Optical frequency divider with division uncertainty at the 10^(-21) level

Optical clocks with unprecedented accuracy of 10^(-18) will lead to innovations in many research areas. All the applications of optical clocks rely on the ability of precisely converting the frequency from one optical clock to another, or particularly to the frequencies in the fiber telecom band for long-distance transmission. Here, we report a low-noise, high precision optical frequency divider. It can realize accurate optical frequency conversion as well as enable precise measurement of optical frequency ratios. By comparing against the frequency ratio between the fundamental and the second harmonic of a 1064 nm laser rather than a second similar system, the optical frequency divider is demonstrated to have a frequency division instability of 6e-19 at 1 s and a fractional frequency division uncertainty of 1.4e-21, nearly three orders of magnitude better than the most accurate optical clocks. It allows optical clocks to be accessible to many precision measurement applications.

Recent progress on optical atomic clocks demonstrates that the fractional frequency instability and uncertainty of optical clocks have been reduced to the 10 −18 level [1−4]. The unprecedented accuracy provided by optical clocks will lead to a revolution in science and technology [5]. Using optical clocks, scientists are able to search for possible variations of fundamental constants in laboratories by precisely measuring the frequency ratios of two different atomic transitions of optical clocks over time [6−8]. In relativistic geodesy, long-distance geopotential difference will be accurately measured by comparing the frequencies of remote optical clocks linked with optical fibers [5,9,10], where the frequencies of optical clocks have to be accurately converted to those in the fiber telecom band for long-distance transmission. In metrology, the second in the International System of Units (SI) will be redefined based on optical atomic clocks [11]. Frequency comparisons between optical clocks based on different atom species have to be performed in order to ascertain the agreement between optical clocks with uncertainty beyond the current SI second, as well as the frequency reproducibility of optical clocks [12]. Moreover, atomic and molecular spectroscopists hope that the frequency accuracy and high frequency stability of optical clocks can be transferred to a wide spectral range for high precision spectroscopy.
All those applications rely on accurate frequency ratio measurement between spectrally-separated optical clocks or frequency conversion of optical clocks. Optical frequency combs [13,14] are employed to link the frequencies of optical clocks. By synchronously counting the frequency of the repetition rate (f r ) and the carrier-envelope offset frequency (f 0 ) of the comb, as well as the beat frequencies between optical clocks and the nearby comb teeth relative to a common hydrogen maser, frequency ratios with statistical uncertainty at the 10 −17 level are obtained [6,15]. Using synchronous counting with a hydrogen maser and the transfer oscillator scheme [16,17], the systematic instability in optical frequency ratio measurement is limited to 2.8 × 10 −16 at 1 s averaging time [18]. With the current experiments on ultrastable lasers which has a frequency instability of 10 −17 and beyond in the near future [19−22], together with the efforts on using correlated atomic samples to overcome the standard quantum limit [23,24], the frequency stability of optical atomic clocks will set a new record, demanding an even more precise way to quickly determine the frequency ratios.
Here we report a low-noise optical frequency divider (OFD) in the visible and near infrared region, capable of linking all the present optical clocks. Using the OFD, the frequency ratios between optical clocks can be precisely measured without synchronous counting the beating frequencies between optical signals and a comb against a hydrogen maser. Moreover, OFD can also accurately convert the frequency from one optical clock with a preset division ratio to another, or to the frequencies in the fiber telecom band [25] or in the microwave region [26] for a wide range of applications. Particularly, using an OFD to convert the frequency of one high-performance clock laser to another at a different wavelength, it improves the frequency ratio measurement by partially cancelling out the laser frequency noise in synchronization operation of optical clocks [15]. By comparing against the frequency ratio between the fundamental and the second harmonic of a 1064 nm laser, the fractional instability of the divisor is demonstrated to be 6 × 10 −19 at 1 s and 2 × 10 −20 at 1000 s, two orders of magnitude better than the most stable lasers [19,27−30]. The fractional uncertainty of the divisor in optical frequency division is characterized to be 1.4 × 10 −21 . It can support frequency division of the most stable lasers and the most accurate optical clocks in the world without degrading the performance, and enables precision measurements at the 10 −21 level.
The output light frequency of an OFD directly relates to the input light frequency with a precise ratio R, f out = f in /R. The experimental schematic of the OFD is shown in Fig. 1. An output laser (f out ) is phase-locked to the input light (f in ) via an optical frequency comb. The frequency of the Nth comb tooth is f N = Nf r + f 0 , where N is an integer. To reduce the comb frequency noise, the comb is optically-referenced to f in by phase-locking the N 1 th comb tooth to f in and stabilizing f 0 to a stable radio frequency (Supplement 1). A beat signal between the input laser light (f in ) and a nearby comb tooth (f N1 ) can be written as b1 in N1 in 0 1 r .
Meanwhile, another beat signal between the output laser (f out ) and a nearby comb tooth (f N2 ) can be written as where N 1 and N 2 are integers associated with the particular comb teeth. By measuring f in and tune .
The residual frequency noise of f 0 is removed by mixing f b1 and f b2 * with f 0 in DBMs.
The signal of f 0 is detected by using a collinear self-referencing 1f-2f set-up [31], in which the .
Then the error signal is sent to a servo to adjust the frequency of the output laser to make ∆ = 0. As a result, f out = (M 2 /M 1 )f in + f tune . In many applications of optical atomic clocks, it requires not only coherence transfer but also precisely setting the ratio between optical frequencies.
In order to set the division ratio precisely, here f tune has to be related to f in only. To achieve this goal, the beat signals of f b1 and f b1 * between f in and the two nearest teeth of the same comb are detected on a photo detector. f b1 removing f 0 and f r using DBMs and DDSs, the resulting signal is directly derived from f in as on f 0 and f r . K 1 and K 2 are the divisors of the DDSs, which are chosen to satisfy K 1 /K 2 = N 1 /(N 1 + 1). In addition, a DDS with the divisor of K 3 is used to synthesize a self-referenced time base signal f time at about 10 MHz, f time = f in /k. Using this time base, a RF tuning frequency f tune is generated from a RF synthesizer (RF SYN) as f tune = f in /K, here K depends on K 1 , K 2 , K 3 and the frequency setting of the RF synthesizer. Benefitting from the self-referenced RF signal, f out is directly divided from f in with a ratio of R = 1/(M 2 /M 1 + 1/K). If f tune is set with a resolution of 1 µHz, the ratio R can be set at the 21th decimal place to an arbitrary pre-determined value when both f in and f out are within the spectrum of the comb.
Meanwhile, R can be precisely tuned [32] by sweeping f tune .
To characterize the performance of the OFD, we measured the divisor R of the OFD against the frequency ratio between the fundamental and the second harmonic of a 1064 nm laser instead of comparing against a second similar OFD. The second harmonic generation can realize optical frequency conversion, however, it is based on a completely different working principle from that of the OFD.
The experimental diagram of measurement is shown in Fig. 2(a). We used the OFD to , the frequency ratio R x can be obtained as The fractional instability of the measured R x is shown in Fig. 2  from a χ 2 analysis [33]. The division uncertainty induced by the OFD is three orders of magnitude better than the most accurate optical clocks [3,4].
The merit of the transfer oscillator scheme is the immunity to comb frequency noise.
Therefore, it is not necessary to phase-lock the comb to f 1064-1 . However, due to the limited response bandwidth of the DDSs, the fast varying signals sent to the DDSs affect the performance of the system. We measured the frequency instability of the beat signal f b between two 1064 nm lasers (f 1064-1 and f 1064-2 ), whose frequencies are linked by the OFD as

TESTING THE IMMUNITY OF COMB FRQUENCY NOISE
If one of the divisors of the DDSs (K 1 , K 2 , M 1 , M 2 ) is set incorrectly, we can easily observe the error by counting the beat frequency between two lasers linked with the optical frequency divider when tuning f 0 or f r . Figure  Moreover, it causes the laser to lose phase lock frequently. When the divisors of the DDSs are set correctly, the beat frequency between two lasers are more stable, as shown in Fig. S3(a).  Allan deviation (red dots) and the modified Allan deviation (black squares) of all the data.