Synchronization of Distant Optical Clocks at the Femtosecond Level

The use of optical clocks/oscillators in future ultra-precise navigation, gravitational sensing, coherent arrays, and relativity experiments will require time comparison and synchronization over terrestrial or satellite free-space links. Here we demonstrate full unambiguous synchronization of two optical timescales across a free-space link. The time deviation between synchronized timescales is below 1 fs over durations from 0.1 s to 6500 s, despite atmospheric turbulence and kilometer-scale path length variations. Over several days, the time wander is 40 fs peak-to-peak. Our approach relies on the two-way reciprocity of a single-spatial-mode optical link, valid to below 225 attoseconds across a turbulent 4-km path. This femtosecond level of time-frequency transfer should enable optical networks using state-of-the-art optical clocks/oscillators.

General synchronization concept. Time information is transmitted between sites across a turbulent air path. Real-time feedback is applied to the clock at site B to synchronize it with the clock at site A. A folded optical path allows for verification of the synchronization by a direct "out-of-loop" measurement. (b) Measured timing deviation, or precision, between the time outputs while synchronized across a 4 km link, based on data acquired over two-days as described in Section IV.

II. Synchronization between two distant optical timescales using two-way time transfer
A first requirement is to create an optical timescale at each site. The name notwithstanding, state-of-the-art atomic optical clocks are operated as frequency standards; they output an optical frequency from a laser stabilized to an optical cavity and atomic transition. Therefore, atomic optical clocks are compared by their frequency ratios, typically via a frequency comb. To create a timescale, we phase lock a self-referenced frequency comb to the cavity-stabilized laser. The optical pulses output by the frequency comb are then analogous to the "ticks" of a conventional clock. To generate a local time, we then define a reference plane and label the comb's optical pulses according to their arrival at the reference plane. In other words, a local controller tracks the pulse number and converts this to a corresponding time using the assumed underlying cavitystabilized laser frequency. This extension to a timescale does require a phase-continuous connection between the comb and the cavity stabilized laser. More generally, this laser could be stabilized to an atomic transition to provide an absolute timescale at a single master site or at both sites for relativity experiments.
We first review the simplest implementation of two-way transfer, before discussing the modified optical two-way transfer demonstrated here. Consider two clocks at separate sites A and B. Suppose site A transmits a pulse at its zero time to site B. Its Averaging time (s) calibration constant, cal  to account for time offsets in the transceivers. Summation of the two signals provides link T and therefore the distance between sites given the speed of light. The optical clock output is a train of very short optical pulses with femtosecond residual timing noise. However, we cannot implement the simple two-way protocol discussed above with direct photodetection of the pulse arrival time, as this immediately introduces picosecond-level uncertainty due to the limited photodetector response time and low light levels (for any reasonable link loss). Instead we implement linear optical sampling between frequency combs with offset repetition rates [35]. The overall setup is illustrated in Fig. 2.
Linear optical sampling requires the transmission of comb pulse trains between sites that differ in repetition rate. On the other hand, synchronization requires optical timescales (comb pulse train) at each site that operate at the same repetition rate. Therefore, we require three combs: a comb at each site with a repetition rate f r , and a third transfer comb at site A, with a repetition rate f r +f r . (This configuration supports not only comparison but the synchronization of site B to the master site A.) The relative evolving time offsets between pulses from the three frequency combs are measured through linear optical sampling via the three balanced photodetectors in Fig . As outlined in Appendix B, we can then derive a "master synchronization equation"the analogy of the simple two-way time transfer equation given earlierfor the time offset between site A and site B as, where link T is the time-of-flight across the link, ADC t  is the time offset in the analog-to-digital converters (ADCs) at the two sites, n  is an integer related to the pulse labelling, and cal  is a calibration offset related to transceiver delays and the selected location of the reference planes. The first three terms of Eq. (1) comprise a generalized two-way time transfer expression. As derived in Appendix B, there are the two additional terms, one proportional to f r and one proportional to n. The latter simply accounts for the   1/ 2 r f ambiguity in the pulse labeling. The former is a small correction accounting for the mismatch in repetition rates between the transfer comb's pulse train and the optical timescales. This mismatch is necessary for the linear optical sampling, but leads to an incomplete cancellation in the two-way comparison of both the path delay and the relative timing of the digitizers used with the photodetectors. The term is small since it is proportional to   2~1 200,000 rr ff  but its inclusion is needed for correct time comparison and synchronization.
The frequency-comb-based measurements cannot provide a value for these last two terms. Instead, we require a "coarse" two-way time transfer measurement that measures . It must also measure ADC t  to below 1/(2f r ) to resolve the integer n  . In our system, the uncertainty of this "coarser" two-way time transfer needs to be below 200 ps, which is well within the capabilities of rf-based systems.
Finally, calculation of Eq. (1) requires information exchange between sites, which in turn requires rapid, real-time communication. Optical communication across a free-space link is well known to suffer from dropouts due to turbulence. Here, however, that problem is moot, as the optical communication channel uses the same single mode spatial link as the comb light. Any turbulence-induced dropouts are correlated and therefore communication is available whenever the timing information is available. Figure 2 shows a high-level view of the physical system. Sites A and B are connected via a free-space single-spatial-mode optical link covering up to 4 km. This link is folded by use of two plane mirrors so that sites A and B are physically adjacent, enabling synchronization verification via an out-of-loop measurement of the actual time offset, T, independent of the "in-loop" predicted time offset AB T  .

III. Experimental implementation
The full system includes two cavity-stabilized lasers, two Doppler cancelled links that carry the light from the cavity-stabilized lasers to the frequency combs, three self-referenced combs, a coherent communication link, coarse two-way phase modulated time transfer, a feedback system to the site B timescale (the synchronized site), field-programmable gate array (FPGA) controllers, and two free-space terminals. We discuss some of the salient details below and in Appendix C.
The cavity-stabilized lasers for both sites are located in an environmentally stable laboratory that is 200 m from the main transceivers. A commercial cw fiber laser is locked to an optical cavity with a ~1 Hz linewidth and a typical environmentally-induced frequency drift ranging from 0 Hz/s to 10 Hz/s. The frequency of the cavity-stabilized laser is 195.297,562 THz for site A and 195.297,364 THz for site B. Two separate Doppler-cancelled fiber links transport these frequencies to the comb-based transceivers located in a rooftop laboratory. The phase-lock of the cavity-stabilized lasers and the Doppler cancelled links are monitored during synchronization to ensure that no phase slips occur.
The three frequency combs are each self-referenced with FGPA-based digital control and can operate for days without phase slips [39]. The 972,920th mode of site A's frequency comb is phase-locked to its cavity-stabilized laser to yield a repetition rate of 200.733,423 MHz. The transfer comb is phase-locked to the same cavity-stabilized laser to yield a repetition rate that differs by 2.27 kHz r f  . At site B, the 972,919th comb mode is locked to the second cavitystabilized laser, yielding a repetition rate very close to the comb at site A. The rf offset of this phase-lock is adjusted to maintain synchronization. For loop stability considerations, the bandwidth of this feedback should be below r / 4 f  = 500 Hz. Here, however, based on the freerunning noise of the cavity-stabilized laser and measurement noise level on AB T  , a 10 Hz feedback bandwidth minimizes the residual jitter. The combs, as well as the heterodyne detection modules, are enclosed in small temperature controlled aluminum boxes within a larger transceiver box, which is loosely temperature controlled. A 16 nm wide (2 THz) section of the comb spectrum, centered at 1555 nm, is transmitted over the link with a power at the transmit aperture of 2.5 mW. FIG 2. (Color, two column) Two optical timescale, at site A and site B, each comprise a cavity-stabilized laser, Doppler-cancelled link, and self-referenced frequency comb. The clock "time" is defined by the arrival of the comb's pulses at the reference plane. (The controller tracks the time label associated with each pulse.) Two-way optical exchange over the free-space link allows comparison, and further synchronization, of the two timesclaes so that their pulses arrive at the reference plane at a common time, as shown in the upper display. An entire unambiguous measurement of the time offset requires only 0.5 ms; therefore, the system operates robustly over the intermittent free-space link.
We implement the "coarse" two-way time transfer, needed to establish the rightmost terms in Eq. (1), as in rf-based two-way time frequency transfer [40], except that the timing signal is carried by phase modulation of an optical carrier (cw laser) with a pseudorandom binary sequence (PRBS), as described in Appendix C. The optical carrier traverses the same singlemode spatial optical link as the two-way comb light and therefore measures the same path delay over the air. The modulated cw light does have a different total path delay than the combs because of non-common mode fiber optic paths in the transceivers, but these are effectively included in the calibration of cal  and their variations are suppressed by in Eq. (1). As implemented here, this coarse two-way time transfer determines the unambiguous value of T link (and t ADC ) with an uncertainty of 40 ps, which is more than sufficient to support femtosecond synchronization via Eq. (1). The real-time optical communications link is implemented across the free-space link using the same hardware as the coarse two-way time transfer, as discussed in Appendix C. The communication is interposed between the coarse twoway time measurements within a single 1/ r f  interval so that Eq. (1) is updated at 1/ 0.5 ms r f  intervals. Finally, the modulated cw light and comb light are combined within the same single mode fiber and launched via free-space optical terminals with tip/tilt control to compensate for turbulence-induced beam wander. As shown in Fig. 1, we verify the time synchronization by direct "out-of-loop" measurements of the time offset, T, that are completely independent of the calculated value, AB T  . The most sensitive measurement of T is achieved by heterodyne detection between the two optical timescale outputsi.e. the 200 MHz pulse trains from the combsat a common reference plane. To do this, the carrier-envelope offset frequency of the frequency comb at Site B is purposefully offset relative to the comb at Site A by 1 MHz. In this case, the heterodyne signal of comb pulses overlapping in time at the reference plane appears at 1 MHz with an amplitude that depends on their time offset, as illustrated in Fig. 3a. The response is measured with a shorted link and cal  is selected so that a nominal zero time offset falls in the linear response region, i.e. the blue dot in Fig. 3a. The system is then operated over the link and the scaled demodulated amplitude provides a direct measurement of T at 0.5 ms intervals, as shown in Fig. 3b. Over the one hour interval, the full standard deviation is 2.4 femtoseconds. The next section provides similar data over a longer time period and for varying path lengths.

IV. Results
This heterodyne detection between the 200 MHz pulse trains does not verify that the timescales are unambiguously synchronized, i.e. that there are no 5-ns slips. Section IV.C provides data on comparison of an optical pulse-per-second output through direct photodetection. It also compares synchronous 1 Hz pulse bursts through direct detection of spatial interference fringes between the optical pulses. For the latter, we are observing optical spatial interference between the ~100 fs optical pulses of two sources that are connected only via a 4 km free space link. A. Synchronization over multiple days FIG 4. Synchronization data across 4 km over a 50 hour time period including, from top to bottom, the measured out-of-loop time offset T; the change in time of flight, T link ; the frequency correction applied to the timescale at site B to maintain synchronization; and the link availability. All data is filtered and downsampled from the 0.5 ms measurement period to 60 s. Figure 4 summarizes an experiment where site A and B were synchronized for over 50 hours across a 4 km free-space link. The system ran without user intervention despite 4 °C rooftop laboratory temperature changes and ended with the arrival of a large snowstorm. (The system is able to operate through light snow and rain but not under heavy precipitation.) The top panel plots the out-of-loop time offset as measured using the technique outlined in Fig. 3. These data are smoothed to 60 seconds. (An expanded view of the unsmoothed performance over short time periods was given in Fig. 3b.) The time-dependent offset is best analyzed by the timing deviation of these data, plotted in Fig. 1b, which is the uncertainty in the time offset as a function of averaging time [41]. From Fig. 1b, this uncertainty is below 1 fs out to 6500 s (1.8 hours), reaching a minimum of 225 attoseconds for a 10 s average. Therefore, we infer that the single spatial-mode link reciprocity over the 4 km air path is verified to below 70 nm at 10 s averaging and below 300 nm out to 6500 s. Fig. 4 shows that over the full 50 hour measurement, the time offset exhibits a larger 40 fs peak-to-peak wander. This time wander does not reflect a breakdown in reciprocity over the free-space link since a shorted link exhibits the same behavior. Instead, it reflects a weak temperature dependence of the system to the 4 °C laboratory temperature variations. Specifically, we attribute most of this wander to temperaturedriven path length variations in the ~ 2 m of fiber patchcords that connect the two sites to the common reference plane and within the transceivers. The second panel of Fig. 4 plots the change in T link over the measurement period. Its average value was ~13 s, corresponding to the 3.942 km path distance. This time-of-flight variation corresponds to an 8.7 cm variation in optical path. The variation is driven by turbulence and building motion on short periods and by atmospheric temperature changes on longer periods.
(Synnchronization under km-scale path variations are shown in the next section.) The third panel of Fig. 4 plots the frequency correction that is applied to the 195.3 THz optical signal underlying the site B timescale. The effective time correction is given by the integral of this curve normalized by the nominal frequency of 195.3 THz and reaches 0.98 ms over the 50 hours reflecting the time wander between the two free-running cavity-stabilized lasers. One of the implicit byproducts of full synchronization is full syntonization, or "frequency lock". The residual frequency uncertainty between the sites is given by the modified Allan deviation, which is simply the timing deviation of Fig. 1b multiplied by √3/t avg , where t avg is the averaging time. As shown in Fig. 5, this Allan deviation is consistent with the earlier 2 km comparison measurement of Ref. [35] despite the additional complexity of full time synchronization and longer distance. Moreover, it extends to longer averaging times reaching as low as 2×10 -19 beyond 10,000 sec. The bottom panel of Fig. 4 shows the "link availability" or the percentage of time per 60second interval when sufficient optical power is transmitted over the link for two-way synchronization. Both the launched comb and communication/PRBS laser power are 2.5 mW, well below the eye safe limit. Atmospheric turbulence causes significant fluctuations in the received laser power in the single-mode fiber. The turbulence structure constant was C n 2 ≈ 10 −14 m −2/3 . With this moderate level of turbulence, the received power varies from 0 nW to ~200 nW, with a median value of 33 nW, which is compared to the detection threshold for the comb transfer of 2 nW, or ~78 photons per pulse (Fig. 6a). There are occasional "dropouts" when the received power is below threshold, leading to less than 100% link availability. These dropouts are typically below 10 ms in duration, as shown in Fig. 6b. During a dropout, the Averaging time (s) synchronization is not active and therefore these periods are excluded from the time offset data. Nevertheless, the cavity-stabilized lasers are well behaved so that the time offset at re-acquisition is typically below 6 fs. For systems that require a continuous output, a Kalman filter could be implemented. This is especially critical for less well behaved oscillators and long dropout durations but would be at the cost of significant added processing complexity.  Fig. 4. Inset shows the ~ 2 nW threshold. (b) Probability density of fades versus duration. 90% of the fades are below 10 ms. Longer durations are typically due to a disruption of the beam from physical objects, re-alignment, or heavy precipitation, rather than turbulence.

B. Synchronization maintained despite kilometer-scale length changes
The synchronization is robust against large changes in link distance. In Fig. 7, the out-of-loop time offset, Δ , is measured while the link distance is alternated between 1 m, 2 km and 4 km by manually adjusting the folding mirrors, as indicated in Fig. 7a. Each adjustment requires about 30 s. The system ran continuously during the link realignment, successfully re-synchronizing within tens of milliseconds of reacquisition of the light across the link. The overall time offset shows less than 2 fs wander with distance which is attributed to either a small systematic shift with distance or simply temperature variations within the laboratory (as mentioned in the previous section). Separate tests found negligible (< 1 fs) systematic shifts with received power.

C. Optical pulse-per-second (PPS)
In conventional time systems, an rf PPS [24] provides unambiguous time markers. Here, we demonstrate femtosecond-level, unambiguous synchronization by generating analogous optical PPS signals. These optical PPS signals are easily generated by gating out a single pulse from the 200 MHz optical pulse train. At each site, the optical pulse train is fiber coupled to a Mach-Zehnder amplitude modulator (MZM) that is driven from a pulse generated by the local FPGA controller. Since this FPGA controller tracks the time associated with each optical pulses, it can gate every 200 millionth pulse (where we define our timescales such that the comb repetition rates are exactly 200 MHz). These pulses still carry the precision and accuracy of the synchronized timescales as they still consist of 150 fs long optical pulses. To verify unambiguous timing, each gated pulse is then photodetected and their arrival compared on a high bandwidth oscilloscope. To verify synchronicity, the common reference plane must be shifted by adjusting  cal from that of Fig. 4 to compensate for relative delays between photodetection and the oscilloscopes. Figure 8a shows an example of synchronization of 1 PPS signals to below ~100 ps, limited by the detector bandwidth. As with Fig. 7, this simultaneity is preserved across large path-length variations.
These data illustrate that the timing is unambiguous, but the uncertainty is limited by the rf bandwidths. As a more sensitive demonstration, we can spatially interfere the optical PPS from the two timescales. To do this, we construct a spatial interference fringe pattern by coupling the two optical outputs into free space and combining them at a slight angle onto an InGaAs focal plane array. A single PPS pair provides insufficient photons across the focal plane array so we increase the gate time to the MZM for a burst of pulses. Spatial interference fringes will be visible only when those bursts occur at the same time and only when the pulses within the burst overlap in time to well within their ~150 fs duration. The presence of the high-contrast spatial interference pattern indicates unambiguous, femtosecond-level synchronization between sites. Figure 8b shows such an interference pattern. The supplemental movie of Appendix A shows the appearance and disappearance of this spatial interference as synchronization is applied or disabled at the site B [42].

V. Discussion
The results of the previous section demonstrate that the reciprocity of single-spatial mode optical links is sufficient to support femtosecond synchronization of distant optical timescales. Moreover, it is possible to achieve this synchronization in a complex, but robust implementation that can operate for days, over turbulent paths, and over path lengths of very different lengths.
In the system here, the two timescale are synchronized relative to each other to below 1 femtosecond for up to 1.8 hours. They are not stabilized to an absolute established timescale, although the master site A could be in principle. This low residual timing is nevertheless useful for a distributed passive or active sensing system or for navigation. For other applications it might be necessary to include an atomic clock at the master site A. For clock-based geodesy or relativity experiments, full atomic clocks are needed at each site for time comparisons. In that case, the residual timing noise associated with any comparison (or synchronization) between sites will be well below the absolute noise of the timescales. Systematic time offsets with distance were below a few fs at 4 km, and no systematics were observed with variations in received optical power. There are however two important systematics. First, there will be temperature-induced path length changes in non-reciprocal optical paths within either the transceivers or in the verification. These effects can be suppressed by appropriate experimental design and by temperature control, down to tens of femtoseconds as shown here. Second, the width of the optical pulses is 100 fs to 1 ps long; the exact definition of the time associated with these pulses depends on how the end user "reads out" the arrival time of the pulse center at the reference plane, which will necessarily depend on the application. Again, this systematic will be on the order of tens of femtoseconds.
Rf-based two-way time-frequency transfer is much more developed and can operate over much longer rangesincluding ground-to-spaceand to moving platforms [24]. Here, our 4-km path is horizontal and therefore suffers equivalent turbulence to a longer vertical ground-tosatellite path, but longer distance operation will have higher transmission loss and path delay, T link . The higher transmission loss will need to be offset by a reduced detection threshold, higher transmit powers, and improved free-space terminals, possibly including adaptive optics. The longer path delay can potentially cause a breakdown in the reciprocity condition, which assumes a "fixed" turbulence over the two-way measurement time of 1/f r . For T link >> 1/f r , the short term turbulence-induced piston noise [38] will not be completely negligible but the long-term piston noise should nevertheless be cancelled via the two-way approach.
Moving platforms present at least two additional problems: point ahead issues and Doppler shifts. For transverse motion between platforms, the "point ahead" effect causes the two signals to traverse slightly different optical paths and therefore will cause a breakdown in reciprocity. As with the impact of a longer path delay, this effect will be strongest in a ground-to-space scenario. These effects have been analyzed recently by Wolf and coworkers, who find an increase in the timing noise over short times below a few seconds but excellent two-way cancellation over longer times [43]. The impact of Doppler shifts will require further study. To lowest order, the technique here is independent of Doppler shifts. However, the exact implementation here is not Doppler insensitive and future work is needed to optimize the system for insensitivity to Doppler shifts and to quantify any performance penalties.

VI. Conclusions
We have demonstrated real-time time transfer and synchronization between remote optical timescales using two-way exchange of light over a reciprocal free-space link. We verify subfemtosecond time synchronization out to 1.8 hours. The long-term wander over two days is 40 fs peak-to-peak, dominated by measurement uncertainty in the out-of-loop verification. The system was operated over a turbulent 4 km free-space path but we have found no fundamental limitations associated with distance. The single-mode free-space path is fully reciprocal to within our measurement uncertainty which reaches 70 nm at 10 second averaging. Provided sufficient received power is available (here equal to 78 photons per pulse), this approach should be scalable to much longer paths. The performance is a thousand times superior to rf based methods and should enable future networks of optical clocks/oscillators that are synchronized in real-time with sub-femtosecond stability.