Demonstration of 75 km-fiber quantum clock synchronization in quantum entanglement distribution network

Abstract

The quantum entanglement distribution network, serviced as the communication infrastructure which distributes quantum information among remote users, enables many applications beyond the reach of classical networks. Recently, the applications such as quantum key distribution and quantum secure direct communication, have been successfully demonstrated in the quantum entanglement distribution network. In this article, we propose a multi-user round-trip quantum clock synchronization (QCS) scheme in the quantum network, which can be implemented with one single entangled photon source located at the server. The server distributes the entangled photons to remote multiple users with the wavelength division multiplexing strategy, and each user feeds partial received photons back to the server. The clock difference between the server and each user is calculated from the one-way and round-trip propagation times, which are determined according to the time correlation of entangled photons. Afterwards, the demonstration has been conducted between the server and a user over a 75-km-long fiber link, where the measured clock difference uncertainty is 4.45 ps, and the time deviation is 426 fs with an average time of 4000 s. Furthermore, the proposed QCS scheme is linearly scalable to many users, with respect to user hardware and number of deployed fibers.

1 Introduction

Quantum networks enable many applications beyond the reach of classical networks [1], such as information-theoretic-security communication [2, 3], high-precision quantum clock synchronization (QCS) [4, 5], distributed quantum computing [6], and distributed quantum sensing [7, 8]. Recently, quantum networks have already stepped into the entanglement distribution network stage [1, 9–14].

Quantum entanglement distribution networks have been successfully achieved by employing the wavelength division multiplexing (WDM) strategies to distribute entangled photons to distant multiple users in multiple degrees [13–17]. Currently, quantum key distribution [15, 16] and quantum secure direct communication [18] have been successfully demonstrated in quantum networks. Additionally, quantum entanglement distribution networks have the potential to implement multi-user QCS, facilitating high-security and high-precision clock synchronization for remote users [19–22]. The QCS scheme usually acquires the high-precision clock difference by calculating the time difference of entangled photons with the references of users’ clocks, which avoids symmetric delay attacks inherently [23, 24]. Furthermore, the security of the QCS scheme can be evaluated by performing the Bell state measurement [22]. Currently, various point-to-point QCS schemes have been successfully demonstrated, including the two-way QCS scheme [23, 25], Hong-Ou-Mandel (HOM) interferometer-based QCS scheme [26, 27], and round-trip QCS scheme [24].

In the two-way QCS scheme, each user (Alice and Bob) has an entangled photon source, the idler photon is detected locally and the signal photon is sent through a fiber and detected on the remote side. The clock difference is calculated by applying an entangled photon cross-correlation algorithm and canceling out the photon propagation time [23, 28]. In 2022, Hong H. et al. demonstrated the two-way QCS scheme over a 50-km-long fiber, achieving a measured clock difference uncertainty of 3 ps and a time deviation (TDEV) of 60 fs with an average time of 25600 s [25]. However, the cost prohibitively will be increased while extending the two-way scheme to multi-user QCS networks.

In the HOM interferometer-based QCS scheme, the third party distributes entangled photons to distant users and performs the HOM interferometer measurement on the photon pairs back from users, whose results are used to control the adjustable fiber delay to ensure the same propagation time of photons toward users’ clocks in real-time transmission [27, 29]. In 2021, Xie M. et al. demonstrated this QCS scheme over a 22-km-long fiber, resulting in a measured clock difference uncertainty of 4 ps and TDEV of 150 fs at an average time of 5500 s [26]. Nevertheless, the critical requirement of equal propagation paths from the third party to users makes this scheme hard to implement in multi-user QCS networks.

In 2022, Lee J. et al. presented a round-trip QCS scheme based on bi-directionally propagating photons generated in a single spontaneous parametric down-conversion (SPDC) source, where the round-trip propagation time can be derived from photons reflected off the end face of the fiber without additional optics [24], resulting a TDEV of 88 ps in 100 s over a 10-km-long fiber. Due to the limited reflection ratio ($\sim 3.5\%$), this scheme has to be improved while extending to long-distance multi-user QCS networks.

In this paper, we propose a multi-user round-trip QCS scheme in the quantum entanglement distribution network, which can be implemented with one single entangled photon source placed at the server. The server distributes the entangled photons to remote multiple users with the WDM strategy, and each user feeds partially received photons back to the server. The clock difference between the server and each user is derived from the one-way and round-trip propagation times, which are determined according to the correlation of entangled photons. Afterward, we demonstrate the multi-user round-trip QCS scheme between the server and a user over a 75-km-long fiber link for approximately 35.6 hours with the frequency dispersion compensation. The experimental results show the clock difference uncertainty is 4.45 ps and the TDEV achieves 426 fs over an average time of 4000 s. Furthermore, our proposed multi-user round-trip QCS scheme is performed with one single entangled photon source and is linearly scalable to many users, with respect to the hardware and deployed fibers.

2 Multi-user round-trip QCS scheme

As the communication infrastructure, the quantum entanglement distribution network has the potential to provide high-precision quantum clock synchronization (QCS) for multiple distant users. In this article, we propose a multi-user round-trip QCS scheme, which can be performed with one single polarization-entangled source and the wavelength division multiplexing (WDM) strategy, illustrated in Fig. 1.

Figure 1

The diagram of the multi-user round-trip QCS scheme in the quantum entanglement distribution network. (a) A demultiplexing equipment separates the entangled signal (idler) photons with the wavelength $\lambda ^{j}_{s}$ and $\lambda ^{j}_{i}$ for user $U_{j}$, $j \in [1,n]$. The signal photons are transmitted to the user, after passing through a round-trip detection (RTD) module. The idler photons are detected in the idler photon detection (IPD) module. (b) RTD module: the input photons pass through an optical circulator (OC) from port 1 to port 2 and the round-trip photons are detected by a detector. (c) IPD module: the entangled idler photons pass through a polarizing beam splitter (PBS), whose output photons $|V_{i}\rangle$ and $|H_{i}\rangle$ are detected, respectively. (d), (e) and (f) show the optical path of user $U_{1}$, $U_{2}$ and $U_{n}$, respectively. The input single photons pass through an OC (from port 2 to port 3) and a PBS, which outputs photons $| H_{s} \rangle$ and $| V_{s} \rangle$. The $| H_{s} \rangle$ photons are detected directly and the $| V_{s} \rangle$ photons are sent back to the server through port 1 to port 2 of the OC. The server’s and user’s detectors are connected to their respective time-to-digital converters (TDC), which are synchronized by their local clocks

In the multi-user QCS network, the polarization-entangled photon source is located at the server, which generates the photons from the nondegenerate spontaneous parametric down-conversion (SPDC) as

$$ |\Phi ^{+}\rangle =\frac{1}{\sqrt{2}} \bigl(| H_{i} H_{s} \rangle + | V_{i} V_{s} \rangle \bigr). $$

(1)

According to the energy conservation during the SPDC process, the frequency ($f_{s}$ and $f_{i}$) of two entangled photons satisfies

$$ f_{p}=f_{s}+f_{i}, $$

(2)

$f_{p}$ is the frequency of pump photons. Therefore, the entangled signal and idler photons can be separated by demultiplexing equipment such as a dense wavelength division multiplexer (DWDM) or a wavelength selective switch.

Assuming the separated wavelengths of the entangled photons are $\lambda _{s} ^{j}$ (signal) and $\lambda _{i} ^{j}$ (idler), $j\in [1,n]$. The entangled signal photons with the wavelength of $\lambda _{s} ^{j}$ are transmitted to user $U_{j}$ through the round-trip detection (RTD) module $\textrm{RTD}_{j}$. The corresponding idler photons (wavelength of $\lambda _{i} ^{j}$) are detected by the idler photon detection (IPD) module $\textrm{IPD}_{j}$. Here, we present the clock synchronization procedure between the user ($U_{1}$) and the server in detail as an example.

For the server, the idler photons with the wavelength of $\lambda _{i} ^{1}$ are transmitted to IPD₁. In IPD₁, the photons are measured by a polarizing beam splitter (PBS), whose output photons $|H_{i} \rangle$ and $|V_{i} \rangle$ are detected by two detectors, respectively. The signal photons with the wavelength of $\lambda _{s} ^{1}$ are transmitted to RTD₁. In RTD₁, the photons pass through the optical circulator (OC) from port 1 to port 2, and the round-trip photons from $U_{1}$ then go through the OC from port 2 to port 3 and are detected by a detector. The outputs of the detectors are all connected to a time-to-digital converter (TDCs) that is synchronized to the server’s clocks.

When user $U_{1}$ receives the entangled signal photons from the server, the photons pass through OC from port 2 to port 3. Subsequently, the photons are measured by a PBS, and the output $|H_{s} \rangle$ photons are detected by a detector. The output $|V_{s} \rangle$ photons are sent back to the server from port 1 to port 2 of OC. The output of $U_{1}$’s detector is connected to TDC1, which is synchronized to $U_{1}$’s clock.

Since the photon pairs emerging from the SPDC are time-correlated [30], the time difference between the signal and idler photons can be determined according to the second-order correlation function $G^{(2)}(\tau )$ [31]. Assuming that the clock difference between $U_{1}$ and the server is $\Delta t_{1}$, the recorded time of photon $|V_{i}\rangle$, $|H_{i}\rangle$, $|V_{s}\rangle$, $|H_{s}\rangle$ is $t_{1}$, $t_{2}$, $t_{3}$, $t_{4}$.

The time difference $\tau _{s-u}$ between $|H_{i}\rangle$ and $|H_{s}\rangle$ contains the photon propagation time $\Delta t_{s-u}$ from the server to the user $U_{1}$ and the clock difference $\Delta t_{1}$, which can be expressed as

$$ \tau _{s-u}=t_{4}-t_{2}=\Delta t_{s-u}+ \Delta t_{1}. $$

(3)

Similarly, the time difference $\Delta t_{s-u-s}$ between $|V_{i}\rangle$ and $|V_{s}\rangle$ contains the photon propagation time from the server to $U_{1}$ and from $U_{1}$ to the server, which can be calculated as

$$ \tau _{s-u-s}=t_{3}-t_{1}=\Delta t_{s-u-s}. $$

(4)

When the asymmetric delay attack [19] is not conducted, the photon propagation time from the server to $U_{1}$ is usually the same as the propagation time from $U_{1}$ to the server. Hence, $\tau _{s-u-s}$ can be represented as

$$ \Delta t _{s-u-s}=2\Delta t_{s-u}. $$

(5)

Therefore, the clock difference $\Delta t_{1}$ can be calculated as

$$ \Delta t_{1}=\tau _{s-u}-\frac{\tau _{s-u-s}}{2}. $$

(6)

The clock difference $\Delta t_{j}$ between user $U_{j}$ and the server can be calculated using the above method, where $j\in [1,n]$. Moreover, the time difference between any pair of users $U_{j}$ and $U_{k}$ can be calculated as

$$ \Delta t_{j-k}=\Delta t_{j} -\Delta t_{k}, $$

(7)

where $j,k \in [1,n]$.

3 Experimental setup

As for the users in our scheme are equivalent, we conducted an experiment demonstration of the proposed scheme over a 75 km fiber with one user and the server. The experimental setup is shown in Fig. 2.

Figure 2

Illustration of the 75 km one-user round-trip QCS experiment. DM: dichroic mirror, HWP: half-wave plate, FPS: fiber polarizing beam splitter, FPC: fiber polarization controller, EPC: electrically driven fiber polarization controller, DCM: dispersion compensation module, DWDM: dense wavelength division multiplexer, SNSPD: superconducting nanowire single-photon detector, RAC: rubidium atomic clock

At the server, a continuous wave polarization-entangled photon pair source is used to generate the entangled photons. The source begins with a continuous wave laser emitting light at a wavelength of 775.06 nm in the vertical polarization state $|V\rangle$. The $|V\rangle$ polarization state is rotated by 45° to become the diagonal polarization state $|D\rangle$ by passing through a half-wave plate. Then, the $|D\rangle$ light is reflected by a dichroic mirror (DM) and directed towards a PBS in a Sagnac loop [32].

The PBS outputs $|H\rangle$ and $|V\rangle$ polarized photons in two different directions, respectively. The $|H\rangle$ polarized photon is rotated to $|V\rangle$ polarized photon by a half-wave plate (HWP) before being injected into a 4 cm-long magnesium oxide-doped periodically poled lithium niobate (MgO:PPLN) crystal. The MgO:PPLN crystal has a poling period of 19.5 μm, which facilitates the SPDC process. The SPDC process generates two entangled photons with orthogonal polarizations: a signal photon $|V_{s}\rangle$ and an idler photon $|V_{i}\rangle$. After the SPDC generation, one direction $|V_{s} \rangle |V_{i} \rangle$ are rotated to $|H_{s} \rangle |H_{i} \rangle$ by passing through the HWP. And the generated photons in two directions are combined at the PBS. The entangled photon pair is separated from the pump 775.06 nm light by the DM, and the central wavelength of the entangled photons is 1550.12 nm, corresponding to the center of International Telecommunication Union’s (ITU, 100 GHz) channel 34.

Then, the signal and entangled idler photons are extracted from ITU C33 (centered at 1549.32 nm) and C35 (centered at 1550.92 nm) by a DWDM. The entangled idler photons first pass through a dispersion compensation module (DCM1), and then are measured by a fiber polarizing beam splitter (FPS1). One output ($| H_{s} \rangle$) of FPS1 is connected to DCM2 and the $| H_{s} \rangle$ photons are detected in channel 1 of a superconducting nanowire single-photon detector (SNSPD). Another output ($| V_{s} \rangle$) photons are detected in channel 2 of SNSPD. The entangled signal photons are sent to the user through the round-trip path with OC1 and OC2, detailly shown in Sect. 2. The user measures the received photons with a fiber polarizing beam splitter (FPS2). The output $| H_{i} \rangle$ photons are detected in channel 4 of SNSPD and the output $| V_{i} \rangle$ photons are fed back to the server through the round-trip path. The server detects the round-trip $| V_{i} \rangle$ photons in channel 3 of SNSPD. And the outputs of SNSPD are connected to a TDC. Furthermore, to compensate for time jitters caused by the crystal oscillator during the photon transmission time, a rubidium atomic clock (RAC) is used to stabilize the TDC clock.

Additionally, three polarization controllers are employed before channels 1, 2, and 4 of the SNSPD, as the detection efficiency of the SNSPD is sensitive to the polarization of photons. To actively compensate for polarization drifting caused by environmental noise, two electrically driven fiber polarization controllers (EPC1 and EPC2) are placed before DCM1 and channel 3 of the SNSPD.

During the experiment, the entangled photon source has a brightness of approximately $3.2 \times 10^{6}$ cps. The insertion loss of both the OC and EPC is about 0.7 dB, and the transmission loss of the 75 km fiber is 15 dB. The SNSPD has a detection efficiency of 80% with a time jitter of 110 ps, and the jitter of TDC is approximately 8 ps. DCM1 and DCM2 are both set to a compensation value of -1230 ps/nm, with an insertion loss of 4 dB.

4 Results and discussion

In the round-trip QCS experiment demonstration, two scenario are considered: one with dispersion compensation (DC) and one without DC. The experiment with DC was carried out for a duration of 35.6 hours, and the experiment without DC was conducted for 28.8 hours. In the experiment without DC, the compensation values of DCMs were set to 0.

During the experiment, correlation analysis is performed based on the arriving timestamps of $| H_{s} \rangle$, $| H_{i} \rangle$, $| V_{s} \rangle$ and $| V_{i} \rangle$, and generate the coincidence count histograms denoted by $G^{(2)}(\tau _{s-u})$ and $G^{(2)}(\tau _{s-u-s})$. The histograms are then Gaussian-fitted to determine the standard deviations and the expectation values of time differences $\tau _{s-u}$ and $\tau _{s-u-s}$.

As an example, Fig. 3 shows the coincidence histograms and the Gaussian-fitted results during the first 100 s experiments. In the experiment without DC, the standard deviations of $G^{(2)}(\tau _{s-u})$ and $G^{(2)}(\tau _{s-u-s})$ are approximately 284 ps and 544 ps, with a total coincidence count of 212.5 kcps and 4.5 kcps, respectively. In contrast, when compensating for the fiber frequency dispersion, the standard deviations are reduced to 174 ps and 201 ps for $G^{(2)}(\tau _{s-u})$ and $G^{(2)}(\tau _{s-u-s})$, respectively, with a total coincidence count of 171.0 kcps and 3.35 kcps. The reduction in the standard deviation indicates a narrower coincidence peak, resulting in the higher precision time difference. However, it should be noted that the increase in the absolute value of DCM results in an increase of the insertion loss, which reduces the coincidence count. In addition, the mathematical expectations of $\tau _{s-u}$ and $\tau _{s-u-s}$ are calculated from the fitted results as $\tau _{s-u}=-359{,}792{,}659$ ps and $\tau _{s-u-s}=-719{,}587{,}252$ ps (with DC) and $\tau _{s-u}=-359{,}816{,}132$ ps and $\tau _{s-u-s}=-719{,}634{,}126$ ps (without DC).

Figure 3

The coincidence count histograms of $G^{(2)}(\tau _{s-u})$ and $G^{(2)}(\tau _{s-u-s})$ in the first 100 s experiment. (a) and (b) [(c) and (d)] show the coincidence histograms $G^{(2)}(\tau _{s-u})$ and $G^{(2)}(\tau _{s-u-s})$ with [without] DC

During the experiment, the time differences $\tau _{s-u}$ and $\tau _{s-u-s}$ are obtained at intervals of 20 s, to ensure the real-time performance of the system. The calculated $\tau _{s-u}$ and $\tau _{s-u-s}$ in the experiment with DC are shown in Fig. 4. $\tau _{s-u}$ exhibited a variation of approximately 3.36 ns, caused by the experimental noise, such as the temperature fluctuations. In contrast, the time difference $\tau _{s-u-s}$ varied twice as fast as $\tau _{s-u}$, with a variation of approximately 6.72 ns during the experiment.

Figure 4

The measured time differences in the experiment with DC. (a) and (b) shows the measured time difference $\tau _{s-u}$ and $\tau _{s-u-s}$ with the interval time of 20 s, respectively

The clock difference Δt is calculated from the time differences $\tau _{s-u}$ and $\tau _{s-u-s}$ using Eq. (6). For the experiments with and without DC, the mean value of the calculated clock differences is 965.54 ps and 939.95 ps, respectively. The non-zero value of Δt is due to the asymmetric transmission times through various system components such as FPS, OC, DCM, PC, EPC, etc. Additionally, the time offset between the two experimental results is attributed to time drifts in the SNSPD, which are affected by the count rate in each channel [33].

In the experiment, the clock difference Δt of each interval is calculated and a clock difference set $S=\{\Delta t_{0},\Delta t_{1},\ldots ,\Delta t_{n-1}\}$ is generated. Then, the standard deviation of these Δt is calculated as 4.45 ps (12.46 ps) for the experiment with (without) DC. The TDEV [34] of the clock difference is calculated as

$$ TDEV(m,S)=\sqrt{\frac{1}{6m^{2}(n-3m+1)}\sum _{j=1}^{n-3m+1} \Biggl( \sum^{m+j-1}_{i=j} \Delta t_{i+2m}-2\Delta t_{i+m}+\Delta t_{i} \Biggr)^{2}}, $$

(8)

m means the averaging time contains m intervals. In the TDEV calculation, the averaging time is from 40 s to 4000 s, and the values obtained are shown in Fig. 5. As the averaging time increases to 4000 s, the TDEV is reduced to 426 fs (933 fs) for the experiment with (without) DC.

Figure 5

The TDEV of the measured clock differences in the round-trip QCS experiment

The experimental results demonstrate the effectiveness of the proposed QCS scheme in achieving high-precision clock synchronization for one user. The same procedure can be applied to synchronize the clocks of other users using other wavelength channels of entangled photons, as the QCS scheme is independent to the wavelengths and optical paths. Moreover, the use of the WDM strategy effectively reduces the spectrum width, leading to high precision even without dispersion compensation, albeit slightly lower than the case with DC.

5 Conclusion

In this paper, we proposed a multi-user round-trip quantum clock synchronization scheme in the quantum entanglement distribution network, which can be implemented with an entangled photon source located at the server. The server distributes the entangled photons to multiple users with the wavelength division multiplexing strategy, and the users feed partial received photons back to the server. Then, the clock difference between the server and each user is calculated from the one-way and round-trip time differences, which are determined according to the time correlation of the entangled photons. In addition, a single-user experiment demonstration was conducted with a 75 km fiber for a duration of 35.6 hours, achieving an uncertainty of 4.45 ps in the measured clock difference and a minimum time deviation of 426 fs at an average time of 4000 s. Furthermore, our proposed scheme can be performed with a single entangled photon source and can scale linearly with respect to the hardware and deployed fibers.