Two-Clock Structure in Optical Metrology

• Two-Clock Structure is a dual-branch fiber network that uses a single ultrastable optical source to perform remote clock comparisons with common-mode drift cancellation.
• It employs cascaded and hybrid branches, utilizing repeater laser stations and combined noise cancellation techniques to achieve precise phase-coherent transfer.
• The architecture attains a combined frequency-transfer uncertainty of 2×10⁻¹⁹ and approximately 90% uptime, supporting extended high-precision clock campaigns.

A two-clock structure, in the context of international optical-frequency metrology, is a dual-branch fiber-link architecture in which a single ultrastable optical source seeds two independent branches, one per remote clock destination, so that two remote clock comparisons can be performed against a common origin with strong common-mode rejection of source drift. In the realization reported for a European network, the local node at SYRTE in Paris distributes a 194.4 THz carrier to a 1410 km cascaded branch and a 43 km hybrid branch through two repeater laser stations (RLS), while a short 5 m inter-branch connection is monitored and removed in post-processing to prevent branching-node noise from contaminating the comparison. The reported combined uptime is approximately 90% over nearly one month, and the combined frequency-transfer uncertainty of the link ensemble is $2\times 10^{-19}$ (Xu et al., 2019).

1. Topology and functional definition

The minimal two-clock configuration is one branch per remote clock. Branch A delivers the Paris optical carrier to a first remote site, where it is compared to a second international link carrying a first remote clock; Branch B delivers the same Paris carrier to a second remote site, where it is compared to a second international link carrying a second remote clock. Because both branches originate from the same ultrastable source, laser drift is common-mode and cancels when the two remote comparisons are differenced. The 5 m path between the two local RLS is separately monitored and subtracted so that non-common fluctuations introduced at the branch point do not appear in the clock comparison (Xu et al., 2019).

At SYRTE, the seed laser is delivered from the laser room to the link room over a 30 m actively stabilized fiber. The two local RLS are interconnected by a short 5 m fiber segment that forms the physical branch point. The architecture is explicitly star-like, and extending it to multiple partners is described as straightforward; the two-clock case is the minimal instance preserving low uncertainty with manageable complexity.

Segment	Realization	Key figures
Local delivery	Actively stabilized in-lab fiber	30 m
Inter-branch node	Short fiber between the two RLS	5 m
Branch A	Paris–Strasbourg–Paris, cascaded bidirectional link	$2\times 705$ km
Branch B	Paris–Villetaneuse, hybrid link	43 km

Branch A is a four-span cascaded bidirectional link on RENATER telecommunication fibers carrying parallel data traffic. It includes 40 optical add-drop multiplexers and 16 bidirectional EDFAs, compensating approximately 410 dB total loss, corresponding to approximately 0.29 dB/km. At Strasbourg, the delivered carrier is compared to a second international link from PTB using a two-way method implemented with RLS at the remote end. Branch B uses a pair of dedicated fibers in a hybrid scheme combining active compensation on one fiber and passive two-way noise cancellation on the return fiber; it includes 4 OADMs, 3 bidirectional EDFAs, and approximately 32 dB total loss, corresponding to approximately 0.37 dB/km. At LPL, the Paris-delivered carrier is compared to a 769 km noise-compensated link from NPL using a two-way method (Xu et al., 2019).

2. Transfer physics and stabilization architecture

Frequency transfer follows phase-coherent optical carrier transfer in the sense of Ma et al. (1994), with active Doppler cancellation of fiber-induced phase noise. In the long-haul cascaded branch, each span uses RLS that regenerate the optical phase while preserving coherence. In the 43 km hybrid branch, active noise cancellation is applied on the forward fiber and passive two-way suppression on the return fiber. This combination permits a common source to feed branches of markedly different scale without changing the underlying phase-coherent transfer principle (Xu et al., 2019).

Each RLS contains an integrated multi-branch Michelson interferometer. Its reference arms sense round-trip phase fluctuations and drive AOM-based phase correction. A known early-generation arm-length imbalance of approximately 15 cm is noted and included in the uncertainty analysis. The remote RLS are engineered so that their local RF references do not degrade the transferred optical frequency.

The 5 m inter-branch stabilization is deliberately simpler. A servo is not used. Instead, light from RLS A is injected into RLS B and reflected by a Faraday mirror at B. An AOM at RLS B shifts the frequency by 37 MHz, so the round-trip beat appears at 74 MHz at RLS A, where it is detected, tracked, divided, and counted. The resulting time series is then subtracted from the remote comparison data in post-processing. Under reciprocal-noise assumptions, this simple two-way variant suppresses inter-branch noise to a level below the long-haul floor if short uncompensated fiber sections are minimized (Xu et al., 2019).

The key hardware stack comprises an ultrastable cavity-stabilized laser at 194.4 THz, a slave laser phase-locked to the master, an optical frequency comb used locally to measure and de-drift the seed laser versus an H-maser, AOMs, Faraday mirrors, OADMs, bidirectional EDFAs, photodetectors for beat-note detection, dead-time-free K+K FXE counters operated in $\Lambda$-mode with 1 s gates, PLLs for laser phase locks, and AOM-based phase-correction loops. Some elements are placed in thermally controlled enclosures.

3. Measurement methodology and data validation

Beat notes are recorded with dead-time-free counters at 1 s gate in $\Lambda$-mode. For the 5 m branch monitor, the 74 MHz round-trip signal is tracked, divided, and counted; the measured fluctuations are subtracted in post-processing from the remote clock-comparison data. End-to-end link assessment is supported by the RLS-embedded Michelson interferometers, while the hybrid branch is assessed using a two-way beat observable such as TWB3 at SYRTE (Xu et al., 2019).

Data selection is not limited to thresholding. The validation procedure includes initial filtering with 10 Hz bandwidth and a quality flag for outlier points, a rolling mean with an approximately 9 s window to detect outliers, a rolling standard deviation with an approximately 2750 s window to flag anomalous noise, and a rolling standard deviation of the quality flag to reject unstable segments. The procedure is described as robust over months of data. Reciprocity and asymmetry were also checked on the 5 m interconnect; the histogram over 12 days was slightly asymmetric and showed daily thermal signatures, but modeling and two-way subtraction indicated residuals below the system floor when short uncompensated segments were minimized.

The stability metrics are expressed in terms of Allan deviation and modified Allan deviation:

$$\sigma_y(\tau)=\sqrt{\frac{1}{2(M-1)}\sum_{i=1}^{M-1}(y_{i+1}-y_i)^2}$$

and

$$\mathrm{Mod}\,\sigma_y(\tau)=\sqrt{\frac{1}{2m^2\tau^2(M-2m+1)}\sum_{i=1}^{M-2m+1}\left(\sum_{j=i}^{i+m-1}(y_{j+m}-y_j)\right)^2 }.$$

The post-processed stability analysis for the short inter-branch path is described using the Newbury coherent-transfer framework under reciprocal-noise assumptions, relating integrated phase-noise PSD to time-domain variance, although no explicit $S_\phi(f)$ is provided (Xu et al., 2019).

4. Reported metrological performance

The 5 m inter-branch segment, before subtraction, exhibits Allan and modified Allan deviations that start near $2.3\times 10^{-16}$ at $\tau=1$ s, peak at approximately $4\times 10^{-16}$ at $2\times 705$0 s, and show a daily bump of approximately $2\times 705$1 at $2\times 705$2 s caused by approximately 1.6 K room-temperature cycling. After two-way post-processing, the modeled residual is negligible relative to the system floor, with practical residuals limited by short uncompensated fiber of approximately 1 m and daily thermal variation, yielding approximately $2\times 705$3 at about half a day and approximately $2\times 705$4 at one day (Xu et al., 2019).

The 1410 km cascaded branch reaches modified Allan deviation below $2\times 705$5 at 1 s and long-term stability of approximately $2\times 705$6 or below. Its reported bias is $2\times 705$7 with uncertainty $2\times 705$8. The 43 km hybrid branch reaches modified Allan deviation of approximately $2\times 705$9 at 1 s and approximately $\Lambda$0 at one day, with reported bias $\Lambda$1 and uncertainty $\Lambda$2.

Element	Stability or bias result	Uptime over $\Lambda$3 days
30 m in-lab fiber	Actively stabilized	100%
5 m inter-RLS segment	Tracked and filtered	98.5%
43 km hybrid branch	$\Lambda$4	95%
1410 km cascaded branch	$\Lambda$5	96.3%

The combined chain uptime is approximately 90%. This level of continuity is operationally significant because the architecture is intended for extended clock-comparison campaigns rather than isolated transfer demonstrations (Xu et al., 2019).

5. Uncertainty budget and dominant limitations

The combined frequency-transfer uncertainty of the full link ensemble is reported as $\Lambda$6, obtained conservatively by root-sum-square combination of the long-haul, short inter-branch, and electronics-related terms (Xu et al., 2019):

$\Lambda$7

Using $\Lambda$8, $\Lambda$9, $\Lambda$0, and small electronics terms gives the reported conservative total of approximately $\Lambda$1.

The dominant uncertainty source is the long-haul link ensemble, at approximately $\Lambda$2 combined. The short 5 m residual, after post-processing and including uncompensated short paths, contributes at most approximately $\Lambda$3 on day timescales. Additional terms come from RLS electronics, interferometer asymmetries such as the 15 cm arm imbalance, and environmental drifts, especially temperature.

Counter and timebase errors at remote ends are explicitly identified as non-negligible when beat notes approach 55 MHz and commercial GPS-disciplined RF references are used. With RF stability of approximately $\Lambda$4 at one day, the contribution is approximately $\Lambda$5. The study notes that in-band timing, for example White Rabbit, can reduce this risk and improve robustness. A plausible implication is that the branching-node problem is not primarily one of geometric complexity, but of how short uncompensated paths and auxiliary RF references are treated within the uncertainty budget.

6. Operational robustness, comparison with point-to-point links, and significance

Operational robustness depends strongly on environmental control and automation. Thermally controlled boxes are used for key components, and daily 1.6 K room-temperature swings are identified as a clear driver of instability in the 5 m segment. The 1410 km cascaded branch reached approximately 96% uptime through automation and remote supervision; remaining downtime was often due to servo unlocks, polarization optimization in RLS, and non-automated loops in laboratory clock and comb systems. On the 43 km link, cycle slips produced approximately $\Lambda$6 frequency hops, but these were easily flagged against short-term noise of approximately $\Lambda$7, and simple 1 Hz filtering with outlier removal was sufficient (Xu et al., 2019).

Relative to a dedicated single point-to-point link, the dual-branch approach offers three stated advantages. First, scalability: one seed feeds multiple branches, enabling multi-partner comparisons without duplicating optical sources. Second, common-mode rejection: a shared origin suppresses source-drift contributions in inter-partner comparisons. Third, network integration: OADMs and bidirectional EDFAs allow operation on RENATER telecom fibers with parallel data traffic and proven high uptime. The trade-offs are also explicit. The 5 m branching segment must be monitored or actively cancelled so that it does not dominate the uncertainty. The larger component count increases the need for automation and supervision. The overall system-level uncertainty budget is slightly higher than that of an optimized dedicated point-to-point link, although still well below current clock uncertainties.

The metrological significance follows directly from the reported figures. A combined transfer uncertainty of $\Lambda$8 is below or comparable to the systematic uncertainties of leading optical lattice clocks and ion clocks in current operation, so the fiber transfer does not limit the two-clock comparison. With approximately 90% uptime over nearly a month, the architecture supports long campaigns involving fountain clocks and optical clocks with high data yield. As clock systematics move toward the low $\Lambda$9 range, further reduction of short uncompensated fibers and more robust in-band timing or RF distribution are identified as the principal paths for preserving the link contribution comfortably below the intrinsic clock budget (Xu et al., 2019).