Brillouin amplification supports $1\times {10}^{-20}$ uncertainty in optical frequency transfer over 1400 km of underground fiber

We investigate optical frequency transfer over a 1400 km loop of underground fiber connecting Braunschweig and Strasbourg. Largely autonomous fiber Brillouin amplifiers are the only means of intermediate amplification used here. This allows phase-continuous measurements over periods up to several days. Over a measurement period of about three weeks we find a weighted mean of the transferred frequency's fractional offset of $(1.1\pm 0.4)\times {10}^{-20}$. In the best case we find an instability of $6.9\times {10}^{-21}$ and a fractional frequency offset of $4.4\times {10}^{-21}$ at an averaging time of around 30 000 s. These results represent an upper limit for the uncertainty over 1400 km when using a chain of remote Brillouin amplifiers, and allow one to compare the world's best optical clocks.

Figure 1

Schematic sketch of the all-fiber measurement setup and the 1400 km Brillouin link PTB-Université de Strasbourg-PTB. The inset shows the remote beat phase noise, calculated from 1 ms phase data (K&K-FXE).

Figure 2

Instability of the optical frequency transfer over the stabilized 1400 km “Brillouin link” Braunschweig-Strasbourg-Braunschweig. The frequency data are reported once per second simultaneously in $\mathrm{\Pi }$ and in $\mathrm{\Lambda }$ mode (FXE K&K); filled circles: Allan deviation of the frequency transfer ($\mathrm{\Pi }$-type data, bandwidth $>10$ kHz); filled triangles: Allan deviation of $\mathrm{\Lambda }$-type remote frequency offset data (${\mathrm{\Lambda }}_{1\mathrm{s}}$-ADEV, bandwidth 1 Hz); open triangles: modified Allan deviation of the transferred frequency; open squares: modified Allan deviation of the in-loop signal; dashed line: instability of the most recent side-by-side or self-comparisons of optical clocks [8, 9]; dotted line: instability without link (“optical shortcut”); the error bars of each ADEV point ${\sigma }_{\mathrm{y}}\left(\tau \right)$ are calculated as ${\sigma }_{\mathrm{y}}\left(\tau \right)/\sqrt{N\left(\tau \right)}$ [27], with $N\left(\tau \right)$ being the number of averaging intervals at $\tau$.

Figure 3

Optical frequency transfer instability (1400 km), different measurement, and demonstration of a numerically extended $\mathrm{\Lambda }$ averaging interval. Filled triangles: Allan deviation of the frequency transfer for $\mathrm{\Lambda }$-type data phase averaged over 1 s; filled diamonds: ${\mathrm{\Lambda }}_{10\mathrm{s}/100\mathrm{s}}$-ADEV of the frequency transfer for $\mathrm{\Lambda }$-type data, ($\mathrm{\Lambda }$-type averaging interval: 10 s; gray circles: 100 s); open triangles: modified Allan deviation of the transferred frequency; dashed line: instability of optical clock comparison as in Fig. 2; dash-dotted line: ModADEV of the fiber link stabilization's correction signal; for frequencies corresponding to the longest averaging time we expect the link stabilization to suppress phase fluctuations by about six orders of magnitude [26].

Figure 4

Overview over all long-term measurements during the period from December 10, 2014 to January 2, 2015. The black squares indicate the unweighted means for numerically extended $\mathrm{\Lambda }\text{−}\mathrm{type}$ averaging intervals of 2000 s; the respective total measurement times are indicated in kiloseconds. The error bars $\pm s$ indicate the last value $s$ of the corresponding ${\mathrm{\Lambda }}_{2000\mathrm{s}}$-ADEV at 1/4 of the total measurement time, yielding a conservative estimate of the statistical uncertainty [29, 30]. The mean of these data weighted by $1/s^2$ is $(1.1\pm 0.4)\times {10}^{-20}$. For comparison, the open diamonds correspond to the unweighted mean of the fiber-induced frequency offset during the respective measurements, multiplied by a factor of $+2\times {10}^{-7}$. This factor corresponds to the relative frequency difference between outgoing and return light.