Demonstration of a time scale based on a stable optical carrier

TL;DR

This paper presents a novel optical time scale using a stable optical carrier and cryogenic silicon cavity, achieving high long-term stability and outperforming existing microwave standards.

Contribution

It introduces a new all-optical time scale architecture that combines a cryogenic silicon cavity with an optical clock, demonstrating superior stability over traditional methods.

Findings

Estimated time error of 48±94 ps over 34 days

Outperforms existing microwave time scales

Capable of reaching stability below 10^-17 after months

Abstract

We demonstrate a time scale based on a phase stable optical carrier that accumulates an estimated time error of $48\pm94$ ps over 34 days of operation. This all-optical time scale is formed with a cryogenic silicon cavity exhibiting improved long-term stability and an accurate $^{87}$Sr lattice clock. We show that this new time scale architecture outperforms existing microwave time scales, even when they are steered to optical frequency standards. Our analysis indicates that this time scale is capable of reaching a stability below $1\times10^{-17}$ after a few months of averaging, making timekeeping at the $10^{-18}$ level a realistic prospect.

Full text

††preprint: APS/123-QED

Accurate and precise timing is critical for a wide array of applications, ranging from navigation and geodesy to studies of fundamental physics Ashby (2003); Roberts et al. (2017); Delva et al. (2018); Takano et al. (2016); Lombardi et al. (2016); Kennedy (2019). The worldwide time standard, Coordinated Universal Time (UTC), is synthesized from a global network of atomic clocks and disseminated at monthly intervals. National metrology institutes bridge the gap between updates of UTC by broadcasting independent time scales derived from ensembles of microwave local oscillators steered to accurate atomic frequency standards Bauch et al. (2012); Rovera et al. (2016). To advance the frontier of precision timekeeping, the development of both improved local oscillators and atomic frequency standards is imperative.

Optical atomic clocks, orders of magnitude more accurate and stable than their microwave counterparts Bloom et al. (2014); Nicholson et al. (2015); McGrew et al. (2018); Takano et al. (2016); Huntemann et al. (2016); Brewer et al. (2019), show promise as frequency standards for time scale applications. Recent efforts to incorporate optical clocks into existing microwave timescales have lead to improved performance Hachisu et al. (2018); Grebing et al. (2016); Yao et al. (2019). However, despite the fact that optical clocks have demonstrated mid-$10^{-17}$ level stability in one second of averaging Oelker et al. (2019); Schioppo et al. (2017), time scales steered to optical standards have thus far required weeks of averaging to reach $10^{-16}$ level precision Yao et al. (2018, 2019). This disparity in performance arises due to down conversion of noise from the local oscillator – a consequence of steering to an atomic standard in the presence of dead time – which degrades the long-term stability of the time scale Yao et al. (2019). This limitation motivates the development of local oscillators with improved stability, particularly at averaging times around the typical interval between clock measurements ($10^{3}$ to $10^{5}$ s). In parallel, improvements in local oscillator stability allow a timescale to maintain a competitive level of performance even when relaxing the requirements on optical clock uptime.

In this Letter, we report on the first realization of an all-optical time scale that outperforms state-of-the-art microwave oscillators steered to either microwave or optical frequency standards. This time scale consists of an optical local oscillator (OLO) based on a cryogenic silicon reference cavity which is steered daily to an accurate 87Sr lattice clock Bothwell et al. (2019) over a month-long campaign. During this period, the frequency stability of the OLO surpasses that of the hydrogen masers in the UTC(NIST) time scale at all averaging intervals up to multiple days Sup , demonstrating the requisite stability for improved time scale performance. Our analysis indicates that daily steering of the OLO frequency with 50% clock uptime allows for a time scale instability below the $10^{-17}$ level within 85 days of operation. Our local oscillator frequency is easily predictable using conventional time scale steering algorithms, allowing us to limit the estimated time error to only $48\pm 94$ ps after $34$ days of operation. The continuous availability of the OLO coupled with the on-demand performance of our optical clock make our system viable for future inclusion in UTC(NIST). This new variant of time scale harnesses both the improved accuracy and stability of optical standards and provides a viable blueprint for the upgrade of time scales worldwide.

After a decade of development Kessler et al. (2012); Zhang et al. (2017), cryogenic silicon reference cavities are now a proven platform for laser stabilization at the mid-$10^{-17}$ level Robinson et al. (2019); Matei et al. (2017). The exceptional short-term stability of these local oscillators has enabled advances in optical clock stability Oelker et al. (2019). These systems outperform all free-running local oscillators at averaging times below $1\times 10^{4}$ seconds Oelker et al. (2019) and exhibit orders-of-magnitude lower frequency drift than other OLOs Hagemann et al. (2014); Robinson et al. (2019). However, achieving a stability commensurate with the best microwave oscillators at longer averaging times has remained an elusive goal, hampering their usefulness as time scale flywheel oscillators. The OLO used in our time scale, based on a $21$ cm long Si cavity operating at $124$ K, was recently optimized to significantly improve its long-term stability. The use of super-polished optics and thermal control of the environment limit parasitic etalons and active optical power stabilization reduces frequency excursions from laser intensity drift Sup .

We combine our local oscillator with an accurate optical frequency standard to form an all-optical time scale. Over a 34 day interval, a strontium lattice clock with systematic uncertainty of $2.0\times 10^{-18}$  Bothwell et al. (2019) is used to track the OLO frequency with 25 percent uptime. Daily measurements of the OLO allow us to build a reliable predictive model of its frequency evolution. As new frequency data become available, the model is updated to better reflect its current behavior. The OLO is steered using the model to correct for changes in its frequency over time, and any residual frequency fluctuations ultimately determine the time scale stability. The analysis required to realize the time scale was carried out in post-processing, though we emphasize that our approach is compatible with real-time implementation.

To track frequency excursions larger than the low-$10^{-16}$ level during intervals when the optical clock is offline, the OLO is compared with two independent ultrastable lasers based on a $6$ cm silicon cavity operated at $4$ K Robinson et al. (2019) and a $40$ cm ultra-low expansion (ULE) cavity Bishof et al. (2013). Because the three systems have comparable short-term stability, one may use a three-cornered hat analysis to identify any significant frequency jumps in the OLO and update the predictive model accordingly Sup .

A schematic of our optical time scale is presented in Fig. 1(a). In order to reference the 87Sr clock laser to the 124 K silicon cavity, we transfer its optical stability from 1542 nm to a prestabilized laser at 698 nm using a femtosecond Er:fiber frequency comb with negligible additive instability Oelker et al. (2019). The frequency corrections applied to AOM1 by the stability transfer servo are recorded to monitor the relative frequency fluctuations between the 40 cm ULE cavity and the OLO. The stabilized 698 nm light is then tuned to resonance for the 87Sr clock transition using AOM2. The AOM2 correction signal is recorded and yields the OLO frequency relative to the 87Sr transition. An optical beatnote at 1542 nm between the OLO and the 6 cm Si cavity serves as a continuous monitor of their frequency difference. Fig. 1(b) depicts AT1, a free running microwave time scale at NIST. Using a hydrogen maser as a transfer oscillator, AT1 is compared remotely with the local oscillator over a stabilized fiber-optic link Sup . To enable this comparison, the OLO is down converted to the RF domain using a frequency comb. This provides an additional record of the long-term performance of the OLO that is nearly continuous (95% uptime) over the measurement campaign. We note that AT1 is chosen rather than UTC(NIST) due to its superior stability over the averaging intervals of relevance to this study.

A record of the OLO frequency during the data campaign spanning from a modified Julian date (MJD) of $58430$ to $58464$ is presented in Fig. 2(a). The clock ran daily with the exception of MJD 58444 and 58447. Three days before the first measurement, the optical power incident on the cavity was changed to reset an intensity noise servo. Consistent with prior silicon cavity drift studies, the frequency evolution of the OLO after adjusting the incident optical power is well modeled by a constant linear drift plus an exponential relaxation term: $a+bt+ce^{-\frac{t}{d}}$ Robinson et al. (2019). Fig. 2(b) shows the residuals of the OLO comparisons with the clock and AT1 after subtracting the modeled drift trend determined by a fit to the 87Sr clock data. Perfect correlation between the two data sets is not expected as both AT1 and the microwave link contribute additional instability to the Si-AT1 record Sup .

During the interval between clock operation on MJD 58441 and 58442, two frequency jumps on the OLO were identified with a combined amplitude of $5.02\times 10^{-15}$. A correction of the same magnitude is applied to all data after this step when performing the analysis presented in this work. No significant change in the long-term drift trend of the local oscillator was observed following these excursions Sup .

To realize a time scale, the OLO frequency record in Fig 2(a) is steered using a predictive model to minimize its offset from the atomic frequency standard. The predictive model utilizes a Kalman filter to estimate the frequency of the OLO at a given time based on prior measurements with the clock. Kalman filtering techniques are commonly used in time scales to model the frequency of hydrogen masers Levine (2012); Yao et al. (2017). These models approximate the hydrogen maser as a linearly drifting oscillator and update the model parameters as new frequency measurements arrive. The drift in the OLO frequency between daily measurements can be modeled using a quadratic function: $k_{0}+k_{1}t+\frac{k_{2}t^{2}}{2}$ and traditional Kalman filtering techniques are applicable. The model prediction is determined by a state vector $[k_{0},k_{1},k_{2}]$ that is updated epoch-by-epoch when the 87Sr clock is running. Further detail on the Kalman filter algorithm is provided in Sup .

To evaluate the performance of a time scale, one typically compares it against a reference time scale with significantly lower timing uncertainty. To our knowledge, no such time scale exists in this case. Instead we treat the 87Sr clock as an ideal frequency reference and examine the fractional frequency offset between the steered OLO record and the clock transition frequency, hereafter referred to as the prediction error. We define the time error of our time scale as the integral of the prediction error over time.

If the frequency record were continuous, the time error could be determined to within the measurement precision of the clock. However, a finite gap of time separates the frequency measurements in Fig. 2(a), ranging from the $5$ second interrogation cycle of our experiment to $24$ hours between daily measurements. Most of the time error accumulates during the longer gaps, when the Kalman filter must accurately predict changes in the OLO frequency without new measurement data from the clock. The time error contribution from a gap is simply the gap duration multiplied by the mean prediction error during this interval. However, the latter quantity cannot be determined exactly from the available data. Instead, we estimate the mean prediction error by averaging the values before and after the gap and multiply by the gap duration to compute an estimated time error. We compute a $1\sigma$ confidence interval for the estimated time error through repeated simulations of the OLO frequency during each gap to determine the uncertainty in the estimation of the prediction error Sup .

An estimate of the integrated time error of our optical time scale is presented in Fig. 3. After 34 days of integration our all-optical time scale accumulates an error of $48\pm 94$ ps. For comparison we simulate time scales consisting of a hydrogen maser steered to a 133Cs fountain for 24 hours/day and a hydrogen maser steered to a 87Sr optical clock for 6 hours/day using the same Kalman filter and noise models for the maser and fountain described in Yao et al. (2018). The typical performance of both time scales is assessed by computing time errors from repeated simulationsSup , and their RMS spread over a 34 window is depicted in Fig. 3. Both exhibit a larger time error than the all-optical time scale.

Because the optical clock is run intermittently, the long-term stability of the time scale will be limited by a $1/\sqrt{\tau(s)}$ slope arising from aliased local oscillator noise akin to the Dick effect Yao et al. (2018, 2019). Determining this stability limit requires an accurate characterization of the OLO. We evaluate the stability of our OLO by analyzing the frequency noise of the residuals in Fig. 2(b). One complicating factor are the gaps in the frequency record during clock downtime. A gap-tolerant Allan variance similar to Sesia and Tavella (2008) is used to compute an estimated stability of the OLO out to multiple day averaging intervals.

The result of this analysis is plotted in Fig. 4. The OLO stability is fit to a noise model that includes the known thermal noise floor Oelker et al. (2019) and a random walk frequency noise term, resulting in an instability at long averaging times consistent with $\sigma_{RW}=1.3\times$ $10^{-18}\sqrt{\tau\textrm{(s)}}$. The OLO maintains an instability below $10^{-15}$ out to $6\times 10^{5}$ s, more than an order of magnitude improvement over the previous characterization of this system Matei et al. (2017). The frequency stability of the Si-AT1 record is presented as well and its value at averaging times past $10^{5}$ s agrees with the clock measurement within statistical uncertainty. At shorter averaging times, the stability is consistent with a noise model Sup accounting for instability from the microwave link, the OLO, and AT1 McGrew et al. (2019).

With an accurate noise model for the OLO in hand, we now consider the anticipated long-term stability of our time scale as a function of optical clock duty-cycle. Similar to Yao et al. (2019, 2018), we simulate a lengthy local oscillator frequency record using the model presented in Fig. 4 with the drift trend from Fig. 2(a) added. This record is then steered to a simulated 87Sr lattice clock for a fixed interval each day using the same Kalman filtering techniques described above. We compute an Allan deviation of the prediction error to determine the stability of the time scale. To quantify the impact of our improved local oscillator we carry out the same analysis for a similar time scale where the OLO has been substituted with a hydrogen maser. The noise model for the simulated hydrogen maser is based on the typical stability of the best performing masers in the UTC(NIST) time scale Yao et al. (2018).

Fig. 5 shows the results of our analysis. As anticipated, the long-term stability of the time scale improves with increased clock uptime and reduced local oscillator noise and is reasonably consistent with the expected instability limit from aliased local oscillator noise past $10^{6}$ s Sup . When the optical clock is run with the same duty cycle, the steered OLO significantly outperforms a steered hydrogen maser at all averaging times. Even when steering one hour per day, our time scale is more stable than a hydrogen maser steered with a 50 percent duty cycle. This capability allows for competitive time scale performance with significantly relaxed uptime requirements. Based on this analysis, we expect a stability of approximately $1.8\times 10^{-17}$ after a 34 day campaign with an average clock uptime of 6 hours/day. This is in good agreement with the observed integrated time error of $48\pm 94$ ps over 34 days, or $1.6\pm 3.2\times 10^{-17}$ in fractional units. When operating the clock 12 hours per day, our all-optical time scale remains at or below the $2\times 10^{-16}$ level at all averaging times and is projected to reach a stability below $10^{-17}$ after only 85 days of operation. Additional effort on automation should allow for a clock duty cycle well above $50\%$. Using an array of $N$ independent silicon cavities would improve the stability by a factor of 1/$\sqrt{N}$ Yao et al. (2019).

By combining an improved local oscillator with an accurate high-uptime optical clock, we have demonstrated a novel time scale architecture with enhanced stability. Additional technical upgrades of our silicon cavity can further improve our optical time scale stability, including greater passive thermal isolation, shorter optical path lengths and operation closer to the silicon coefficient of thermal expansion zero crossing. In addition, reducing the optical power incident on the cavity offers the capability to reduce the linear drift Robinson et al. (2019).

Future efforts will leverage existing time transfer infrastructure in Boulder, CO to incorporate this optical technology into the UTC(NIST) time scale. An underground fiber network is in place to support phase-stabilized optical signal transfer from JILA to NIST with negligible excess noise Foreman et al. (2007a, b). Using a femtosecond frequency comb Leopardi et al. (2017); Fortier et al. (2006), our optical time scale signal will be linked to UTC.

W.R.M. and E.O. contributed equally to this work.