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Quantum-limited optical time transfer for futuregeosynchronouslinks
未来地球同步链路的量子限制光时间传输

https://doi.org/10.1038/s41586-023-06032-5
Received: 15 December 2022
收到：2022 年 12 月 15 日
Accepted: 30 March 2023 接受：接受： 2023 年 3 月 30 日
Published online: 21 June 2023
在线出版：2023 年 6 月 21 日
Check for updates 检查更新

Emily D. Caldwell 1 , 2 , 5 1 , 2 , 5 ^(1,2,5){ }^{1,2,5}, Jean-Daniel Deschenes 3 , 5 3 , 5 ^(3,5){ }^{3,5}, Jennifer Ellis', William C. Swann 1 1 ^(1){ }^{1}, Benjamin K. Stuhl 4 4 ^(4){ }^{4}, Hugo Bergeron 3 3 ^(3){ }^{3}, Nathan R. Newbury 1 , 5 ] ] 1 , 5 ] ] ^(1,5]]){ }^{1,5 \rrbracket} & Laura C. Sinclair 1 , 5 ⋈ 1 , 5 ⋈ ^(1,5|><|){ }^{1,5 \bowtie}
Emily D. Caldwell 1 , 2 , 5 1 , 2 , 5 ^(1,2,5){ }^{1,2,5} , Jean-Daniel Deschenes 3 , 5 3 , 5 ^(3,5){ }^{3,5} , Jennifer Ellis', William C. Swann 1 1 ^(1){ }^{1} , Benjamin K. Stuhl 4 4 ^(4){ }^{4} , Hugo Bergeron 3 3 ^(3){ }^{3} , Nathan R. Newbury 1 , 5 ] ] 1 , 5 ] ] ^(1,5]]){ }^{1,5 \rrbracket} & Laura C. Sinclair 1 , 5 ⋈ 1 , 5 ⋈ ^(1,5|><|){ }^{1,5 \bowtie}

The combination of optical time transfer and optical clocks opens up the possibility of large-scale free-space networks that connect both ground-based optical clocks and future space-based optical clocks. Such networks promise better tests of general relativity 1 − 3 1 − 3 ^(1-3){ }^{1-3}, dark-matter searches 4 4 ^(4){ }^{4} and gravitational-wave detection 5 5 ^(5){ }^{5}. The ability to connect optical clocks to a distant satellite could enable space-based very long baseline interferometry 6 , 7 6 , 7 ^(6,7){ }^{6,7}, advanced satellite navigation 8 8 ^(8){ }^{8}, clock-based geodesy 29 , 10 29 , 10 ^(29,10){ }^{29,10} and thousandfold improvements in intercontinental time dissemination 11 , 12 11 , 12 ^(11,12){ }^{11,12}. Thus far, only optical clocks have pushed towards quantum-limited performance 13 13 ^(13){ }^{13}. By contrast, optical time transfer has not operated at the analogous quantum limit set by the number of received photons. Here we demonstrate time transfer with near quantum-limited acquisition and timing at 10,000 times lower received power than previous approaches 14 − 24 14 − 24 ^(14-24){ }^{14-24}. Over 300 km between mountaintops in Hawaii with launched powers as low as 40 μ W 40 μ W 40 muW40 \mu \mathrm{~W}, distant sites are synchronized to 320 attoseconds. This nearly quantum-limited operation is critical for long-distance free-space links in which photons are few and amplification costly: at 4.0 mW transmit power, this approach can support 102 dB link loss, more than sufficient for future time transfer to geosynchronous orbits.
光时间传输和光学时钟的结合为连接地面光学时钟和未来天基光学时钟的大规模自由空间网络提供了可能。这种网络有望更好地测试广义相对论 1 − 3 1 − 3 ^(1-3){ }^{1-3} 、暗物质搜索 4 4 ^(4){ }^{4} 和引力波探测 5 5 ^(5){ }^{5} 。将光学时钟连接到遥远卫星的能力可以实现天基甚长基线干涉测量 6 , 7 6 , 7 ^(6,7){ }^{6,7} 、先进的卫星导航 8 8 ^(8){ }^{8} 、基于时钟的大地测量 29 , 10 29 , 10 ^(29,10){ }^{29,10} 以及洲际时间传播的千倍改进 11 , 12 11 , 12 ^(11,12){ }^{11,12} 。迄今为止，只有光学时钟的性能达到了量子极限 13 13 ^(13){ }^{13} 。相比之下，光学时间传输还没有达到由接收光子数量设定的类似量子极限。在这里，我们展示了接近量子限制的时间传输，其采集和计时的接收功率比以前的方法低 10,000 倍 14 − 24 14 − 24 ^(14-24){ }^{14-24} 。在发射功率低至 40 μ W 40 μ W 40 muW40 \mu \mathrm{~W} 的情况下，在夏威夷山顶之间 300 多公里的距离上，遥远的站点同步至 320 阿秒。这种近乎量子限制的操作对于光子数量少、放大成本高的长距离自由空间链路至关重要：在发射功率为 4.0 mW 的情况下，这种方法可以支持 102 dB 的链路损耗，足以满足未来向地球同步轨道进行时间传输的需要。
Comb-based optical time transfer (OTT) follows previous microwave two-way time-frequency transfer 25 25 ^(25){ }^{25}. Optical pulses from coherent frequency combs located at remote sites are exchanged across a two-way free-space link. The difference in the detected pulse time-of-arrival between sites yields their clock offset, independent of the time-of-flight (assuming full reciprocality). Previous comb-based OTT used linear optical sampling (LOS) against a local frequency comb with an offset repetition rate to scan across the incoming comb pulses and measure their timing 14 − 24 14 − 24 ^(14-24){ }^{14-24}. This approach is photon inefficient and requires signals of a few nanowatts, 40 dB above the quantum limit. Despite this, with a combination of 40 − cm 40 − cm 40-cm40-\mathrm{cm} aperture telescopes, adaptive optics and watt-level amplifiers, Shen et al. 24 24 ^(24){ }^{24} achieved a working range of 113 km . The alternative approach of conventional optical frequency transfer (OFT) using continuous wave lasers achieves high performance 26 − 28 26 − 28 ^(26-28){ }^{26-28} but is unable to measure the elapsed time between sites-the quantity of interest to many applications-in the presence of link disruption due to atmospheric turbulence, weather or multiplexed operation.
基于梳状结构的光时域传输（OTT）沿用了以前的微波双向时频传输 25 25 ^(25){ }^{25} 。来自远程站点相干频率梳的光脉冲通过双向自由空间链路进行交换。站点之间检测到的脉冲到达时间的差异会产生时钟偏移，而与飞行时间无关（假设完全互易）。以前基于梳状结构的 OTT 使用线性光学采样（LOS），针对具有偏移重复率的本地频率梳状结构扫描传入的梳状脉冲，并测量它们的时间 14 − 24 14 − 24 ^(14-24){ }^{14-24} 。这种方法的光子效率较低，需要几毫微瓦的信号，比量子极限高出 40 dB。尽管如此，Shen 等人结合使用了 40 − cm 40 − cm 40-cm40-\mathrm{cm} 孔径望远镜、自适应光学和瓦特级放大器， 24 24 ^(24){ }^{24} 实现了 113 千米的工作距离。使用连续波激光器进行传统光频率传输 (OFT) 的替代方法实现了高性能 26 − 28 26 − 28 ^(26-28){ }^{26-28} ，但在大气湍流、天气或多路复用操作导致链路中断的情况下，无法测量站点之间的经过时间（这是许多应用所关心的数量）。
In close analogy with optical clocks, the quantum-limited uncertainty for time transfer by means of an optical pulse of width τ p τ p tau_(p)\tau_{\mathrm{p}} (here roughly 350 fs ) is simply
与光学时钟近似，通过宽度为 τ p τ p tau_(p)\tau_{\mathrm{p}} （此处约为 350 fs）的光脉冲进行时间传输的量子极限不确定性简单地表示为
σ t = γ τ p n σ t = γ τ p n sigma_(t)=gamma(tau_(p))/(sqrtn)\sigma_{\mathrm{t}}=\gamma \frac{\tau_{\mathrm{p}}}{\sqrt{n}}
where n n nn is the number of detected photons in the measurement interval and γ γ gamma\gamma is a constant of order unity. Here, we demonstrate
其中， n n nn 是测量区间内检测到的光子数量， γ γ gamma\gamma 是一个数量级为一的常数。在此，我们证明
OTT at this quantum limit by exploiting the precision and agility of a time-programmable frequency comb (TPFC) 29 29 ^(29){ }^{29} in conjunction with Kalman filter-based signal processing. The improvement over previous LOS-based OTT is large: the minimum received power decreases 10,000-fold from a few nanowatts to a few hundred femtowatts, which means only one out of 100 received frequency comb pulses contains a photon.
通过利用时间可编程频率梳（TPFC） 29 29 ^(29){ }^{29} 的精确性和敏捷性，并结合基于卡尔曼滤波器的信号处理，在这一量子极限上实现了 OTT。与之前基于 LOS 的 OTT 相比，改进幅度很大：最小接收功率从几纳瓦特降低到几百飞瓦特，降低了 10,000 倍，这意味着 100 个接收到的频率梳脉冲中只有一个包含光子。
We demonstrate this quantum-limited OTT by synchronizing two optical timescales across two different free-space links: a 2-km link with low turbulence in Boulder, CO, USA and a 300 km link with strong turbulence between two mountaintops in Hawaii. Under low turbulence, where the free-space path is indeed reciprocal, the two-way time transfer is nearly quantum-limited; the clocks are synchronized to 246 as / P τ / P τ //sqrt(P tau)/ \sqrt{P \tau} in time deviation and to 4.3 × 10 − 16 / P τ 3 4.3 × 10 − 16 / P τ 3 4.3 xx10^(-16)//sqrt(Ptau^(3))4.3 \times 10^{-16} / \sqrt{P \tau^{3}} in fractional frequency (modified Allan deviation), where P P PP is the received power in picowatts and τ τ tau\tau is the averaging time in seconds, with respective floors of roughly 35 attoseconds and below 10 − 18 10 − 18 10^(-18)10^{-18}. Over the 300 km horizontal link, the one-way timing signals are still measured at nearly the quantum limit with a power threshold of 270 fW . However, the strong integrated turbulence leads to excess non-reciprocal time-of-flight noise, attributed to multipath effects. Nevertheless, the clocks remain synchronized to 1.6 fs τ − 1 / 2 1.6 fs τ − 1 / 2 1.6fstau^(-1//2)1.6 \mathrm{fs} \tau^{-1 / 2} reaching a floor of 320 attoseconds in time and 2.8 × 10 − 15 τ − 3 / 2 2.8 × 10 − 15 τ − 3 / 2 2.8 xx10^(-15)tau^(-3//2)2.8 \times 10^{-15} \tau^{-3 / 2} reaching a floor of 3.1 × 10 − 19 3.1 × 10 − 19 3.1 xx10^(-19)3.1 \times 10^{-19} in frequency. Finally, synchronization is achieved even at attenuated comb powers of 40 μ W 40 μ W 40 muW40 \mu \mathrm{~W} with a median received power of 150 fW . This low-power performance is enabled by the robust Kalman filter approach that can tolerate the greater than
我们通过在两个不同的自由空间链路上同步两个光学时标来演示这种量子限制的 OTT：一个是美国科罗拉多州博尔德市湍流较少的 2 千米链路，另一个是夏威夷两个山顶之间湍流较强的 300 千米链路。在低湍流条件下，自由空间路径确实是互惠的，双向时间传输几乎是量子限制的；时钟同步到 246，时间偏差为 / P τ / P τ //sqrt(P tau)/ \sqrt{P \tau} ，分数频率（修正的阿伦偏差）为 4.3 × 10 − 16 / P τ 3 4.3 × 10 − 16 / P τ 3 4.3 xx10^(-16)//sqrt(Ptau^(3))4.3 \times 10^{-16} / \sqrt{P \tau^{3}} ，其中 P P PP 是以皮瓦特为单位的接收功率， τ τ tau\tau 是以秒为单位的平均时间，各自的下限约为 35 阿秒和 10 − 18 10 − 18 10^(-18)10^{-18} 以下。在 300 千米的水平链路上，单向定时信号的测量仍然接近量子极限，功率阈值为 270 fW。然而，强烈的综合湍流导致了过多的非互易飞行时间噪声，这归因于多径效应。尽管如此，时钟仍能保持同步， 1.6 fs τ − 1 / 2 1.6 fs τ − 1 / 2 1.6fstau^(-1//2)1.6 \mathrm{fs} \tau^{-1 / 2} 在时间上达到 320 阿秒的底限， 2.8 × 10 − 15 τ − 3 / 2 2.8 × 10 − 15 τ − 3 / 2 2.8 xx10^(-15)tau^(-3//2)2.8 \times 10^{-15} \tau^{-3 / 2} 在频率上达到 3.1 × 10 − 19 3.1 × 10 − 19 3.1 xx10^(-19)3.1 \times 10^{-19} 的底限。最后，即使在中值接收功率为 150 fW 的 40 μ W 40 μ W 40 muW40 \mu \mathrm{~W} 的衰减梳状功率下，也能实现同步。这种低功耗性能得益于稳健的卡尔曼滤波方法，该方法可以承受大于 40 μ W 40 μ W 40 muW40 \mu \mathrm{~W}的频率。

Article 文章

Fig. 1|Quantum-limited OTT. a, The system was tested on a 300 km folded free-space link established between two sites, colocated at the Mauna Loa Observatory (elevation 3 , 400 m 3 , 400 m 3,400m3,400 \mathrm{~m} ) and a cat-eye retroflector located on the summit of Haleakala (elevation 3,050 m). Colocation of the sites allows a direct, out-of-loop timing verification.b, Detailed schematic for site B. (Site A is identical except without the synchronization (Synch) lock.) Each site uses two TPFCs phase locked to a local optical reference. The clock TPFC output defines the local timescale. It is also transmitted across the bidirectional link through a 10-cm aperture free-space optical (FSO) terminal (Extended Data Fig.1). The
图 1：量子限制 OTT。a, 系统在两个站点之间建立的 300 公里折叠自由空间链路上进行了测试，这两个站点分别位于毛纳罗亚天文台（海拔 3 , 400 m 3 , 400 m 3,400m3,400 \mathrm{~m} ）和哈雷阿卡拉山顶（海拔 3,050 米）的猫眼式逆反射器上。站点 B 的详细示意图（站点 A 除没有同步锁外完全相同）。每个站点使用两个 TPFC，相位锁定到本地光学基准。时钟 TPFC 输出定义了本地时标。它还通过一个 10 厘米孔径的自由空间光学（FSO）终端在双向链路上传输（扩展数据图 1）。图 1
tracking TPFC is used to acquire, track and measure the timing of the incoming clock comb pulse train. Two-way combination of the measured timing signals generates an error signal that is applied at site B B BB for synchronization (text). The filtered comb output powers are 4.0 mW and 5.9 mW for sites A A AA and B B BB (Extended Data Fig. 2), but can be attenuated by the in-line attenuator to mimic links with higher loss. The fibre spools before the FSO terminals compensate for the 300 km of air dispersion. (See also Extended Data Fig. 3 and Methods.) The underlying map in a is from Google Earth with image data from Landsat and Copernicus. RF, radio frequency.
跟踪 TPFC 用于获取、跟踪和测量输入时钟梳状脉冲序列的定时。测量定时信号的双向组合会产生一个误差信号，应用于站点 B B BB 进行同步（文本）。站点 A A AA 和 B B BB 的滤波梳状输出功率分别为 4.0 mW 和 5.9 mW（扩展数据图 2），但可通过在线衰减器进行衰减，以模拟损耗较高的链路。FSO 终端前的光纤线轴可补偿 300 公里的空气色散。(另见扩展数据图 3 和方法）a 中的底图来自谷歌地球，图像数据来自 Landsat 和 Copernicus。RF：射频。
70% signal fades. In comparison to the longest previously reported range for LOS-based OTT 24 24 ^(24){ }^{24}, quantum-limited OTT operates across three times the distance, at 20 times improved update rate using 200 times less comb power and at four times lower aperture diameter, with more than 14 dB greater tolerable link loss. The tolerable link loss of 102 dB exceeds that of future ground-to GEO links with similar 10 − cm 10 − cm 10-cm10-\mathrm{cm} apertures and milliwatt comb powers.
70% 的信号衰减。与之前报道的基于 LOS 的 OTT 24 24 ^(24){ }^{24} 的最远距离相比，量子限界 OTT 的运行距离是其三倍，更新率提高了 20 倍，梳状功率降低了 200 倍，孔径直径缩小了四倍，可容忍链路损耗增加了 14 分贝以上。102 dB 的可容忍链路损耗超过了采用类似 10 − cm 10 − cm 10-cm10-\mathrm{cm} 孔径和毫瓦级梳状功率的未来地对地地球同步轨道链路。

Quantum-limited time transfer using a TPFC
利用 TPFC 进行量子限时传输

The quantum-limited OTT was demonstrated first over a 2 km link at the NIST campus in Boulder and then over the 300 km link between the Hawaiian Islands shown in Fig. 1a. For both, a folded-link geometry enabled direct out-of-loop verification of the synchronization, at the cost of added link loss, but the system could ultimately be used in a point-to-point 21 , 24 21 , 24 ^(21,24){ }^{21,24} or multi-node geometry 20 20 ^(20){ }^{20}. For the Hawaii link, operation was mainly limited to overnight and early morning hours because of daytime clouds in the interisland convergence zone.
量子限制 OTT 首先在博尔德 NIST 校园的 2 千米链路上进行了演示，然后在图 1a 所示的夏威夷群岛之间的 300 千米链路上进行了演示。在这两条链路上，折叠链路几何结构实现了同步的直接环外验证，但代价是增加了链路损耗，但该系统最终可用于点对点 21 , 24 21 , 24 ^(21,24){ }^{21,24} 或多节点几何结构 20 20 ^(20){ }^{20} 。对于夏威夷链路，由于岛际会聚区白天有云，因此主要限于在夜间和清晨运行。
The system is centred around fibre-based, 200-MHz repetition frequency, TPFCs that provide real-time attosecond-level digital control of the pulse timing. A heterodyne timing discriminator 29 29 ^(29){ }^{29}, as shown in Extended Data Fig. 4, measures the time offset between a local tracking TPFC and the incoming clock comb pulse signal with shot-noise limited sensitivity. This time offset acts as an error signal to adjust the digital control of the tracking TPFC to follow the incoming clock
该系统以基于光纤的 200 兆赫重复频率 TPFC 为中心，可对脉冲定时进行实时的等秒级数字控制。如扩展数据图 4 所示，外差式定时鉴别器 29 29 ^(29){ }^{29} 测量本地跟踪 TPFC 与输入时钟梳状脉冲信号之间的时间偏移，其灵敏度受射频噪声的限制。该时间偏移作为误差信号，用于调整跟踪 TPFC 的数字控制，以跟踪输入时钟。
comb pulses (Methods). The commanded tracking-TPFC timing then replicates the timing of the incoming comb pulse train at each site, t A t A t_(A)t_{\mathrm{A}} or t B t B t_(B)t_{\mathrm{B}}, whose difference,
梳状脉冲（方法）。然后，指令跟踪-TPFC 时序在每个部位复制传入梳状脉冲串的时序， t A t A t_(A)t_{\mathrm{A}} 或 t B t B t_(B)t_{\mathrm{B}} ，其差值为 t A t A t_(A)t_{\mathrm{A}} 、
Δ t = t A − t B 2 = ( Δ T osc − Δ T cntrl ) + ε NR , turb + ε qn + ε combs Δ t = t A − t B 2 = Δ T osc − Δ T cntrl + ε NR , turb + ε qn + ε combs Delta t=(t_(A)-t_(B))/(2)=(DeltaT_(osc)-DeltaT_(cntrl))+epsi_(NR,turb)+epsi_(qn)+epsi_(combs)\Delta t=\frac{t_{\mathrm{A}}-t_{\mathrm{B}}}{2}=\left(\Delta T_{\mathrm{osc}}-\Delta T_{\mathrm{cntrl}}\right)+\varepsilon_{\mathrm{NR}, \mathrm{turb}}+\varepsilon_{\mathrm{qn}}+\varepsilon_{\mathrm{combs}}
is a measure of the time offset between the two clock combs, ( Δ T osc − Δ T cntr Δ T osc  − Δ T cntr DeltaT_("osc ")-DeltaT_(cntr)\Delta T_{\text {osc }}-\Delta T_{\mathrm{cntr}} ), where Δ T osc Δ T osc  DeltaT_("osc ")\Delta T_{\text {osc }} is the time offset between the local reference oscillators and Δ T cntrl Δ T cntrl  DeltaT_("cntrl ")\Delta T_{\text {cntrl }} is the synchronization feedback to the clock TPFC at site B ( Δ T cntrl = 0 B Δ T cntrl = 0 B(DeltaT_(cntrl)=0:}B\left(\Delta T_{\mathrm{cntrl}}=0\right. for open-loop operation). The fundamental reciprocity of a single spatial mode link 30 30 ^(30){ }^{30} means that the time-of-flight, including turbulence effects, should cancel in this two-way comparison of equation (2). Nevertheless, we include a non-reciprocal, turbulence noise term, ϵ NR , turb ϵ NR ,  turb  epsilon_(NR," turb ")\epsilon_{\mathrm{NR}, \text { turb }}, for reasons discussed later. The quantum noise term, ϵ q n ϵ q n epsilon_(qn)\epsilon_{q n}, has a standard deviation following equation (1). The system noise, ϵ combs ϵ combs  epsilon_("combs ")\epsilon_{\text {combs }}, is typically negligible at short averaging times and low powers, but leads to the flicker floor at long averaging times.
是两个时钟振荡器之间的时间偏移量（ Δ T osc − Δ T cntr Δ T osc  − Δ T cntr DeltaT_("osc ")-DeltaT_(cntr)\Delta T_{\text {osc }}-\Delta T_{\mathrm{cntr}} ），其中 Δ T osc Δ T osc  DeltaT_("osc ")\Delta T_{\text {osc }} 是本地参考振荡器之间的时间偏移量， Δ T cntrl Δ T cntrl  DeltaT_("cntrl ")\Delta T_{\text {cntrl }} 是对站点 B ( Δ T cntrl = 0 B Δ T cntrl = 0 B(DeltaT_(cntrl)=0:}B\left(\Delta T_{\mathrm{cntrl}}=0\right. 的时钟 TPFC 的同步反馈（开环运行）。单空间模式链路 30 30 ^(30){ }^{30} 的基本互易性意味着飞行时间（包括湍流效应）应在方程 (2) 的双向比较中抵消。尽管如此，我们还是加入了一个非对等的湍流噪声项 ϵ NR , turb ϵ NR ,  turb  epsilon_(NR," turb ")\epsilon_{\mathrm{NR}, \text { turb }} ，原因稍后讨论。量子噪声项 ϵ q n ϵ q n epsilon_(qn)\epsilon_{q n} 的标准偏差与公式 (1) 一致。系统噪声 ϵ combs ϵ combs  epsilon_("combs ")\epsilon_{\text {combs }} 在短平均时间和低功率时通常可以忽略，但在长平均时间时会导致闪烁底限。
To track the incoming pulses, the incoming clock comb and local tracking comb pulses must overlap in time to within the roughly 2 τ p 2 τ p 2tau_(p)2 \tau_{\mathrm{p}} dynamic range of the timing discriminator. This initial alignment is accomplished by sweeping the tracking comb pulse position in time 29 29 ^(29){ }^{29} and searching for peaks in the heterodyne signal (Fig. 2a). Scintillation from turbulence, however, causes random 100 % 100 % 100%100 \% intensity fluctuations that complicate mapping the peak heterodyne voltage to the incoming pulse location. Using a Kalman filter to aggregate intermittent
为了跟踪进入的脉冲，进入的时钟梳状脉冲和本地跟踪梳状脉冲必须在时间上重叠，大致在定时鉴别器的 2 τ p 2 τ p 2tau_(p)2 \tau_{\mathrm{p}} 动态范围内。在时间 29 29 ^(29){ }^{29} 内扫描跟踪梳脉冲位置，并搜索外差信号中的峰值，即可实现初始对准（图 2a）。然而，湍流产生的闪烁会导致随机 100 % 100 % 100%100 \% 强度波动，从而使将外差电压峰值映射到输入脉冲位置变得复杂。使用卡尔曼滤波器汇总间歇性的
Fig. 2 ∣ 2 ∣ 2∣2 \mid Low-power acquisition and quantum-limited performance. a, Demonstrated signal acquisition over the 300 km Hawaii link. Initially, the local tracking TPFC is swept over its full 5 ns non-ambiguity range in a triangular waveform. At roughly 3 s into the acquisition, a peak in the heterodyne signal indicates a transient temporal overlap between the tracking TPFC and the incoming clock comb (inset). On the basis of the observation of a heterodyne signal above the 135 fW threshold, the signal processor steers the tracking comb back to this location for finer search before initiating the tracking lock at
图： 2 ∣ 2 ∣ 2∣2 \mid 低功耗采集和量子限制性能。a, 在 300 公里夏威夷链路上演示信号采集。最初，本地跟踪 TPFC 以三角波形扫过其整个 5 毫微秒的非模糊范围。大约在采集开始 3 秒时，外差信号中出现一个峰值，表明跟踪 TPFC 和输入时钟梳之间出现了短暂的时间重叠（插图）。在观测到高于 135 fW 门限的外差信号时，信号处理器会将跟踪梳引导回该位置进行更精细的搜索，然后再启动跟踪锁定。
observations of pulse overlap, we can track the estimated temporal position of the incoming pulses, and the associated uncertainty, despite fades. As the estimated position uncertainty decreases, the search space narrows. When the estimated uncertainty reaches 500 fs , the tracking lock is engaged. To detect the weakest possible incoming comb light, the detection bandwidth for the heterodyne timing discriminator should be as narrow as possible given the constraints of atmospheric turbulence phase noise and platform and fibre vibration. We settled on 26 kHz here as a conservative compromise.
通过对脉冲重叠的观测，我们可以跟踪到输入脉冲的估计时间位置，以及相关的不确定性，尽管有衰减。随着估计位置不确定性的减小，搜索空间也随之缩小。当估计的不确定性达到 500 fs 时，跟踪锁定就会启动。考虑到大气湍流相位噪声以及平台和光纤振动的限制，为了探测到尽可能弱的入射梳状光，外差定时鉴别器的探测带宽应尽可能窄。作为一种保守的折衷方案，我们在此确定了 26 kHz 的带宽。
For robust operation through signal fades, the timing samples, t A t A t_(A)t_{\mathrm{A}} and t B t B t_(B)t_{\mathrm{B}}, are input into the Kalman filter to generate optimal estimates of the timing with 10 − 25 Hz 10 − 25 Hz 10-25Hz10-25 \mathrm{~Hz} effective bandwidth. These Kalman-filtered values are used in equation (2) and input to a 15 − Hz 15 − Hz 15-Hz15-\mathrm{Hz} bandwidth synchronization lock to steer the site B clock comb. Here, the timing signals used in the two-way combining are communicated from sites A to B by coaxial cable but an optical communications link could be implemented as in ref. 15. The transceiver time delays are calibrated so the pulses from the two clock combs overlap at the out-of-loop verification reference plane located within site B when Δ t → 0 Δ t → 0 Delta t rarr0\Delta t \rightarrow 0 (Extended Data Figs. 5 and 6).
为了在信号衰减时仍能稳健运行，将定时采样 t A t A t_(A)t_{\mathrm{A}} 和 t B t B t_(B)t_{\mathrm{B}} 输入卡尔曼滤波器，以生成具有 10 − 25 Hz 10 − 25 Hz 10-25Hz10-25 \mathrm{~Hz} 有效带宽的最佳定时估计值。这些卡尔曼滤波值用于公式 (2) 并输入 15 − Hz 15 − Hz 15-Hz15-\mathrm{Hz} 带宽同步锁，以引导站点 B 的时钟梳。在这里，双向合成中使用的定时信号是通过同轴电缆从 A 站传送到 B 站的，但也可以像参考文献 15 中那样使用光通信链路。15.收发器的时间延迟经过校准，因此当 Δ t → 0 Δ t → 0 Delta t rarr0\Delta t \rightarrow 0 时，两个时钟梳的脉冲在位于 B 站点内的环外验证参考平面上重叠（扩展数据图 5 和 6）。
At low power and weak turbulence, the two-way time transfer is nearly quantum limited following equation (1) until it reaches the system noise floor (Fig. 2b,c). We apply a roughly 270 fW threshold on the received power for a valid timing measurement, chosen such that the quantum-limited timing noise standard deviation was roughly one-sixth the full timing discriminator dynamic range (Supplementary Information). For comparison, the power threshold for signal acquisition is lower, at roughly 135 fW , selected to limit false detections to fewer than one per day (Methods). These thresholds correspond to n = 40 n = 40 n=40n=40 and 20 photons per signal integration time ( 19 μ s ) ( 19 μ s ) (19 mus)(19 \mu \mathrm{~s}), respectively, or 0.01 and 0.005 mean photons per comb pulse. The values of n > 1 n > 1 n > 1n>1 reflect the conservatively chosen threshold to ensure low probability of false detection. Both the acquisition and timing measurements operate at roughly twice the quantum limit because of detector noise power penalty and differential chirp between the tracking comb and incoming comb pulses. There is a negligible contribution from daylight; reflected
在低功率和弱湍流条件下，双向时间传输在达到系统噪声本底（图 2b、c）之前，几乎是受等式（1）限制的量子传输。我们对有效定时测量的接收功率设定了约 270 fW 的阈值，选择该阈值时，量子限定时噪声标准偏差约为整个定时鉴别器动态范围的六分之一（补充信息）。相比之下，信号采集的功率阈值更低，约为 135 fW，以限制误检测每天少于一次（方法）。这些阈值分别对应于每个信号积分时间 n = 40 n = 40 n=40n=40 和 20 个光子，或每个梳状脉冲 0.01 和 0.005 个平均光子。 n > 1 n > 1 n > 1n>1 的值反映了为确保低误检概率而保守选择的阈值。由于探测器噪声功率惩罚以及跟踪梳状脉冲和进入的梳状脉冲之间的差分啁啾，采集和定时测量的运行速度大约是量子极限的两倍。日光的影响可以忽略不计；反射光的影响可以忽略不计。

roughly 5 sinto the acquisition. b b b\mathbf{b}, The timing noise (standard deviation over 600 s ) in Δ t Δ t Delta t\Delta t measured over a shorted link (open circles) and a 2 km free-space link (closed circles). Colours correspond to traces in cbelow. The timing follows the quantum limit from equation (1) for γ = 2.1 γ ql γ = 2.1 γ ql gamma=2.1gamma_(ql)\gamma=2.1 \gamma_{\mathrm{ql}} (grey line), where γ q ≈ 0.6 γ q ≈ 0.6 gamma_(q)~~0.6\gamma_{\mathrm{q}} \approx 0.6 is the quantum limit for Gaussian pulses (ref. 29 and Extended Data Table 1). c, Time deviations (TDEV) over the 2 km free-space link at received powers of 800 fW and 20.6 pW follow the quantum limit with γ = 2.1 γ ql γ = 2.1 γ ql gamma=2.1gamma_(ql)\gamma=2.1 \gamma_{\mathrm{ql}} (dotted lines), until reaching the system noise floor at received powers above 10 nW .
大约 5 sinto 采集。 b b b\mathbf{b} ，在短路链路（开圆圈）和 2 千米自由空间链路（闭圆圈）上测量的 Δ t Δ t Delta t\Delta t 中的定时噪声（600 秒内的标准偏差）。颜色与下面 cbel 中的轨迹相对应。c, 在接收功率为 800 fW 和 20.6 pW 时，2 km 自由空间链路上的时间偏差（TDEV）与 γ = 2.1 γ ql γ = 2.1 γ ql gamma=2.1gamma_(ql)\gamma=2.1 \gamma_{\mathrm{ql}} 的量子极限一致（虚线），直到接收功率超过 10 nW 时达到系统噪声本底。
sunlight would contribute only 10 attowatts within the single-mode heterodyne detection bandwidth.
在单模外差探测带宽内，太阳光的功率只有 10 瓦。
Figure 3 compares the performance over the 300 and 2 km links in terms of timing synchronization, timing instability and frequency instability. Further data are provided in the Extended Data Figs. 7-10. Although the one-way timing measurements are quantum limited over the strongly turbulent 300 km link, unlike the shorted and 2 km link, their two-way subtraction, Δ t Δ t Delta t\Delta t, does not reach the quantum limit for reasons discussed below. Nevertheless, it drops below state-of-the-art transportable optical atomic clocks 9 9 ^(9){ }^{9} after only 6 s of averaging time and laboratory optical atomic clocks after 17 s of averaging time 21 21 ^(21){ }^{21}.
图 3 比较了 300 公里和 2 公里链路在定时同步性、定时不稳定性和频率不稳定性方面的性能。扩展数据图 7-10 提供了更多数据。虽然单向定时测量在强湍流 300 公里链路上受到量子限制，但与短路和 2 公里链路不同，其双向减法 Δ t Δ t Delta t\Delta t 并未达到量子限制，原因将在下文讨论。尽管如此，在平均时间仅为 6 秒钟后，它就低于最先进的可运输光学原子钟 9 9 ^(9){ }^{9} ，在平均时间为 17 秒钟后，低于实验室光学原子钟 21 21 ^(21){ }^{21} 。
To demonstrate operation at extreme link loss, the comb power from site B B BB was attenuated to 40 μ W 40 μ W 40 muW40 \mu \mathrm{~W} leading to a median received power of 150 fW at site A. Despite signal fading 73 % 73 % 73%73 \% time below threshold, timing acquisition and synchronization were still achieved with minimal 2.8 times performance degradation at short times. This attenuation of the site B B BB power is done in the two-way path and is equivalent to operation with 4.0 mW of comb power over a total link loss of 106 dB .
尽管信号衰减 73 % 73 % 73%73 \% 时间低于阈值，但在短时间内仍能实现定时采集和同步，性能下降幅度最小为 2.8 倍。站点 B B BB 功率的衰减是在双向路径中完成的，相当于在总链路损耗为 106 dB 的情况下使用 4.0 mW 的梳状功率运行。

Effects of turbulence on timing
湍流对时间的影响

The increased timing noise across the 300 km link is attributed to the strong integrated turbulence and a breakdown in the expected reciprocity in time-of-flight over the single-mode link 30 30 ^(30){ }^{30}. Previous combbased OTT has not seen clear violations in reciprocity even at 100 km (refs. 14-24). However, the enhanced sensitivity of quantum-limited OTT means we can probe timing fluctuations at the attosecond-level during deep signal fades when the effects of multipath interference are at their strongest.
300 公里链路上时序噪声的增加归因于强大的综合湍流和单模链路 30 30 ^(30){ }^{30} 上飞行时间互易性的破坏。以前基于组合的 OTT 即使在 100 公里处也没有发现明显的互易性破坏（参考文献 14-24）。然而，量子限 OTT 灵敏度的提高意味着我们可以在多径干扰影响最强的深层信号衰减过程中探测阿秒级的定时波动。
This excess noise is illustrated best in the power spectral densities (PSDs) of Δ t Δ t Delta t\Delta t and its counterpart t ¯ = ( t A + t B ) / 2 t ¯ = t A + t B / 2 bar(t)=(t_(A)+t_(B))//2\bar{t}=\left(t_{\mathrm{A}}+t_{\mathrm{B}}\right) / 2 (Fig.4). The latter measures the time-of-flight and shows the expected piston noise
Δ t Δ t Delta t\Delta t 及其对应的 t ¯ = ( t A + t B ) / 2 t ¯ = t A + t B / 2 bar(t)=(t_(A)+t_(B))//2\bar{t}=\left(t_{\mathrm{A}}+t_{\mathrm{B}}\right) / 2 的功率谱密度 (PSD) 最能说明这种过量噪声（图 4）。后者测量的是飞行时间，显示的是预期的活塞噪声
Fig. 3|OTT measured by out-of-loop timing comparison. a, Clock time difference (out-of-loop verification) across the 300 km link for 4.0 mW comb power and 14 pW median received power. b,c, Time deviation (b) and fractional frequency instability © (modified Allan deviation) across the 300 km link at 4.0 mW comb power with 14 pW median received power (blue circles) and 40 μ W 40 μ W 40 muW40 \mu \mathrm{~W} attenuated comb power with 150 fW median received power (green triangles). Also shown are data from the 2 km link at 30 pW received power (yellow stars) and shorted (grey dots). At low turbulence over the 2 km link, the performance is nearly quantum limited (yellow dashed line for γ = 3.4 γ ql γ = 3.4 γ ql gamma=3.4gamma_(ql)\gamma=3.4 \gamma_{\mathrm{ql}} ). Over 300 km , performance is limited by non-reciprocal multipath atmospheric turbulence at short times. The time deviations follow 1.6 fs / τ 1.6 fs / τ 1.6fs//sqrttau1.6 \mathrm{fs} / \sqrt{\tau} and 49 as / τ 49 as / τ 49as//sqrttau49 \mathrm{as} / \sqrt{\tau} for the 300 km and 2 km links, plateauing at 475 as . The corresponding modified Allan deviations follow 2.8 × 10 − 15 τ − 3 / 2 2.8 × 10 − 15 τ − 3 / 2 2.8 xx10^(-15)tau^(-3//2)2.8 \times 10^{-15} \tau^{-3 / 2} and 8.5 × 10 − 17 τ − 3 / 2 8.5 × 10 − 17 τ − 3 / 2 8.5 xx10^(-17)tau^(-3//2)8.5 \times 10^{-17} \tau^{-3 / 2}, reaching a floor of 3.1 × 10 − 19 3.1 × 10 − 19 3.1 xx10^(-19)3.1 \times 10^{-19}. For context, instability curves are provided for comparisons of physically separated clocks involving transportable 9 9 ^(9){ }^{9} (light pink) and laboratory optical clocks 21 21 ^(21){ }^{21} (dark pink).
a, 4.0 mW 梳状功率和 14 pW 中值接收功率下 300 km 链路上的时钟时差（环外验证）。b,c, 4.0 mW 梳状功率和 14 pW 中值接收功率下 300 km 链路上的时间偏差 (b) 和分数频率不稳定性©（修正的阿伦偏差）（蓝色圆圈）和 40 μ W 40 μ W 40 muW40 \mu \mathrm{~W} 衰减梳状功率和 150 fW 中值接收功率下 300 km 链路上的时间偏差（b）和分数频率不稳定性©（修正的阿伦偏差）（绿色三角形）。同时显示的还有接收功率为 30 pW 的 2 km 链路数据（黄星）和短路数据（灰点）。在 2 千米链路的低湍流条件下，性能几乎受到量子限制（ γ = 3.4 γ ql γ = 3.4 γ ql gamma=3.4gamma_(ql)\gamma=3.4 \gamma_{\mathrm{ql}} 的黄色虚线）。在 300 km 以上，短时间内的性能受到非互易多径大气湍流的限制。300 千米和 2 千米链路的时间偏差分别为 1.6 fs / τ 1.6 fs / τ 1.6fs//sqrttau1.6 \mathrm{fs} / \sqrt{\tau} 和 49 as / τ 49 as / τ 49as//sqrttau49 \mathrm{as} / \sqrt{\tau} ，在 475 时趋于平稳。相应的修正阿伦偏差为 2.8 × 10 − 15 τ − 3 / 2 2.8 × 10 − 15 τ − 3 / 2 2.8 xx10^(-15)tau^(-3//2)2.8 \times 10^{-15} \tau^{-3 / 2} 和 8.5 × 10 − 17 τ − 3 / 2 8.5 × 10 − 17 τ − 3 / 2 8.5 xx10^(-17)tau^(-3//2)8.5 \times 10^{-17} \tau^{-3 / 2} ，最低点为 3.1 × 10 − 19 3.1 × 10 − 19 3.1 xx10^(-19)3.1 \times 10^{-19} 。为便于理解，还提供了不稳定曲线，用于比较物理上分离的时钟，包括可运输的 9 9 ^(9){ }^{9} （浅粉色）和实验室光学时钟 21 21 ^(21){ }^{21} （深粉色）。
(time-of-flight pulse wander) with its f − 8 / 3 f − 8 / 3 f^(-8//3)f^{-8 / 3} Kolmogorov scaling 31 , 32 31 , 32 ^(31,32){ }^{31,32}. A fit to this piston noise, assuming a 10 m s − 1 10 m s − 1 10ms^(-1)10 \mathrm{~m} \mathrm{~s}^{-1} wind speed on the basis of typical meteorological data, generates an estimate of the integrated turbulence strength, L C n 2 L C n 2 LC_(n)^(2)L C_{n}^{2}, where C n 2 C n 2 C_(n)^(2)C_{n}^{2} is the turbulence structure function and L L LL is the link distance. The values of L C n 2 L C n 2 LC_(n)^(2)L C_{n}^{2} range over two orders of magnitude, from 1 × 10 − 11 1 × 10 − 11 1xx10^(-11)1 \times 10^{-11} to 1 × 10 − 9 m 1 / 3 1 × 10 − 9 m 1 / 3 1xx10^(-9)m^(1//3)1 \times 10^{-9} \mathrm{~m}^{1 / 3}, for measurements overnight and into the early morning hours (Fig. 4 inset). Note that the use of a folded-link geometry only increases, rather than cancels, the piston-induced time-of-flight noise.
(飞行时间脉冲漂移）及其 f − 8 / 3 f − 8 / 3 f^(-8//3)f^{-8 / 3} 柯尔莫哥洛夫缩放 31 , 32 31 , 32 ^(31,32){ }^{31,32} 。在典型气象数据的基础上，假设风速为 10 m s − 1 10 m s − 1 10ms^(-1)10 \mathrm{~m} \mathrm{~s}^{-1} ，对活塞噪声进行拟合，可得到综合湍流强度的估计值 L C n 2 L C n 2 LC_(n)^(2)L C_{n}^{2} ，其中 C n 2 C n 2 C_(n)^(2)C_{n}^{2} 为湍流结构函数， L L LL 为链距。 L C n 2 L C n 2 LC_(n)^(2)L C_{n}^{2} 的值在两个数量级以上，从 1 × 10 − 11 1 × 10 − 11 1xx10^(-11)1 \times 10^{-11} 到 1 × 10 − 9 m 1 / 3 1 × 10 − 9 m 1 / 3 1xx10^(-9)m^(1//3)1 \times 10^{-9} \mathrm{~m}^{1 / 3} ，用于夜间和凌晨的测量（图 4 插图）。请注意，使用折叠链路几何结构只会增加而不是消除活塞引起的飞行时间噪声。
The PSD of t ¯ t ¯ bar(t)\bar{t} shows excess time-of-flight noise (shaded region), beyond the piston noise, out to the Greenwood frequency of roughly 300 Hz before dropping to the quantum-limited white noise floor. The PSD of Δ t Δ t Delta t\Delta t (open loop) follows the expected f − 3 f − 3 f^(-3)f^{-3} phase noise of the reference oscillators but shows similar excess timing noise (shaded purple region), although suppressed by roughly 7 − 11 dB 7 − 11 dB 7-11dB7-11 \mathrm{~dB} from t ¯ t ¯ bar(t)\bar{t}. This excess noise on Δ t Δ t Delta t\Delta t limits the out-of-loop synchronization, yielding the white noise floor shown in Fig. 4 and the corresponding elevated instabilities in Fig. 3. We attribute it to multipath interference due to strong atmospheric turbulence across the 300 km horizontal link. As the turbulence strength increases, the pulse shape itself can distort and spread after transmission 31 , 33 , 34 31 , 33 , 34 ^(31,33,34){ }^{31,33,34}. We speculate these multipath related distortions
t ¯ t ¯ bar(t)\bar{t} 的 PSD 显示出超出活塞噪声的过量飞行时间噪声（阴影区域），最高可达大约 300 Hz 的格林伍德频率，然后下降到量子限白噪底。 Δ t Δ t Delta t\Delta t （开环）的 PSD 遵循参考振荡器的预期 f − 3 f − 3 f^(-3)f^{-3} 相位噪声，但也显示出类似的过量定时噪声（紫色阴影区域），尽管从 t ¯ t ¯ bar(t)\bar{t} 大约抑制了 7 − 11 dB 7 − 11 dB 7-11dB7-11 \mathrm{~dB} 。 Δ t Δ t Delta t\Delta t 上的过量噪声限制了环外同步，从而产生了图 4 中显示的白噪声本底和图 3 中相应的升高不稳定性。我们将其归因于横跨 300 千米水平链路的强大气湍流造成的多径干扰。随着湍流强度的增加，脉冲形状本身会在传输 31 , 33 , 34 31 , 33 , 34 ^(31,33,34){ }^{31,33,34} 后发生扭曲和传播。我们推测这些与多径相关的畸变
Fig. 4 ∣ 4 ∣ 4∣4 \mid PSDs measured over the strong turbulence of the 300 k m 300 k m 300km300 \mathbf{k m} link.
图：在 300 k m 300 k m 300km300 \mathbf{k m} 链路的强湍流上测量到的 4 ∣ 4 ∣ 4∣4 \mid PSD。
Timing PSD for a representative 90 min measurement for Δ t = ( t A − t B ) / 2 Δ t = t A − t B / 2 Delta t=(t_(A)-t_(B))//2\Delta t=\left(t_{\mathrm{A}}-t_{\mathrm{B}}\right) / 2 (purple solid line), the time-of-flight term given by t ¯ = ( t A + t B ) / 2 t ¯ = t A + t B / 2 bar(t)=(t_(A)+t_(B))//2\bar{t}=\left(t_{\mathrm{A}}+t_{\mathrm{B}}\right) / 2 (green solid line) and out-of-loop synchronization verification (blue solid line). Shaded regions indicate excess noise beyond the expected reference oscillator noise and piston noise values, for Δ t Δ t Delta t\Delta t and t ¯ t ¯ bar(t)\bar{t}, respectively, as well as the shared quantum noise floor (dashed lines). The excess timing noise on Δ t Δ t Delta t\Delta t sets the floor for the out-of-loop timing below the 15 Hz synchronization bandwidth. Inset shows the non-reciprocal excess versus L C n 2 L C n 2 LC_(n)^(2)L C_{n}^{2} for 10 min intervals over many runs (grey circles). The solid red line illustrates the linear trend with integrated turbulence strength.
对 Δ t = ( t A − t B ) / 2 Δ t = t A − t B / 2 Delta t=(t_(A)-t_(B))//2\Delta t=\left(t_{\mathrm{A}}-t_{\mathrm{B}}\right) / 2 （紫色实线）、 t ¯ = ( t A + t B ) / 2 t ¯ = t A + t B / 2 bar(t)=(t_(A)+t_(B))//2\bar{t}=\left(t_{\mathrm{A}}+t_{\mathrm{B}}\right) / 2 给出的飞行时间项（绿色实线）和环外同步验证（蓝色实线）进行 90 分钟代表性测量的定时 PSD。阴影区域分别表示 Δ t Δ t Delta t\Delta t 和 t ¯ t ¯ bar(t)\bar{t} 超出预期参考振荡器噪声和活塞噪声值的过量噪声，以及共享量子噪声本底（虚线）。 Δ t Δ t Delta t\Delta t 上的过量定时噪声设定了低于 15 Hz 同步带宽的环外定时下限。插图显示了多次运行中 10 分钟间隔的非互易过量与 L C n 2 L C n 2 LC_(n)^(2)L C_{n}^{2} 的关系（灰色圆圈）。红色实线表示综合湍流强度的线性趋势。
will depend on the pulse chirp, which differs between clock combs leading to the limited 7 − 11 dB 7 − 11 dB 7-11dB7-11 \mathrm{~dB} suppression in the reciprocal measurement. Whereas such strong turbulence effects are challenging to analyse theoretically, the noise should increase with integrated turbulence, which is indeed the case as illustrated in the inset of Fig. 4.
将取决于脉冲啁啾，不同的时钟梳之间存在差异，从而导致在倒易测量中出现有限的 7 − 11 dB 7 − 11 dB 7-11dB7-11 \mathrm{~dB} 抑制。虽然从理论上分析这种强烈的湍流效应具有挑战性，但噪声应该随着综合湍流的增加而增加，如图 4 的插图所示，情况确实如此。
Figure 5 puts the performance and size, weight, power and cost (SWaP-C) of quantum-limited OTT in context with previous work and future space-based OTT. The current 10 cm apertures and 4.0 mW comb powers bode well for future low-SWaP-C space instruments 35 35 ^(35){ }^{35}. Moreover, the use of 10 cm apertures for the ground as well as satellite terminals permits compact ground stations. For applications such as global time distribution, clock-based geodesy or very long baseline interferometry, the satellite would include OTT terminals and a high-performance oscillator 36 − 38 36 − 38 ^(36-38){ }^{36-38}. Relativistic tests or dark-matter searches would require an atomic optical clock onboard the satellite, along with the OTT terminals.
图 5 将量子限幅 OTT 的性能以及尺寸、重量、功率和成本（SWaP-C）与以前的工作和未来的天基 OTT 相结合。目前的 10 厘米孔径和 4.0 mW 梳状功率对于未来的低 SWaP-C 空间仪器 35 35 ^(35){ }^{35} 来说是个好兆头。此外，在地面和卫星终端上使用 10 厘米孔径可实现紧凑型地面站。对于全球时间分布、基于时钟的大地测量或超长基线干涉测量等应用，卫星将包括 OTT 终端和高性能振荡器 36 − 38 36 − 38 ^(36-38){ }^{36-38} 。相对论测试或暗物质搜索需要在卫星上安装一个原子光学时钟和 OTT 终端。
Whereas GEO-to-ground links are 100 times farther than the Hawaii link, the path integrated turbulence is roughly 100 times lower. As a result, for paired 10 − cm 10 − cm 10-cm10-\mathrm{cm} aperture telescopes, the total link loss is similar to that shown in Fig. 5 (Supplementary Information) and the conservative 102 dB tolerable loss exceeds the estimated ground-to-GEO link loss by 11 dB . A fourfold increase in the ground aperture should increase the reach to cis-lunar orbits. Because the integrated turbulence is about 100 times lower than for the Hawaii link, ground-to-space OTT should not suffer from the same level of non-reciprocal timing noise from multipath effects. However, the turbulence-induced piston noise will no longer perfectly cancel as the up- and down-going comb pulses traverse the turbulent atmosphere with a 0.12-s time offset, roughly the time-of-flight to GEO. As a result, there is a differential piston noise
虽然地球同步轨道到地面的链路比夏威夷链路远 100 倍，但路径综合湍流大约低 100 倍。因此，对于成对的 10 − cm 10 − cm 10-cm10-\mathrm{cm} 孔径望远镜来说，总链路损耗与图 5（补充信息）所示的相似，保守的 102 dB 可容忍损耗比估计的地面到地球同步轨道链路损耗高出 11 dB。地面孔径增加四倍应能增加到达顺月轨道的距离。由于综合湍流比夏威夷链路低约 100 倍，地对空 OTT 不应受到多径效应产生的同等程度的非互易定时噪声的影响。然而，湍流引起的活塞噪声将不再完全抵消，因为上行和下行梳状脉冲穿越湍流大气层时会有 0.12 秒的时间偏移，大致相当于飞往地球同步轨道的时间。因此，会出现不同的活塞噪声
Fig. 5 ∣ 5 ∣ 5∣5 \mid Ground-to-space time transfer. Future free-space OTT will require operation at long distance, set by tolerable link loss, and low SWaP-C, largely set by the power-aperture product. The current demonstration (large yellow star) achieves the highest tolerable loss of 102 dB at the very low poweraperture product of 0.3 W cm 2 0.3 W cm 2 0.3Wcm^(2)0.3 \mathrm{~W} \mathrm{~cm}^{2} as compared to previous comb-based timefrequency transfer or OFT (magenta circles) 20 , 24 , 27 , 28 20 , 24 , 27 , 28 ^(20,24,27,28){ }^{20,24,27,28}. The linear trade-off (white dashed lines) between tolerable loss or maximum (Max.) distance and poweraperture product is set by the received power threshold, as shown for the LOS comb-based OTT ( 2.5 nW ) and quantum-limited OTT ( 250 fW ) with both curves assuming a 10 cm aperture. For an additional comparison, several optical communication missions are shown (orange circles for GEO and orange crosses for LEO) 44 − 49 44 − 49 ^(44-49){ }^{44-49}. The lower bottom and far right axes assume matched 10 cm apertures, 2 dB channel loss, 6 dB total transceiver loss and 6 dB additional coupling loss (Supplementary Information). The maximum projected distance for the current OTT exceeds the 35 , 786 km 35 , 786 km 35,786km35,786 \mathrm{~km} GEO altitude and even reaches the 10 times farther cis-lunar distances if the ground aperture is increased to 40 cm , while leaving the space-based aperture at 10 cm (small yellow star).
图： 5 ∣ 5 ∣ 5∣5 \mid 地对空时间传输。未来的自由空间 OTT 需要在长距离和低 SWaP-C 下运行，前者取决于可容忍的链路损耗，后者主要取决于功率-孔径乘积。与以前的梳状时频传输或 OFT（洋红色圆圈） 20 , 24 , 27 , 28 20 , 24 , 27 , 28 ^(20,24,27,28){ }^{20,24,27,28} 相比，目前的演示（黄色大星）在 0.3 W cm 2 0.3 W cm 2 0.3Wcm^(2)0.3 \mathrm{~W} \mathrm{~cm}^{2} 的极低功率孔径乘积下实现了 102 dB 的最高可容忍损耗。如图所示，基于 LOS 梳状 OTT（2.5 nW）和量子限幅 OTT（250 fW）的接收功率阈值决定了可容忍损耗或最大（Max.）距离与功率孔径乘积之间的线性权衡（白色虚线），两条曲线均假设孔径为 10 cm。为了进行更多比较，图中显示了几个光通信任务（橙色圆圈表示 GEO，橙色十字表示 LEO） 44 − 49 44 − 49 ^(44-49){ }^{44-49} 。下方和最右边的坐标轴假定有匹配的 10 厘米孔径、2 dB 信道损耗、6 dB 收发器总损耗和 6 dB 附加耦合损耗（补充信息）。如果将地面孔径增加到 40 厘米，同时将天基孔径保持在 10 厘米（小黄星），则当前 OTT 的最大预测距离将超过 35 , 786 km 35 , 786 km 35,786km35,786 \mathrm{~km} 地球同步轨道高度，甚至达到 10 倍的顺月球距离。
contribution that limits the time deviation to ≈ 2 fs / τ ≈ 2 fs / τ ~~2fs//sqrttau\approx 2 \mathrm{fs} / \sqrt{\tau} and modified Allan deviation to ≈ 4 × 10 − 15 τ − 3 / 2 ≈ 4 × 10 − 15 τ − 3 / 2 ~~4xx10^(-15)tau^(-3//2)\approx 4 \times 10^{-15} \tau^{-3 / 2} (Supplementary Information). Purely coincidentally, these values are close to the data of Fig. 3 although of very different origin. Nevertheless, the residual instability is below state-of-the-art optical clocks for τ > 10 τ > 10 tau > 10\tau>10 s of seconds. OTT could then compare two optical clocks in different locations without degradation by means of common view, or a ground clock to a space-based clock for tests of relativity 1 − 3 1 − 3 ^(1-3){ }^{1-3}. Active clock synchronization would be limited to a bandwidth of roughly 1 / ( 8 × 0.12 s ) = 1 Hz 1 / ( 8 × 0.12 s ) = 1 Hz 1//(8xx0.12s)=1Hz1 /(8 \times 0.12 \mathrm{~s})=1 \mathrm{~Hz}, or roughly ten times lower than the Hawaii link. Therefore, applications that require high coherence between ground and space might combine active synchronization (for observation times beyond 2 s ) with near-real-time postprocessing of the time samples (for observation times below 2 s ).
的贡献，将时间偏差限制在 ≈ 2 fs / τ ≈ 2 fs / τ ~~2fs//sqrttau\approx 2 \mathrm{fs} / \sqrt{\tau} ，将修正的阿伦偏差限制在 ≈ 4 × 10 − 15 τ − 3 / 2 ≈ 4 × 10 − 15 τ − 3 / 2 ~~4xx10^(-15)tau^(-3//2)\approx 4 \times 10^{-15} \tau^{-3 / 2} （补充信息）。纯属巧合的是，这些数值与图 3 中的数据非常接近，尽管它们的来源截然不同。尽管如此，残余不稳定性还是低于 τ > 10 τ > 10 tau > 10\tau>10 秒的最先进光学时钟。这样，OTT 就可以通过共同视图对不同地点的两个光学时钟进行比较，而不会出现劣化，也可以将地面时钟与天基时钟进行比较，以测试相对论 1 − 3 1 − 3 ^(1-3){ }^{1-3} 。主动时钟同步将被限制在大约 1 / ( 8 × 0.12 s ) = 1 Hz 1 / ( 8 × 0.12 s ) = 1 Hz 1//(8xx0.12s)=1Hz1 /(8 \times 0.12 \mathrm{~s})=1 \mathrm{~Hz} 的带宽内，或者说比夏威夷链路低大约十倍。因此，需要地面和空间高度一致性的应用可能会将主动同步（观测时间超过 2 秒）与时间采样的近实时后处理（观测时间低于 2 秒）结合起来。
Although the link margin is more than sufficient for OTT to low to mid-Earth orbits (LEO/MEO), these involve up to 8 km s − 1 8 km s − 1 8kms^(-1)8 \mathrm{~km} \mathrm{~s}^{-1} velocities with significant resulting Doppler shifts, other relativistic effects and point-ahead. Relativistic effects are in principle calculable and non-reciprocity from point-ahead have been shown to be negligible after correction 39 − 41 39 − 41 ^(39-41){ }^{39-41}. However, the strong Doppler shifts will significantly complicate signal acquisition and signal processing, while potentially introducing systematics. For example, signal acquisition within the 26 − kHz 26 − kHz 26-kHz26-\mathrm{kHz} detection bandwidth requires orbit velocity knowledge to ± 2 cm s − 1 ± 2 cm s − 1 +-2cms^(-1)\pm 2 \mathrm{~cm} \mathrm{~s}^{-1}. In addition, coupling of Doppler shifts to optical dispersion can mimic time shifts, causing systematic errors. For these reasons, quantum-limited OTT to LEO/MEO orbits will require an even more advanced system. It is likely such a system will require a hybrid approach that combines comb-based OTT with continuous wave laser-based OFT to assist with the frequency acquisition and tracking of the large Doppler shifted signals.
虽然链路裕度对于中低地球轨道（LEO/MEO）的 OTT 来说绰绰有余，但这涉及到高达 8 km s − 1 8 km s − 1 8kms^(-1)8 \mathrm{~km} \mathrm{~s}^{-1} 的速度，以及由此产生的显著多普勒频移、其他相对论效应和超前点。相对论效应原则上是可以计算的，而且经过 39 − 41 39 − 41 ^(39-41){ }^{39-41} 校正后，证明超前点产生的非互惠性可以忽略不计。然而，强烈的多普勒频移将使信号采集和信号处理变得非常复杂，同时还可能带来系统误差。例如，在 26 − kHz 26 − kHz 26-kHz26-\mathrm{kHz} 探测带宽内采集信号，需要了解 ± 2 cm s − 1 ± 2 cm s − 1 +-2cms^(-1)\pm 2 \mathrm{~cm} \mathrm{~s}^{-1} 的轨道速度。此外，多普勒频移与光色散的耦合会模拟时移，造成系统误差。由于这些原因，对低地球轨道/地球同步轨道的量子限制 OTT 将需要更先进的系统。这种系统很可能需要一种混合方法，将基于连续波激光的 OTT 与基于连续波激光的 OFT 结合起来，以协助频率采集和跟踪大的多普勒频移信号。
Figure 5 includes several optical communication missions. Quantum-limited OTT might piggyback on future coherent optical communication links, which would be facilitated by its low power threshold of -96 dBm either by use of a shared aperture or separately steered subaperture. It also includes several demonstrations of OFT by means of two-way transmission of a continuous wave laser, which reaches extremely low instabilities over low turbulence, kilometre-scale links 26 − 28 26 − 28 ^(26-28){ }^{26-28}. The tolerable loss (and therefore distance) for OFT could be significantly extended by operation near the shot-noise limit depending on the tolerance for cycle slips 42 42 ^(42){ }^{42}. However, signal interruptions from turbulence, weather, platform motion and/or multiplexed operation to different sites will result in penalties in the frequency transfer 43 43 ^(43){ }^{43} and, critically, prevent measurement of the relative time offset between clocks, the quantity of interest for many applications. Quantum-limited OTT, on the other hand, can track the relative time with femtosecond precision even if the link availability is limited to seconds interspersed across hours, days or longer.
图 5 包括几项光通信任务。量子限幅 OTT 可借助于未来的相干光通信链路，其-96 dBm 的低功率阈值可通过使用共享光圈或单独转向的子光圈来实现。报告还通过连续波激光器的双向传输演示了几次 OFT，在低湍流、千米级链路 26 − 28 26 − 28 ^(26-28){ }^{26-28} 上达到了极低的不稳定性。根据对周期滑动 42 42 ^(42){ }^{42} 的容忍度，OFT 的可容忍损耗（以及距离）可通过在射噪极限附近运行而显著延长。然而，湍流、天气、平台移动和/或不同站点的多路复用操作造成的信号中断将导致频率传输 43 43 ^(43){ }^{43} 的损失，关键是无法测量时钟之间的相对时间偏移，而这正是许多应用所关心的数量。而量子限制 OTT 则可以飞秒级精度跟踪相对时间，即使链路可用性仅限于数秒，间隔时间为数小时、数天或更长。

Conclusion 结论

By operating near the quantum limit, this OTT requires 10,000 times less power than previous LOS approaches. Here, we show attosecond time transfer over record distances of 300 km and record link losses of 102 dB , while transmitting only the 4.0 mW output from a compact, unamplified frequency comb. At low turbulence strengths, the time deviation is as low as 35 attoseconds at 3 s averaging. Under strong turbulence, the time deviation drops as low as 220 attoseconds at roughly 300 s of averaging: well below what is needed for distributed coherent sensing, redefinition of the second and tests of fundamental physics 1 − 12 1 − 12 ^(1-12){ }^{1-12}.
通过在量子极限附近运行，这种 OTT 所需的功率比以前的 LOS 方法低 10,000 倍。在这里，我们展示了阿秒级时间传输，传输距离达到创纪录的 300 千米，链路损耗达到创纪录的 102 分贝，而传输功率仅为紧凑型非放大频率梳输出的 4.0 毫瓦。在湍流强度较低的情况下，3 秒平均时间偏差低至 35 阿秒。在强湍流条件下，平均时间约为 300 秒时，时间偏差低至 220 阿秒：远低于分布式相干传感、秒的重新定义和基础物理学 1 − 12 1 − 12 ^(1-12){ }^{1-12} 测试所需的时间偏差。
The combination of this level of performance, low-power aperture product and the ability to operate at 102 dB link loss without adaptive optics will enable future low SWaP-COTT from geosynchronous orbits (GEO) to portable ground-based clocks for time transfer, geodesy, relativity tests or distributed sensing. Indeed, with a four times increase in ground terminal aperture, future OTT could even reach cis-lunar orbits.
这种性能水平、低功耗孔径产品以及在没有自适应光学器件的情况下以 102 dB 链路损耗运行的能力相结合，将使未来的低 SWaP-COTT 能够从地球同步轨道（GEO）到便携式地基时钟，用于时间传送、大地测量、相对论测试或分布式传感。事实上，如果地面终端孔径增加四倍，未来的 OTT 甚至可以达到顺月轨道。

Online content 在线内容

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-023-06032-5.
任何方法、其他参考文献、《自然》组合报告摘要、源数据、扩展数据、补充信息、致谢、同行评审信息；作者贡献和竞争利益详情；以及数据和代码可用性声明，均可在https://doi.org/10.1038/s41586-023-06032-5上查阅。
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Methods 方法

Overview of OTT operation
OTT 运营概览

The steps necessary for operation of the time transfer system are as follows (Extended Data Fig. 3):
时空传送系统的操作步骤如下（扩展数据图 3）：
(1) Initial calibration (done once after transceiver construction):
(1) 初始校准（在收发器制造完成后进行一次）：
(a) Calibrate the transceiver time delay between the tracking comb reference plane (used to measure the time offset between the local tracking and clock combs at the tracking comb initialization step below) and the clock comb reference plane (used as out-of-loop verification plane). This calibrated time delay depends only on fibre path lengths in the transceiver, which are minimized and temperature controlled to reduce drift (Extended Data Fig. 5).
(a) 校准跟踪梳参考平面（用于在下面的跟踪梳初始化步骤中测量本地跟踪梳和时钟梳之间的时间偏移）与时钟梳参考平面（用作环外验证平面）之间的收发器时延。这一校准时间延迟仅取决于收发器中的光纤路径长度，而光纤路径长度已被最小化并进行了温度控制，以减少漂移（扩展数据图 5）。
(b) Calibrate the timing discriminator for both optical timing error and incoming optical power (Extended Data Fig. 4).
(b) 根据光学定时误差和输入光功率校准定时鉴别器（扩展数据图 4）。
© Calibrate the out-of-loop verification channel voltage-to-time scaling.
校准环外验证通道电压-时间比例。
(2) Pre-operation steps (done before every time transfer measurement):
(2) 操作前步骤（在每次时间转移测量前完成）：
(a) Lock the clock comb and tracking comb to the local reference oscillator (portable cavity-stabilized laser in this experiment).
(a) 将时钟梳和跟踪梳锁定到本地参考振荡器（本实验中为便携式腔稳激光器）。
(b) Initialize the tracking comb time offset by adjusting its zero-time offset to coincide with its pulse overlapping the local clock comb at the tracking comb reference plane.
(b) 初始化跟踪梳时间偏移，调整其零时间偏移，使其与跟踪梳参考平面的本地时钟梳重叠脉冲相吻合。
© Generate an initial estimate of the carrier frequency offset between the local tracking comb and incoming clock comb. Here, this estimate is based on measured cavity-stabilized laser frequencies and comb locking parameters.
生成本地跟踪梳状和输入时钟梳状之间载波频率偏移的初始估计值。在此，这一估计值基于测量到的腔稳激光频率和梳状锁定参数。
(3) Optical link acquisition. Establish bidirectional coupling of the transmitted clock comb light into single-mode optical fibre after propagation across the free-space link (Extended Data Fig.1).
(3) 光链路采集。将传输的时钟梳状光在自由空间链路上传播后，双向耦合到单模光纤中（扩展数据图 1）。
(4) Tracking comb timing acquisition. Search for timing overlap between the incoming clock comb and local tracking comb at each site (Fig. 2a and later discussion in Methods).
(4) 跟踪梳定时采集。在每个站点搜索传入时钟梳和本地跟踪梳之间的时序重叠（图 2a 和后面 "方法 "中的讨论）。
(5) Tracking comblock. Actively feedback to the local tracking combtiming to maintain overlap with the incoming clock comb pulses to within the dynamic range of the optical timing discriminator (Extended Data Fig. 4). In addition, actively feedback to the tracking comb frequency to keep the heterodyne signal between the tracking and local clock combs centred within the 26 kHz detection bandwidth.
(5) 跟踪组合块。对本地跟踪梳状频率进行主动反馈，以保持与传入时钟梳状脉冲的重叠在光学定时鉴别器的动态范围内（扩展数据图 4）。此外，主动反馈跟踪梳状频率，以保持跟踪和本地时钟梳状之间的异频信号在 26 kHz 检测带宽内居中。
(6) Note that the Kalman filter continuously tracks of the timing uncertainty between the local tracking and incoming clock comb. In the event of a signal fade from turbulence, if this uncertainty is too large, the tracking comb timing is re-acquired as discussed below and in the main text.
(6) 请注意，卡尔曼滤波器会持续跟踪本地跟踪和输入时钟梳之间的时间不确定性。在湍流造成信号衰减时，如果这种不确定性过大，就会按照下文和正文中的讨论重新获取跟踪梳状时间。
(7) Communication between sites (site A to B). Site A continuously transmits back the received timing to site B B BB so that the clock timing offset can be calculated by means of the two-way timing calculation.
(7) 站点之间的通信（站点 A 至 B）。站点 A 不断向站点 B B BB 发回接收到的定时，以便通过双向定时计算来计算时钟定时偏移。
(8) Synchronization (site B). The clock timing offset is input to the synchronization loop to adjust the clock comb at site B. At the initiation of the synchronization loop, the tracking comb timing at site A A AA is adjusted to avoid loss of tracking (Methods).
(8) 同步（站点 B）。时钟定时偏移被输入同步环，以调整 B 站点的时钟梳。在同步环启动时，站点 A A AA 的跟踪梳定时被调整，以避免跟踪丢失（方法）。
(9) Out-of-loop verification. Once synchronization has been activated, measure the timing overlap of the two clock combs at the out-of-loop verification channel (Extended Data Fig. 6).
(9) 环外验证。启动同步后，在环外验证通道测量两个时钟梳的时序重叠（扩展数据图 6）。

Timing discriminator operation
定时判别器运行

As shown in Extended Data Fig. 5, the optical timing discriminator uses the birefringence of polarization-maintaining (PM) optical fibre to generate two interferograms in which the lead and lag positions of the incoming clock comb pulse and local tracking comb pulses are switched 29 29 ^(29){ }^{29}. The signals that are output from each balanced detector sit at a heterodyne frequency set by the offset between the two comb’s carrier frequencies. Because we demodulate the signals, this frequency offset must be known a priori to ± 13 kHz ± 13 kHz +-13kHz\pm 13 \mathrm{kHz} during acquisition, but after acquisition the frequency offset is tracked and adjusted to centre the
如扩展数据图 5 所示，光学定时鉴别器利用偏振保持（PM）光纤的双折射产生两个干涉图，其中输入时钟梳状脉冲和本地跟踪梳状脉冲的前导和滞后位置进行了 29 29 ^(29){ }^{29} 切换。每个平衡探测器输出的信号位于由两个梳状脉冲载波频率之间的偏移量设定的外差频率上。由于我们对信号进行了解调，因此在采集过程中，必须事先知道 ± 13 kHz ± 13 kHz +-13kHz\pm 13 \mathrm{kHz} 的频率偏移，但在采集之后，频率偏移会被跟踪和调整，以将信号集中到 29 29 ^(29){ }^{29}的中心。
demodulation frequency. Initial knowledge of the carrier frequency offset to within ± 13 kHz ± 13 kHz +-13kHz\pm 13 \mathrm{kHz} (fractional frequency knowledge of 6.8 × 10 − 11 6.8 × 10 − 11 6.8 xx10^(-11)6.8 \times 10^{-11} at the optical carrier frequency) can be provided by (1) measuring the reference oscillator frequencies against a GPS disciplined oscillator, (2) use of a coarser frequency transfer over a comms link as has been done previously or (3) past estimates from a Kalman filter. Alternatively, the acquisition could include a search over frequency, but this additional search has not been implemented here. The demodulation frequency is set to differ between sites, nominally at values of 10 and 12 MHz to avoid cross talk from internal reflections within the free-space optical terminals.
解调频率。载波频率偏移在 ± 13 kHz ± 13 kHz +-13kHz\pm 13 \mathrm{kHz} 范围内（光载波频率为 6.8 × 10 − 11 6.8 × 10 − 11 6.8 xx10^(-11)6.8 \times 10^{-11} 的分数频率知识）的初始知识可通过以下方式提供：(1) 根据 GPS 标准振荡器测量参考振荡器频率；(2) 通过通信链路使用较粗的频率传输，如之前所做的那样；或 (3) 过去通过卡尔曼滤波器进行的估计。另外，还可以在采集过程中进行频率搜索，但这里没有进行这种额外的搜索。为了避免自由空间光学终端内部反射造成的串扰，解调频率在不同站点之间设置不同，名义上分别为 10 和 12 兆赫。
The output of the timing discriminator is then the magnitude of the two demodulated signals, | V 1 | V 1 |V_(1)|\left|V_{1}\right| and | V 2 | V 2 |V_(2)|\left|V_{2}\right|. After calibration, the sum of their squares provides a measure of the incoming clock comb power, in watts, and their normalized relative values provides a measure of the relative timing of the incoming clock comb pulse train with respect to the local tracking comb, in femtoseconds.
定时鉴别器的输出是两个解调信号 | V 1 | V 1 |V_(1)|\left|V_{1}\right| 和 | V 2 | V 2 |V_(2)|\left|V_{2}\right| 的幅度。校准后，这两个信号的平方和就可以测量输入时钟梳功率（单位：瓦），而它们的归一化相对值则可以测量输入时钟梳脉冲序列与本地跟踪梳的相对定时（单位：飞秒）。

Timing discriminator signal power calibration
定时鉴别器信号功率校准

As with previous comb-based OTT using LOS, the input power is measured by calibrating the heterodyne signal amplitude 15 , 24 15 , 24 ^(15,24){ }^{15,24} but in this case the signals from the timing discriminator are used. (Direct detection of the incoming power at the 100 fW level would otherwise be challenging). For this heterodyne detection, the power is given by P = C p [ ( | V 1 | 2 + | V 2 | 2 ) / 2 ] P = C p V 1 2 + V 2 2 / 2 P=C_(p)[(|V_(1)|^(2)+|V_(2)|^(2))//2]P=C_{\mathrm{p}}\left[\left(\left|V_{1}\right|^{2}+\left|V_{2}\right|^{2}\right) / 2\right], where the calibration factor, C p C p C_(p)C_{\mathrm{p}} has units of W V − 2 W V − 2 WV^(-2)\mathrm{W} \mathrm{V}^{-2} and is a function of the local tracking comb power on the detector, pulse shape and dispersion, and timing imbalance of the optical timing discriminator. To measure this calibration constant, the link is shorted, attenuated to low powers and synchronized. We then measure the value of ( | V 1 | 2 + | V 2 | 2 ) / 2 V 1 2 + V 2 2 / 2 (|V_(1)|^(2)+|V_(2)|^(2))//2\left(\left|V_{1}\right|^{2}+\left|V_{2}\right|^{2}\right) / 2. Next, we measure the total clock comb optical power at the input of the four balanced detector input ports using a calibrated power meter. On the basis of repeated calibrations, we expect this calibration constant to be accurate to within ± 20 % ± 20 % +-20%\pm 20 \%.
与之前使用 LOS 的基于梳状信号的 OTT 一样，输入功率是通过校准外差信号振幅 15 , 24 15 , 24 ^(15,24){ }^{15,24} 来测量的，但这次使用的是来自定时鉴别器的信号。(否则，直接检测 100 fW 级的输入功率将非常困难）。对于这种外差探测，功率由 P = C p [ ( | V 1 | 2 + | V 2 | 2 ) / 2 ] P = C p V 1 2 + V 2 2 / 2 P=C_(p)[(|V_(1)|^(2)+|V_(2)|^(2))//2]P=C_{\mathrm{p}}\left[\left(\left|V_{1}\right|^{2}+\left|V_{2}\right|^{2}\right) / 2\right] 给出，其中校准因子 C p C p C_(p)C_{\mathrm{p}} 的单位为 W V − 2 W V − 2 WV^(-2)\mathrm{W} \mathrm{V}^{-2} ，是探测器上的本地跟踪梳功率、脉冲形状和色散以及光学定时鉴别器的定时不平衡的函数。要测量这个校准常数，需要将链路短接、衰减到低功率并同步。然后测量 ( | V 1 | 2 + | V 2 | 2 ) / 2 V 1 2 + V 2 2 / 2 (|V_(1)|^(2)+|V_(2)|^(2))//2\left(\left|V_{1}\right|^{2}+\left|V_{2}\right|^{2}\right) / 2 的值。接下来，我们使用校准功率计测量四个平衡检测器输入端口输入端的总时钟梳光功率。在反复校准的基础上，我们预计这一校准常数的精确度将在 ± 20 % ± 20 % +-20%\pm 20 \% 范围内。
This power calibration applies to the normal operation in which the demodulated signals are around half their peak values (Extended Data Fig. 5). During acquisition, the two demodulated squared signals are added together with a temporal delay that reflects the linearly scanned tracking comb. As a result, the peak demodulated voltages are roughly twice higher for the same optical power, and the calibration factor is ideally four times lower. In reality, for the delays here, the increase is closer to a factor of two as the operation point is not fully half of the peak voltage giving C p , acq ≈ C p / 2 C p , acq ≈ C p / 2 C_(p,acq)~~C_(p)//2C_{\mathrm{p}, \mathrm{acq}} \approx C_{\mathrm{p}} / 2.
这种功率校准适用于正常操作，此时解调信号约为其峰值的一半（扩展数据图 5）。在采集过程中，两个解调后的平方信号通过一个反映线性扫描跟踪梳的时间延迟加在一起。因此，在光功率相同的情况下，解调电压的峰值大约要高出两倍，而校准因子理想情况下要低四倍。实际上，就此处的延迟而言，由于操作点并不完全是 C p , acq ≈ C p / 2 C p , acq ≈ C p / 2 C_(p,acq)~~C_(p)//2C_{\mathrm{p}, \mathrm{acq}} \approx C_{\mathrm{p}} / 2 峰值电压的一半，因此增加的系数更接近于两倍。

Timing discriminator timing error calibration
定时判别器定时误差校准

The timing error is calculated on the basis of the normalized difference of the two demodulated voltages by δ t = f td ( E ) δ t = f td ( E ) delta t=f_(td)(E)\delta t=f_{\mathrm{td}}(E), where E = ( | V 1 | − | V 2 | ) / ( | V 1 | + | V 2 | ) E = V 1 − V 2 / V 1 + V 2 E=(|V_(1)|-|V_(2)|)//(|V_(1)|+|V_(2)|)E=\left(\left|V_{1}\right|-\left|V_{2}\right|\right) /\left(\left|V_{1}\right|+\left|V_{2}\right|\right) and f td ( E ) f td ( E ) f_(td)(E)f_{\mathrm{td}}(E) is the calibration function. f td ( E ) f td ( E ) f_(td)(E)f_{\mathrm{td}}(E) is determined in a separate shorted calibration by deliberately offsetting the clock and tracking comb pulse and then measuring the corresponding error signal, as shown in Extended Data Fig. 4. Because the error signal is normalized by the sum of the two demodulated signals, the timing error is insensitive to common-mode (for example, turbulence-induced) amplitude fluctuations between the channels. Furthermore, the performance of the overall system is insensitive to small changes in the calibration function, for example because of changes in the differential pulse dispersion, as the feedback seeks to lock this error signal δ t δ t delta t\delta t to zero (Extended Data Fig. 4). Moreover, the fact that the timing discriminator’s differential delay arises from the difference in propagation constant of the two fibre polarization modes means that the temperature stability requirements are loose.
时序误差是根据两个解调电压的归一化差值 δ t = f td ( E ) δ t = f td ( E ) delta t=f_(td)(E)\delta t=f_{\mathrm{td}}(E) 计算得出的，其中 E = ( | V 1 | − | V 2 | ) / ( | V 1 | + | V 2 | ) E = V 1 − V 2 / V 1 + V 2 E=(|V_(1)|-|V_(2)|)//(|V_(1)|+|V_(2)|)E=\left(\left|V_{1}\right|-\left|V_{2}\right|\right) /\left(\left|V_{1}\right|+\left|V_{2}\right|\right) 和 f td ( E ) f td ( E ) f_(td)(E)f_{\mathrm{td}}(E) 是校准函数。如扩展数据图 4 所示， f td ( E ) f td ( E ) f_(td)(E)f_{\mathrm{td}}(E) 是在单独的短路校准中通过故意偏移时钟和跟踪梳状脉冲确定的，然后测量相应的误差信号。由于误差信号通过两个解调信号之和归一化，因此定时误差对通道之间的共模（例如湍流引起的）振幅波动不敏感。此外，整个系统的性能对校准函数的微小变化也不敏感，例如，由于差分脉冲频散的变化，反馈会将误差信号 δ t δ t delta t\delta t 锁定为零（扩展数据图 4）。此外，时序鉴别器的差分延迟来自两种光纤极化模式的传播常数差异，这意味着对温度稳定性的要求很宽松。

Quantum-limited operation
量子限制运行

Both quantum-limited acquisition and quantum-limited operation depend on the signal-to-noise ratio of the timing discriminator. If that signal-to-noise ratio is shot-noise limited, then the timing discriminator
量子限制采集和量子限制运行都取决于定时鉴别器的信噪比。如果信噪比受到射噪限制，那么定时鉴别器
is operating at the standard quantum limit. We consider that limit under the assumption of heterodyne detection.
是在标准量子极限下运行的。我们将在外差探测的假设下考虑这一极限。
We first consider acquisition. Acquisition uses the average mean-squared voltage detected by the timing discriminator, ( | V 1 | 2 + | V 2 | 2 ) / 2 V 1 2 + V 2 2 / 2 (|V_(1)|^(2)+|V_(2)|^(2))//2\left(\left|V_{1}\right|^{2}+\left|V_{2}\right|^{2}\right) / 2 to search for pulse overlap. The detector noise level is 1.7 dB above shot noise set by the local tracking comb. On the basis of this noise level, we very conservatively set a threshold value of roughly ( 98 μ V ) 2 ( 98 μ V ) 2 (98 muV)^(2)(98 \mu \mathrm{~V})^{2}, which yields a false trigger rate of less than once per day. Given C p , acq C p , acq C_(p,acq)C_{\mathrm{p}, \mathrm{acq}}, this threshold corresponds to P thresh,acq ≈ 135 fW P thresh,acq  ≈ 135 fW P_("thresh,acq ")~~135fWP_{\text {thresh,acq }} \approx 135 \mathrm{fW}, or roughly 20 photons for the sample time of ( 2 × 26 kHz ) − 1 = 19 μ s ( 2 × 26 kHz ) − 1 = 19 μ s (2xx26kHz)^(-1)=19 mus(2 \times 26 \mathrm{kHz})^{-1}=19 \mu \mathrm{~s} (or 0.005 mean photons per comb pulse) corresponding to the 26 kHz heterodyne detection bandwidth.
我们首先考虑采集。采集使用定时鉴别器检测到的平均均方电压 ( | V 1 | 2 + | V 2 | 2 ) / 2 V 1 2 + V 2 2 / 2 (|V_(1)|^(2)+|V_(2)|^(2))//2\left(\left|V_{1}\right|^{2}+\left|V_{2}\right|^{2}\right) / 2 来搜索脉冲重叠。检测器的噪声水平比本地跟踪梳设定的发射噪声高 1.7 分贝。在此噪声水平的基础上，我们非常保守地将阈值大致设定为 ( 98 μ V ) 2 ( 98 μ V ) 2 (98 muV)^(2)(98 \mu \mathrm{~V})^{2} ，这样每天的误触发率不到一次。考虑到 C p , acq C p , acq C_(p,acq)C_{\mathrm{p}, \mathrm{acq}} ，该阈值对应于 P thresh,acq ≈ 135 fW P thresh,acq  ≈ 135 fW P_("thresh,acq ")~~135fWP_{\text {thresh,acq }} \approx 135 \mathrm{fW} ，即 ( 2 × 26 kHz ) − 1 = 19 μ s ( 2 × 26 kHz ) − 1 = 19 μ s (2xx26kHz)^(-1)=19 mus(2 \times 26 \mathrm{kHz})^{-1}=19 \mu \mathrm{~s} （或每个梳状脉冲平均 0.005 个光子）采样时间内大约 20 个光子，对应于 26 kHz 外差探测带宽。
Next, we consider quantum-limited operation of the timing discriminator in its estimate of the timing error, δ t δ t delta t\delta t, between the clock comb and tracking comb pulses, for example, the value of γ γ gamma\gamma in equation (1). To find the quantum limit, we consider heterodyne detection between two unchirped, Gaussian pulses offset in time. (The Gaussian shape avoids sidelobes that can lead to false timing error values and also corresponds to the Gaussian filters used in our apparatus). This simple configuration is equivalent to considering the output from only one arm of the timing discriminator. For timing measurements, we define δ t = 0 δ t = 0 delta t=0\delta t=0 to correspond to a pulse separation equal to the full-width at half-maximum, τ p τ p tau_(p)\tau_{\mathrm{p}}, of the pulse intensity. As discussed in ref. 29 , this yields a timing error of equation (3) with γ q , 1 = ( 2 ln ⁡ ( 2 ) η ) − 1 γ q , 1 = ( 2 ln ⁡ ( 2 ) η ) − 1 gamma_(q,1)=(2ln(2)sqrteta)^(-1)\gamma_{\mathrm{q}, 1}=(2 \ln (2) \sqrt{\eta})^{-1}, where η η eta\eta is the detector quantum efficiency. For the two-way time transfer, assuming an equal number of received photons at each site, there is an additional factor of 2 2 sqrt2\sqrt{2}, giving
接下来，我们将考虑定时鉴别器在估计时钟梳状脉冲和跟踪梳状脉冲之间的定时误差 δ t δ t delta t\delta t 时的量子限制操作，例如方程 (1) 中的 γ γ gamma\gamma 值。为了找到量子极限，我们考虑在两个未啁啾的高斯脉冲之间进行时间偏移的外差检测。(高斯形状避免了可能导致错误时间误差值的侧晃，同时也与我们仪器中使用的高斯滤波器相对应）。这种简单的配置相当于只考虑时序鉴别器一个臂的输出。对于定时测量，我们将 δ t = 0 δ t = 0 delta t=0\delta t=0 定义为相当于脉冲强度的半最大全宽 τ p τ p tau_(p)\tau_{\mathrm{p}} 的脉冲间隔。如参考文献 29 所述，这会产生等式 (3) 的时间误差 γ q , 1 = ( 2 ln ⁡ ( 2 ) η ) − 1 γ q , 1 = ( 2 ln ⁡ ( 2 ) η ) − 1 gamma_(q,1)=(2ln(2)sqrteta)^(-1)\gamma_{\mathrm{q}, 1}=(2 \ln (2) \sqrt{\eta})^{-1} ，其中 η η eta\eta 是探测器量子效率。对于双向时间传输，假设每个站点接收到的光子数量相等，则还有一个额外的系数 2 2 sqrt2\sqrt{2} ，即
γ ql = 1 2 2 ln ⁡ ( 2 ) n ≈ 0.6 γ ql = 1 2 2 ln ⁡ ( 2 ) n ≈ 0.6 gamma_(ql)=(1)/(2sqrt2ln(2)sqrtn)~~0.6\gamma_{\mathrm{ql}}=\frac{1}{2 \sqrt{2} \ln (2) \sqrt{n}} \approx 0.6
In the actual system, we use the dual outputs of the unbalanced Mach-Zehnder interferometer. In that case, n n nn is the total number of received photons. Furthermore, there is a roughly 1.7 dB penalty, D D DD, owing to detector noise, as discussed above. Finally, in our system, the largest penalty results from differential dispersion between the incoming clock comb pulses and the local tracking comb pulse. To lowest order, this causes broadening of the temporal signal by a factor C C CC and a corresponding reduction in peak height by C C CC, leading to a C 2 C 2 C^(2)C^{2} increase in γ γ gamma\gamma. The differential dispersion contains higher orders that lead to distortions of the pulse shape beyond broadening, but this first-order model is sufficient to characterize the degradation in the factor γ γ gamma\gamma, as
在实际系统中，我们使用不平衡马赫-泽恩德干涉仪的双输出。在这种情况下， n n nn 是接收到的光子总数。此外，如上文所述，由于探测器的噪声，会产生大约 1.7 dB 的损耗 D D DD 。最后，在我们的系统中，输入时钟梳状脉冲和本地跟踪梳状脉冲之间的差分色散造成了最大的损耗。就最低阶而言，这会导致时间信号拓宽 C C CC 倍，峰高相应降低 C C CC 倍，从而导致 C 2 C 2 C^(2)C^{2} 增加 γ γ gamma\gamma 。微分色散包含更高的阶数，会导致脉冲形状扭曲，而不仅仅是展宽，但这个一阶模型足以描述因数 γ γ gamma\gamma 的衰减，如图所示
γ = ( D C 2 ) γ q 1 γ = D C 2 γ q 1 gamma=(DC^(2))gamma_(q1)\gamma=\left(D C^{2}\right) \gamma_{\mathrm{q} 1}
Experimental values for γ , C γ , C gamma,C\gamma, C and other relevant parameters are given in Extended Data Table 1. We set a power threshold for a ‘valid’ timing discriminator measurement of 264 fW at site A and 283 fW at site B (Extended Data Fig. 2). These are chosen conservatively to remain in the linear regime of the timing discriminator curve (Extended Data Fig.4) despite the additive shot noise in the detected signal (and including the 1.7 dB detector noise penalty). At this threshold, additive noise leads to a noise level in E E EE of ± 0.3 ± 0.3 +-0.3\pm 0.3 (two sigma), which covers 40 % 40 % 40%40 \% of the shaded region in Extended Data Fig. 4. This power level corresponds to roughly 40 photons in a sample time of ( 2 × 26 kHz ) − 1 = 19 μ s ( 2 × 26 kHz ) − 1 = 19 μ s (2xx26kHz)^(-1)=19 mus(2 \times 26 \mathrm{kHz})^{-1}=19 \mu \mathrm{~s} (or 0.01 mean photons per comb pulse).
γ , C γ , C gamma,C\gamma, C 和其他相关参数的实验值见扩展数据表 1。我们将 "有效 "定时鉴别器测量的功率阈值设定为：A 点 264 fW，B 点 283 fW（扩展数据图 2）。尽管探测到的信号中存在加性射电噪声（包括 1.7 分贝的探测器噪声惩罚），但我们还是保守地选择了这一阈值，以保持定时鉴别器曲线的线性状态（扩展数据图 4）。在此阈值下，加性噪声会导致 E E EE 中的噪声水平达到 ± 0.3 ± 0.3 +-0.3\pm 0.3 （两个西格玛），涵盖扩展数据图 4 中阴影区域的 40 % 40 % 40%40 \% 。这一功率水平大约相当于 ( 2 × 26 kHz ) − 1 = 19 μ s ( 2 × 26 kHz ) − 1 = 19 μ s (2xx26kHz)^(-1)=19 mus(2 \times 26 \mathrm{kHz})^{-1}=19 \mu \mathrm{~s} 采样时间内的 40 个光子（或每个梳状脉冲 0.01 个平均光子）。

Free-space optical terminals
自由空间光学终端

The terminals are low loss, fibre coupled, single-transversal-mode and fully reciprocal for the frequency comb light following the design of ref. 50. For the pair of terminals used here (Extended Data Fig. 1), the total ‘fibre-to-fibre’ loss for comb light between them over a short distance is measured to be 3.3 dB , which includes roughly 2 dB of loss from the optical components and an additional roughly 1.3 dB of coupling loss into the single-mode fibre.
这些终端是低损耗、光纤耦合、单横模和完全对等的梳状频率光，采用了参考文献 50 的设计。50.对于这里使用的这对终端（扩展数据图 1），测量到它们之间短距离的梳状光 "光纤到光纤 "总损耗为 3.3 dB，其中包括来自光学元件的约 2 dB 损耗和进入单模光纤的额外约 1.3 dB 耦合损耗。
Unlike ref. 50, we use a larger 10 cm beam expander and use a focal plane array (FPA) for detection of the beacon laser for the much longer span of 300 km . The 10 cm aperture was chosen on the basis of commercially available optics and to be larger than the maximum expected Fried parameter. The received beacon is directed to the FPA of an InGaAs camera (Extended Data Fig.1). For operation at low received powers, the beacon is synchronously modulated and detected at half the camera frame rate, allowing for real-time background subtraction. The beam position on the FPA is maintained at a precalibrated location through feedback to the galvo mirror. Additionally, the beacon power integrated over the region of interest and recorded.
与参考文献 50 不同的是，我们使用了更大的 10 厘米扩束器，并使用焦平面阵列（FPA）对 300 公里的更长跨度进行探测。与参考文献 50 不同的是，我们使用了更大的 10 厘米扩束器，并使用焦平面阵列（FPA）来探测跨度更长的 300 公里范围内的信标激光。选择 10 厘米孔径的依据是市场上可买到的光学器件，并大于最大预期弗里德参数。接收到的信标激光被引向 InGaAs 摄像机的 FPA（扩展数据图 1）。为了在低接收功率下运行，信标被同步调制，并以摄像机帧频的一半进行检测，从而实现实时背景减除。通过对振镜的反馈，FPA 上的光束位置保持在预校准位置。此外，信标功率在感兴趣的区域内整合并记录。
The use of galvo mirrors as in ref. 50 and the FPA provides means of a broad search for the beacon enabling link acquisition after coarse initial pointing. A visible camera with telescopic lens provides the gross initial pointing, aided by an 830 nm , broad angle beacon transmitted through the cat-eye reflector and a matched 830 nm bandpass filter in front of the camera.
参考文献 50 中使用的振镜和 FPA 提供了广泛搜索信标的手段，使其能够在粗略的初始指向后获取链路。50 和 FPA 提供了对信标进行大范围搜索的手段，从而能够在粗略的初始指向后获取链路。通过猫眼反射镜和相机前的匹配 830 纳米带通滤波器传输的 830 纳米宽角度信标，一台带望远镜镜头的可见光相机提供了粗略的初始指向。
As dispersion amounts to a decrease in both the detection sensitivity and the timing sensitivity, we must compensate for the additional group delay dispersion of the 300 km of air (and also to maintain the accuracy of the power and timing slope calibrations). Using 70,000 pascal for air pressure, 10 ∘ C 10 ∘ C 10^(@)C10^{\circ} \mathrm{C} for air temperature, 25 % 25 % 25%25 \% relative humidity and 450 ppm for CO 2 CO 2 CO_(2)\mathrm{CO}_{2} mole fraction, we estimate the group velocity dispersion of the air in Hawaii to be β 2 = 7.43 × 10 − 30 s 2 m − 1 β 2 = 7.43 × 10 − 30 s 2 m − 1 beta_(2)=7.43 xx10^(-30)s^(2)m^(-1)\beta_{2}=7.43 \times 10^{-30} \mathrm{~s}^{2} \mathrm{~m}^{-1}. In comparison, at 1 , 560 nm 1 , 560 nm 1,560nm1,560 \mathrm{~nm}, PM1550 fibre has dispersion of β 2 = − 2.18 × 10 − 26 s 2 m − 1 β 2 = − 2.18 × 10 − 26 s 2 m − 1 beta_(2)=-2.18 xx10^(-26)s^(2)m^(-1)\beta_{2}=-2.18 \times 10^{-26} \mathrm{~s}^{2} \mathrm{~m}^{-1}. Therefore, we can compensate for the 297 km of air by adding 101 m of PM1550 fibre to the fibre path length already in place for a shorted measurement. As a final fine-tuning step, we measure the pulse widths across the atmosphere and minimize the width of the interference signal. This optimization yields a total PM1550 fibre length of 105 m .
由于弥散会降低探测灵敏度和定时灵敏度，我们必须对 300 千米空气中的额外群延迟弥散进行补偿（同时还要保持功率和定时斜率校准的准确性）。使用 70,000 帕斯卡气压、 10 ∘ C 10 ∘ C 10^(@)C10^{\circ} \mathrm{C} 气温、 25 % 25 % 25%25 \% 相对湿度和 450 ppm 的 CO 2 CO 2 CO_(2)\mathrm{CO}_{2} 分子分数，我们估计夏威夷空气的群速度弥散为 β 2 = 7.43 × 10 − 30 s 2 m − 1 β 2 = 7.43 × 10 − 30 s 2 m − 1 beta_(2)=7.43 xx10^(-30)s^(2)m^(-1)\beta_{2}=7.43 \times 10^{-30} \mathrm{~s}^{2} \mathrm{~m}^{-1} 。相比之下，在 1 , 560 nm 1 , 560 nm 1,560nm1,560 \mathrm{~nm} 条件下，PM1550 纤维的分散度为 β 2 = − 2.18 × 10 − 26 s 2 m − 1 β 2 = − 2.18 × 10 − 26 s 2 m − 1 beta_(2)=-2.18 xx10^(-26)s^(2)m^(-1)\beta_{2}=-2.18 \times 10^{-26} \mathrm{~s}^{2} \mathrm{~m}^{-1} 。因此，我们可以通过在短路测量的光纤路径长度上增加 101 米的 PM1550 光纤来补偿 297 千米的空气。作为最后的微调步骤，我们测量整个大气层的脉冲宽度，并尽量减小干扰信号的宽度。经过优化后，PM1550 光纤总长度为 105 米。

Tracking comb timing acquisition and lock
跟踪梳时间采集和锁定

At the onset of operation or after a long fade, the system must acquire the tracking lock between the local tracking comb and the incoming clock comb. This acquisition is accomplished with a Kalman filter-based search. As discussed in the main text, this search is based on the timing discriminator signal power output value of ( | V 1 ( t ) | 2 + | V 2 ( t − r τ p ) | 2 ) / 2 V 1 ( t ) 2 + V 2 t − r τ p 2 / 2 (|V_(1)(t)|^(2)+|V_(2)(t-rtau_(p))|^(2))//2\left(\left|V_{1}(t)\right|^{2}+\left|V_{2}\left(t-r \tau_{\mathrm{p}}\right)\right|^{2}\right) / 2, where r r rr is the scanning rate of the tracking comb. This signal is proportional to the input clock comb pulse power. A voltage threshold is applied that corresponds to roughly 135 fW at zero pulse time offset (Timing discriminator operation).
在开始运行或经过长时间衰减后，系统必须获取本地跟踪梳和输入时钟梳之间的跟踪锁定。这种获取是通过基于卡尔曼滤波器的搜索完成的。如正文所述，这种搜索基于定时鉴别器信号功率输出值 ( | V 1 ( t ) | 2 + | V 2 ( t − r τ p ) | 2 ) / 2 V 1 ( t ) 2 + V 2 t − r τ p 2 / 2 (|V_(1)(t)|^(2)+|V_(2)(t-rtau_(p))|^(2))//2\left(\left|V_{1}(t)\right|^{2}+\left|V_{2}\left(t-r \tau_{\mathrm{p}}\right)\right|^{2}\right) / 2 ，其中 r r rr 是跟踪梳的扫描速率。该信号与输入时钟梳状脉冲功率成正比。在脉冲时间偏移为零时（定时鉴别器工作），电压阈值大致相当于 135 fW。
The search for temporal overlap is complicated by turbulence and by differential clock drift between the unsynchronized sites. Intensity scintillation from turbulence causes the signal intensity to fluctuate randomly, complicating the mapping between peak heterodyne signals and the location of the incoming clock comb pulses in time. Additionally, differential drift between the two cavity-stabilized lasers will cause the incoming clock comb pulses to move in time relative to the local comb pulses on the timescale of the search. To overcome these challenging conditions, we tightly couple the acquisition algorithm with a Kalman filter that keeps track of the estimated position of the incoming clock comb time and most critically, its associated uncertainty.
湍流和非同步点之间的差分时钟漂移使寻找时间重叠变得复杂。湍流造成的强度闪烁会使信号强度随机波动，从而使异频信号峰值与进入的时钟梳状脉冲在时间上的位置之间的映射变得复杂。此外，两个腔体稳定激光器之间的差分漂移会导致传入的时钟梳状脉冲在搜索时间尺度上相对于本地梳状脉冲的时间移动。为了克服这些挑战性条件，我们将采集算法与卡尔曼滤波器紧密结合在一起，卡尔曼滤波器可以跟踪估计的输入时钟梳时间位置，最关键的是，还可以跟踪其相关的不确定性。
Starting from a completely unknown received comb timing with respect to the tracking comb, the search controller sweeps the tracking comb through the full 5 ns non-ambiguity range until a signal is observed above threshold, indicating a momentary coincidence between the tracking comb and the received clock comb. This signal is then used to update the Kalman filter’s state and uncertainty estimates.
从一个完全未知的相对于跟踪梳状信号的接收梳状信号时序开始，搜索控制器在整个 5 ns 非模糊范围内扫描跟踪梳状信号，直到观察到一个高于阈值的信号，表明跟踪梳状信号和接收时钟梳状信号之间出现了瞬间重合。然后利用该信号更新卡尔曼滤波器的状态和不确定性估计值。
The effect of the Kalman filter (KF) is to aggregate these intermittent observations of the temporal overlap into a consistent picture for the estimated trajectories of t a KF t a KF t_(a)^(KF)t_{\mathrm{a}}^{\mathrm{KF}} and σ KF , t a 2 σ KF , t a 2 sigma_(KF,ta)^(2)\sigma_{\mathrm{KF}, t \mathrm{a}}^{2} (for the measurement of the received clock comb pulse time from site B B BB at site A A AA ). These are in turn
卡尔曼滤波器 (KF) 的作用是将这些断断续续的时间重叠观测结果汇总到一个一致的图像中，用于估计 t a KF t a KF t_(a)^(KF)t_{\mathrm{a}}^{\mathrm{KF}} 和 σ KF , t a 2 σ KF , t a 2 sigma_(KF,ta)^(2)\sigma_{\mathrm{KF}, t \mathrm{a}}^{2} 的轨迹（用于测量站点 B B BB 在站点 A A AA 接收到的时钟梳脉冲时间）。反过来
used by the search controller to decide the search pattern swept by the tracking comb. Specifically, the tracking comb timing is swept as a triangle waveform through a search space set by t a KF ± 3 σ KF , t a t a KF ± 3 σ KF , t a t_(a)^(KF)+-3sigma_(KF,ta)t_{\mathrm{a}}^{\mathrm{KF}} \pm 3 \sigma_{\mathrm{KF}, t \mathrm{a}}. Each individual observation of the interference signal improves the accuracy of t a KF t a KF t_(a)^(KF)t_{\mathrm{a}}^{\mathrm{KF}} and reduces its estimated uncertainty σ KF , t a 2 σ KF , t a 2 sigma_(KF,ta)^(2)\sigma_{\mathrm{KF}, t \mathrm{a}}^{2}, narrowing the size of the search span and rapidly converging on the true position of the received comb pulses. Finally, once the position uncertainty is small enough ( σ KF , t a < 2 τ p σ KF , t a < 2 τ p sigma_(KF,t_(a)) < 2tau_(p)\sigma_{\mathrm{KF}, t_{\mathrm{a}}}<2 \tau_{\mathrm{p}} ), the tracking comb control switches to tracking t a t a t_(a)t_{\mathrm{a}} continuously.
搜索控制器用于决定跟踪梳扫描的搜索模式。具体来说，跟踪梳定时以三角波形扫过由 t a KF ± 3 σ KF , t a t a KF ± 3 σ KF , t a t_(a)^(KF)+-3sigma_(KF,ta)t_{\mathrm{a}}^{\mathrm{KF}} \pm 3 \sigma_{\mathrm{KF}, t \mathrm{a}} 设定的搜索空间。对干扰信号的每次单独观测都会提高 t a KF t a KF t_(a)^(KF)t_{\mathrm{a}}^{\mathrm{KF}} 的精确度，并降低其估计不确定性 σ KF , t a 2 σ KF , t a 2 sigma_(KF,ta)^(2)\sigma_{\mathrm{KF}, t \mathrm{a}}^{2} ，从而缩小搜索跨度的大小，并迅速收敛到接收到的梳状脉冲的真实位置上。最后，一旦位置不确定性足够小（ σ KF , t a < 2 τ p σ KF , t a < 2 τ p sigma_(KF,t_(a)) < 2tau_(p)\sigma_{\mathrm{KF}, t_{\mathrm{a}}}<2 \tau_{\mathrm{p}} ），跟踪梳状控制就会切换到连续跟踪 t a t a t_(a)t_{\mathrm{a}} 。
In practice, given the good performance of the cavity-stabilized continuous wave reference lasers and the high timing accuracy of each detection of temporal overlap between the tracking and incoming clock comb pulses, the first observation event collapses the uncertainty of the prediction from 5 ns to roughly 1 ps . This leaves almost exclusively a contribution from the unknown initial frequency offset between the sites, which produces a ramp in time offset. As soon as a second observation is produced, the frequency offset is determined and the uncertainty becomes very small, dominated by the random link delay fluctuations. Depending on the received link fades, acquisition either completes almost immediately after these initial observations, or the tracking comb sweeps over a small range centred on the predicted location until the received power crosses threshold again.
在实践中，由于腔稳连续波参考激光器的性能良好，而且每次跟踪和传入时钟梳脉冲之间的时间重叠检测都具有很高的定时精度，因此第一次观测事件将预测的不确定性从 5 ns 降低到大约 1 ps。这几乎完全是由站点之间未知的初始频率偏移造成的，它产生了一个时间偏移斜坡。一旦进行第二次观测，频率偏移就会确定，不确定性也会变得非常小，主要由随机链路延迟波动造成。根据接收到的链路衰减情况，要么在这些初始观测之后立即完成采集，要么在以预测位置为中心的小范围内进行跟踪梳扫描，直到接收功率再次越过阈值。
However, at the lower received powers, there is a significant probability that the signal is not observable (below threshold) at the moment in the sweep that the tracking comb and incoming clock comb pulses overlap temporally. This lack of signal does not have a direct effect on the Kalman filter’s estimates, but it does have an indirect one: the uncertainty in the clock time offset grows over time in the absence of observations due to the inherent statistics of the reference oscillator noises. Thus, if too much time passes between observations, the search space will naturally increase, up to the maximum of the full 5 ns search space, essentially resetting the search process.
然而，在较低的接收功率下，在扫描过程中跟踪梳状脉冲和输入时钟梳状脉冲在时间上重叠的时刻，信号不可观测（低于阈值）的可能性很大。信号的缺失不会直接影响卡尔曼滤波器的估计值，但会产生间接影响：由于参考振荡器噪声的固有统计特性，在没有观测到信号的情况下，时钟时间偏移的不确定性会随着时间的推移而增加。因此，如果观测时间间隔过长，搜索空间自然会增大，最大可达 5 ns 的搜索空间，从而重置搜索过程。
Similarly, when the system is in tracking mode but experiences a fade, the Kalman filter’s uncertainty can be used to decide whether a new search is necessary ( σ KF , ta > τ p σ KF , ta > τ p sigma_(KF,ta) > tau_(p)\sigma_{\mathrm{KF}, \mathrm{ta}}>\tau_{\mathrm{p}} ), and if so, the search can occur over a fraction of the 5 ns non-ambiguity range if 6 σ KF , t a < 5 ns 6 σ KF , t a < 5 ns 6sigma_(KF,ta) < 5ns6 \sigma_{\mathrm{KF}, t \mathrm{a}}<5 \mathrm{~ns}, greatly speeding up the process.
同样，当系统处于跟踪模式但出现衰减时，卡尔曼滤波器的不确定性可用于决定是否有必要进行新的搜索（ σ KF , ta > τ p σ KF , ta > τ p sigma_(KF,ta) > tau_(p)\sigma_{\mathrm{KF}, \mathrm{ta}}>\tau_{\mathrm{p}} ），如果有必要，则可在 6 σ KF , t a < 5 ns 6 σ KF , t a < 5 ns 6sigma_(KF,ta) < 5ns6 \sigma_{\mathrm{KF}, t \mathrm{a}}<5 \mathrm{~ns} 时在 5 ns 非模糊范围的一小部分进行搜索，从而大大加快搜索过程。

Synchronization 同步

On initiation of synchronization, the time offset of the site B clock comb can suddenly change by up to the full 5 ns non-ambiguity range. As a consequence, the tracking comb at site A (the guide site) can lose acquisition and an entire re-acquisition cycle would need to be initiated. To avoid this issue, on initiation of synchronization site B transmits a short data packet by means of the digital communication channel indicating the size of the timing jump it is about to commit to its local clock comb along with the magnitude of the frequency error. When site A receives this packet, it temporarily pauses the active feedback to the site A tracking comb and instead implements exactly the same timing jump and timing slope in an open-loop manner. Once this open-loop time jump is completed, the site A tracking comb re-acquires the incoming clock comb but using a very short and restricted temporal search to account for any accumulated timing error during this open-loop operation either from atmospheric time-of-flight fluctuations or the relative reference oscillator drift. This restricted search is implemented by resetting the Kalman’s filter’s uncertainty to roughly 50 ps. After
在启动同步时，B 站点时钟梳的时间偏移会突然发生变化，变化幅度可达整个 5 ns 非模糊范围。因此，A 站点（引导站点）的跟踪梳可能会丢失采集，需要启动整个重新采集周期。为避免这一问题，在启动同步时，B 站点会通过数字通信通道发送一个短数据包，说明其即将对本地时钟梳进行的定时跳变的大小以及频率误差的大小。站点 A 收到该数据包后，会暂时停止对站点 A 跟踪梳的主动反馈，转而以开环方式实施完全相同的定时跳变和定时斜率。开环跳时完成后，A 站点跟踪梳状器重新获取传入的时钟梳状器，但使用非常短且受限的时间搜索，以消除开环操作期间因大气飞行时间波动或相对参考振荡器漂移而累积的任何定时误差。这种受限搜索是通过将卡尔曼滤波器的不确定性重置为大约 50 ps 来实现的。之后
re-acquisition, site B B BB is fully synchronized to site A A AA, as shown by the out-of-loop verification.
重新采集时，站点 B B BB 与站点 A A AA 完全同步，如环外验证所示。

Out-of-loop verification 回路外验证

The folded-link geometry used here allows for out-of-loop verification of the synchronization by measuring the relative timing of the pulses from the clock comb on site A and the synchronized clock on site B. The clock comb from site A A AA was routed to site B B BB by means of a fibre and combined with the synchronized clock comb on site B, filtered with a 12 -nm wide filter at 1 , 560 nm 1 , 560 nm 1,560nm1,560 \mathrm{~nm}, and detected on a balanced photodetector. The resulting signal is centred at the frequency offset between the two clock combs. The demodulated signal voltage depends on the relative timing between pulses. (Because there is no scintillation present, the full timing discriminator approach is not necessary.) The calibration between the demodulated voltage and the time offset of the clock combs is shown in Extended Data Fig. 6, which was generated by digitally stepping the offset applied to the synchronization feedback loop. For normal operation, we set the synchronization calibration offset so that zero-time offset falls at the half-way level so as to provide maximum dynamic range for measurements of the out-of-loop timing. From a fifth-order polynomial fit to the curve in Extended Data Fig. 6, we generate a calibration function that yields the out-of-loop time offset as a function of measured voltage.
站点 A A AA 的时钟梳通过光纤传送到站点 B B BB 并与站点 B 的同步时钟梳结合，在 1 , 560 nm 1 , 560 nm 1,560nm1,560 \mathrm{~nm} 处使用 12 纳米宽滤波器进行滤波，然后在平衡光电探测器上进行检测。由此产生的信号以两个时钟梳之间的频率偏移为中心。解调后的信号电压取决于脉冲之间的相对时间。(由于不存在闪烁，因此无需采用完全定时鉴别器方法）。解调电压与时钟梳的时间偏移之间的校准如扩展数据图 6 所示，它是通过对同步反馈环路的偏移进行数字步进而产生的。在正常运行时，我们设置同步校准偏移量，使零时偏移量落在半程水平，从而为测量环外定时提供最大的动态范围。通过对扩展数据图 6 中的曲线进行五阶多项式拟合，我们生成了一个校准函数，该函数可将环外时间偏移量转换为测量电压的函数。

Data availability 数据可用性

All data for the figures in this manuscript are available at https://data. nist.gov/od/id/mds2-2967.
本手稿中数字的所有数据可在https://data。nist.gov/od/id/mds2-2967。

Code availability 代码可用性

The algorithms necessary to perform this experiment are described between the main text and the Methods.
本实验所需的算法在正文和方法中均有描述。
50. Swann, W. C. et al. Low-loss reciprocal optical terminals for two-way time-frequency transfer. Appl. Opt. 56, 9406-9413 (2017).
50.用于双向时频传输的低损耗互易光学终端。56, 9406-9413 (2017)。
Acknowledgements We acknowledge comments from T. Bothwell, F. Giorgetta and B. Washburn. We acknowledge technical assistance from S. Syed, M. Bodine, H. Leopardi, T. Wright, M. Martinsen, D. Kuniyuki, S. Baumann, the NOAA Mauna Loa Observatory and the Haleakala MEES Observatory. We acknowledge the Air Force Office of Scientific Research (grant no. MIPR F4FGA02152G001), Air Force Research Laboratory (grant no. FA9453-16-D-0004), NSF QLCI Award (grant no. OMA - 2016244), NSF GRFP (grant no. DGE 1650115), OSD and DARPA DSO through a CRADA with Vector Atomic and NIST for funding. Approved for public release; distribution is unlimited. Approval no. AFRL-2022-5993.
致谢 我们感谢 T. Bothwell、F. Giorgetta 和 B. Washburn 的意见。我们感谢 S. Syed、M. Bodine、H. Leopardi、T. Wright、M. Martinsen、D. Kuniyuki、S. Baumann、NOAA Mauna Loa 天文台和 Haleakala MEES 天文台的技术协助。我们感谢空军科学研究办公室（资助编号：MIPR F4FGA02152G001）、空军研究实验室（资助编号：FA9453-16-D-0004）、美国国家科学基金会 QLCI 奖（资助编号：OMA - 2016244）、美国国家科学基金会 GRFP（资助编号：DGE 1650115）、OSD 和 DARPA DSO 通过与 Vector Atomic 和 NIST 的 CRADA 提供的资助。批准公开发布；发布不受限制。批准号AFRL-2022-5993。
Author contributions J.-D.D., L.C.S. and N.R.N. conceived the experiment. E.D.C., J.-D.D., L.C.S. and N.R.N. acquired and analysed the synchronization data from Hawaii, and wrote the paper. J.-D.D. and H.B. developed the signal acquisition processing. J.E., E.D.C. and L.C.S. constructed the optical system and acquired 2 km data. W.C.S. and B.K.S. designed and built the free-space optical terminals and contributed to the writing.
作者贡献 J.-D.D.、L.C.S. 和 N.R.N. 构思了该实验。E.D.C.、J.-D.D.、L.C.S.和 N.R.N.获取并分析了夏威夷的同步数据，并撰写了本文。J.-D.D. 和 H.B. 开发了信号采集处理。J.E.、E.D.C. 和 L.C.S. 构建了光学系统并获取了 2 公里数据。W.C.S.和B.K.S.设计并建造了自由空间光学终端，并参与了论文撰写。
Competing interests The authors declare no competing interests.
利益冲突 作者声明不存在利益冲突。

Additional information 其他信息

Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41586-023-06032-5.
补充信息 在线版本包含补充材料，可在https://doi.org/10.1038/s41586-023-06032-5获取。
Correspondence and requests for materials should be addressed to Nathan R. Newbury or Laura C. Sinclair.
来信和索取资料请寄给 Nathan R. Newbury 或 Laura C. Sinclair。
Peer review information Nature thanks David Gozzard and Jungwon Kim and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
同行评审信息 Nature 感谢 David Gozzard 和 Jungwon Kim 以及其他匿名评审人对本作品的同行评审做出的贡献。可查阅同行评审报告。
Reprints and permissions information is available at http://www.nature.com/reprints.
转载和许可信息请访问http://www.nature.com/reprints。

Article 文章

Extended Data Fig. 1|Free space optical terminal design. Each terminal transmits both the comb light (at a few mW ) and a beacon laser signal at similar low power ( -2 mW at the aperture) through a 10 cm aperture with an 8 cm 1 / e 2 8 cm 1 / e 2 8cm1//e^(2)8 \mathrm{~cm} 1 / \mathrm{e}^{2} beam diameter. To avoid sacrificing comb light to active tip/tilt stabilization, a separate beacon beam at 1532 nm or 1542 nm is polarization multiplexed directly onto the comb beam. (The beacon wavelengths differ for the two terminals, allowing wavelength demultiplexing of the transmitted and received beacon beams). A terminal’s transmitted beacon beam, originating from a polarization maintaining fiber-coupled laser, is collimated through a
扩展资料 图 1|自由空间光学终端设计。每个终端通过一个 10 厘米的孔径（ 8 cm 1 / e 2 8 cm 1 / e 2 8cm1//e^(2)8 \mathrm{~cm} 1 / \mathrm{e}^{2} 光束直径）同时发射梳状光（几毫瓦）和信标激光信号，信标激光信号的功率同样很低（孔径处为-2 毫瓦）。为了避免因主动尖端/倾斜稳定而牺牲梳状光束，波长为 1532 nm 或 1542 nm 的单独信标光束被直接偏振复用到梳状光束上。(两个终端的信标波长不同，因此可以对发射和接收的信标光束进行波长解复用）。终端发射的信标光束来自偏振保持光纤耦合激光器，通过一个准直光束接收。
fiber collimator, reflected off the face of an interference filter (which acts as a mirror), and directed to the polarization coupler. The combined, ∼ 3.3 − mm ∼ 3.3 − mm ∼3.3-mm\sim 3.3-\mathrm{mm} diameter beam is directed to a galvo mirror that provides tip/tilt control, and then passes through a 24:1,10 cm aperture beam expander. The beacon light is detected at the far end by a focal plane array. The images are processed, and the beam position fed back to adjust the tip/tilt of the outgoing combined comb light and beacon light. This corrects for atmospheric turbulence and optimizes coupling of the incoming frequency comb light into the polarization maintaining single-mode optical fiber.
光束经光纤准直器，从干涉滤波器（起镜面作用）表面反射后，进入偏振耦合器。直径为 ∼ 3.3 − mm ∼ 3.3 − mm ∼3.3-mm\sim 3.3-\mathrm{mm} 的组合光束被导向一个提供尖端/倾斜控制的振镜，然后通过一个 24:1,10 厘米孔径的扩束器。信标光在远端由焦平面阵列检测。对图像进行处理，并反馈光束位置，以调整出射的组合光和信标光的尖端/倾角。这样可以校正大气湍流，优化进入的频率组合光与偏振保持单模光纤的耦合。

b
Extended Data Fig. 2 ∣ 2 ∣ 2∣2 \mid Total link loss. (a) Diagram of link loss. The aperture-to-aperture link loss, which includes the channel loss ℓ channel ℓ channel  ℓ_("channel ")\ell_{\text {channel }}, is reciprocal as is any excess loss from coupling into single-mode fiber, ℓ s m f ℓ s m f ℓ_(smf)\ell_{s m f}, whereas the transceiver-specific losses ℓ r c v ℓ r c v ℓ_(rcv)\ell_{r c v} and ℓ x m i t ℓ x m i t ℓ_(xmit)\ell_{x m i t} are not. While ℓ s m f ℓ s m f ℓ_(smf)\ell_{s m f} is reciprocal for both sites, the loss is unidirectional, that is it only occurs when
扩展数据 图 2 ∣ 2 ∣ 2∣2 \mid 总链路损耗。(a) 链路损耗示意图。包括信道损耗 ℓ channel ℓ channel  ℓ_("channel ")\ell_{\text {channel }} 在内的孔径到孔径链路损耗与耦合到单模光纤产生的多余损耗 ℓ s m f ℓ s m f ℓ_(smf)\ell_{s m f} 是对等的，而收发器特有损耗 ℓ r c v ℓ r c v ℓ_(rcv)\ell_{r c v} 和 ℓ x m i t ℓ x m i t ℓ_(xmit)\ell_{x m i t} 则不是对等的。虽然 ℓ s m f ℓ s m f ℓ_(smf)\ell_{s m f} 对两个站点都是互易的，但损耗是单向的，即只有在以下情况下才会发生
the incoming light is coupled into the fiber. (b) Measured median total loss over different runs plotted against the integrated turbulence, as measured by the piston noise and assuming a wind velocity of 10 m / s 10 m / s 10m//s10 \mathrm{~m} / \mathrm{s}. The mean loss across runs is 91 dB giving 11 − dB 11 − dB 11-dB11-\mathrm{dB} margin over the tolerable loss of 102 dB .
入射光耦合到光纤中。(b) 根据活塞噪声测量的综合湍流，并假设风速为 10 m / s 10 m / s 10m//s10 \mathrm{~m} / \mathrm{s} ，绘制出不同运行的总损耗测量中值。各次运行的平均损耗为 91 dB，与 102 dB 的可容忍损耗相比有 11 − dB 11 − dB 11-dB11-\mathrm{dB} 的余量。

Article 文章

Extended Data Fig. 3|System diagram emphasizing signal processing. Each site has a local reference oscillator (cavity stabilized laser), to which both the clock comb and tracking comb are self-referenced. As shown for site A, after both the clock and tracking combs are phase-locked to the reference oscillator, they have an arbitrary local time offset between each other. This local time offset is directly zeroed by digitally adjusting the tracking comb timing until there is a maximum heterodyne overlap with the clock comb pulses. Subsequent timing values for the local tracking comb are then referenced to the local clock comb. After acquisition, the timing discriminator measures the (small) time offset δ t a δ t a deltat_(a)\delta t_{a} between the incoming clock comb and local tracking comb. This timing difference is summed with the tracking comb time offset, to generate an estimate of the incoming clock comb pulse time, t a t a t_(a)t_{a}. This estimate, along with the corresponding measured incoming power, is input to the Kalman filter, whose output provides an optimized, filtered estimate of the incoming clock comb pulse time, t a K F t a K F t_(a)^(KF)t_{a}^{K F}. This value is fed into a final feedback controller for the tracking comb, G L O G L O G_(LO)G_{L O}. Due to the feedback loop, the tracking comb output itself (both the physical optical pulse time and the corresponding
扩展资料 图 3：强调信号处理的系统图。每个站点都有一个本地参考振荡器（腔稳激光器），时钟梳和跟踪梳都以其为自参考。如站点 A 所示，时钟梳和跟踪梳与参考振荡器锁相后，它们之间有一个任意的局部时间偏移。通过数字调整跟踪梳定时，可直接将局部时间偏移归零，直到与时钟梳脉冲的异频重叠达到最大值。随后，本地跟踪梳的定时值将参考本地时钟梳。采集完成后，定时鉴别器会测量输入时钟梳和本地跟踪梳之间的（小）时间偏移 δ t a δ t a deltat_(a)\delta t_{a} 。该时间差与跟踪梳时间偏移相加，生成输入时钟梳脉冲时间的估计值 t a t a t_(a)t_{a} 。该估计值与相应的测量输入功率一起输入卡尔曼滤波器，其输出提供了输入时钟梳脉冲时间的优化滤波估计值 t a K F t a K F t_(a)^(KF)t_{a}^{K F} 。该值被输入跟踪梳状脉冲的最终反馈控制器 G L O G L O G_(LO)G_{L O} 。通过反馈回路，跟踪梳输出本身（物理光脉冲时间和相应的
digital value), is now a filtered, estimated value of the incoming clock comb pulse timing with respect to the local clock comb pulse time. This value is transmitted over a communication link from site A A AA to site B B BB, where it is combined with the corresponding local value of t b t b t_(b)t_{b} to generate an error signal for the site B clock comb. When both sites have acquired their lock onto the incoming clock comb pulses, the final synchronization feedback controller G s y G s y G_(sy)G_{s y} is activated and the site B B BB clock comb is actively synchronized to the site A A AA clock comb, at their local reference point. The effective bandwidth of the Kalman Filter depends on the input power, and ranges from 10 Hz to 25 Hz . The bandwidth of the subsequent lock of the tracking comb ( H track ) H track  (H_("track "))\left(H_{\text {track }}\right) is ∼ 450 Hz ∼ 450 Hz ∼450Hz\sim 450 \mathrm{~Hz}. The bandwidth of the synchronization lock ( H synch ) H synch  (H_("synch "))\left(H_{\text {synch }}\right) is ∼ 15 Hz ∼ 15 Hz ∼15Hz\sim 15 \mathrm{~Hz}. The communication link is over rf coaxial cable here. For a future point-to-point link, it would be over free space by either rf or optical as in ref.15. The message rate (for update of t a t a t_(a)t_{a} ) is 400 Hz and the total bit rate is 26 kbps . red solid lines: optical comb pulses, red dashed line: CW optical laser light, black lines: digital values, P P PP : input optical power.
数字值），现在是输入时钟梳状脉冲时间相对于本地时钟梳状脉冲时间的滤波估计值。该值通过通信链路从站点 A A AA 传输到站点 B B BB ，并与相应的本地值 t b t b t_(b)t_{b} 相结合，生成站点 B 时钟梳的误差信号。当两个站点都锁定了输入的时钟梳脉冲后，最终同步反馈控制器 G s y G s y G_(sy)G_{s y} 将被激活，站点 B B BB 的时钟梳将主动同步到站点 A A AA 的时钟梳上的本地参考点。卡尔曼滤波器的有效带宽取决于输入功率，范围为 10 Hz 至 25 Hz。跟踪梳 ( H track ) H track  (H_("track "))\left(H_{\text {track }}\right) 的后续锁定带宽为 ∼ 450 Hz ∼ 450 Hz ∼450Hz\sim 450 \mathrm{~Hz} 。同步锁定 ( H synch ) H synch  (H_("synch "))\left(H_{\text {synch }}\right) 的带宽为 ∼ 15 Hz ∼ 15 Hz ∼15Hz\sim 15 \mathrm{~Hz} 。这里的通信链路是通过射频同轴电缆进行的。对于未来的点对点链路，它将通过自由空间的射频或光纤实现，如参考文献 15 所述。信息传输速率（用于更新 t a t a t_(a)t_{a} ）为 400 Hz，总比特率为 26 kbps：红色实线：光学梳状脉冲，红色虚线：CW 光学激光，黑色实线：数字值， P P PP ：输入光功率。
Extended Data Fig. 4 ∣ 4 ∣ 4∣4 \mid Tracking comb timing measurement using an optical timing discriminator. (a) Subset of system diagram from Extended Data Fig. 4 showing how the timing of the incoming clock comb is measured with the local tracking comb. (b) System diagram of the optical timing discriminator and subsequent signal processing. The optical timing discriminator generates two measurement channels with the lead and lag positions between the two combs switched between channels. After measuring these channels on balanced photodiodes (BPD), the heterodyne output voltages are demodulated to generate IQ (complex) signals, which are then low pass filtered (LPF). The phase of one signal is used in a phase-locked loop to adjust the demodulation
扩展资料 图 4 ∣ 4 ∣ 4∣4 \mid 使用光学定时鉴别器进行跟踪梳定时测量。(a) 扩展数据图 4 系统图的子集，显示如何使用本地跟踪梳测量输入时钟梳的定时。 (b) 光学定时鉴别器和后续信号处理的系统图。光学授时鉴别器生成两个测量通道，两个梳状信号之间的前导和滞后位置在通道间切换。在平衡光电二极管（BPD）上测量这些通道后，对外差输出电压进行解调，生成 IQ（复数）信号，然后对其进行低通滤波（LPF）。一个信号的相位被用于锁相环，以调整解调信号。
frequency, f D D S f D D S f_(DDS)f_{D D S}, and centre the baseband signals at DC. The magnitudes of the signals are combined to generate an estimate of the power (from their mean squared values) and a timing offset value, δ t b δ t b deltat_(b)\delta t_{b}, from their normalized difference. BS, beam splitter; ADC, analog to digital converter; NCO, numerically controlled oscillator. © Absolute value of the timing discriminator output voltages, | V 1 | V 1 |V_(1)|\left|V_{1}\right| and | V 2 | V 2 |V_(2)|\left|V_{2}\right|, as a function of the time offset between the local tracking comb and incoming clock comb. (d) The error signal generated from the two channels along with the polynomial fit used in the digital processing to generate the timing error value based on the normalized error signal, E E EE.
频率， f D D S f D D S f_(DDS)f_{D D S} ，并将基带信号置于直流中心。合并信号的大小，根据它们的均方值得出功率估计值，并根据它们的归一化差值得出定时偏移值 δ t b δ t b deltat_(b)\delta t_{b} 。BS：分光器；ADC：模数转换器；NCO：数控振荡器。(c) 时间鉴别器输出电压 | V 1 | V 1 |V_(1)|\left|V_{1}\right| 和 | V 2 | V 2 |V_(2)|\left|V_{2}\right| 的绝对值与本地跟踪梳状信号和输入时钟梳状信号之间的时间偏移量的函数关系。 (d) 两个通道产生的误差信号，以及数字处理中用于根据归一化误差信号 E E EE 生成时间误差值的多项式拟合。

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Extended Data Fig. 5 ∣ 5 ∣ 5∣5 \mid Optical time transfer transceiver design for a single site. This transceiver routes the local clock comb to the FSO terminals and the incoming remote clock comb to the optical timing discriminator for mixing and detection with the local tracking comb. To minimize excess fiber optic delays, the transceiver also includes the necessary fiber optic components to generate the RF optical beat signals, f opt f opt  f_("opt ")f_{\text {opt }}, used for locking the two clock and tracking combs to the local cavity stabilized laser (CW in) reference, and the out-of-loop verification beat signals. The reference plane for the out-of-loop verification is defined by the point at which the two clock combs overlap within
扩展资料 图： 5 ∣ 5 ∣ 5∣5 \mid 单站点光授时收发器设计。该收发器将本地时钟梳路由至 FSO 终端，并将输入的远程时钟梳路由至光学授时鉴别器，以便与本地跟踪梳进行混合和检测。为了最大限度地减少多余的光纤延迟，收发器还包括必要的光纤组件，用于生成射频光学节拍信号 f opt f opt  f_("opt ")f_{\text {opt }} （用于将两个时钟和跟踪梳锁定到本地腔稳激光器（CW in）基准）和环外验证节拍信号。用于环外验证的参考平面由两个时钟梳在以下范围内的重叠点确定
the 50:50 splitter. A calibration step with a fiber-shorted link determines the time offset for the two clock combs due to path delays in the transceivers. This time offset is included in the overall synchronization loop so that the clock pulses remain overlapped when the system is operated over the link. All fiber is PM1550.50:50,50/50 splitter; 90:10, 90/10 splitter; BPF, band-pass filter; iso, isolator; DWDM, dense wavelength division multiplexer at the cavity-stabilized laser wavelength; 45 ∘ BS 45 ∘ BS 45^(@)BS45^{\circ} \mathrm{BS}, polarization beam splitter with the input fiber rotated 45 ∘ 45 ∘ 45^(@)45^{\circ}.
50:50 分光器。使用光纤短路链路的校准步骤可确定收发器路径延迟导致的两个时钟梳的时间偏移。该时间偏移包含在整个同步环路中，以便系统在链路上运行时时钟脉冲保持重叠。所有光纤均为 PM1550。50:50，50/50 分光器；90:10，90/10 分光器；BPF，带通滤波器；iso，隔离器；DWDM，腔稳激光波长的密集波分复用器； 45 ∘ BS 45 ∘ BS 45^(@)BS45^{\circ} \mathrm{BS} ，偏振分光器，输入光纤旋转 45 ∘ 45 ∘ 45^(@)45^{\circ} 。
Extended Data Fig. 6 ∣ 6 ∣ 6∣6 \mid Out-of-loop verification. Calibration curve of heterodyne voltage vs time offset between the two clock combs used for out-of-loop verification. A fifth order polynomial is fit to the curve to generate a mapping from the measured demodulated heterodyne voltage to the time offset.
扩展数据 图 6 ∣ 6 ∣ 6∣6 \mid 环外验证。用于环外验证的两个时钟梳之间的外差电压与时间偏移的校准曲线。对曲线进行五阶多项式拟合，以生成从测量的解调外差电压到时间偏移的映射。
Extended Data Fig. 7 | Traces for synchronization over 300 km for 4.0 mW of comb power at site B B BB (left, blue traces) and for reduced, 40 μ W 40 μ W 40 muW40 \mu \mathrm{~W} comb power at site B (right, green curves). (a) Time trace of received power, P r e c P r e c P_(rec)P_{r e c}, measured at the output of the timing discriminator for site A A AA with the applied threshold shown as a dashed black line. (b) The control effort, Δ T cntrl Δ T cntrl  DeltaT_("cntrl ")\Delta T_{\text {cntrl }}, applied on site B to maintain synchronization between the two site’s clock combs. As such, it is also a measurement of the time offset between the two cavity-stabilized reference
扩展数据 图 7：站点 B B BB 的 4.0 mW 梳状功率（左图，蓝色轨迹）和站点 B 的 40 μ W 40 μ W 40 muW40 \mu \mathrm{~W} 梳状功率（右图，绿色曲线）在 300 公里范围内的同步轨迹。(a) 在站点 A A AA 的定时鉴别器输出端测量到的接收功率 P r e c P r e c P_(rec)P_{r e c} 的时间轨迹，应用阈值显示为黑色虚线。(b) 在站点 B 上应用的控制功率 Δ T cntrl Δ T cntrl  DeltaT_("cntrl ")\Delta T_{\text {cntrl }} ，用于保持两个站点时钟梳之间的同步。因此，它也是对两个空腔稳定基准之间时间偏移的测量。
lasers. © Changes in the time-of-flight. These changes are due to temperature drifts in the 300 km of air and in the fiber paths up to the terminals, atmospheric turbulence, and mechanical movement in the terminals. (d) The out-of-loop timing verification or ‘Truth’ data indicates constant temporal overlap between the clock combs at both sites despite 100’s of ps changes in the time-of-flight and 100’s of ns changes in Δ T cntrl Δ T cntrl  DeltaT_("cntrl ")\Delta T_{\text {cntrl }}. This truth data is used to generate the instability deviations of Fig. 3.
激光。飞行时间的变化。这些变化是由于 300 千米空气中和通往终端的光纤路径中的温度漂移、大气湍流以及终端的机械运动造成的。(d) 环外定时验证或 "真实 "数据表明，尽管飞行时间发生了 100's ps 的变化， Δ T cntrl Δ T cntrl  DeltaT_("cntrl ")\Delta T_{\text {cntrl }} 也发生了 100's ns 的变化，但两个站点的时钟梳之间仍然存在时间上的重叠。该真实数据用于生成图 3 的不稳定偏差。
Extended Data Fig. 8 ∣ 8 ∣ 8∣8 \mid Additional fractional frequency instabilities.
扩展数据图 8 ∣ 8 ∣ 8∣8 \mid 其他分数频率不稳定性。
Modified Allan deviations of the out-of-loop verification data (‘truth’ data) for 4 different days across the 300 km link. Data from June 27 th 27 th  27^("th ")27^{\text {th }} and early on June 28 th 28 th  28^("th ")28^{\text {th }} (purple and pink curves) have an elevated MDEV at long averaging times due to
300 公里链路上 4 个不同日期的环外验证数据（"真实 "数据）的修正阿伦偏差。6 月 27 th 27 th  27^("th ")27^{\text {th }} 和 6 月初 28 th 28 th  28^("th ")28^{\text {th }} 的数据（紫色和粉色曲线）在较长的平均时间内具有较高的 MDEV，这是由于
a malfunctioning temperature controller on the out-of-loop verification fiber and both transceivers. The data fromJune 29 th 29 th  29^("th ")29^{\text {th }} (blue curve) also appears in Fig. 3 of the main text.
这是因为超出环路验证光纤和两台收发器上的温度控制器出现故障。6 月 29 th 29 th  29^("th ")29^{\text {th }} 的数据（蓝色曲线）也出现在正文图 3 中。

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a

b

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d
Extended Data Fig. 9|Received power and fade statistics over the 300 km link at site A A AA for 4.0 mW of comb power from site B B BB. (a-c) Received power over 600 -seconds, 6 s , and 0.06 s . Red line indicates the 270 fW detection threshold. (d) Normalized histogram (i.e., probability density function, PDF) of
扩展数据 图 9|来自站点 B B BB 的 4.0 mW 梳状功率在站点 A A AA 的 300 公里链路上的接收功率和衰减统计。(a-c) 600 秒、6 秒和 0.06 秒内的接收功率。 红线表示 270 fW 检测阈值。(d) B B BB的归一化直方图（即概率密度函数，PDF）。

the received power for the 600 -second segment. (e) PDF of the fade durations. For the 4.0 mW comb power sent from site B, no fades exceeded a duration of 10 ms - a direct consequence of the low detection threshold at site A A AA.
600 秒段的接收功率。(e) 衰减持续时间的 PDF。对于从站点 B 发送的 4.0 mW 梳状功率，没有任何衰减持续时间超过 10 毫秒--这是站点 A A AA 检测阈值较低的直接结果。
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Extended Data Fig. 10 ∣ 10 ∣ 10∣10 \mid Received power and fade statistics over the 300 k m 300 k m 300km300 \mathbf{k m} link at site A A AA for 40 μ W 40 μ W 40 mu W40 \boldsymbol{\mu W} of comb power from site B B BB. (a-c) Received power over 600 -seconds, 6 s , and 0.06 s . Red line indicates the 270 fW detection threshold. (d) Normalized histogram (i.e., probability density function, PDF) of the
扩展数据 图： 300 k m 300 k m 300km300 \mathbf{k m} 站点 A A AA 对来自站点 B B BB 的 40 μ W 40 μ W 40 mu W40 \boldsymbol{\mu W} 梳状功率的 300 k m 300 k m 300km300 \mathbf{k m} 链路上的接收功率和衰减统计。(a-c) 600 秒、6 秒和 0.06 秒内的接收功率。 红线表示 270 fW 检测阈值。(d) B B BB 的归一化直方图（即概率密度函数，PDF）。
received power for the 600-second segment. (e) PDF of the fade durations. For the greatly reduced launch power, fades are more frequency and of longer duration than the data of Extended Data Fig. 9.
600 秒段的接收功率。(e) 衰减持续时间的 PDF。与扩展数据图 9 的数据相比，在发射功率大幅降低的情况下，衰减频率更高，持续时间更长。

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Extended Data Table 1|Experimental parameters for the data in Figs. 2-4 related to quantum-limited operation
扩展数据 表 1|图 2-4 中与量子限制操作有关的数据的实验参数
Quantity 数量

图 2 （方形过滤器）
Figure 2
(square filters)
Figure 2 (square filters)| Figure 2 | | :--- | | (square filters) |

图 3 和图 4 （高斯滤波器）
Figures 3 and 4
(Gaussian filters)
Figures 3 and 4 (Gaussian filters)| Figures 3 and 4 | | :--- | | (Gaussian filters) |
Pulse width for zero chirp, τ p τ p tau_(p)\tau_{p}
零啁啾脉冲宽度， τ p τ p tau_(p)\tau_{p} 		245 fs 355 fs
Detector quantum efficiency, η η eta\eta
探测器量子效率， η η eta\eta 		0.80 0.80
Detector noise penalty, D D DD
检测器噪声罚款， D D DD 		1.2 ( 1.7 dB ) 1.2 ( 1.7 dB ) 1.2(1.7dB)1.2(1.7 \mathrm{~dB}) 1.2 ( 1.7 dB ) 1.2 ( 1.7 dB ) 1.2(1.7dB)1.2(1.7 \mathrm{~dB})
Pulse broadening, C C CC
脉冲展宽， C C CC 		1.35 1.7 ( 2 − km 1.7 ( 2 − km 1.7(2-km1.7(2-\mathrm{km} link)  1.7 ( 2 − km 1.7 ( 2 − km 1.7(2-km1.7(2-\mathrm{km} 链接)
1.5 ( 300 − km 1.5 ( 300 − km 1.5(300-km1.5(300-\mathrm{km} link)  1.5 ( 300 − km 1.5 ( 300 − km 1.5(300-km1.5(300-\mathrm{km} 链接)
Quantum-limit, γ q l γ q l gamma_(ql)\gamma_{q l} 量子极限， γ q l γ q l gamma_(ql)\gamma_{q l} 0.6 0.6
Actual γ γ gamma\gamma 实际 γ γ gamma\gamma 2.1 γ q l 2.1 γ q l 2.1gamma_(ql)2.1 \gamma_{q l} 3.5 γ q l ( 2 − km 3.5 γ q l ( 2 − km 3.5gamma_(ql)(2-km3.5 \gamma_{q l}(2-\mathrm{km} link)  3.5 γ q l ( 2 − km 3.5 γ q l ( 2 − km 3.5gamma_(ql)(2-km3.5 \gamma_{q l}(2-\mathrm{km} 链接)
2.7 γ q l ( 300 − km 2.7 γ q l ( 300 − km 2.7gamma_(ql)(300-km2.7 \gamma_{q l}(300-\mathrm{km} link ) ) ))
2.7 γ q l ( 300 − km 2.7 γ q l ( 300 − km 2.7gamma_(ql)(300-km2.7 \gamma_{q l}(300-\mathrm{km} 链接 ) ) ))
Quantity "Figure 2 (square filters)" "Figures 3 and 4 (Gaussian filters)" Pulse width for zero chirp, tau_(p) 245 fs 355 fs Detector quantum efficiency, eta 0.80 0.80 Detector noise penalty, D 1.2(1.7dB) 1.2(1.7dB) Pulse broadening, C 1.35 1.7(2-km link) 1.5(300-km link) Quantum-limit, gamma_(ql) 0.6 0.6 Actual gamma 2.1gamma_(ql) 3.5gamma_(ql)(2-km link) 2.7gamma_(ql)(300-km link )| Quantity | Figure 2 <br> (square filters) | Figures 3 and 4 <br> (Gaussian filters) | | :--- | :--- | :--- | | Pulse width for zero chirp, $\tau_{p}$ | 245 fs | 355 fs | | Detector quantum efficiency, $\eta$ | 0.80 | 0.80 | | Detector noise penalty, $D$ | $1.2(1.7 \mathrm{~dB})$ | $1.2(1.7 \mathrm{~dB})$ | | Pulse broadening, $C$ | 1.35 | $1.7(2-\mathrm{km}$ link) | | | | $1.5(300-\mathrm{km}$ link) | | Quantum-limit, $\gamma_{q l}$ | 0.6 | 0.6 | | Actual $\gamma$ | $2.1 \gamma_{q l}$ | $3.5 \gamma_{q l}(2-\mathrm{km}$ link) | | | | $2.7 \gamma_{q l}(300-\mathrm{km}$ link $)$ |
Pulse width is defined as the full-width half-maximum value. The values for C C CC vary between sites, but the effective averaged value is provided here.
脉冲宽度定义为全宽半最大值。不同站点的 C C CC 值各不相同，但这里提供的是有效的平均值。
  1. 1 1 ^(1){ }^{1} National Institute of Standards and Technology (NIST), Boulder, CO, USA. 2 2 ^(2){ }^{2} Department of Electrical, Energy and Computer Engineering, University of Colorado, Boulder, CO, USA. 3 3 ^(3){ }^{3} Octosig Consulting, Quebec City, Quebec, Canada. 4 4 ^(4){ }^{4} Space Dynamics Laboratory, North Logan, UT, USA. 5 5 ^(5){ }^{5} These authors contributed equally: Emily D. Caldwell, Jean-Daniel Deschenes, Nathan R. Newbury, Laura C. Sinclair. ⊠ ⊠ ^(⊠){ }^{\boxtimes} e-mail: nathan.newbury@nist.gov; laura.sinclair@nist.gov
    1 1 ^(1){ }^{1} 美国科罗拉多州博尔德美国国家标准与技术研究院（NIST）。 2 2 ^(2){ }^{2} 美国科罗拉多州博尔德市科罗拉多大学电气、能源和计算机工程系。 3 3 ^(3){ }^{3} 加拿大魁北克省魁北克市 Octosig 咨询公司。 4 4 ^(4){ }^{4} 空间动力学实验室，美国犹他州北洛根。 5 5 ^(5){ }^{5} 这些作者的贡献相同：Emily D. Caldwell、Jean-Daniel Deschenes、Nathan R. Newbury、Laura C. Sinclair。 ⊠ ⊠ ^(⊠){ }^{\boxtimes} 电子邮件：nathan.newbury@nist.gov; laura.sinclair@nist.gov
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