><td>2.6 × 10<sup>-16</sup></td></tr><tr><td>14/11/2018</td><td>29</td><td>464</td><td>24</td><td>24</td><td>2.3 × 10<sup>-16</sup></td></tr><tr><td>24/11/2018</td><td>29</td><td>445</td><td>23</td><td>29</td><td>2.5 × 10<sup>-16</sup></td></tr><tr><td>04/12/2018</td><td>29</td><td>471</td><td>23</td><td>33</td><td>2.2 × 10<sup>-16</sup></td></tr><tr><td>15/12/2018</td><td>29</td><td>457</td><td>22</td><td>26</td><td>1.6 × 10<sup>-16</sup></td></tr><tr><td>25/12/2018</td><td>29</td><td>476</td><td>24</td><td>22</td><td>2.0 × 10<sup>-16</sup></td></tr><tr><td>15/01/2019</td><td>29</td><td>479</td><td>25</td><td>24</td><td>1.5 × 10<sup>-16</sup></td></tr><tr><td>25/01/2019</td><td>28</td><td>506</td><td>23</td><td>26</td><td>1.8 × 10<sup>-16</sup></td></tr><tr><td>14/02/2019</td><td>36</td><td>786</td><td>21</td><td>29</td><td>1.9 × 10<sup>-16</sup></td></tr></tbody></table>

The delay residuals are the root-mean-square of the residuals of the fit using the node-hub-style data. The relative frequency uncertainty of the transfer is derived from the fit and corresponds to 1s.d.

the transportable stations (nodes) can be calculated as the difference of the two delays with the large antenna (hub) after applying a small correction factor (Methods). This measurement benefits from the improvement of the equivalent dish area, which scales as the geometrical mean, without the penalties specific to large antennas because the delay variations due to gravitational distortion of the large dish or temperature dependence in long cables are largely reduced as a common-mode noise.

Here, one node was installed at NICT headquarters in Koganei (Japan) while the other was transported to the Radio Astronomical Observatory operated by INAF in Medicina (Italy), forming an intercontinental baseline of 8,700 km (Fig. 1). Observational data at Medicina and Koganei were stored on hard-disk drives at each station and transferred over high-speed Internet networks to the correlation centre in Kashima for analysis. Ten frequency measurements were performed via VLBI between October 2018 and February 2019, and from these we calculated the frequency difference between the reference clocks at the two stations: the local hydrogen masers in Medicina and Koganei. Each session lasted from 28 h to 36 h and included at least 400 scans observing between 16 and 25 radio sources in the ICRF list. A least-squares fit of each run was used to calculate the observed delay, with typical residuals of 30 ps and a formal uncertainty in the frequency comparison as low as $1.5 \times 10^{-16}$ for a single run (Table 1). At this level of uncertainty, accurate modelling of atmospheric delays is a crucial part of the analysis (Methods).

We used this VLBI frequency link to compare two optical clocks: the ytterbium-171 optical lattice clock$^{34}$ IT-Yb1 at INRIM in Torino (Italy) and the strontium-87 optical lattice clock$^{35,36}$ NICT-Sr1 in Koganei. Both optical clocks are realizations of the unit of time, formally recognized within the Metre Convention. This is possible because the frequency values of ytterbium and strontium clock transitions are recommended as secondary representations of the second in the list of standard reference frequency values$^{6,37}$ of the International Committee for Weights and Measures (CIPM), with frequencies $f(\text{Yb}) \approx 518$ THz and $f(\text{Sr}) \approx 429$ THz and relative uncertainties of $5 \times 10^{-16}$ and $4 \times 10^{-16}$, respectively. Therefore, the frequency ratio between IT-Yb1 and NICT-Sr1 can be compared to the officially recommended values in addition to previous measurements$^{13,22,38,39}$.

A frequency comparison between IT-Yb1 and NICT-Sr1 followed the chain shown in Fig. 1. In Torino, the ytterbium clock was

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

**Fig. 2 | Comparison of Yb/Sr frequency ratio measurements.** a, Ratio measurements for each VLBI run at a given modified Julian date (MJD) measured via VLBI or precise point positioning with integer ambiguity resolution (IPPP). Red bars represent the statistical uncertainty of the VLBI link only, while blue bars represent the total uncertainty for the VLBI comparisons (including optical clocks, optical combs and extrapolations). The blue shaded region is the weighted mean of the VLBI comparison with its total uncertainty. Green points represent the IPPP comparisons with their total uncertainty. The green shaded region is the weighted mean of the IPPP comparison with its total uncertainty. The horizontal offset between the two datasets is for clarity. b, Optical frequency ratio $f(\text{Yb})/f(\text{Sr})$ as measured in this work (red/blue and green points as in a) or in previous measurements, where purple squares represent local measurements at the RIKEN centre$^{38,39}$ and by PTB and INRIM$^{19}$, and the purple diamond denotes a remote measurement performed by NICT and KRISS$^{22}$. All uncertainties correspond to 1s.d.

compared to a local hydrogen maser using an optical frequency comb$^{40}$, which allows precise optical-to-microwave frequency comparisons. The comb in Torino also measured a narrow-linewidth laser at telecommunication wavelengths that was sent on a 535-km compensated optical-fibre link$^{41}$ to the Radio Observatory in Medicina. Here, another optical frequency comb compared the laser fibre-delivered from Torino with the hydrogen maser referencing the VLBI station. In Koganei, a third optical frequency comb was used for comparison of the strontium clock with local hydrogen masers, including the maser used as reference for the VLBI station.

Because the optical clocks, optical link and VLBI link operated intermittently throughout this campaign we extrapolate the hydrogen maser frequencies$^{42,43}$ to calculate a frequency ratio for each VLBI run. Figure 2a shows the results of the ratio
<table><caption>Table 2 | Uncertainty budgets for the ratio and closure measurements</caption><thead><tr><th colspan="2">Yb/Sr ratio via VLBI</th><th colspan="2">Yb/Sr ratio via IPPP</th><th colspan="2">Closure</th></tr><tr><th>Contribution</th><th>Uncertainty (×10<sup>-16</sup>)</th><th>Contribution</th><th>Uncertainty (×10<sup>-16</sup>)</th><th>Contribution</th><th>Uncertainty (×10<sup>-16</sup>)</th></tr></thead><tbody><tr><td>VLBI</td><td>0.9</td><td>IPPP</td><td>2.6</td><td>VLBI</td><td>0.7</td></tr><tr><td>Clocks</td><td>0.9</td><td>Clocks</td><td>0.9</td><td>IPPP</td><td>2.3</td></tr><tr><td>Combs (microwave/optical)</td><td>1.1</td><td>Combs (microwave/optical)</td><td>1.1</td><td>Combs (microwave/optical)</td><td>1.1</td></tr><tr><td>Extrapolation</td><td>2.2</td><td>Extrapolation</td><td>2.6</td><td>Extrapolation</td><td>1.9</td></tr><tr><td>Optical link</td><td>&lt;0.1</td><td></td><td></td><td>Optical link</td><td>&lt;0.1</td></tr><tr><td>Total</td><td>2.8</td><td>Total</td><td>4.0</td><td>Total</td><td>3.2</td></tr></tbody></table>

The listed uncertainties are calculated for the weighted average of the points in Fig. 2a and the weighted average of their difference. The VLBI and IPPP uncertainties are the statistical contribution for the frequency transfer. The clock contributions consist of the systematic uncertainties of both clocks, including the gravitational redshift. The extrapolation uncertainty is a statistical uncertainty due to the intermittent operations of the optical clocks and the optical link. The uncertainty for the combs arises from their accuracy and from the instability in the microwave-to-optical conversions. The optical link contribution is negligible for the present measurement. All uncertainties correspond to 1s.d.

measurements, with uncertainties evaluated for each step of the frequency chain including the extrapolations (Methods). The weighted mean for the campaign is the frequency ratio $R=f(Yb)/f(Sr)=1.20$ 750703934333805(34), measured via VLBI. This corresponds to a fractional deviation from the ratio $R_0$ of the recommended frequencies for ytterbium and strontium of $y(Yb/Sr)=R/R_0-1=2.5(2.8) \times 10^{-16}$, in agreement with previous measurements (Fig. 2b). Over the course of the campaign, the overall VLBI measurement contributes a 1-s.d. uncertainty of only $9 \times 10^{-17}$ for a total measurement time of 300 h. The optical link, conversion to microwave signals, maser frequency extrapolation and systematic effects in the optical clocks contribute to the overall uncertainty of $2.8 \times 10^{-16}$ reported in Table 2. The gravitational redshift induced by the different gravitational potential at the position of the optical clocks$^{34,35}$ has been accounted for in the clock systematic shifts, bringing a relative correction of $-1.765(2) \times 10^{-14}$ to the frequency ratio.

To confirm the results, we established an independent frequency link using GPS satellites between receivers at INRIM and at NICT (Fig. 1). To achieve low uncertainty on this link, we calculated a precise point positioning solution with integer ambiguity resolution (IPPP)$^{44}$. For this baseline, IPPP has an uncertainty of $9 \times 10^{-16}$ at 1 day of averaging time. A comparison of the ytterbium and strontium optical clocks using the IPPP solution over the same time intervals as the VLBI runs is also shown in Fig. 2a. The weighted mean of the frequency deviation of the ratio is $y(Yb/Sr) = -3.2(4.0) \times 10^{-16}$ measured via IPPP. As shown in Table 2, the GPS link contributes an overall uncertainty for the campaign of $2.6 \times 10^{-16}$. We can also compare the VLBI and IPPP results directly, closing the loop via the optical link, and the weighted mean of the differences for the entire campaign is $y(VLBI - IPPP) = 3.5(3.2) \times 10^{-16}$.

By leveraging broadband observations, our VLBI system surpasses the best satellite techniques and what is possible with traditional geodetic VLBI, resulting in a typical formal uncertainty of $2 \times 10^{-16}$ at $10^5$ s of averaging time. It has the potential to reach uncertainty in the low $10^{-17}$ region at 10 days of averaging time provided variations in instrumentation delay are minimized, radio sources are carefully selected to minimize structure effects and the atmosphere is accurately modelled. Although VLBI still requires optical-to-microwave conversions, this is an attractive technique for optical clocks comparisons. With increased operational duty times of the optical clocks and improved characterization of the combs$^{45}$ and of the frequency chains, it will allow for link uncertainties that approach the systematic uncertainties of optical clocks for manageable measurement times.

Our measurements show agreement between the VLBI link, the optical clocks and the GPS link at the low $10^{-16}$ level. There is still some uncertainty on the optimal treatment for the

calibration of the tropospheric delay in a VLBI experiment with a long single baseline, and further investigation should clarify the correlation with the estimated clock frequency. We have estimated that thermal sensitivity of the cables and of the antennas contributed an uncertainty of less than $1 \times 10^{-17}$. The microwave signal acquired by the antenna is transferred to the data acquisition system by amplitude modulation of an infrared laser over fibre. In Medicina, a 600-m-long underground fibre is used, with frequency shifts of less than $8 \times 10^{-17}$ in a day. In our experiment, we observed excess delay residuals for 3 of the 35 observed sources caused by the source structure effect$^{46,47}$. The increased residuals are accounted for in the fit procedure, but a careful selection of radio sources will help to avoid systematic effects in future clock comparisons. On the other hand, transportable broadband antennas may play a significant role in increasing the density and extending the baselines in large networks of antennas, with the aim to improve investigations of the source with structure.

The endpoint VLBI stations are transportable and can be deployed close to time-and-frequency laboratories, just like traditional satellite antennas, supported by the node-hub topology. They can be operated without the need for a radio transmission licence and, with a higher data transmission capability, they may provide a continuous link instead of the separate sessions used in our measurements. Where laboratories lack the facilities or sky coverage to house a VLBI station, they can be connected by local optical-fibre links, as demonstrated here. It is envisioned that optical clock comparisons will increasingly rely on local area fibre networks connected by non-fibre intercontinental links, with applications in precision timing, searches for new physics and clock-based geodesy$^{48}$. Proposals for improved intercontinental links includes IPPP$^{44}$, carrier-phase two-way frequency transfer$^{49}$ and the use of transportable optical clocks$^{13}$ or dedicated orbital systems such as the planned Atomic Clock Ensemble in Space (ACES) mission$^{50}$. Our experiment realizes such a hybrid network with a high-performance VLBI link, which is useful for primary metrology and fundamental physics.

## Online content

Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41567-020-01038-6.

Received: 16 March 2020; Accepted: 10 August 2020; Published online: 5 October 2020
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Methods

**Broadband VLBI observation system.** In VLBI observations, the standard deviation of the observed group delay is given by

$$\sigma_{\tau} = \frac{1}{2\pi R_{\text{sn}} \Delta f_{\text{rms}}} \qquad (2)$$

where $\Delta f_{\text{rms}}$ is the root-mean-square deviation of the observation frequencies $f_i$, while the signal-to-noise ratio, $R_{\text{sn}}$, is given by

$$R_{\text{sn}} = \eta \sqrt{\frac{A_1 A_2 B t}{T_1 T_2}} \frac{S}{2k_B} \qquad (3)$$

where $A_1$ are the effective areas of the two antennas, $T_1$ are the system noise temperatures, $B$ is the observation bandwidth, $t$ is the integration time, $S$ is the correlated flux density of the radio source, $k_B$ is Boltzmann's constant and $\eta$ is a digital processing loss factor$^{1}$. Delay observations can thus be improved by increasing the spread of observation frequency $\Delta f_{\text{rms}}$ and the bandwidth $B$ of the observations.

The VLBI observation system used in this experiment, GALA-V$^{52,53}$, implements data acquisition according to the VGOS standard, which aims at making broadband observations spread over the frequency range 2–14 GHz. NICT has developed a unique broadband feed$^{54}$ named NINJA, with constant narrow beam size for a wide frequency range, which enables broadband observation with existing Cassegrain antennas. The NINJA broadband feeds were installed in the two transportable 2.4-m-diameter antennas and the Kashima 34-m-diameter antenna. The transportable antennas acquire a single linear (V) polarization signal. To avoid a loss of correlation from the different polarizations observed by distant stations due to parallactic angle, the hub station in Kashima acquires both (V and H) polarizations. The VH and VV datasets are then processed coherently.

The data acquisition we adopted uses four channels with $B=1,024$ MHz bandwidth at frequencies $f_1=6.0$ GHz, $f_2=8.5$ GHz, $f_3=10.4$ GHz and $f_4=13.3$ GHz. This system is simpler than the prototype VGOS proof-of-concept implementation$^{15}$, which acquires data on more than 100 channels with 32-MHz frequency bandwidth. Our array of selected frequencies provides a sharply peaked delay resolution function (Extended Data Fig. 1), and the large bandwidth of each channel allows the unambiguous determination of the absolute group delay using the technique of bandwidth synthesis$^{13}$. Because the group delay has a linear phase slope over frequency, a stable phase relation between adjacent channels is essential and it is assured by a fully digital signal processing. High-speed radiofrequency direct sampling$^{55}$ is used to digitize the signal without any analogue frequency conversions: the received signal is amplified and then first converted to digital data with a 16,384-MHz sampling rate via the high-speed sampler K6/GALAS$^{53}$. A digital filter implemented on a field programmable gate array (FPGA) extracts the four signal channels with 1-bit sampling at 2,048 megasamples per second for each polarization. The observational data are stored on hard-disk drives at each station, and then transferred to the station at Kashima over high-speed research networks for correlation processing.

**Node-hub-style VLBI observations.** Transportable VLBI systems have previously been investigated for geodetic VLBI$^{56,57}$, although their sensitivity is limited by the physical size of the collecting area.

Our broadband observation system allows for smaller antennas by increasing the observation bandwidth (equation (3)). Yet, even in this case, the transportable antenna pair does not have enough sensitivity for the reduced correlated flux density on an intercontinental baseline. This is overcome by combining the small transportable antennas (nodes) with a high-sensitivity antenna (hub). The baseline between the smaller stations is then deduced by a closure delay relation$^{58}$. For two node stations A and B and a hub station R making an observation of a point-like radio source (Extended Data Fig. 2), the time delay differences at time $t$ for the arrival of an identical wave are related by the linear combination

$$\tau_{AB}(t) = \tau_{RB}(t) - \tau_{RA}(t) - \tilde{\tau}_{AB}(t) \cdot \tau_{RA}(t) \qquad (4)$$

Higher-order terms in this relation are negligible for any terrestrial baselines when the radio source structure effect is negligible. The uncertainty of the delay $\tau_{AB}$ calculated in this way is still dominated by the contributions of instruments, atmosphere and source structure at the nodes as if the delay had been measured directly. In obtaining the delay observable of the AB baseline, the delay properties of station R such as atmospheric delay, cable length change and gravitational deformation of the dish are cancelled out. In our case, with both the hub and a transportable station separated by only 110 km, equation (4) is still valid even in the presence of radio source structure, because the short baseline is insensitive while only the effect on the baseline AB is observed.

**VLBI signal processing and analysis.** The data reduction procedure for the VLBI observation consists of three steps for each of the node-hub baselines: first, the individual VH and VV polarization pairs for each frequency band are cross-correlated, and time integration is made using the software correlator GICO3$^{59}$ running on a cluster of computers. Second, synthetic cross-correlation

data are generated by combining the VH and VV data such that correlation amplitudes are maximized for each scan and frequency band. Third, wideband bandwidth synthesis$^{53}$ is applied to derive wideband group delay observables. Here we account for phase shifts of the propagating electromagnetic signal introduced by the ionospheric dispersive medium in the form of $\phi=A\delta_{\text{TEC}}/f$, where $\delta_{\text{TEC}}$ is the difference in the total electron content (TEC) in the line of sight from each observation station, $f$ is the radiofrequency of the observations and $A$ is a constant given by $e^2/(4\pi c\epsilon_0 m)$, with $\epsilon_0$ being the vacuum electric permittivity and $m$ the mass of the electron. The wideband bandwidth synthesis procedure calibrates ionospheric delay by simultaneously estimating $\delta_{\text{TEC}}$ and the broadband group delay. Group delay data for all scans and baselines is stored in a Mark3 VLBI database. The virtual delays on the baseline between transportable antennas are analysed by a least-squares fit using the Calc/Solve software$^{60}$. The parameters of the fit are the coordinates of one station, the atmospheric delay parameters for both stations as piecewise linear functions with 60-min intervals and a single clock offset and clock rate.

The observed radio sources were selected from the recently proposed ICFR3 list$^{61}$, which provides accurately known coordinates that predict geometrical delays. The a priori delay calculation includes relativistic coordinate transformation from the solar barycentric frame to a geocentric frame co-moving with the Earth, accounting also for the gravitational time delay$^{62}$. The excess path caused by a dry atmosphere can be accurately predicted from the air pressure on the ground and the height and latitude of the station$^{63}$. However, the distribution of water vapour is hard to predict from ground weather data, and the anisotropy of the atmosphere is difficult to estimate directly from the VLBI data because the east-west long baseline results in a limited sky coverage. Thus, we use the Vienna Mapping Functions 3 (VMF3)$^{64}$ for a priori delay calibration. The VMF3 are computed by ray tracing within a numerical weather model of the European Centre for Medium-Range Weather Forecasts (ECMWF). They include dry and wet components, as well as gradients, in the model data. Their data are publicly available and regularly computed for most space geodetic observation sites, including Kashima, Koganei and Medicina. Residual atmospheric contributions were estimated in the least-squares analysis of the VLBI data.

**Uncertainty of the optical clocks, optical link and combs.** The operation of IT-Yb1 and NICT-Sr1 is similar and presented in detail in refs. $^{34,35,42,65}$. In both systems, an ultrastable clock laser is stabilized to the clock transition of atoms cooled to microkelvin temperatures and trapped in an optical lattice at the magic wavelength$^{16}$ (759 nm for ytterbium and 813 nm for strontium). Frequency combs measure the clock laser relative to the local hydrogen masers.

For this campaign, the systematic frequency shifts of the clocks have been characterized with an uncertainty of $3.1 \times 10^{-17}$ for IT-Yb1 and $8 \times 10^{-17}$ for NICT-Sr1. The lattice light shift, density shift and static Stark shift are evaluated by interleaving measurements in different conditions. The blackbody radiation shift is calculated from the temperature of the physics package of each clock using known atomic parameters$^{67,68}$, where the uncertainty is dominated by temperature inhomogeneity. The lattice hyperpolarizability and multipolar shift$^{69-71}$, the quadratic Zeeman shift$^{72}$, the background gas shift$^{73}$ and the probe light shift$^{74}$ are calculated from known atomic coefficients.

At both laboratories, we determined the gravitational redshift with respect to the conventionally adopted gravity potential$^{75}$, $W_0=62,636,856.0\,\text{m}^2\,\text{s}^{-2}$. At INRIM, the redshift is calculated from a gravitational potential difference measured with a global navigation satellite system/geoid approach from the geoid model of Europe EGG2015$^{76}$, with an uncertainty of $3 \times 10^{-18}$ corresponding to ~3 cm in height. At NICT, the redshift is calculated from the geoid model of Japan GSIGEO2011$^{77}$, with an uncertainty of $2 \times 10^{-17}$, which includes oscillations from tidal effects whose 10-day mean is less than 15 cm in height.

The operation of the combs in Torino and Medicina and of the connecting optical link was monitored by redundantly tracking beatnotes to detect and remove cycle slips in the phase-locked loops. For this task, acquisition in Torino and Medicina was synchronized using the Network Time Protocol at better than 100 ms. The average number of cycle slips during the up time was less than one per hour. The fibre-delivered laser is loosely phase locked to the hydrogen maser in Torino using the comb to avoid frequency drifts. In these conditions, the optical link contributes less than $1 \times 10^{-18}$ after 1,000 s of averaging time$^{41,78}$.

For this measurement, the optical-to-microwave comparison at the combs in Torino, Medicina and Koganei is evaluated at an uncertainty of $8 \times 10^{-17}$ each, while optical-to-optical comparisons are better characterized$^{79}$ at less than $1 \times 10^{-18}$. We also conservatively estimated that each optical-to-microwave comparison at the comb introduces phase noise with a corresponding instability of $2 \times 10^{-14}$ at 1 s of averaging time. The instabilities of the optical clocks are negligible compared to those of the maser frequency measurements, at $\sim 2 \times 10^{-15}$ at 1 s for the ytterbium clock and $\sim 7 \times 10^{-15}$ at 1 s for the strontium clock.

**Uncertainty in the frequency chain.** The calculation of the frequency chain for the VLBI measurement of optical clocks follows the measurement model

$$\frac{f(\text{Yb})}{f(\text{Sr})} = \frac{f(\text{Yb}) f(\text{H}_T) f(\text{H}_M) f(\text{H}_K)}{f(\text{H}_T) f(\text{H}_M) f(\text{H}_K) f(\text{Sr})} \qquad (5)$$
where $f(\text{Yb})/f(\text{H}_T)$ is the frequency ratio between the ytterbium optical clock and the hydrogen maser in Torino measured at the comb, $f(\text{H}_T)/f(\text{H}_M)$ is the frequency ratio between the masers in Torino and in Medicina measured using the optical-fibre link, $f(\text{H}_M)/f(\text{H}_K)$ is the frequency ratio between the masers in Medicina and in Koganei measured using the VLBI comparison, and $f(\text{H}_K)/f(\text{Sr})$ is the frequency ratio between the hydrogen maser in Koganei and the strontium optical clock measured at the comb.

In the data analysis, each ratio $r$ in equation (5) is expressed as a fractional deviation $y=r/r_0-1$, where $r_0$ is an arbitrary reference ratio. We chose the reference ratios for the clocks to be consistent with the recommended frequencies of ytterbium and strontium as secondary representations of the second$^{37}$: $f_0(\text{Yb})=518,295,836,590,863.6(3)$ Hz and $f_0(\text{Sr})=429,228,004,229,873.00(17)$ Hz. After linearization, the fractional correction of the result $y(f(\text{Yb}))/f(\text{Sr})$ can be calculated as the sum of the fractional corrections of each ratio.

The four frequency ratios in equation (5) were not measured simultaneously and need to be extrapolated to common intervals using the hydrogen masers as flywheel oscillators$^{34,35}$. After accounting for a linear drift, the dominant contribution to the extrapolation uncertainty is due to the stochastic maser behaviour$^{32,43,80}$. We model the maser noise with a power-law model$^{81}$ described by a Hadamard variance, $\sigma_{\text{H}}^2 = h_{-2}(\tau/s)^{-2} + h_{-1}(\tau/s)^{-1} + h_0 + h_2(\tau/s)^2$, where $\tau$ is the averaging time. The coefficients $h_n$ conventionally identified with white phase noise, white frequency noise, flicker frequency noise and flicker-walk frequency noise, respectively, are deduced from comparisons with the optical clocks or within the maser ensembles at INRIM and at NICT (Extended Data Fig. 3b). The $h_n$ coefficient provides the best description of the long-term behaviour observed at NICT and it predominantly affects extrapolations over datasets longer than 10 days. There are no significant changes in the overall result if using an alternative set of coefficients that sets $h_2=0$ or replaces it with a best-fit $h_1$ coefficient associated with random-walk noise. We calculated the uncertainty due to dead time by Monte Carlo methods, using an algorithm for the generation of power-law noise$^{82}$ and simulating 500 repeated averages over the time periods involved$^{33,34,35}$. The numerical calculations agree with analytic results based on the power spectral density of the modelled noise. The drift of each maser was determined from comparison with the optical clocks and extrapolations were calculated as $d\Delta t$, where $d$ is the measured drift and $\Delta t$ is the time difference of the barycentres of constituent data for the ratio under investigation. The averaging intervals were carefully selected to only include periods of well-characterised maser behaviour. We note that we chose the averaging interval for each step of the chain aiming to minimize the final uncertainty for the ratio via VLBI (Extended Data Fig. 3a).

The extrapolations have a typical uncertainty of $1 \times 10^{-15}$ for each VLBI measurement in Fig. 2a. For the last point on MJD 58529, this contribution was reduced to $2.8 \times 10^{-16}$ by operating all parts of the experiment simultaneously, with residual extrapolation uncertainties only due to down times of the optical clocks and of the optical link.

At NICT, the VLBI station and the strontium optical clock were compared to different masers in the available ensemble. Using the continuously operated Japan Standard Time System$^{83}$, the conversion between masers can be calculated with a negligible uncertainty of $2 \times 10^{-17}$ after $10^5$ s of measurement time.

The measurement models for the frequency ratio via IPPP and for the closure difference between VLBI, the optical link and IPPP employ a similar sequence of calculations, but require different extrapolations. The closure calculation does not rely on optical clock data except through the extracted maser drifts.

**Averages and statistics of the frequency comparison.** In our analysis, we consider the covariance matrix between the various measurements and calculate the weighted averages of Yb/Sr and closure ratios with the Gauss–Markov theorem or generalized least-squares fit$^{3,85}$. We assume the systematic uncertainties of each clock and each comb as fully correlated over time. In a limited number of cases, the same optical clock data contribute to the ratio evaluation of more than one VLBI run (Extended Data Fig. 3a) and we account for this correlation.

For our measurements we averaged 190 h of ytterbium clock data, 790 h of the optical link data, 300 h of the VLBI sessions and GPS link and 130 h of strontium clock data. The Birge ratios$^{86}$ for the weighted averages over 10 VLBI sessions are $\sqrt{\chi^2}/n = 1.07$ for the VLBI measurement, $\sqrt{\chi^2}/n = 0.90$ for the IPPP measurement and $\sqrt{\chi^2}/n = 1.32$ for the closure, where $\chi^2$ is the chi-squared test for each fit, all with $n=9$ degrees of freedom.

The Gauss–Markov method is used to establish optimal weights of the data points for each of these measurements. The changing allocation of weights results in a variation of overall uncertainty contributions (for example, for VLBI and IPPP in Table 2 and for the extrapolation in Extended Data Fig. 4) because the weights are calculated from the total uncertainty of each measurement.

**Yb/Sr ratio via satellite link.** The ratio between the ytterbium clock and the strontium clock with the satellite link reported in the main text only used data coinciding with VLBI runs. We can average the GPS data to calculate a Yb/Sr frequency ratio without this limitation, looking for the best overlap of ytterbium and strontium data in the 130 days of the campaign. We calculated this ratio using both a precise point positioning (PPP) solution and the IPPP solution over the entire campaign (Extended Data Fig. 4), resulting in frequency deviations

of $\gamma(\text{Yb/Sr}) = -2.4(4.0) \times 10^{-16}$ using PPP and $\gamma(\text{Yb/Sr}) = -3.1(2.5) \times 10^{-16}$ using IPPP. For these measurements we averaged 180 h of ytterbium clock data, 580 h of satellite transfer and 140 h of strontium clock data. The Birge ratios for the weighted averages of these measurements are $\sqrt{\chi^2}/n = 0.53$ for PPP and $\sqrt{\chi^2}/n = 0.29$ for IPPP, both with $n=9$ degrees of freedom. As expected, IPPP results in a better uncertainty corresponding to a frequency ratio of $f(\text{Yb})/f(\text{Sr}) = 1.207,507,039,343,337,38(30)$. The significantly longer measurement time ultimately allows this measurement to achieve the same uncertainty as the VLBI result.

## Data availability

Data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

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## Acknowledgements

Development of the broadband feed system was supported by K. Fujisawa of Yamauchi University. We acknowledge S. Hasegawa and Y. Miyauchi of NICT/Kashima Space Technology Center, F. Nakagawa and H. Ishijima of NICT headquarters and T. Ikeda of KDDI Research Laboratory for their contribution to this work. We thank A. Tabellini of IRA/INAF, A. Dondi and M. Alberghini of the Bologna Customs Agency for the management of the temporary importation of the transportable VLBI antenna from Japan to Italy. The broadband feeds were realized through a grant of Joint Development Research on 'Development of Broadband Receiver System of Kashima 34m' (2013–2015) supported by the Research Coordination Committee, the National Astronomical Observatory of Japan (NAOJ), the National Institutes of Natural Sciences (NINS). The development of the ytterbium optical clock and the optical link was supported by the European Metrology Program for Innovation and Research (EMPIR) projects 15SIB03 OC18, 15SIB05 OFTEN, 17IND14 WRITE and 18SIB05 ROCIT, by the Italian Ministry

of Education, University and Research (LIFT and Metgesp projects) and by the Horizon 2020 Marie Skłodowska-Curie Research and Innovation Staff Exchange (MSCA-RISE) project Q-SENSE (grant agreement no. 691156). The EMPIR initiative is co-funded by the European Union's Horizon 2020 research and innovation programme and the EMPIR Participating States. J.L. was supported by the French National Research Agency under the LABEX Cluster of Excellence FIRST-TF (ANR-10-LABX-48-01). Fast data transfer from Italy to Japan for quick turnaround of the experiments was realized with the support of the high-speed research networks JGN, Internet2, TransPAC, APAN, GEANT and GARR. IPPP results make use of the GINS software developed by the French Space Agency, CNES.

## Author contributions

D.C., T.I. and M.S. designed, prepared and coordinated the experiment with contributions from F.P. and F.L. M.S., K.T., H.U., M.T., J.K. and R.I. designed and built the broadband VLBI system. M.S., K.T., H.U., M.T., E.K., K.N., Y.O. and R.T. set up and operated the VLBI stations in Koganei and Kashima. M.S., K.T., M.T., H.U., K.N., Y.O. and R.T. set up and operated the VLBI station in Medicina with support from G.M., M.R., C.B., G.Z., J.R. and F.P. M.P., F.B. and P.B. evaluated and operated the ytterbium clock and the comb in Torino. C.C. and A.M. evaluated and operated the optical-fibre link and the combs in Torino and Medicina with support from M.R. and C.B. H.H., N.N. and T.I. evaluated and operated the strontium clock and the comb in Koganei. M.S., K.T. and T.K. carried out the correlation analysis of the VLBI delay data. M.S., M.N. and R.R. carried out the geodetic VLBI analysis. E.C., G.C., R.I., J.L. and G.P. carried out the data analysis of the satellite links. M.P. and M.S. performed the data analysis for frequency comparisons, with contributions from N.N., C.C., J.L. and G.P. M.P. wrote the manuscript with support from M.S., N.N. and C.C. All authors discussed the results and commented on the manuscript.

## Competing interests

The authors declare no competing interests.

## Additional information

**Extended data** is available for this paper at https://doi.org/10.1038/s41567-020-01038-6.

**Supplementary information** is available for this paper at https://doi.org/10.1038/s41567-020-01038-6.

**Correspondence and requests for materials** should be addressed to M.P. or M.S.

**Peer review information** *Nature Physics* thanks Jacob Covey, Rüdiger Haas and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

**Reprints and permissions information** is available at www.nature.com/reprints.
a

<figure><img src="image_6.png" /></figure>

b

<figure><img src="image_7.png" /></figure>

**Extended Data Fig. 1** | Delay resolution function of GALA-V and standard geodetic VLBI. **a**, Delay resolution functions for the frequency array of GALA-V described in the text and for standard geodetic VLBI (IVS-T2 session, experiment code T2126). GALA-V delay resolution function shows a single peak and fine delay resolution that allows the derivation of the absolute group delay without ambiguity. **b**, The delay resolution function formed from the frequency array of standard geodetic VLBI shows a delay ambiguity with a period equal to the reciprocal of the greatest common denominator of the frequency interval.
a

<figure><img src="image_8.png" /></figure>

b

<figure><img src="image_9.png" /></figure>

**Extended Data Fig. 2** | Node-hub style observations with transportable broadband VLBI stations. **a**, Transportable small VLBI stations A and B can be used for intercontinental baseline by using a Node-hub style VLBI scheme with the station R. Delay data of AB baseline is computed by linear combination of delay data on RA and RB baselines. **b**, Block diagram of the Node-hub measurement with the broadband VLBI system.
a

<figure><img src="image_10.png" /></figure>

b

<table><thead><tr><th>Maser</th><th>h<sub>-2</sub><sup>1/2</sup></th><th>h<sub>-1</sub><sup>1/2</sup></th><th>h<sub>0</sub><sup>1/2</sup></th><th>h<sub>2</sub><sup>1/2</sup></th></tr></thead><tbody><tr><td>H<sub>T</sub></td><td>1.5 × 10<sup>-13</sup></td><td>3.5 × 10<sup>-14</sup></td><td>5 × 10<sup>-16</sup></td><td>2.5 × 10<sup>-22</sup></td></tr><tr><td>H<sub>M</sub></td><td>1.5 × 10<sup>-13</sup></td><td>2.8 × 10<sup>-14</sup></td><td>4 × 10<sup>-16</sup></td><td>2.5 × 10<sup>-22</sup></td></tr><tr><td>H<sub>K</sub></td><td>1.1 × 10<sup>-13</sup></td><td>4.5 × 10<sup>-14</sup></td><td>8 × 10<sup>-17</sup></td><td>2.5 × 10<sup>-22</sup></td></tr></tbody></table>

**Extended Data Fig. 3** | Extrapolations over dead times using the masers as a flywheel. **a**, Uptimes of IT-Yb1 (yellow), optical link (green), VLBI link (blue) and NICT-Sr1 (red) marked as colored regions as a function of MJD. Gray horizontal bars represent evaluation intervals corresponding to each VLBI session, as marked by a vertical notch. For each evaluation interval the extrapolation was calculated for the masers at INRIM (between IT-Yb1 and the optical link), INAF (between the optical link and VLBI), and NICT (between VLBI and NICT-Sr1). **b**, Noise models used for extrapolations for the masers at INRIM, INAF, and NICT, describing the Hadamard variance $\sigma_H^2 = h_{-2}(\tau/s)^{-2} + h_{-1}(\tau/s)^{-1} + h_0 + h_2(\tau/s)^2$.
a

<figure><img src="image_11.png" /></figure>

b

<table><thead><tr><th colspan="2">Yb/Sr ratio via PPP</th><th colspan="2">Yb/Sr ratio via IPPP</th></tr><tr><th>Contribution</th><th>Uncer. (×10<sup>-16</sup>)</th><th>Contribution</th><th>Uncer. (×10<sup>-16</sup>)</th></tr></thead><tbody><tr><td>PPP satellite transfer</td><td>3.3</td><td>IPPP satellite transfer</td><td>1.4</td></tr><tr><td>Clocks</td><td>0.9</td><td>Clocks</td><td>0.9</td></tr><tr><td>Combs (microwave/optical)</td><td>1.1</td><td>Combs (microwave/optical)</td><td>1.1</td></tr><tr><td>Extrapolation</td><td>1.8</td><td>Extrapolation</td><td>1.4</td></tr><tr><td>Total</td><td>4.0</td><td>Total</td><td>2.5</td></tr></tbody></table>

**Extended Data Fig. 4** | Yb/Sr frequency ratio measurements using IPPP or PPP. **a**, Plot of the frequency ratio measurements. Green points represent the IPPP comparison with its total uncertainty. Green shaded region is the weighted mean of the IPPP comparison with its total uncertainty. Purple points represent the PPP comparison with its total uncertainty. Purple shaded region is the weighted mean of the PPP comparison with its total uncertainty. Horizontal offset between the two dataset is just for clarity. **b**, Uncertainty budget for the satellite comparisons. All uncertainties correspond to one standard deviation.