All-polarization-maintaining, single-port Er:fiber comb for high-stability comparison of optical lattice clocks

All-polarization-maintaining, single-port Er:fiber combs offer long-term robust operation as well as high stability. We have built two such combs and evaluated the transfer noise for linking optical clocks. A uniformly broadened spectrum over 135-285 THz with a high signal-to-noise ratio enables the optical frequency measurement of the subharmonics of strontium, ytterbium, and mercury optical lattice clocks with the fractional frequency-noise power spectral density of $(1-2)\times 10^{-17}$ Hz$^{-1/2}$ at 1 Hz. By applying a synchronous clock comparison, the comb enables clock ratio measurements with $10^{-17}$ instability at 1 s, which is one order of magnitude smaller than the best instability of the frequency ratio of optical lattice clocks.


E-mail: ohmae@riken.jp
All-polarization-maintaining, single-port Er:fiber combs offer long-term robust operation as well as high stability. We have built two such combs and evaluated the transfer noise for linking optical clocks. A uniformly broadened spectrum over 135-285 THz with a high signal-to-noise ratio enables the optical frequency measurement of the subharmonics of strontium, ytterbium, and mercury optical lattice clocks with the fractional frequency-noise power spectral density of (1-2)×10 -17 Hz -1/2 at 1 Hz. By applying a synchronous clock comparison, the comb enables clock ratio measurements with 10 -17 instability at 1 s, which is one order of magnitude smaller than the best instability of the frequency ratio of optical lattice clocks.
Template for APEX (Jan. 2014) 2 Optical frequency combs have become indispensable tools for precision measurements including atomic/molecular spectroscopy, low-phase-noise microwave generation, and ranging. 1) In optical clocks, linking different atomic clocks 2,3) or distant clocks via optical fibers 4) with a fractional uncertainty of 10 -18 is of significant concern with a future redefinition of the second in the International System of Units (SI) 5) . Such endeavors, in turn, offer intriguing opportunities for testing the constancy of the fundamental constants 6) and for relativistic geodesy 4,7) . These applications necessitate ultralow-noise optical frequency combs that allow long-term and robust operation. Titanium-sapphire-based frequency combs have demonstrated outstanding stability. 8,9) However, their bulky optical setups require regular maintenance, thus hampering long-term and robust operation, which limits their potential applications. In contrast, erbium (Er) fiber combs enable all-fiber architecture for robust operation. Although a typical Er:fiber comb uses nonlinear polarization rotation (NPR) to acquire mode-locking with excellent noise performance [9][10][11][12][13] , the operational condition for such NPR-based Er:fiber oscillators can be sensitive to environmental conditions. For practical applications, such as the long-term operation of clocks to generate the optical second 14) , and field and space applications, the all-polarization-maintaining (PM) architecture is preferred 15,16) . However, such architecture has shown relatively large phase noise. Low intrinsic phase noise with PM architecture is demonstrated by applying a nonlinear amplifying loop mirror (NALM) 17,18) .
In linking multiple optical frequencies, Er:fiber combs with a multibranch configuration, where each port consists of an Er-doped fiber amplifier (EDFA) and a highly nonlinear fiber (HNLF), have been employed 3,19) , as it allows sufficient output power per comb tooth optimized for the single frequency. In such a multi-branch comb, the phase noise in different branches introduces the instability of ~10 -16 at 1 s 10,11) . A record high instability of 4×10 -16 (/s) -1/2 has been demonstrated for synchronous clock comparison between strontium (Sr) and ytterbium (Yb)-based optical lattice clocks 3) , in which the instability is mainly limited by the Dick effect 20) due to the frequency noise of the multibranch comb.
The single-port architecture 12,13) is advantageous for suppressing such interbranch relative phase noise that is caused by the optical path length fluctuation. Moreover, in order to access multiple optical clocks with different frequencies, an octave-spanning supercontinuum (SC) output with a sufficient signal-to-noise ratio (SNR) is favored. In this work,  Figure 2 shows the optical spectra of the Er:fiber oscillator and the SC output.
This oscillator has a single-sideband phase-noise floor of about -100 dBc/Hz, which is measured by beating with a kHz-linewidth laser at 192 THz. Applying an EDFA and a HNLF, we obtain a uniformly broadened octave-spanning spectrum (red line in Fig. 2) with the average power of about 80 mW. To evaluate the SNR of this SC output, we simultaneously monitor the beat signals with lasers at 215, 259, and 282 THz, which respectively correspond to the subharmonics of Sr, Yb, and Hg clock frequencies 21) . Figure 1 The carrier-envelop offset frequency fCEO is obtained using a self-referencing interferometer with the phase-noise floor of -100 dBc/Hz, as shown in the lower panel in Fig. 1(b).
The spectral shape and bandwidth of the SC output is optimized by the current supplied to pumping laser diodes (LDs) for the EDFA. Once optimized, the beat signals, as shown in NPR-based Er:fiber combs 10) . This residual phase noise is sufficiently small to remain locked for longer than a few days 17) .
To evaluate the frequency spectral transfer noise, we prepare two identical combs, both of which are stabilized to the 'clock laser' at 215 THz, which is locked to a 40-cm-long reference cavity 21) . Lasers at 259 and 282 THz are stabilized to the respective teeth of Er comb (1) in Fig. 1(a). In the following, we analyze the beat signals of these lasers and Er comb (2) with the measurement bandwidth of 2 MHz by using in-and quadrature-phase demodulators based on analog frequency mixers and the RF reference. Most of the optical paths connecting the two combs with PM fibers and free-space optics are stabilized using interferometer-based Doppler noise cancellers (DNCs) 22) , where an approximately 10-cmlong optical path remains uncompensated. Since the frequency noise of the spectral transfer via the comb typically shows white phase noise characteristics, we use the modified Allan deviation to indicate the instability. Figure 3 shows the modified Allan deviation of the Template for APEX (Jan. 2014) 5 fractional frequency noise observed in the spectral transfer from 215 to 259 THz (blue) and to 282 THz (red). This shows instabilities of (5-7)×10 -18 at  = 1 s and 1×10 -19 at  = 100 s, which is similar to those described in Refs. 12 and 13. Compared with the instability using a multi-branch NPR-based Er:fiber comb 10,21) as shown by a black line, the spectral transfer instability at  = 1 s is improved by more than 30 times. At  > 100 s, the instability reaches the floor of around 10 -19 , which is most likely caused by the fluctuation of the residual uncompensated optical path length. Since this measurement utilizes an RF reference at 10 MHz with a fractional frequency uncertainty of 10 -12 , this corresponds to a sub-10 -19 uncertainty. Within this statistical uncertainty, there is no obvious frequency offset. For f > 100 Hz, the spectral transfer noise is close to the limit estimated by the SNR of the beat signals except for a bump at around 1 kHz, which is caused by the insufficient control gain of DNCs. On the other hand, excess noise above the SNR limit is observed for f < 100 Hz. This is possibly due to the fluctuation of the uncompensated optical path length.
Finally, we discuss the fractional instability of the frequency ratio = 1 2 ⁄ in clock comparison, which offers a benchmark for testing the short-term stability of the frequency comb used in atomic clocks. The fractional instability is limited by the quantum projection noise (QPN) 23) QPN , the Dick effect Dick 20) , and the spectral transfer noise of the comb Comb to bridge the two clock frequencies ν1 and ν2. The overall instability is described as The QPN-limited instability for respective clocks j=1,2 is given by QPN ∼ ( i ) −1 ( / where ( ) is the fractional frequency noise PSD in units of 1/Hz , 0 is the 1-cycle average of a sensitivity function ( ), and c and s are the cosine and sine components of the n-th Fourier series expansion of ( ) , respectively 20) . As the sensitivity [( c 0 ⁄ ) 2 + ( s 0 ⁄ ) 2 ] 1 2 ⁄ rapidly decreases for f >> 1/Ti, frequency noise with low Fourier components of a few Hz solely affects the clock instability Dick . Note that the frequency noise of the clock laser with the state-of-the-art instability σy~1×10 -16 at  = 1 s 25,26,27) [green line in Fig. 4(a)] is an order of magnitude larger than the frequency transfer noise of the comb for f < 100 Hz, where the Dick effect plays a decisive role. Assuming this laser noise, the Dick effect limit of comparison of Yb (Hg) and Sr optical lattice clocks is calculated to be Dick ~10 -16 (/s) -1/2 for each clock j=1 and 2, as shown by the black circles in Fig. 4(b). This indicates that the thermal noise of an optical cavity 28) used to stabilize the clock laser severely degrades the short-term instability of optical clocks, and the superb transfer instability of the comb is not fully utilized.
The ratio measurement beyond the Dick effect limit of the 'clock laser' is possible by applying synchronous interrogation to reject the laser frequency noise 3,24) . When the two clocks share a single cavity by transferring its spectral characteristics via the comb, the Dick effect term Dick in Eq. (1) is given by where the Dick effect due to the cavity-induced laser noise is partially rejected and reduced to Dick(Cavity) , and the spectral transfer via the comb adds an extra Dick effect Dick(Comb) .
Employing a comb with lower frequency noise than the cavity thermal noise will allow of magnitude (red circles) smaller than Dick and the comb frequency noise sets the Dick effect limit Dick(comb) , allowing total instability of low 10 -17 (/s) -1/2 , which improves the stability by an order of magnitude compared with asynchronous operation (black circles).
We assume = 10 5 atoms to allow QPN ≤ Dick Sync (orange line), which will be affordable with optical lattice clocks. The application of a cryogenic monocrystalline silicon cavity 27) or crystalline-coated mirrors 29) with reduced thermal noise of frequency noise PSD of ~3×10 -17 (f/Hz) -1/2 1/Hz 1/2 will allow further reduction of the instability. As for the last term in Eq.
In summary, we have developed all-PM and single-port Er:fiber combs with an octavespanning high-SNR optical spectrum, which is, in particular, suitable for the high-stability ratio measurement of optical lattice clocks. By synchronously operating the clocks, we show that the Dick-effect contribution of the comb instability to low 10 -17 (/s) -1/2 is achievable, which is even better than the state-of-the-art laser instability. Low-noise and all-PM architecture facilitates robust comparison of highly stable optical lattice clocks 4) , offering new applications in cm-level 7) relativistic geodesy and a search for the variation of fundamental constants 30) and the Lorentz invariance 31) at shorter time scales.