Vernier microcombs for integrated optical atomic clocks

Feb. 20 , 2025SJ_Zhang

Kerr microcombs have drawn substantial interest as mass-manufacturable, compact alternatives to bulk frequency combs. This could enable the deployment of many comb-reliant applications previously confined to laboratories. Particularly enticing is the prospect of microcombs performing optical frequency division in compact optical atomic clocks. Unfortunately, it is difficult to meet the self-referencing requirement of microcombs in these systems owing to the approximately terahertz repetition rates typically required for octave-spanning comb generation. In addition, it is challenging to spectrally engineer a microcomb system to align a comb mode with an atomic clock transition with a sufficient signal-to-noise ratio. Here we adopt a Vernier dual-microcomb scheme for optical frequency division of a stabilized ultranarrow-linewidth continuous-wave laser at 871 nm to an ~235 MHz output frequency. This scheme enables shifting an ultrahigh-frequency (~100 GHz) carrier-envelope offset beat down to frequencies where detection is possible and simultaneously placing a comb line close to the 871 nm laser—tuned so that, if frequency doubled, it would fall close to the clock transition in ¹⁷¹Yb⁺. Our dual-comb system can potentially combine with an integrated ion trap towards future chip-scale optical atomic clocks.

Main

By virtue of their extreme long-term frequency stability, optical atomic clocks promise to revolutionize timing systems, enable fundamental tests of physics^1,2 and allow chronometric levelling and geodesy^2,3,4. Bulky optical lattice clocks^2,4,5,6 and ion trap clocks^6,7 can provide state-of-the-art frequency stability. However, scaling these clocks to a low size, weight and power architecture is an important and challenging hurdle preventing ubiquitous deployment of the highest-performing optical atomic clock technology. Efforts such as refs. ^8,9 towards integrated ion traps and optical lattices illustrate recent progress towards high-stability, chip-scale optical atomic clocks.

Microcombs are an essential part of future chip-scale optical atomic clocks^10,11 as they establish a coherent link from atomic references at hundreds of terahertz all the way down to the radio frequency (RF) domain (analogous to a gear set; Fig. 1a)¹² while preserving the stability of an atom-referenced clock laser. The realization of this coherent link is referred to as optical frequency division (OFD). The comb’s optical modes f_m (m is the mode order) are equally spaced by the repetition rate f_rep (usually at RF), and the whole comb has a spectral offset (the carrier-envelope offset (CEO) frequency, f_CEO) so that f_m = mf_rep + f_CEO. Historically, frequency combs in optical atomic clocks have often been generated by mode-locked lasers^13,14,15 that have volumes on the order of many litres and generally require at least a measure of hand assembly, complicating mass production and increasing costs. Shrinking down these combs utilizing Kerr microresonators on photonic chips will be helpful towards miniaturizing optical atomic clocks and other systems requiring stabilized combs¹⁶.

**Fig. 1: Overview of Vernier dual-microcomb scheme for OFD.**

Unfortunately, Kerr microcombs possessing the octave span needed for self-referencing typically have ~1 THz repetition rates. This leads to f_CEO in the range of ±500 GHz, which is difficult to design because of microresonator fabrication imperfections^17,18; hence, the f_CEO beat is frequently too high for electronic detection. Similarly, targeting a particular clock transition frequency through microresonator design and fabrication is challenging. Terahertz microcombs with their coarsely spaced lines have low chances of hitting the targeted frequency. In part due to these issues, microcomb-based OFD remains a challenging task. Although there are recent demonstrations on chip-based two-point OFD using a narrowband microcomb^19,20, such architecture is not applicable for single-point OFD from a single laser with f-2f self-referencing—a requirement for an optical atomic clock. The only two demonstrations^10,11 of microcomb-based OFD of atom-referenced lasers, to our knowledge, relied on low repetition rate (tens of gigahertz) and narrowband combs generated from millimetre-size silica disk resonators, either spectrally broadened in a nonlinear fibre¹⁰ or paired with a secondary broadband silicon nitride (SiN) terahertz repetition rate microcomb¹¹. Both of these works utilized thermal vapours for their atomic references with limited frequency stabilities compared with cooled or trapped atoms^8,9.

In this work, we demonstrate an integrated photonic platform based on Vernier dual-microcomb that overcomes some of the fundamental challenges of previous microcomb-based systems to realize chip-scale optical clocks. Specifically, by pairing a main octave-spanning terahertz repetition rate microcomb with a secondary broadband terahertz repetition rate Vernier microcomb, both on a SiN platform, we successfully frequency divide a continuous-wave laser at 871 nm (ref. ²¹) to an RF clock output at ~235 MHz. This laser is designed for frequency doubling to reach the ytterbium ion (¹⁷¹Yb⁺) clock transition at 435.5 nm, which supports better frequency stability²² than thermal atomic references. By virtue of its dual broadband combs, the Vernier scheme used here presents substantial advantages over single broadband comb schemes. One of the advantages is the freedom of picking comb lines from either or both of the combs for sum-frequency generation (SFG) to aid in reaching a much greater variety of wavelengths²³. Here, this is adopted both to help circumvent the high-frequency f_CEO detection problem and to reach 871 nm. Although our focus is on this transition, the technique is broadly applicable to other atomic transitions for future optical clocks. For example, SFG between the high-power pump and a comb line can cover ~600 nm to ~870 nm, which is viable for Ca⁺ (729.1 nm) and Sr⁺ (674.0 nm) clock transitions. In addition, with both terahertz microcombs fabricated in SiN films of the same thickness, there is potential to integrate our scheme on a single chip. Since the repetition rate of a microcomb scales as inverse of the microring circumference, the footprint of a dual ~1 THz microcomb structure could be several orders of magnitude smaller than architectures using ~20 GHz microcombs, enabling a higher manufacturing yield in mass production. Furthermore, the all-planar geometry used here avoids the complexity of coupling to suspended whispering gallery mode resonators or integrating them²⁴. These results demonstrate a versatile and general microcomb platform to realize chip-scale optical atomic clocks.

Results

Dual-microcomb scheme

The overview of our experiment configuration is visualized in Fig. 1b. An ultranarrow-linewidth 871 nm (~344 THz) local oscillator (LO) laser developed by OEwaves^21,25 is phase-locked to an external self-referenced fibre comb (FC), which serves as a frequency-stable proxy for a potential Yb⁺ frequency reference (which is not available in our laboratory). The Vernier dual-microcomb scheme^23,26 is leveraged here for full OFD for the first time. Two broadband microcombs with large repetition rates (~900 GHz), slightly offset from one another (δf_rep ≈ 20 GHz), are generated with a shared continuous-wave pump. This configuration has been shown to allow the detection of both large f_rep and f_CEO (ref. ²³). We aim to transfer the LO laser stability to an RF clock output using a heterodyne beat in the dual-microcomb system.

We utilize a pair of microrings²⁷ (~25 μm radius) to generate microcombs. The main comb is designed to have dispersive waves at ~1 μm and ~2 μm wavelengths to assist the f-2f process, whereas the Vernier comb should have a resonance wavelength close to that of the main comb to allow a shared pump. Based on these criteria, we select microrings fabricated on separate chips but the same wafer. A single ~1,550 nm laser is amplified and split to pump both the main and Vernier devices simultaneously, generating an octave-spanning soliton microcomb in each. Figure 2a,b shows the spectra for the main and Vernier combs. The main (Vernier) comb possesses a repetition rate f_rep1 ≈ 896 GHz (f_rep2 ≈ 876 GHz). The pump frequency f_pump is ~193.54 THz, which can be written as f_pump = f_CEO1 + 216f_rep1 = f_CEO2 + 220f_rep2. When the first sidebands from each comb are heterodyned together on a photodetector (Fig. 1b), they create an electronic signal oscillating at

?clock=?rep1−?rep2≈19.7GHz. (1)

Through appropriate feedback to our two combs, we seek to transfer the stability of the 344 THz LO laser to this ~19.7 GHz dual-comb beat.

To accomplish OFD from the LO laser to f_clock, at a minimum, we need to stabilize the two repetition rates to the LO laser. We separate the 1 μm, 1.55 μm and 2 μm spectral components and combine the dual-comb spectra at 1.55 μm and 2 μm wavelengths for the detection of various heterodyne beats. The spectral domain schematic is illustrated in Fig. 1b (the detailed experimental setup is described in Methods). Due to their different repetition rates, the two combs walk off from each other as one moves away from the pump frequency, eventually coming back together at the so-called Vernier overlap point. The comb modes at this point can be photodetected (here, around 1,945 nm) to produce a relatively low-frequency beat

?Vernier=44?rep1−45?rep2≈−8GHz. (2)

With this signal phase-locked through feedback to the Vernier comb, f_rep2 will follow f_rep1 (ref. ²⁸). We still need to stabilize the two repetition rates to the LO laser through feedback to the main comb to achieve a fully stabilized f_clock (equation (1)).

We perform a novel f-2f process via SFG²³ in a periodically poled lithium niobate (PPLN) waveguide, which sums one comb line from each comb at around 2 μm (f_CEO1 + 166f_rep1 and f_CEO2 + 179f_rep2). The sum products are combined with the short-wavelength side of the main comb around 1 μm (f_CEO1 + 342f_rep1) to create an f-2f-like beat, f_f-2f. Figure 2c shows the sample spectra of the sum products and the short wavelength line. This f_f-2f contains contributions from both the f_CEO and the repetition rates of the combs

?f−2f=?CEO2−176?rep1+179?rep2≈11GHz. (3)

Finally, to relate the dual microcombs to the LO laser, which is outside the span of both comb spectra, we use a second PPLN waveguide for SFG between the 1,550 nm pump (f_CEO2 + 220f_rep2) and a long-wavelength Vernier comb line (f_CEO2 + 171f_rep2 ≈ 1,990 nm) to create a nonlinear product. For OFD architecture demonstration, the LO laser is tuned to ~871.042 nm (a few gigahertz away from twice the Yb⁺ clock transition wavelength of 871.035 nm) to beat against the SFG product to create f_comb-LO

?comb−LO=2?CEO2+391?rep2−?871≈−19GHz. (4)

Figure 2d shows the sample spectra for the SFG lines and the LO laser. This SFG process benefits from the high power of the pump and the optically amplified 2 μm comb line, leading to a reasonable SFG power of ~−36 dBm. We note that the Vernier scheme provides extra candidate lines in the SFG process by summing comb lines from different combs, where an SFG line right around 871.035 nm can be obtained (Fig. 2d and Supplementary Section 5.2).

As both f_f-2f and f_comb-LO contain the f_CEO2 term, but with a fixed ratio of 1:2, we can cancel the CEO term through electronic frequency division and frequency mixing

?xCEO=?f−2f−?comb−LO2=?8712−176?rep1−16.5?rep2. (5)

We refer to the frequency of the resulting signal as f_xCEO because, ideally, contributions from both combs’ f_CEO are removed. Instead, f_xCEO depends on the two combs’ repetition rates and relates them to the LO laser. If we lock this signal through feedback to the main comb and simultaneously lock f_Vernier by feedback to the Vernier comb, the repetition rates will each be stabilized to the LO laser. As a consequence, f_clock should be stabilized, completing the clock frequency division. Importantly, this is irrespective of the free-running CEO frequencies of both combs. This simplifying scheme allows us to construct our clock system with two servos, removing requirements for a third servo to stabilize the offset frequency. The idea of eliminating a certain frequency comb instability to simplify a complex system is somewhat similar to the transfer oscillator concept for comparing two frequency standards²⁹.

Microcomb stabilization

To achieve the clock frequency division from the optical reference, we first stabilize the 871 nm LO laser to one optical line from a stable FC f_FC through an offset phase lock referenced to an RF synthesizer, where the offset frequency f_LO-FC ≈ ±910 MHz (the sign is variable in our system) is defined as

?LO−FC=?871−?FC. (6)

We then phase-lock the f_xCEO and f_Vernier beats to two other RF synthesizers by feedback to the pump power of the main and Vernier combs, respectively, using intensity modulators before input coupling to the two microrings. This stabilization scheme without CEO frequency locking avoids tuning the pump frequency and intensity simultaneously, which may introduce crosstalk between repetition rate and CEO frequency control³⁰. Figure 3a–f shows the electrical spectrum analyser traces of the three beats before and after phase locking. The three RF synthesizers for the offset locks, the FC and the frequency stability testing utilities are all synchronized to a common GPS-disciplined oscillator (GPSDO) to eliminate relative drifting.

To verify the repetition rates for both microcombs are stabilized to the optical reference (i.e., FC), we conduct out-of-loop measurements using an electro-optic frequency comb³¹ to downshift the repetition rates to an electronically detectable range. The time traces of the repetition rates are recorded by a zero dead-time frequency counter running at 1 ms gate time. We calculate the fractional Allan deviation of the two repetition rates, as shown in Fig. 3g. The frequency stability of the FC relative to the GPSDO is also obtained by measuring its ~250-MHz-repetition-rate beat note using a phase noise test set (PNTS) (Fig. 3g, blue trace). This is an estimate of the fractional frequency instability of the optical modes of the FC (Supplementary Section 2).

Both microcomb repetition rates follow the optical reference (Fig. 3g, orange and yellow traces), indicating that the stability of the LO laser has been successfully transferred to the two terahertz repetition rates while the combs’ CEO frequencies remain free-running. We demonstrate this by beating the pump with a FC line at ~1,550 nm and recording the unstable output with a frequency counter (Fig. 3g, purple trace). Frequency division from f₈₇₁ to f_clock is complete, following the expression (see further discussion in Supplementary Section 1)

?clock=117,292(?871−2?xCEO+385?Vernier)≈117,292?871. (7)

RF clock performance

We characterized the fractional frequency instability of the RF clock output with the PNTS. The results are shown in Fig. 4a. Although, initially, the fractional instability (orange line, ‘initial clock’) decreases approximately inversely with the averaging time τ, as expected, it shows excess noise with ~5× higher fractional instability than our system floor (blue line, FC) and plateaus at a value of ~10⁻¹² for τ ≈ 1 s and beyond. A similar effect has been reported in other recent works^32,33 but was not investigated therein. We attribute this excess noise to time-varying phase perturbations in the fibre leads that bring light in and out of the individual microrings, as illustrated in Fig. 4b. Related noise effects are well known in the distribution of ultrastable frequency references and coherent optical carriers over fibre links and are usually addressed by feedback control to an acousto-optic frequency shifter to stabilize a heterodyne beat involving two-way propagation through both the acousto-optic and the fibre link^34,35,36. Here, we adopt an open-loop electronic mixing technique that reduces the excess noise from our clock signal without the need for two-way propagation and brings the frequency instability down to a level near our system floor, explained below. The idea is similar to the transfer oscillator scheme to remove residual fibre noise³⁷.

Because f_clock can be generated by photodetecting the first sidebands at either side of the pump (f_clock+ and f_clock−), which carry frequency noise with equal amplitude but opposite sign, we can suppress the frequency noise by summing these two clocks (see Methods for details). We implement this differential noise suppression scheme by detecting f_clock+ and f_clock− on separate photodetectors in parallel and frequency mixing the resulting heterodyne beats (Fig. 4c). In practice, we mix a frequency-divided f_clock+/168 and a copy of f_clock− upshifted by 40 MHz and subsequently frequency divided by 168 (Fig. 4c and Methods). The result is a noise-suppressed RF clock output at ~235 MHz.

To demonstrate OFD, we measure the frequency of the optical reference f₈₇₁ and the RF clock output as a function of time using two synchronized counters with the two repetition rate locks in place. The results are plotted in Fig. 4d,e. The y-axis span of the optical reference is exactly 17,292 × 84 times that of the RF clock, corresponding to the OFD factor of 17,292 (equation (7)) and the electronic division factor of 84 (= 168/2) for bringing down the clock frequency to within counter and electronic mixer ranges. The LO laser is initially free-running, and we switch on the lock to the FC midway through the measurement. The RF clock frequency follows that of the LO laser, both when the laser is free-running and when it is locked. The larger frequency fluctuations in the clock trace are due to the counter measurement limitations (Methods and Supplementary Section 7).

The fractional Allan deviation of our ~235 MHz noise-suppressed RF clock output, measured using the PNTS, is plotted in Fig. 4a (green trace). The frequency instability is substantially reduced, now essentially overlaying with that of the optical reference. These results signify that the frequency stability of the 344 THz LO laser has been successfully transferred to our ~235 MHz RF clock. This suggests that our Vernier OFD system supports a fractional frequency instability of at least ~3 × 10⁻¹³/τ. The frequency instability with longer averaging time reaches ~2 × 10⁻¹⁵ at 1,000 s (Fig. 4a, purple trace; see Supplementary Section 6 for more information). The performance of our OFD system is sufficient to support an atomic reference with stability better than the commercially available caesium clock with <8.5 × 10⁻¹³ at 100 s.

Our method also allows us to extract the differential-mode interferometric frequency noise directly (Methods), which we also plot in Fig. 4a (yellow trace).

Discussion

We have demonstrated the use of a Vernier dual-microcomb system to frequency divide an ultranarrow-linewidth 871 nm laser down to an RF output of ~235 MHz using only two feedback servos, enabled in part by an interferometric noise suppression scheme. The LO laser is tuned to within a few gigahertz of being able to be frequency doubled to a clock transition of a ytterbium ion.

The Vernier dual-microcomb OFD platform used here is useful for detecting high-frequency f_CEO and potentially for performing OFD on a variety of atomic species through a dual-comb sum-frequency process. Our simple dual-comb OFD architecture involving only one pump and two servo locks helps to save auxiliary photonic and electronic components, which is also of paramount importance towards ultimate development of a low size, weight and power package. With the continued advancement of microcomb system integration capabilities³⁸, in the future, our dual combs can potentially be integrated onto a single SiN chip, together with on-chip thermal heaters for spectral alignment and microcomb feedback, and spectral filtering for separation and routing of different wavelength bands. Furthermore, there has been exciting progress in the heterogeneous integration of III–V^39,40 and thin-film lithium niobate^41,42 materials on the SiN platform. These may enable the integration of III–V lasers for microcomb pumping and PPLN for second-order nonlinear frequency conversion towards a fully integrated dual-comb system in the future. In addition, through the elimination of many of the fibres used in our current experiment, such a system should be much less sensitive to environmental perturbations.

With further advances in compact ion traps and optical lattices^8,9,43,44, we anticipate that our dual-microcomb system, paired with an integrated atomic reference and compact narrow linewidth laser, may one day enable the development of a fully integrated high-performance optical atomic clock.

Methods

Experimental setup

The experimental setup is shown in Extended Data Fig. 1. We use a single-sideband modulator to rapidly frequency sweep⁴⁵ a ~1,550 nm external cavity diode laser (Toptica CTL 1550) pump into resonance for both microrings simultaneously, generating microcombs in each. The two resonators are frequency aligned using thermo-electric temperature control units. After comb generation is initiated, the pump frequency is adjusted through piezoelectric control of the external cavity diode laser towards a longer wavelength to optimize the dispersive wave conditions of the microcombs, where we obtain a high-power (~−20 dBm) short-wavelength dispersive wave emission from the main comb at ~978 nm. In this dispersive wave state, we also make use of the long-wavelength dispersive wave of the main comb, which benefits the SFG process for f-2f. A network of fibre optical filters and couplers are used to separate spectral bands at ~1,550 nm, ~2,000 nm and ~1,000 nm and combine the dual-comb spectra at ~1,550 nm and ~2,000 nm. For the results shown in the main text, we have introduced foam partitions to cover most of our setup to reduce noise effects due to air currents and the like (for further discussion, see Supplementary Section 3).

The combined 2,000 nm spectral components are amplified using a thulium-doped fibre amplifier and then distributed in three arms for (1) the SFG process for f-2f, (2) the SFG process to hit 871 nm via combining with the 1,550 nm residual pump from the Vernier comb and (3) the Vernier beat detection. The SFG products for f-2f are subsequently amplified by a semiconductor optical amplifier (Innolume) and filtered by a 1-nm-bandwidth bandpass filter (Photonwares) to suppress the broadband amplified spontaneous emission noise. It is then combined with the ~1,000 nm light from the main comb to generate the f-2f beat f_f-2f using a balanced photodetector (BPD). The SFG products at 871 nm from the Vernier comb are combined with 25% of the 871 nm LO laser using a 75:25 coupler to generate f_comb-LO, with ~−36 dBm of SFG (spectrum shown in Fig. 2d) and ~−4.5 dBm of laser power measured at the photodetector. Seventy-five per cent of the LO laser is combined with the FC (bandpass-filtered at ~871 nm) using a 50:50 coupler and f_LO-FC is detected on another BPD, with ~3 dBm of LO power and ~−56 dBm power per FC line at one arm of the BPD (where the laser and FC spectra in Fig. 2d are measured). The clocks at both sides of the pump are selected by a programmable optical filter (Finisar), optically amplified in the same erbium-doped fibre amplifier and separated by a dense wavelength-division multiplexing filter for detection on two distinct photodetectors.

The Vernier beat f_{V ernier} detected by an amplified photodetector is divided by 8 and sent to an offset phase lock servo (Vescent D2-135). We obtain f_xCEO/8 by mixing f_f-2f divided by 8 and f_comb-LO divided by 16. This beat is sent to another phase lock servo. We use two stable RF synthesizers (Agilent E8257D and Keysight 33600A) as the frequency references for f_Vernier/8 and f_xCEO/8, synchronized to a 10 MHz GPSDO (EndRun Meridian). Finally, for our frequency instability measurements, we use frequency counters (Keysight 53230A) and a PNTS (Microsemi 5125A), which will be discussed in the Noise-suppressed RF clock section.

Interferometric noise suppression scheme

In Fig. 4b, we show the phase fluctuations in the upper and lower arms as ?₁(t) and ?₂(t), respectively. ?₁(t) is composed of the time-varying phase experienced by the pump in the interferometer arm upstream of the main ring plus the phase picked up by the comb line of interest downstream from the main ring; ?₂(t) is defined similarly but refers to the leads connecting the Vernier ring. As illustrated in Fig. 4c, we term the two clocks detected at different sides of the pump f_clock+ and f_clock−, corresponding to the higher- and lower-frequency side, respectively,

ππ?clock+=[?pump+?rep1+?1′(?)2π]−[?pump+?rep2+?2′(?)2π], (8)

ππ?clock−=[?pump−?rep2+?2′(?)2π]−[?pump−?rep1+?1′(?)2π], (9)

where ?′(?) represents the time derivative of ?(t). These expressions take into account the frequency fluctuations that arise in proportion to the derivative of the time-varying phases⁴⁶. Because ?₁(t) and ?₂(t) may be uncorrelated, they give rise to frequency noise on the clock signal. Critically, the frequency noise on f_clock+ and f_clock− is equal and opposite in the expressions above. By summing f_clock+ and f_clock− with an electronic mixer

?clock++?clock−=2(?rep1−?rep2), (10)

we can ideally suppress the frequency noise completely. Note that the formulation above ignores any noise contributed by the electronics and approximates the phase shifts incurred by comb lines ±1 in the fibre leads downstream of the resonators as equal. Because the frequencies of the ±1 modes differ from the pump frequency by only a small fractional amount, the error involved in this approximation is small. Further analysis on the noise suppression scheme can be found in Supplementary Section 4.

Noise-suppressed RF clock

Our noise suppression scheme relies on mixing f_clock+ and f_clock− to cancel differential-mode noise in the RF clock output. We mix frequency-divided versions of these two signals in practice (f_clock+/168 and f_clock−/168). Importantly, the sum of these two signals is almost exactly equal to the second harmonic of either (that is, f_clock+/168 + f_clock−/168 ≈ 2f_clock+/168 ≈ 2f_clock−/168). This means that it is impossible to filter out the undesirable second harmonics originating in the mixer and get the pure sum-frequency signal, thus still resulting in excess noise. To avoid this problem, we frequency shift one of the clocks by mixing with the 40 MHz signal (generated by twice-doubling the 10 MHz GPS-disciplined reference) to obtain f_clock− + 40 MHz, filtered by a 30-MHz-bandwidth yttrium iron garnet bandpass filter. This frequency-upshifted clock is being divided and electronically mixed with the other divided clock to realize f_clock+/168 + (f_clock− + 40 MHz)/168 at the output. The desired sum-frequency output is then distinguishable from harmonics of either of the mixer inputs. Although the second harmonic and other products still exist and are spaced by ~238 kHz (40 MHz/168), the desired noise-suppressed output is ~20 dB higher than the other products and the PNTS can track the desired signal (see Supplementary Fig. 8 for the RF spectrum). Importantly, we note that the 10 MHz reference used here to shift one of the clock products should in principle be able to be derived from an optical reference if our system were to perform OFD on an atomic specimen (somewhat similar to ref. ¹³). This would remove the need for a GPS-referenced 10 MHz sync signal to operate the OFD system.

For Fig. 4d,e, we measure the frequency traces of the optical reference and the RF clock output simultaneously on two synchronized frequency counters running at 100 ms gate time with the same trigger signal and external gate. The optical reference frequency f₈₇₁ is obtained by recording f_LO-FC and calculating f₈₇₁ using equation (6). We attribute the larger frequency fluctuations in the RF clock trace compared with the optical reference to the spurious frequency content in the output signal in addition to the typical counter timing jitter-related resolution limit. Although the weak spurs cause no notable problem for the PNTS (because the noise-equivalent bandwidth of PNTS at 1 ms gate time is 500 Hz and spurs fall outside the band), they do result in a few-fold increase in the frequency fluctuations reported by the frequency counter (see Supplementary Sections 7 and 8 for further discussion).

The Allan deviation for the RF clock output and the FC (Menlo Systems) reference were taken several times over multiple days. The symbols and error bars plotted in Fig. 4a represent the averages and standard deviations from multiple measurement results. The error bars are relatively tight, indicating good repeatability.

Interferometric noise extraction

The noise suppression scheme can be modified to extract the interferometric noise for direct measurement. As explained in the main text, by adding f_clock+ and f_clock− the differential frequency noise can be made to cancel out. By measuring the difference between f_clock+ and f_clock−, the clock term is cancelled and we obtain the frequency noise term δf_N(t). To implement the noise measurement, we frequency upshift one of the clocks by 200 MHz and divide it by 168, then frequency mix with the other clock also divided by 168 (Extended Data Fig. 2). This creates a frequency difference of ~1.2 MHz out of the mixer, which is within the frequency range of our PNTS (1–400 MHz). The fractional instability of this signal filtered using a low-pass filter (normalized to the ~235 MHz noise-suppressed clock frequency) is shown as the yellow curve in Fig. 4a and is similar to the curve for the clock output without noise suppression.

Data availability

The data used to produce the plots within this Article are available via Figshare at https://doi.org/10.6084/m9.figshare.23971425 (ref. ⁴⁷).

Code availability

The code used to produce the plots within this Article is available via Figshare at https://doi.org/10.6084/m9.figshare.23971425 (ref. ⁴⁷).