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
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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}$${}^{14-24}$^(14-24){ }^{14-24}. Over 300 km between mountaintops in Hawaii with launched powers as low as $40\mu \mathrm{W}$$40\mu \mathrm{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}$${}^{14-24}$^(14-24){ }^{14-24} 。在发射功率低至 $40\mu \mathrm{W}$$40\mu \mathrm{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-\mathrm{cm}$$40-\mathrm{cm}$40-cm40-\mathrm{cm} aperture telescopes, adaptive optics and watt-level amplifiers, Shen et al. ${}^{24}$${}^{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}$${}^{14-24}$^(14-24){ }^{14-24} 。这种方法的光子效率较低，需要几毫微瓦的信号，比量子极限高出 40 dB。尽管如此，Shen 等人结合使用了 $40-\mathrm{cm}$$40-\mathrm{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 ${\tau }_{\mathrm{p}}$${\tau }_{\mathrm{p}}$tau_(p)\tau_{\mathrm{p}} (here roughly 350 fs ) is simply
与光学时钟近似，通过宽度为 ${\tau }_{\mathrm{p}}$${\tau }_{\mathrm{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

where $n$$n$nn is the number of detected photons in the measurement interval and $\gamma$$\gamma$gamma\gamma is a constant of order unity. Here, we demonstrate
其中， $n$$n$nn 是测量区间内检测到的光子数量， $\gamma$$\gamma$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 $/\sqrt{P\tau }$$/\sqrt{P\tau }$//sqrt(P tau)/ \sqrt{P \tau} in time deviation and to $4.3\times {10}^{-16}/\sqrt{P{\tau }^3}$$4.3\times {10}^{-16}/\sqrt{P{\tau }^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$$P$PP is the received power in picowatts and $\tau$$\tau$tau\tau is the averaging time in seconds, with respective floors of roughly 35 attoseconds and below ${10}^{-18}$${10}^{-18}$10^(-18)10^{-18}. Over the 300 km horizontal link, the one-way timing signals are still measured at nearly the quantum limit with a power threshold of 270 fW . However, the strong integrated turbulence leads to excess non-reciprocal time-of-flight noise, attributed to multipath effects. Nevertheless, the clocks remain synchronized to $1.6\mathrm{fs}{\tau }^{-1/2}$$1.6\mathrm{fs}{\tau }^{-1/2}$1.6fstau^(-1//2)1.6 \mathrm{fs} \tau^{-1 / 2} reaching a floor of 320 attoseconds in time and $2.8\times {10}^{-15}{\tau }^{-3/2}$$2.8\times {10}^{-15}{\tau }^{-3/2}$2.8 xx10^(-15)tau^(-3//2)2.8 \times 10^{-15} \tau^{-3 / 2} reaching a floor of $3.1\times {10}^{-19}$$3.1\times {10}^{-19}$3.1 xx10^(-19)3.1 \times 10^{-19} in frequency. Finally, synchronization is achieved even at attenuated comb powers of $40\mu \mathrm{W}$$40\mu \mathrm{W}$40 muW40 \mu \mathrm{~W} with a median received power of 150 fW . This low-power performance is enabled by the robust Kalman filter approach that can tolerate the greater than
我们通过在两个不同的自由空间链路上同步两个光学时标来演示这种量子限制的 OTT：一个是美国科罗拉多州博尔德市湍流较少的 2 千米链路，另一个是夏威夷两个山顶之间湍流较强的 300 千米链路。在低湍流条件下，自由空间路径确实是互惠的，双向时间传输几乎是量子限制的；时钟同步到 246，时间偏差为 $/\sqrt{P\tau }$$/\sqrt{P\tau }$//sqrt(P tau)/ \sqrt{P \tau} ，分数频率（修正的阿伦偏差）为 $4.3\times {10}^{-16}/\sqrt{P{\tau }^3}$$4.3\times {10}^{-16}/\sqrt{P{\tau }^3}$4.3 xx10^(-16)//sqrt(Ptau^(3))4.3 \times 10^{-16} / \sqrt{P \tau^{3}} ，其中 $P$$P$PP 是以皮瓦特为单位的接收功率， $\tau$$\tau$tau\tau 是以秒为单位的平均时间，各自的下限约为 35 阿秒和 ${10}^{-18}$${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\mathrm{fs}{\tau }^{-1/2}$$1.6\mathrm{fs}{\tau }^{-1/2}$1.6fstau^(-1//2)1.6 \mathrm{fs} \tau^{-1 / 2} 在时间上达到 320 阿秒的底限， $2.8\times {10}^{-15}{\tau }^{-3/2}$$2.8\times {10}^{-15}{\tau }^{-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\times {10}^{-19}$$3.1\times {10}^{-19}$3.1 xx10^(-19)3.1 \times 10^{-19} 的底限。最后，即使在中值接收功率为 150 fW 的 $40\mu \mathrm{W}$$40\mu \mathrm{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\mu \mathrm{W}$$40\mu \mathrm{W}$40 muW40 \mu \mathrm{~W}的频率。

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\mathrm{m}$$3,400\mathrm{m}$3,400m3,400 \mathrm{~m} ) and a cat-eye retroflector located on the summit of Haleakala (elevation 3,050 m). Colocation of the sites allows a direct, out-of-loop timing verification.b, Detailed schematic for site B. (Site A is identical except without the synchronization (Synch) lock.) Each site uses two TPFCs phase locked to a local optical reference. The clock TPFC output defines the local timescale. It is also transmitted across the bidirectional link through a 10-cm aperture free-space optical (FSO) terminal (Extended Data Fig.1). The
图 1：量子限制 OTT。a, 系统在两个站点之间建立的 300 公里折叠自由空间链路上进行了测试，这两个站点分别位于毛纳罗亚天文台（海拔 $3,400\mathrm{m}$$3,400\mathrm{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}$${}^{24}$^(24){ }^{24}, quantum-limited OTT operates across three times the distance, at 20 times improved update rate using 200 times less comb power and at four times lower aperture diameter, with more than 14 dB greater tolerable link loss. The tolerable link loss of 102 dB exceeds that of future ground-to GEO links with similar $10-\mathrm{cm}$$10-\mathrm{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}$${}^{24}$^(24){ }^{24} 的最远距离相比，量子限界 OTT 的运行距离是其三倍，更新率提高了 20 倍，梳状功率降低了 200 倍，孔径直径缩小了四倍，可容忍链路损耗增加了 14 分贝以上。102 dB 的可容忍链路损耗超过了采用类似 $10-\mathrm{cm}$$10-\mathrm{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。

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_{\mathrm{A}}$$t_{\mathrm{A}}$t_(A)t_{\mathrm{A}} or $t_{\mathrm{B}}$$t_{\mathrm{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_{\mathrm{A}}$$t_{\mathrm{A}}$t_(A)t_{\mathrm{A}} 或 $t_{\mathrm{B}}$$t_{\mathrm{B}}$t_(B)t_{\mathrm{B}} ，其差值为 $t_{\mathrm{A}}$$t_{\mathrm{A}}$t_(A)t_{\mathrm{A}} 、

is a measure of the time offset between the two clock combs, ( $\mathrm{\Delta }{T}_{\text{osc}}-\mathrm{\Delta }{T}_{\mathrm{cntr}}$$\mathrm{\Delta }{T}_{\text{osc }}-\mathrm{\Delta }{T}_{\mathrm{cntr}}$DeltaT_("osc ")-DeltaT_(cntr)\Delta T_{\text {osc }}-\Delta T_{\mathrm{cntr}} ), where $\mathrm{\Delta }{T}_{\text{osc}}$$\mathrm{\Delta }{T}_{\text{osc }}$DeltaT_("osc ")\Delta T_{\text {osc }} is the time offset between the local reference oscillators and $\mathrm{\Delta }{T}_{\text{cntrl}}$$\mathrm{\Delta }{T}_{\text{cntrl }}$DeltaT_("cntrl ")\Delta T_{\text {cntrl }} is the synchronization feedback to the clock TPFC at site $B(\mathrm{\Delta }{T}_{\mathrm{cntrl}}=0$$B\left(\mathrm{\Delta }{T}_{\mathrm{cntrl}}=0\right.$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}$${}^{30}$^(30){ }^{30} means that the time-of-flight, including turbulence effects, should cancel in this two-way comparison of equation (2). Nevertheless, we include a non-reciprocal, turbulence noise term, ${ϵ}_{\mathrm{NR},\text{turb}}$${ϵ}_{\mathrm{NR},\text{ turb }}$epsilon_(NR," turb ")\epsilon_{\mathrm{NR}, \text { turb }}, for reasons discussed later. The quantum noise term,${ϵ}_{qn}$${ϵ}_{qn}$epsilon_(qn)\epsilon_{q n}, has a standard deviation following equation (1). The system noise, ${ϵ}_{\text{combs}}$${ϵ}_{\text{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（ $\mathrm{\Delta }{T}_{\text{osc}}-\mathrm{\Delta }{T}_{\mathrm{cntr}}$$\mathrm{\Delta }{T}_{\text{osc }}-\mathrm{\Delta }{T}_{\mathrm{cntr}}$DeltaT_("osc ")-DeltaT_(cntr)\Delta T_{\text {osc }}-\Delta T_{\mathrm{cntr}} ），其中 $\mathrm{\Delta }{T}_{\text{osc}}$$\mathrm{\Delta }{T}_{\text{osc }}$DeltaT_("osc ")\Delta T_{\text {osc }} 是本地参考振荡器之间的时间偏移量， $\mathrm{\Delta }{T}_{\text{cntrl}}$$\mathrm{\Delta }{T}_{\text{cntrl }}$DeltaT_("cntrl ")\Delta T_{\text {cntrl }} 是对站点 $B(\mathrm{\Delta }{T}_{\mathrm{cntrl}}=0$$B\left(\mathrm{\Delta }{T}_{\mathrm{cntrl}}=0\right.$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}$${}^{30}$^(30){ }^{30} 的基本互易性意味着飞行时间（包括湍流效应）应在方程 (2) 的双向比较中抵消。尽管如此，我们还是加入了一个非对等的湍流噪声项 ${ϵ}_{\mathrm{NR},\text{turb}}$${ϵ}_{\mathrm{NR},\text{ turb }}$epsilon_(NR," turb ")\epsilon_{\mathrm{NR}, \text { turb }} ，原因稍后讨论。量子噪声项 ${ϵ}_{qn}$${ϵ}_{qn}$epsilon_(qn)\epsilon_{q n} 的标准偏差与公式 (1) 一致。系统噪声 ${ϵ}_{\text{combs}}$${ϵ}_{\text{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{\tau }_{\mathrm{p}}$$2{\tau }_{\mathrm{p}}$2tau_(p)2 \tau_{\mathrm{p}} dynamic range of the timing discriminator. This initial alignment is accomplished by sweeping the tracking comb pulse position in time ${}^{29}$${}^{29}$^(29){ }^{29} and searching for peaks in the heterodyne signal (Fig. 2a). Scintillation from turbulence, however, causes random $100\mathrm{\%}$$100\mathrm{\%}$100%100 \% intensity fluctuations that complicate mapping the peak heterodyne voltage to the incoming pulse location. Using a Kalman filter to aggregate intermittent
为了跟踪进入的脉冲，进入的时钟梳状脉冲和本地跟踪梳状脉冲必须在时间上重叠，大致在定时鉴别器的 $2{\tau }_{\mathrm{p}}$$2{\tau }_{\mathrm{p}}$2tau_(p)2 \tau_{\mathrm{p}} 动态范围内。在时间 ${}^{29}$${}^{29}$^(29){ }^{29} 内扫描跟踪梳脉冲位置，并搜索外差信号中的峰值，即可实现初始对准（图 2a）。然而，湍流产生的闪烁会导致随机 $100\mathrm{\%}$$100\mathrm{\%}$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\mid$$2\mid$2∣2 \mid Low-power acquisition and quantum-limited performance. a, Demonstrated signal acquisition over the 300 km Hawaii link. Initially, the local tracking TPFC is swept over its full 5 ns non-ambiguity range in a triangular waveform. At roughly 3 s into the acquisition, a peak in the heterodyne signal indicates a transient temporal overlap between the tracking TPFC and the incoming clock comb (inset). On the basis of the observation of a heterodyne signal above the 135 fW threshold, the signal processor steers the tracking comb back to this location for finer search before initiating the tracking lock at
图： $2\mid$$2\mid$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_{\mathrm{A}}$$t_{\mathrm{A}}$t_(A)t_{\mathrm{A}} and $t_{\mathrm{B}}$$t_{\mathrm{B}}$t_(B)t_{\mathrm{B}}, are input into the Kalman filter to generate optimal estimates of the timing with $10-25\mathrm{Hz}$$10-25\mathrm{Hz}$10-25Hz10-25 \mathrm{~Hz} effective bandwidth. These Kalman-filtered values are used in equation (2) and input to a $15-\mathrm{Hz}$$15-\mathrm{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 $\mathrm{\Delta }t\to 0$$\mathrm{\Delta }t\to 0$Delta t rarr0\Delta t \rightarrow 0 (Extended Data Figs. 5 and 6).
为了在信号衰减时仍能稳健运行，将定时采样 $t_{\mathrm{A}}$$t_{\mathrm{A}}$t_(A)t_{\mathrm{A}} 和 $t_{\mathrm{B}}$$t_{\mathrm{B}}$t_(B)t_{\mathrm{B}} 输入卡尔曼滤波器，以生成具有 $10-25\mathrm{Hz}$$10-25\mathrm{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-\mathrm{Hz}$$15-\mathrm{Hz}$15-Hz15-\mathrm{Hz} 带宽同步锁，以引导站点 B 的时钟梳。在这里，双向合成中使用的定时信号是通过同轴电缆从 A 站传送到 B 站的，但也可以像参考文献 15 中那样使用光通信链路。15.收发器的时间延迟经过校准，因此当 $\mathrm{\Delta }t\to 0$$\mathrm{\Delta }t\to 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$$n=40$n=40n=40 and 20 photons per signal integration time $(19\mu \mathrm{s})$$(19\mu \mathrm{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. $\mathbf{b}$$\mathbf{b}$b\mathbf{b}, The timing noise (standard deviation over 600 s ) in $\mathrm{\Delta }t$$\mathrm{\Delta }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 $\gamma =2.1{\gamma }_{\mathrm{ql}}$$\gamma =2.1{\gamma }_{\mathrm{ql}}$gamma=2.1gamma_(ql)\gamma=2.1 \gamma_{\mathrm{ql}} (grey line), where ${\gamma }_{\mathrm{q}}\approx 0.6$${\gamma }_{\mathrm{q}}\approx 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 $\gamma =2.1{\gamma }_{\mathrm{ql}}$$\gamma =2.1{\gamma }_{\mathrm{ql}}$gamma=2.1gamma_(ql)\gamma=2.1 \gamma_{\mathrm{ql}} (dotted lines), until reaching the system noise floor at received powers above 10 nW .
大约 5 sinto 采集。 $\mathbf{b}$$\mathbf{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 $\mathrm{\Delta }t$$\mathrm{\Delta }t$Delta t\Delta t 中的定时噪声（600 秒内的标准偏差）。颜色与下面 cbel 中的轨迹相对应。c, 在接收功率为 800 fW 和 20.6 pW 时，2 km 自由空间链路上的时间偏差（TDEV）与 $\gamma =2.1{\gamma }_{\mathrm{ql}}$$\gamma =2.1{\gamma }_{\mathrm{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, $\mathrm{\Delta }t$$\mathrm{\Delta }t$Delta t\Delta t, does not reach the quantum limit for reasons discussed below. Nevertheless, it drops below state-of-the-art transportable optical atomic clocks ${}^9$${}^9$^(9){ }^{9} after only 6 s of averaging time and laboratory optical atomic clocks after 17 s of averaging time ${}^{21}$${}^{21}$^(21){ }^{21}.
图 3 比较了 300 公里和 2 公里链路在定时同步性、定时不稳定性和频率不稳定性方面的性能。扩展数据图 7-10 提供了更多数据。虽然单向定时测量在强湍流 300 公里链路上受到量子限制，但与短路和 2 公里链路不同，其双向减法 $\mathrm{\Delta }t$$\mathrm{\Delta }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$$B$BB was attenuated to $40\mu \mathrm{W}$$40\mu \mathrm{W}$40 muW40 \mu \mathrm{~W} leading to a median received power of 150 fW at site A. Despite signal fading $73\mathrm{\%}$$73\mathrm{\%}$73%73 \% time below threshold, timing acquisition and synchronization were still achieved with minimal 2.8 times performance degradation at short times. This attenuation of the site $B$$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\mathrm{\%}$$73\mathrm{\%}$73%73 \% 时间低于阈值，但在短时间内仍能实现定时采集和同步，性能下降幅度最小为 2.8 倍。站点 $B$$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。

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 $\mathrm{\Delta }t$$\mathrm{\Delta }t$Delta t\Delta t and its counterpart $\overline{t}=(t_{\mathrm{A}}+t_{\mathrm{B}})/2$$\overline{t}=\left(t_{\mathrm{A}}+t_{\mathrm{B}}\right)/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
$\mathrm{\Delta }t$$\mathrm{\Delta }t$Delta t\Delta t 及其对应的 $\overline{t}=(t_{\mathrm{A}}+t_{\mathrm{B}})/2$$\overline{t}=\left(t_{\mathrm{A}}+t_{\mathrm{B}}\right)/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\mu \mathrm{W}$$40\mu \mathrm{W}$40 muW40 \mu \mathrm{~W} attenuated comb power with 150 fW median received power (green triangles). Also shown are data from the 2 km link at 30 pW received power (yellow stars) and shorted (grey dots). At low turbulence over the 2 km link, the performance is nearly quantum limited (yellow dashed line for $\gamma =3.4{\gamma }_{\mathrm{ql}}$$\gamma =3.4{\gamma }_{\mathrm{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\mathrm{fs}/\sqrt{\tau }$$1.6\mathrm{fs}/\sqrt{\tau }$1.6fs//sqrttau1.6 \mathrm{fs} / \sqrt{\tau} and $49\mathrm{as}/\sqrt{\tau }$$49\mathrm{as}/\sqrt{\tau }$49as//sqrttau49 \mathrm{as} / \sqrt{\tau}