Geodesy and metrology with a transportable optical clock

The advent of novel measurement instrumentation can lead to paradigm shifts in scientific research. Optical atomic clocks, due to their unprecedented stability and uncertainty, are already being used to test physical theories and herald a revision of the International System of units (SI). However, to unlock their potential for cross-disciplinary applications such as relativistic geodesy, a major challenge remains. This is their transformation from highly specialized instruments restricted to national metrology laboratories into flexible devices deployable in different locations. Here we report the first field measurement campaign performed with a ubiquitously applicable $^{87}$Sr optical lattice clock. We use it to determine the gravity potential difference between the middle of a mountain and a location 90 km apart, exploiting both local and remote clock comparisons to eliminate potential clock errors. A local comparison with a $^{171}$Yb lattice clock also serves as an important check on the international consistency of independently developed optical clocks. This campaign demonstrates the exciting prospects for transportable optical clocks.

and secondly the area exhibits long-term land uplift (Alpine orogeny) accompanied by a secular gravity potential variation. Furthermore, LSM lacks the metrological infrastructure and environmental control on which the operation of optical clocks usually relies. The selected location thus constitutes a challenging but realistic testbed with practical relevance.
The transportable 87 Sr lattice clock was operated in both locations, LSM and INRIM, to eliminate the need for a priori knowledge of the clock's frequency. The schematic outline of the experiment is given in Fig. 1

. LSM and INRIM
were connected by a 150 km noise-compensated optical fibre link (see Supplement). At LSM, a transportable frequency comb measured the optical frequency ratio between a laser resonant with the Sr clock transition at 698 nm and 1.542 µm radiation from an ultrastable link laser transmitted from INRIM. In this way, the frequency of the optical clock at LSM could be directly related to the frequency of the link laser even without a highly accurate absolute frequency reference. In addition to the optical carrier, the fibre link was used to disseminate a 100 MHz radio frequency reference signal from INRIM for the frequency comb, frequency counters, and acousto-optic modulators at LSM (see Supplement). At INRIM, a cryogenic Cs fountain clock 22 and a 171 Yb optical lattice clock 16 served as references. The connection between the clocks at INRIM and the link laser is provided by a second frequency comb.
Ten days after arriving at LSM in early February 2016, the first spectra of motional sidebands on the 1 S0 -3 P0 clock transition were recorded from the 87 Sr transportable clock. The operation of the lattice clock (see Methods) was was placed in the LSM underground laboratory close to the France-Italy border in the Fréjus tunnel (top left). The clock was connected by a noise-compensated fibre link to the Italian national metrology institute INRIM in Torino (red line). There, a primary Cs fountain clock and a 171 Yb optical lattice clock were operated (right). At both sites, frequency combs were used to relate the frequencies of the 1 S0 -3 P0 optical clock transitions and the 1.5 µm laser radiation transmitted through the link. After the remote frequency comparison, the transportable clock was moved to INRIM for a side-by-side frequency ratio measurement. (b) Frequency of the transportable Sr clock as seen by the INRIM Cs fountain clock (black circles, uncertainties are 1σ). The potential difference ΔU is based on the geodetic measurement. The red line shows the expected variation of the Sr clock transition frequency due to the relativistic redshift. (c) The potential difference between LSM and INRIM was also determined independently by a combination of GNSS, spirit levelling and gravimetric geoid modelling (see Methods). similar to the procedure described in previous works. All laser systems required for laser cooling, state preparation and trapping of the Sr atoms in the optical lattice were operated, together with the vacuum system and the control electronics, in an air-conditioned car trailer. The ultrastable interrogation laser for the Sr clock transition was placed next to the trailer in the underground laboratory to avoid its performance being degraded by vibrations induced in the trailer by its air conditioning system. The frequency comb was also operated next to the trailer.
During this first run of the apparatus in particularly challenging environmental conditions at LSM, the transportable clock operated less reliably than in initial tests before transport. 13 Interruptions were mainly caused by degradation of the light delivery setup for the first cooling stage of the magneto-optical trap (MOT) on the 461 nm 1 S0 -1 P1 transition, which was based on a commercial semi-monolithic fibre-based light distribution system.  22 ), the frequency of the Sr lattice clock at LSM was measured by the fountain clock at INRIM with an uncertainty of 18×10 16 (see Fig. 1). With these changes, the availability of the Sr clock was improved significantly, allowing for several hours of data taking per day after the initial setup phase was completed. With systematic uncertainties comparable to the first campaign and a fountain instability of 3.6×10 13 τ 1/2 , the total uncertainty was reduced by a factor of two to 9×10 16 (see Supplement for Sr transitions frequencies). In this chronometric levelling demonstration, we resolved a relativistic redshift of the optical clock lattice clock of 47.92(83) Hz * (Fig. 1), from which we infer a potential difference of 10 034(174) m 2 /s 2 . This is in excellent agreement with the value of 10 032.1(16) m 2 /s 2 determined independently by geodetic means (Methods). Though our result does not yet challenge the classical approach in accuracy, it is a strong first demonstration of chronometric levelling using a transportable optical clock.
With the increased reliability of the transportable Sr clock, we were also able to measure its optical frequency ratio with the Yb lattice clock 16 operated on the 1 S0 -3 P0 transition at 578 nm. In total, 31 000 s of common operation of the two optical clocks and the frequency comb were achieved over a period of 7 days. This optical-optical comparison (Fig. 2) shows much higher stability than the optical-microwave one. Consequently, the optical frequency ratio measurement is limited by the systematic uncertainty of the clocks (Tab. 1), rather than by their instability. This demonstrates the key advantage of optical frequency standards: they are able to achieve excellent uncertainties in short averaging times even though they may operate less reliably than their microwave counterparts.
The 171 Yb/ 87 Sr frequency ratios measured on different days are summarized in Fig. 3, which also shows previous measurements of this ratio. After averaging (Supplement), we determine the ratio to be R = νYb / νSr = * The number in parentheses is the uncertainty referred to the corresponding last digits of the quoted result.   (34), which is within two standard deviations of the most accurate previous measurement 24 (Fig. 3). To our knowledge, this is the only optical frequency ratio that has been measured directly by three independent groups. 24,27,28 It therefore constitutes an important contribution to verifying the consistency of optical clocks worldwide. 25 Such measurements are key to establishing more accurate secondary representations of the second 26 as provided by the International Committee for Weights and Measures (CIPM) as a step towards a future redefinition of the SI second.
Note that even with the only slightly improved transportable Sr apparatus as used at INRIM, chronometric levelling against the Yb lattice clock with considerably improved resolution would be possible. We expect that the transportable clock will be able to achieve an uncertainty of 1×10 17 or better after a revised evaluation. This uncertainty will enable height differences of 10 cm to be resolved, which is a relevant magnitude for geodesy in regions such as islands, which are hard to access using conventional approaches. As metrological fibre links become more common, chronometric levelling along their paths 29 will become a realistic prospect.

Operation of lattice clocks
The realization and operation of the 171 Yb (I = 1/2) and 87 Sr (I = 9/2) clocks are very similar and have been presented in detail. 16,13,30 Ytterbium and strontium atoms are cooled to microkelvin temperatures in two-stage magnetooptical traps (MOTs), exploiting the strong 1 S0 -1 P1 and weaker 1 S0 -3 P1 transitions (at 399 nm and 556 nm for Yb and 461 nm and 689 nm for Sr, respectively). The atoms are then trapped in one-dimensional optical lattices operating at the magic wavelengths 31  Yb magic ≈ 759 nm and  Sr magic ≈ 813 nm.
Finally, the atoms are prepared for spectroscopy in a single magnetic sublevel mf by optical pumping. As a result, shifts due to cold collisions and line pulling are reduced. The two π-transitions from the mf = 1/2 sublevels in Yb (mf = 9/2 in Sr) are probed alternately at approximately halfwidth detunings so that the interrogation laser is locked to their average transition frequency. This effectively removes the linear Zeeman shift.

Uncertainties of lattice clocks
Here, we discuss the most important uncertainty contributions listed in Tab We measured the linear shift near the magic wavelength while the nonlinear induced lattice light shift can be calculated using data from Nemitz et al. 24 For the Sr lattice clock, the typical lattice depth was about 100 Er as measured from sideband spectra. These also yielded an atomic temperature of about 3.5 µK. The light shift cancellation frequency was determined earlier; a reference resonator served as a wavelength reference during the experiments discussed here. The uncertainty of the linear lattice light shift allows for a resonator drift of 50 MHz and changes due to variations of the scalar and tensor light shift. 32 Higher order light shifts were calculated using the coefficients in the same reference. As a check, three of the measurements in Fig. 2 were performed with a deeper lattice of about 160 Er which resulted in uncertainties for the linear lattice light shift and higher order shifts of 29×10 17 and 1×10 17 , respectively. No significant variation of the measured frequency ratio R was observed.
Density shift: The density shift was evaluated in both lattice clocks by varying the interrogated atom number.
Corrections for changes of the atomic temperature have been applied for the Sr clock.

Blackbody radiation (BBR) shift:
The influence of BBR on the clock frequency has been discussed elsewhere. 33,2,34 Temperatures of the atomic environment were measured with calibrated platinum resistance thermometers. The uncertainty of the BBR shift is mostly related to temperature inhomogeneity.

H-maser as flywheel
A flywheel oscillator with good stability and high reliability, such as a H-maser, can be used to extend the averaging time between a less reliable system such as our Sr lattice clock and a Cs primary clock with lower stability. 23 The frequency ratio νSr/νCs was thus determined from the frequency ratios νSr/νH and νH/νCs using datasets with different length. The noise of the flywheel means that it had different average frequencies for these two intervals, but the additional uncertainty can be calculated 23 if the noise is well characterized, as it often is for masers. We modelled the maser noise by a superposition of flicker phase noise 6×10 14 τ 1 (1×10 13 τ 1 ), white frequency noise 5×10 14 τ 1/2 (4.5×10 14 τ 1/2 ), and a flicker noise 1.7×10 15 (1×10 15 ) in March and (May) 2016, respectively.

Gravity potential determination
To provide an accurate reference for the chronometric levelling, we performed a state-of-the-art determination of the gravity (gravitational plus centrifugal) potential with the best possible uncertainty at each clock site. Here, besides the global long-wavelength and eventually the temporal variations 35 of the Earth's gravity potential, the local spatial influence of the gravity potential on the clock frequency needs to be considered. To refine the gravity field modelling around the clock sites and to improve the reliability and uncertainty of the derived geoid model, measurements; the larger uncertainty for the clocks is due to the simple method used to determine the local height differences between the clocks and the reference markers.

Supplement: Frequency transfer INRIM -LSM
The remote clock comparison was performed by comparing the frequency of a link laser at 1542.14 nm, sent from INRIM to LSM by a telecom optical fibre, to the frequencies of the clocks operated in the two locations. Two fibre frequency combs spanned the spectral gaps between the link laser and the clock interrogation lasers. The combs employed the transfer oscillator principle, 36 making the measurements of the optical frequency ratios immune to the frequency noise of the combs. The frequency of the link laser was stabilized using a high-finesse cavity, whose long term drift is removed by a loose phase-lock to a H-maser via a fibre frequency comb. As a result, the beat notes with the combs remained within a small frequency interval, facilitating long-term operation and reducing potential errors arising from any counter de-synchronization between INRIM and LSM. 18 The link laser used a multiplexed channel in the telecom fibre. Its path was equipped with two dedicated bidirectional Erbium-doped fibre amplifiers that allowed a phase stable signal to be generated at LSM through the Doppler noise cancellation technique. 18,19 The contribution of fibre frequency transfer to the total fractional uncertainty was assessed to be 3×10 -19 by looping back the signal from LSM using a parallel fibre. The occasional occurrence of cycle slips was detected by redundant counting of the beat note at INRIM. At LSM, the signal was regenerated by a diode laser phase locked to the incoming radiation with a signal to noise ratio >30 dB in 100 kHz bandwidth; this ensured robust and cycle-slip-free operation.
In addition to the optical reference, a high-quality radio frequency (RF) signal was needed at LSM to operate the Sr clock apparatus (frequency shifters and counters) and the frequency comb. Given the impossibility of having a GNSS-disseminated signal in the underground laboratory, a 100 MHz RF signal was delivered there by amplitude modulation of a second 1.5 µm laser that was transmitted through an optical fibre parallel to the first. At LSM, the amplitude modulation was detected on a fast photodiode, amplified and regenerated by an oven-controlled quartz oscillator (OCXO) at 10 MHz to improve the signal-to-noise ratio. The inherent stability of the free-running fibre link is in this case enough to deliver the RF signal with a long-term instability and uncertainty smaller than 10 -13 .
This resulting uncertainty contribution to the optical frequency ratio measurement is below 1×10 -19 .

Averaging of the optical frequency ratio data
We made eight different optical frequency ratio measurements with a total measurement time of 15 h over a period of one week in May 2016 (Fig. 3). The data acquired on different days have different statistical and systematic uncertainties. We applied a statistical analysis that considers the correlations between the measurements coming from the different systematic shifts where the covariance matrix of the eight daily measurements is used to calculate a generalized least squares fit for the average. 25,37 We regarded the systematic uncertainties of the clocks (Tab. 1) as fully correlated, while the statistics related to the measurement duration were uncorrelated.

Absolute Frequency of the Sr lattice clock
The chronometric levelling can be viewed from an alternative perspective: If we assume the conventional measurement of the gravity potential difference is correct then we can deduce an average absolute frequency The open diamonds at the bottom of the graph show the results from the campaign discussed here. For the LSM data, a correction for the gravitational redshift of -48.078 Hz as derived from the geodetic data has been applied. The other data have been compiled from various references (38, 39,40,41,42,43,44,45,46,47,48,49,50,23,51,52,53). The vertical line indicates the frequency recommended for the secondary representation of the second by Sr lattice clocks 26 and its uncertainty (dashed lines).