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来源机构： Optics Express Optics Express Aug 22, 2021 801.95KB 科技论文

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## Implementation of a twin-beam state-based clock synchronization system with dispersion-free HOM feedback

1. Introduction . Since the beginning of the 21st century, clock synchronization schemes based on quantum entanglement and compressed states have been widely investigated because of their improvement in accuracy and potential applications in secure clock transfer. In terms of clock accuracy, the number-squeezed states $\left n\right \rangle$ in the $m$ -fold entangled state can enhance the accuracy limit by $\sqrt {mn}$ times [ 1 ]. For the security of clock transfer, the non-locality of quantum sources shows the potential to find possible eavesdroppers in the transmission process [ 2 ]. So far, the schemes of quantum clock synchronization based on biphoton entanglement mostly belong to two categories. One relies on the evolution time of the entangled two-level atom systems deposited in two parties [ 3 – 7 ]. The other uses the second-order coherence of entangled photon pairs for obtaining the time difference between two parties [ 8 , 9 ]. Our subsequent discussion will focus mainly on the second implementation method. For most of the clock synchronization schemes based on the coherence of frequency-entangled state, current implementations still focus on calibrating the clock between two parties using entangled photon pairs [ 10 – 15 ]. In 2004, V. Giovannetti et al. put forward the conveyor belt protocol [ 8 ]. In the same year, A. Valencia et al. proposed an unidirectional clock synchronization protocol based on the coincidence measurement of entangled photon pairs [ 10 ]. Besides, T. B. Bahder et al. proposed a clock synchronization protocol that has equal transmission paths for signal and idler photons [ 9 ]. This scheme was experimentally verified by R. Quan et al. in 2016 [ 12 ]. In 2018, L. Antia et al. reported a bidirectional secure quantum clock synchronization scheme based on polarization-entangled photon pairs [ 14 ]. In 2019, B. Li et al. provided a new idea for achieving secure clock synchronization using frequency-entangled photon source through non-local dispersion compensation verification experiment [ 2 ]. An experiment of bidirectional clock transfer based on frequency-entangled source was also implemented in the same year [ 16 ]. Among most of the above experimental schemes, the second-order correlation function of the separated optical fields at two remote parities directly determines the accuracy and stability of the clock measurement. To improve the clock property, we consider the engineering of the entangled photon source, whose intensity and spectral properties are closely related to the peak position and width of the second-order correlation distribution. When V. Giovannetti et al. firstly proposed the scheme of quantum-enhanced clock synchronization, they pointed out that the photon source was assumed to be a perfect coincident-in-frequency light source, i.e., $\omega _s=\omega _i$ , where $\omega _{s(i)}$ is the frequency of the signal (idler). However, because of the finite crystal length, the equality of photon pair frequency is not strictly satisfied in a type-II phase-matching process. The two-photon joint spectrum generally possesses a considerable spectral distribution around the line $\omega _s=\omega _i$ . In this paper, we choose the scheme described by Ref. [ 9 ] to explore the performance of the clock synchronization using the maximum frequency-entangled source generated by continuous-wave (CW) pumping and the “imperfect” coincident-in-frequency light source generated by pulsed pumping. Based on the properties of the photon pairs in the time domain, the above two sources are referred to as twin-beam state and difference-beam state, respectively [ 17 ]. It is noted that the chosen scheme covers both the application of HOM interference and coincidence measurement in clock transfer and has strong practicability in the experimental implementations. Therefore, the analysis in this paper offers instructions for the theoretical design and the selection of experimental apparatus. Given the simulation results, we use the twin-beam state in our experimental system to measure the initial clock difference between two parties with a 22 km transmission distance. It is verified that the application of the twin-beam state and the introduction of dispersion-compensated materials can provide higher accuracy and better stability for quantum clock synchronization. We arrange this paper as follows: Section 2 is the theoretical comparison of the twin-beam and difference-beam states in long-distance clock transfer. More precisely, in section 2.1 , we exhibit the phase-matching conditions for generating the above two sources. In sections 2.2 and 2.3 , we simulate the result of HOM interference and coincidence measurement after the long-distance transmission. Section 3 is the experimental implementation for a clock synchronization system using the twin-beam state. In section 3.1 , we describe the experimental setup in detail. In section 3.2 , we show the theoretical and experimental comparison for the HOM interference and coincident measurement. Finally, in section 3.3 , we evaluate the precision of the time offset and long-time stability of the clock synchronization process between two parties. 2. Comparison of frequency-entangled photon source . A strong pump light propagating through a second-order nonlinear medium drives the spontaneous parametric down-conversion (SPDC) process and produces photon pairs with different degrees of frequency entanglement. Two photon joint spectrum, which could be used for describing the frequency entanglement, is expressed by the combination of pump distribution function and phase-matching function [ 18 , 19 ], (1) $$A(\omega_s, \omega_i)= \exp{\left[-\frac{{(\omega_s+\omega_i-\omega_{p0})}^2}{2\sigma^2}\right]}\textrm{sinc}\left(\frac{\Delta \beta L}{2}\right).$$ Here, $\omega _{j0}$ and $\omega _{j}$ ( $j = p,s,i$ ) stand for the phase-matched central frequency and real frequency of pump ( $p$ ), signal ( $s$ ), and idler ( $i$ ), respectively. $\sigma$ is related to the full width half maximum (FWHM) of the pump pulse $B_p$ by $\sigma =B_p/\left (2\sqrt {2\ln 2}\right )$ . $\Delta \beta$ is the phase mismatch between the pump and generated photon pairs while $L$ is the length of the nonlinear medium. To simplify the calculations in the following sections, we make a Gaussian approximation by $\textrm{sinc}(x)\approx e^{-\gamma _0x^2}$ ( $\gamma _0=0.193$ ) and take the first-order Taylor expansion $\Delta \beta \approx \tau _s(\omega _s-\omega _{s0})+\tau _i(\omega _i-\omega _{i0})$ , where $\tau _{s(i)}$ stands for the first-order differential of propagation constant $\beta _j$ between signal(idler) and the pump [ 20 ]. In this work, we mainly focus on two special cases in frequency entanglement: twin-beam state and difference-beam state. Considering the imperfect frequency equality of the difference-beam state, the joint spectrum of the two states are: (2) \begin{aligned} A_{\textrm{TB}}(\omega_s, \omega_i) & =\delta(\omega_s+\omega_i-\omega_{p0}) \exp{\left[-\frac{\gamma_0L^2}{4}{(\tau_s(\omega_s-\omega_{s0})+\tau_i(\omega_i-\omega_{i0}))}^2\right]},\\ A_{\textrm{DB}}(\omega_s, \omega_i) & =\exp{\left[-\frac{{(\omega_s+\omega_i-\omega_{p0})}^2}{2\sigma^2}\right]}\exp{\left[-\frac{\gamma_0L^2}{4}{\tau_s}^2{(\omega_s-\omega_i-\omega_{s0}+\omega_{i0})}^2\right]}.\\ \end{aligned} For twin-beam states, the frequency of photon pairs are symmetric about half of the pump frequency $\omega _s+\omega _i=\omega _{p0}$ and generated simultaneously in the time-domain [ 21 ]. For difference-beam states, the phase-matching function doesn’t show a form of Dirac-delta function with a finite crystal length. However, the group velocity matching $\tau _s=-\tau _i$ allows a simplification of the phase-matching function in Eq.?( 1 ). 2.1 Comparison of phase-matched conditions . Generally speaking, the phase mismatch $\Delta \beta$ of an SPDC process that happened inside a quasi-phase-matched periodically poled crystal is determined by the central frequency $\omega _{j0}$ , poled period $\Lambda$ , and crystal temperature $T_0$ . Here, we consider a type-II SPDC process in PPKTP that generates photon pairs with the same central frequency and orthogonal polarization. Combining the energy conservation $2\omega _{s(i)0}=\omega _{p0}$ and phase-matched condition $\Delta \beta (\lambda _{p0},\Lambda ,T_0)=\beta _{p}(\lambda _{p0},T_0)-\beta _{s}(\lambda _{s0},T_0)-\beta _{i}(\lambda _{i0},T_0)+2\pi /\Lambda$ ( Here $\lambda _{j0}$ are the central wavelengths of pump, signal and idler), we show the conditions for generating degenerate twin-beam and difference-beam sources in Fig.? 1 . For the difference-beam state, the additional condition $\tau _s=-\tau _i$ needs to be satisfied first in order to generate a coincident-in-frequency state [ 22 ], which strictly tighten the generation conditions in Fig.? 1 (b) compared with the twin-beam state in Fig.? 1 (a). ? Fig. 1. Phase-matched conditions for generating degenerate photon pairs. (a) Twin-beam state (b) Difference-beam state. The points in the figure shows the corresponding crystal period and temperature for generating 1550 nm photon pairs (thermal expansion of poled period is not considered here). Download Full Size PPT Slide PDF The above phase-matching conditions show that the confinement from group velocity matching significantly reduces the capability of generating designated photon wavelengths with a fixed crystal period or temperature. Therefore the twin-beam state is advantageous for its accessible phase-matched conditions and flexible wavelength selection under room temperature. 2.2 Comparison of HOM interference with long-distance transmission . In Fig.? 2 , we show the principle of clock synchronization schemes based on two equal transmission paths [ 9 ]. Two distant parties have separate clocks A and B that need to be synchronized. The third-party generates frequency-entangled photon pairs and takes control of the fiber length feedback system. In the beginning, the frequency-entangled degenerate photon pairs are separated and distributed through the equal paths $l_s$ and $l_i$ to A and B, respectively. A and B then measure the arrival time of part of the photons and return the rest of them. The HOM interference measurement is performed on the photon pairs back from A and B, whose result is applied to control the adjustable fiber delay to ensure the separate signal and idler photons experience the same propagation time in the fiber toward clock A and B in real-time transmission. The information of the arrival time is recorded by A and B and transferred through a classical channel from one to the other. Note that we suppose the frequency references of the time-stamping units at A and B are the same, which means the time offset between the two parties is stable. Therefore, A(or B) can calibrate its clock based on the time offset between their measurement results. ? Fig. 2. Clock synchronization scheme based on HOM feedback and coincidence measurement. Download Full Size PPT Slide PDF In the above process, the HOM interference result serves as an indicator to control the feedback system and therefore resist the ambient noise and ensure the real-time equilength of the transmission distance. In this part, we compare the HOM curves between twin-beam states and difference-beam states under different transmission distances. When the signal and idler photons pass through the optical fiber with lengths of $l_s$ and $l_i$ , respectively, the expression of the HOM curve is [ 22 ]: (3) \begin{aligned} P(\tau)= & \iint d\omega_sd\omega_i\leftA(\omega_i,\omega_s)\right.\\ & \left.-A(\omega_s,\omega_i)\exp{\left[2i(k_s(\omega_s)l_s+k_i(\omega_i)l_i-k_s(\omega_i)l_s-k_i(\omega_s)l_i)\right]}\exp{\left[2i(\omega_s-\omega_i)\tau\right]}\right^2. \end{aligned} Here, $k_{s(i)}$ represents the propagation constant of signal (idler) photons when passing through the fiber. $\tau$ is the time offset introduced by the adjustable fiber delay. For twin-beam states, the pump spectral function is regarded as a Dirac-delta function. Combining the two-photon joint spectrum with Eq.?( 3 ), the expression of the HOM curve formed by twin-beam states is: (4) \begin{aligned} P_{\textrm{TB}}(\tau) & =\frac{2\sqrt{2\pi}}{\sqrt{\gamma_0}L\tau_s-\tau_i}\left\{1-\exp{\left[-\frac{\gamma_0L^2{(\tau_s-\tau_i)}^2{(\omega_{s0}-\omega_{i0})}^2}{8}\right]}\right.\\ & \left.\times\exp{\left[-\frac{8{[(k^{'}_sl_s-k^{'}_il_i)-\frac{1}{2}(k^{\prime\prime}_sl_s+k^{\prime\prime}_il_i)(\omega_{s0}-\omega_{i0})+\tau]}^2}{\gamma_0L^2{(\tau_s-\tau_i)}^2}\right]}\right\}. \end{aligned} Here we make second-order Taylor expansion for the fiber propagation constants $k_{s(i)}$ , whose first- and second-order differential are shown as $k^{'}_{s(i)}$ and $k^{''}_{s(i)}$ , respectively. Similarly, combining the two-photon joint spectrum of difference-beam states with Eq.?( 3 ), the expression of the HOM curve is: (5) \begin{aligned} P_{\textrm{DB}}(\tau) & =\frac{\sqrt{2}\pi \sigma}{\sqrt{\gamma_0}L\tau_s}-\frac{\sqrt{2}\pi \sigma}{\sqrt{\gamma_0L^2{\tau_s}^2+\frac{1}{2}{(k^{\prime\prime}_sl_s-k^{\prime\prime}_il_i)}^2\sigma^2}}\exp{\left[-\frac{\gamma_0L^2{\tau_s}^2{(\omega_{s0}-\omega_{i0})}^2}{2}\right]}\\ & \times \exp{\left[-\frac{{[(k^{'}_sl_s-k^{'}_il_i)-\frac{1}{2}(k^{\prime\prime}_sl_s+k^{\prime\prime}_il_i)(\omega_{s0}-\omega_{i0})+\tau]}^2}{\frac{\gamma_0L^2{\tau_s}^2}{2}+\frac{1}{4}{(k^{\prime\prime}_sl_s-k^{\prime\prime}_il_i)}^2\sigma^2}\right]}. \end{aligned} Combining Eq.?( 4 ) and Eq.?( 5 ), we compare the specific features of the two HOM curves in Table? 1 , including the dip position $\tau _0$ , visibility $V$ , and dip width $\Delta \tau$ . Table 1. Comparison of HOM features. View Table View all tables in this article According to Table? 1 , the dip positions of the two sources are the same while the visibility and dip width of the two sources show different features, especially with nondegenerate photon frequency. For either degenerate or nondegenerate photon pairs, it is noted that the visibility and width of the twin-beam states are independent of the fiber properties. Therefore, the shape of HOM curves remains unchanged with varying transmission distances. However, the increasing transmission distance influences the visibility and width of the nondegenerate difference beam state, especially for the biphotons generated from the pulsed pump with broad spectral bandwidth, which will affect the precision of fiber length feedback. In conclusion, nondegenerate twin-beam states have absolute predominance in long-distance clock transfer because it is free from fiber dispersion effect. 2.3 Comparison of coincidence measurement in long-distance transmission . In the scheme described in Fig.? 2 , the final readout of the clock time offset requires A and B to make a coincidence measurement using the signal and idler photons they collected. The coincidence counting is closely related to the optical field second-order correlation function $G^{(2)}$ . Here we calculate the $G^{(2)}$ function of the two sources and compare their performance under long-distance transmission. When the signal and idler photons pass through the optical fiber with respective length of $l_s$ and $l_i$ , the relationship between the second-order interference curve $G^{(2)}$ and the two-photon joint spectrum is: (6) $$G^{(2)}(\tau_A, \tau_B)= {\left\iint d\omega_sd\omega_i A(\omega_i,\omega_s)e^{i\left(k_s(\omega_s)l_s+k_i(\omega_i)l_i\right)}e^{{-}i\omega_s\tau_A}e^{{-}i\omega_i\tau_B}\right}^2.$$ $\tau _A$ and $\tau _B$ are the arrival time of signal and idler photons after passing through $l_s$ and $l_i$ , respectively. The $G^{(2)}$ function of the twin-beam state is calculated to be (7) $$G_{\textrm{TB}}^{(2)}(\tau_A, \tau_B) \propto \exp{\left[-\frac{{(\tau_B-\tau_A+k^{'}_sl_s-k^{'}_il_i)}^2}{\frac{\gamma_0L^2{(\tau_s-\tau_i)}^2}{2}+\frac{{2(k^{\prime\prime}_sl_s+k^{\prime\prime}_il_i)}^2}{\gamma_0L^2{(\tau_s-\tau_i)}^2}}\right]},$$ while for the difference-beam state is (8) \begin{aligned} & G_{\textrm{DB}}^{(2)}(\tau_A, \tau_B) \propto \exp{\left[-\frac{A+B+C+D}{{\left(\frac{2\gamma_0L^2\tau_s^2}{\sigma^2}-k^{\prime\prime}_sk^{\prime\prime}_il_sl_i\right)}^2+{\left(\frac{1}{\sigma^2}+\frac{\gamma_0L^2\tau_s^2}{2}\right)}^2{\left(k^{\prime\prime}_sl_s+k^{\prime\prime}_il_i\right)}^2}\right]},\\ A & =\frac{2\gamma_0L^2{\tau_s}^2}{\sigma^4}{\left(\tau_B-\tau_A+k^{'}_sl_s-k^{'}_il_i\right)}^2,\quad B=\frac{1}{\sigma^2}{\left[k^{\prime\prime}_sl_s(\tau_B-k^{'}_il_i)+k^{\prime\prime}_il_i(\tau_A-k^{'}_sl_s)\right]}^2,\\ C & =\frac{{\gamma_0}^2L^4{\tau_s}^4}{\sigma^2}{\left(\tau_B+\tau_A-k^{'}_sl_s-k^{'}_il_i\right)}^2,\quad D=\frac{\gamma_0L^2{\tau_s}^2}{2}{\left[k^{\prime\prime}_sl_s(\tau_B-k^{'}_il_i)-k^{\prime\prime}_il_i(\tau_A-k^{'}_sl_s)\right]}^2. \end{aligned} If we perform the coincidence measurement between the arrival time at A and B, which means we calculate the statistic distribution of the time offset $\tau '=\tau _B-\tau _A$ . Then the relationship between the distribution of coincidence count and $G^{(2)}$ function is $G^{(2)'}(\tau ')=\iint d\tau _Ad\tau _BG^{(2)}\left (\tau _A, \tau _B\right )\delta \left (\tau _B-\tau _A-\tau '\right )$ . Therefore, the coincidence distributions of two states are shown as: (9) \begin{aligned} G_{\textrm{TB}}^{(2)'}(\tau') & \propto \exp{\left[-\frac{{\left(\tau'+k^{'}_sl_s-k^{'}_il_i\right)}^2}{\frac{\gamma_0L^2{(\tau_s-\tau_i)}^2}{2}+\frac{{2(k^{\prime\prime}_sl_s+k^{\prime\prime}_il_i)}^2}{\gamma_0L^2{(\tau_s-\tau_i)}^2}}\right]},\\ G_{\textrm{DB}}^{(2)'}(\tau') & \propto \exp{\left[-\frac{{\left(\tau'+k^{'}_sl_s-k^{'}_il_i\right)}^2}{2\gamma_0L^2{\tau_s}^2+\frac{{\left(k^{\prime\prime}_sl_s+k^{\prime\prime}_il_i\right)}^2}{2\gamma_0L^2{\tau_s}^2}+ \frac{{\left(k^{\prime\prime}_sl_s-k^{\prime\prime}_il_i\right)}^2\sigma^2}{4}}\right]}. \end{aligned} The peak position $\tau '_0$ and width $\Delta \tau '$ of the coincidence distribution for the two states are compared in the following Table? 2 . Table 2. Comparison of coincidence measurement features. View Table View all tables in this article According to Table? 2 , the peaks of the coincidence distribution count are the same for the two states. Considering the feedback of HOM interference, which adds an extra time delay $\tau _0$ at the signal arm, we further modify the peaks to $\tau _{0\textrm{tot}}=\tau _B-(\tau _A+\tau _0)=-\frac {1}{2}\left (k^{''}_sl_s+k^{''}_il_i\right )(\omega _{s0}-\omega _{i0})$ . It is noted that the feedback plays a considerable role when the length of signal and idler paths are not the same. However, the result after feedback is still related to the transmission distance unless the photon pairs are perfectly degenerate. Besides, the peak widths of the two states are broadened with the increase of transmission distance. Compared with the twin-beam state, the difference-beam state has an unique positive term corresponding to the spectral broadening of the pulsed pump. It implies that the width of the coincidence count distribution for a difference-beam state is always wider than a twin-beam state though the value of this term is usually negligible in practical applications. Based on the above discussion, the group velocity dispersion of the generated biphotons induces the shift of the peak position as well as the broadening of the distribution width, which induce the dependence of clock properties on the transmission distance. In this part, we compare the performance of the twin-beam state generated by CW pumping and the difference-beam state generated by pulsed pumping in three aspects: the phase matching conditions, HOM interference curves, and distribution of coincidence measurement. The result shows that the twin-beam state has flexible wavelength selection under room temperature and keeps high visibility and narrow width in the HOM interference under long-distance transmissions. At the same time, the width of the coincidence count distribution is narrower compared with the difference-beam state generated by a broadband pulsed pump light. Given these advantages, in the following section, we use the twin-beam biphoton source to carry out the quantum clock synchronization experiment. 3. Experiment . 3.1 Experimental setup . The experimental setup is shown in Fig.? 3 . A linearly polarized CW laser (DLC DLpro780, Toptica) with a center frequency at 775 nm and average power of 60 mW was employed in the photon pair generation part (part (a)). After shaped by the cylindrical lens, the linearly polarized pump light passed through the 1 mm $\times$ 2 mm $\times$ 20 mm PPKTP crystal (Raicol) and generated orthogonally polarized photon pairs with central wavelengths at 1549.9 nm and 1550.1 nm. The photon pairs were then separated by the polarization beam splitter (PBS) at the left side of the part (b). ? Fig. 3. Experimental setup. Lx, Ly: Cylindrical lens, M: Mirror, HWP1(2): half-wave plate at 775 nm (1550 nm), FL: focusing lens, F: filter, PBS: polarization beam splitter (1550 nm), PMFC: polarization maintaining fiber coupler (splitting ratio 1:1), BS:beam splitter (splitting ratio 9:1), MDL: modulated optical delay line, SMF: single-mode fiber (10 m, 1 km, 10 km), DCF: dispersion compensated fiber, FR: faraday mirror, FPC: fiber polarization controller, SNSPD: superconducting nanowire single-photon detector, TCSPC: time-correlated single photon counting. In part(b), red lines represents the polarization maintaining fiber, blue lines represents the single-mode fiber, and orange lines represents dispersion compensated fiber. Download Full Size PPT Slide PDF In part (b), the divided signal and idler photons reached their separate destinations A and B by the upper and lower paths, respectively. Both parties employed a 9:1 beam splitter (BS) and Faraday mirror (FR) to reflect most photons to the feedback system. These photons were guided to perform a HOM interference and then collected by a four-channel superconducting nanowire single-photon detector (SNSPD, Scontel) through port 1 and port 2. The statistic distribution of HOM interference was exported by a time-correlated single-photon counting system (TCSPC, QuTag, standard type) and the dip positions were extracted to control the modulated optical delay line (MDL, MODL-1000, Conquer). The remaining 10 $\%$ photons at A and B were directly measured by the SNSPD and guided to two different channels of the TCSPC to perform the coincidence measurement. The position of the coincident peak showed the time offset between the two parties. (Here is the time offset between the two selected TCSPC channels.) 3.2 Comparison between theoretical and experimental result . According to the theoretical analysis, the width and visibility of HOM curves based on twin-beam state remain unchanged after long-distance transmission. Here in Fig.? 4 (a), we compare the normalized HOM curves with transmission distances of 20 m and 20 km ( $l_s = l_i = 10 m, 10 km$ ). The exposure time of each point is 100 ms, and the time interval between each point is 0.033 ps. The collection is repeated five times at each distance. As seen in the figure, the experimental result is in good accordance with the theoretical simulation. The FWHMs of the two HOM curves are maintained around 1 ps while the visibilities are 97.3 $\% \pm$ 0.5 $\%$ and 96.1 $\% \pm$ 2 $\%$ under 20 m and 20 km lengths, allowing an accurate estimation for the dip position in a fiber-length feedback system. ? Fig. 4. Experimental and theoretical comparison. (a) HOM curves. Orange lines are theoretical calculation based on Eq.?( 4 ). (b) Coincidence measurement. Orange line shows the normalized coincidence count distribution calculated from Eq.?( 9 ). The resolution of each collection point is 1 ps. Download Full Size PPT Slide PDF Similarly, we show the normalized coincidence measurement comparison between theoretical calculation and experimental measurement in Fig.? 4 (b). Considering the finite time resolution of the SNSPD, we calculate the total width of the coincidence count distribution by $\Delta \tau _{\textrm{tot}}=\sqrt {{\Delta \tau '}^2+{\Delta \tau _{\textrm{SNSPD}}}^2}$ , where $\Delta \tau _{\textrm{SNSPD}}$ = 80 ps is the time resolution of the SNSPD. As seen in the figure, the FWHMs of the coincidence count distributions are 80.1 ps and 399.2 ps with a transmission distance of 10 m and 10 km, respectively, which shows a good agreement with simulation results. In this section, we show good consistency between the experimental data and the theoretical simulations. Our experiment demonstrates a high constantness of HOM curves under long-distance transmission and gives predictable width broadening for coincidence measurement. In the following section, we will analyze the clock synchronization properties based on these experimental results. 3.3 Analysis of clock synchronization property . To show the clock properties of our system comprehensively, we transmitted the signal and idler photons with 20 m-, 2 km-, and 20 km-long distances. For each situation, the clock accuracy and stability were carefully analyzed and compared. 3.3.1 Time offset and its precision . In Fig.? 5 (a), we show the coincidence count under different transmission distances. Each data point is collected with a time window of 10 ps and an acquisition time of 10 s. Specially, we regard the peak position measured with 20 m SMF as the reference clock offset between two parties. Then the differences between other peak positions and this designated reference stand for their respective clock accuracy. As shown in the inset, the average clock offsets of 2 km and 20 km fiber have 4 ps and 20 ps derivation from the reference. And the deviation would be further enlarged with increasing transmission distance because of the fiber dispersion. To suppress the degradation of clock accuracy, we introduce dispersion compensated materials to counteract the fiber dispersion in long-distance transmission. The black line and spot show the results of a 20 km SMF distribution and 2 km DCF compensation. The derivation of the time offset from the reference is reduced to 4 ps, which shows a significant improvement in clock accuracy. ? Fig. 5. Time offset and its precision. (a) Coincidence measurement. The inset at the top right shows the peak positions of each curve. (b) Relation between the precision of time offset and transmission distance. The x-axis is a logarithmic coordinate axis. (c) Relation between the precision of time offset and acquisition time. Both x- and y-axis are logarithmic coordinates. Download Full Size PPT Slide PDF The precision of clock offset is determined by the number of collected photon pairs and the width of coincident distribution, which is defined as [ 15 ]: (10) $$\delta t=\frac{\sigma_{\textrm{tot}}}{\sqrt{N}}=\frac{\sqrt{\frac{\gamma_0L^2{\left(\tau_s-\tau_i\right)}^2}{4}+\frac{{\left(k^{\prime\prime}_sl_s+k^{\prime\prime}_il_i\right)}^2}{\gamma_0L^2{\left(\tau_s-\tau_i\right)}^2}+{(\frac{\Delta \tau_{\textrm{SNSPD}}}{2\sqrt{2\ln2}})}^2}}{\sqrt{N_0T10^{-\frac{0.2l_s}{10000}}10^{-\frac{0.2l_i}{10000}}}}.$$ Here, $N$ is the total coincidence counts collected by the SNSPD. $N_0$ is the coincidence counting rate without fiber transmission. $\Delta \sigma _{\textrm{tot}} =\Delta \tau _{\textrm{tot}}/\left (2\sqrt {2\ln 2}\right )$ describes the width of coincidence distribution. $T$ is the acquisition time. Equation?( 10 ) shows that the precision of time offset degrades with the increase of transmission distance since the 0.2 dB/km fiber loss reduces the number of coincidence counts $N$ and the fiber dispersion increase the width of coincident curves $\Delta \sigma _{\textrm{tot}}$ . The precisions of time offset with 10 s acquisition time under different fiber lengths are shown in Fig.? 5 (b). The precisions of time offset are 0.56 ps,0.58 ps, and 5.8 ps with propagation distances of 20 m, 2 km, and 20 km, respectively. Besides, the dispersion compensation from DCF also improves the precision of time offset by reducing the width broadening. As the black point shows, the precision of time offset with a total 22 km distribution is improved to be 1.8 ps. Another factor that influences the precision of time offset is the acquisition time $T$ . In Fig.? 5 (c), we show a linear dependence on the acquisition time with a 20 km transmission distance in a logarithmic coordinate. In the experiment, we calculate the precision of time offset with the acquisition time of 5 s, 10 s, 20 s, 60 s, and 120 s. Each data point is in good agreement with the theoretical prediction. 3.3.2 Clock synchronization stability . Stability is another important indicator to evaluate the clock synchronization property. In our experiment, we carried out over ten hours of continuous measurement for each transmission distance. The time resolution of each point was set to be 1 ps. The time offset for each coincidence measurement is recorded as $x_{i}$ (i=1,2,3,…). Here we use the time deviation (TDEV) as the standard to evaluate clock stability, the definition of which is [ 23 ]: (11) $$\textrm{TDEV}(T_{\textrm{ave}})=\sqrt{\frac{1}{6n^2\left(N_{\textrm{peak}}-3n+1\right)}\sum_{j=1}^{N_{\textrm{peak}}-3n+1}{\left[\sum_{i=j}^{n+j-1}\left(x_{i+2n}-2x_{i+n}+x_{i}\right)\right]}^2}.$$ Here $N_{\textrm{peak}}$ is the number of peak positions over the collection period. $T = 10 s$ is the acquisition time for each coincidence measurement. $T_{\textrm{ave}}=nT$ (n=1,2,3 $\cdots$ ) is the averaging time. For the accuracy of the experimental result, the maximum averaging time is 1/8 of the total collection time. In Fig.? 6 , the slopes of the four lines are approximate to -1/2 in a short period and stay between -1/2 and 0 at a long averaging time. With 4000 s averaging time, the values of TDEV are 95 fs, 110 fs, and 633 fs with transmission distances of 20 m, 2 km, and 20 km, respectively. After dispersion compensation of the DCF, the time stability is improved to 170 fs at 4000 s with a 22 km-long transmission. In this case, we conduct an extra two hours of measurement. The TDEV described by the black line still shows a downward trend with the increase of the averaging time, and the stability achieves 150 fs near 5500 s. ? Fig. 6. Clock synchronization stability. The x- and y-axis are logarithmic coordinates. The error bars indicate 90 $\%$ confidence intervals for the numerical results. Download Full Size PPT Slide PDF Compared with other works based on the same scheme [ 12 , 24 ], we achieve better clock properties both in accuracy and stability at the same averaging time (5500 s) with a longer transmission distance. This enhancement is mainly because of the introduction of dispersion-compensated material. Furthermore, compared with other schemes based on second-order interference of quantum sources [ 13 ], we also have a five-fold enhancement in the accuracy while ten-fold improvement in the stability under the same transmission distance. However, the two-way quantum time transfer in Ref. [ 16 ] has a stability of 90 fs with 5000 s averaging time, which is 1.5 times better than our result. To improve the properties of the current system, we can further focus on introducing high nonlinearity materials with easy access to on-chip technology [ 25 , 26 ] to enhance the photon production rate of CW pumping biphoton source compared with a pulsed pumping, which also benefits for the integration and miniaturization of the system. In this section, we analyze the accuracy and stability of our clock synchronization system. By introducing dispersion-compensated materials in propagation paths, we achieve a clock accuracy at 4 ps with 22 km transmission distance, the precision of time offset is 1.8 ps with 10 s acquisition time, and the TDEV is 150 fs with 5500 s averaging time. Our experiment exhibits excellent properties compared with other experimental results using the same synchronization scheme. 4. Conclusion . In this paper, approximate maximum frequency-anticorrelated biphotons pumping by CW light were applied to a quantum clock synchronization scheme based on second-order interference. We compared the performance of the twin-beam state and the difference-beam state in phase-matching conditions, HOM curves, and coincidence count distributions under long-distance transmission. The result showed that the twin-beam state had flexible wavelength selection under room temperature, unaltered HOM width and visibility in long-distance transmission, as well as a narrower coincidence count distribution width. In the experiment, we verified the above theoretical simulation results. And for the first time, employed the twin-beam state to realize the second-order coherence-based quantum clock synchronization scheme. With a distribution distance of 22 km (20 km SMF plus 2 km DCF), the average deviation of clock offset from the reference was 4 ps with a time offset precision of 1.8 ps. The time stability was 150 fs with an averaging time of 5500 s. Besides, we exhibited the importance of dispersion compensation materials in the long-distance transmission of the quantum clock synchronization system. With the introduction of DCF, we decreased the time offset deviation from 20 ps to 4 ps in a 20 km transmission. The corresponding precision of the time offset was improved from 5.8 ps to 1.8 ps with 10 s acquisition time. And the TDEV with 4000 s averaging time was reduced from 633 fs to 170 fs. Future works can be focused on choosing the condensed dispersion-compensated materials only for coincidence measurement and enhancing the brightness of the photon pair source with high nonlinearity materials to further improve the clock synchronization properties. In conclusion, by employing the twin-beam source and dispersion-compensated materials in a second-order interference-based clock synchronization scheme, we use the CW pump instead of a well-designated pulsed pump to achieve an improvement on both the clock accuracy and stability in the experimental implementation compared with others using the same scheme. This work is especially helpful for choosing an appropriate frequency-entangled source, applying dispersion-compensated materials, and potential miniaturization and integration in future clock synchronization systems. Funding . National Natural Science Foundation of China (11974205); Tsinghua Initiative Scientific Research Program ; Beijing Innovation Center for Future Chip ; Special Project for Research and Development in Key Areas of Guangdong Province (2018B030325002); National Key Research and Development Program of China (2017YFA0303700). Disclosures . The authors declare no conflicts of interest. Data availability . Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. References . 1. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412 (6845), 417–419 (2001). [ CrossRef ] ? 2. B. Li, F. Hou, R. Quan, R. Dong, L. You, H. Li, X. Xiang, T. Liu, and S. Zhang, “Nonlocality test of energy-time entanglement via nonlocal dispersion cancellation with nonlocal detection,” Phys. Rev. A 100 (5), 053803 (2019). [ CrossRef ] ? 3. R. Jozsa, D. S. Abrams, J. P. 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