Clock synchronization with pulsed single photon sources

Strong coherent pulses are very straightforward to use for timing applications (figure 1(a)). Every pulse consists of a large number of photons which simplifies detecting the rising edge of the pulse to give it some timing. On the contrary, weak coherent pulses have their intensity level attenuated to the single-photon level (figure 1(b)). As a consequence, the location of the single photon becomes uncertain as the arrival time is described by a probability distribution. This simple comparison already tells us that single photons are not easy to handle for time transfer due to an increased uncertainty.

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Quantum communication systems that are based on correlated photon pair sources that are driven by a continuous-wave pump have some practical downsides, as the random photon arrival time from the source cannot be used directly as frequency carrier. The source operates entirely passively and the system requires substantial communication between the receivers to find the interdependent clock frequency and timing offset (figure 2(a)). To be more precise, the receivers process their coincident correlation events of the arriving photons. This dependency has the consequence that there is no way to find the absolute timing offset if the clock frequencies of the receivers were much different in low signal scenarios. A reliable method to still match the clock frequencies between receivers is to vary the frequency of one receiver that increases the signal-to-noise ratio [23]. However, this method is very demanding with respect to computation power, as it requires several Fourier-based cross-correlations with large array sizes.

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This changes when the pump for photon-pair generation is pulsed, as the arrival time of the photon pairs is no longer random. This method allows one to synchronize the clock frequency at the receiver with a common master clock on the source side (figure 2(b)). The procedure has two important consequences:

• (i)  
  The clock frequency at the receiver is derived independently of the other receiver, which means that it is only necessary to process the arrival time of photons locally at one receiver. This avoids the use of the much lower number of coincident correlation events, which increases the signal-to-noise ratio significantly.
• (ii)  
  The clock frequency is found without the necessity of an absolute timing offset. This breaks the interdependence of timing offset and clock frequency that exists in sources driven by a continuous-wave pump. In consequence, it drastically increases the efficiency of the algorithm for synchronization.

Quantum communication setups based on weak coherent sources have a similar synchronization character to schemes that rely on pulsed photon pairs. As with pulsed-photon pair sources, the clock frequency can be derived first and the absolute timing offset second (figure 2(c)). The upcoming sections describe the underlying methods in more detail with the example of weak coherent sources.

Our exemplary experimental setup consists of a single-photon source of faint pulses that transmits the timing and frequency information to our receiver. We consider this to be an add-on to the decoy-state BB84 quantum key distribution protocol (figure 3(a)). The source consists of an arbitrary waveform generator (with the intrinsic clock Alice) to transfer the digital information bits to an analog waveform that drives an amplitude and phase modulator and generates a typical sequence for time-bin encoded quantum key distribution. This is in the time basis the states $|{\mathrm{Early}}\rangle$/$|{\mathrm{Late}}\rangle$ and in the superposition basis $|{+}\rangle$ (the phase difference is zero) and $|{\mathrm{-}}\rangle$ (the phase difference is π). The signal is carried to the receiver via approximately 10 m of optical fiber. As the setup is in an air-conditioned laboratory environment, we expect the fiber to introduce a time deviation smaller than 1 ps at longer averaging times of larger 10 s (appendix A.2). The impact on short averaging times is minor. The receiver consists of nanowire single-photon detectors and an analog-to-digital converter that generates time stamps according to its internal clock Bob. The timing jitter budget is summarized in table 1. The clocks can be synchronized through a 10 MHz signal carried by an RF cable—this will provide our reference measurement (ground truth). The clocks of both the source and the receiver are free-running quartz oscillators. In consequence, achieving high stability is challenging due to the clock's

• (i)  
  low precision, because of the considerable frequency differences that complicate the initialization of the synchronization (addressed in section 2.3) and,
• (ii)  
  low stability that comes with strong time-dependent changes of their frequency. Subsequently, these will affect the synchronization during the communication session. This is addressed by fast frequency update times in section 2.4).

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The source repetition rate acts as a mediator of the clock frequency and becomes part of the initialization of the synchronization. The underlying assumption is that the repetition rate is locked to the sender clock frequency and acts as a derivative. The timing offset between the source and receiver also serves as an indicator of the clock frequency and is exploited during the communication session for frequency tracking and update purposes.

The initialization of the synchronization requires first the processing of the arrival times of photons at the receiver. This is accomplished by correlating the arrival time to the time that a photon has been sent out. Such processing algorithms are commonly based on the start-stop method with a summary in histograms or based on a full cross-correlation through Fourier transforms [23]. In this method, every time of the Nth transmitted photon is compared with all Mth received photons. When zero-padding the shorter array, the array size is $n = \mathrm{max}(N, M)$. This gives rise to a computational complexity of $\mathcal{O}(n^2)$ for a standard Fourier transformation and $\mathcal{O}(n \log_2 n)$ for the fast Fourier transformation [48]. The computational cost can be reduced further to $\mathcal{O}(n \log_2(\log_2 n))$ at the expense of synchronization strings [31]. Although our algorithm uses a single fast Fourier transformation to find the absolute timing offset during initialization (complexity $\mathcal{O}(n \log_2 n)$), all the other steps (including frequency locking) use the much more efficient modulo operator. The operator allocates the photon arrival time t_i within a time window $1/f_\textrm c$, given by the source clock rate $f_\textrm c$. This reduces the complexity to only the number of photons received with computational cost $\mathcal{O}(n)$, since the modulo operation applied to a single number is a $\mathcal{O}(1)$ operation (it is just the remainder after division).

At the beginning of the initialization of the synchronization, the source clock rate is just an estimate value because the exact source clock rate is not known due to the low accuracy and stability of the clock. Such an estimate could be the clock rate set in the protocol (e.g. 500  MHz). The photon's arrival time T_i within the time window is calculated through the modulo operation (mod) as

$$\begin{equation} T_i = \bmod_{1/f_\textrm c} \left( t_i \right).\end{equation} \tag{ 1 }$$

The arrival time statistics $c(\tau)$ are given by

$$\begin{equation} c\left(\tau\right) = \mathrm{Histogram}\left[T_i,\tau\right].\end{equation} \tag{ 2 }$$

The nature of this operation provides only a relative timing offset within the source clock rate window ($0 \ldots 1/f_\textrm c$). The absolute timing offset is derived by comparing sent and received data bits in a subsequent step.

The modulo operation is especially beneficial under low signal conditions due to its lower computational complexity (see appendix A.1). Let us consider a typical scenario of long-distance quantum communication with high losses on the link. As a consequence, the signal-to-noise ratio is already low. If network nodes are not equipped with high-performance clocks, this will further reduce the signal-to-noise ratio, so that correlation features can no longer be observed. In such cases, it is beneficial to vary the receiver clock frequency until there is a closer frequency match between the sender and receiver clocks [23, 37]. In practice, this is performed through time tag post-processing (to vary the source clock rate in equation (1)) or with hardware modification of the clock frequency. Both options allow for an improvement of the signal-to-noise ratio and allow the initialization of the synchronization (figure 3(b)). However, for the frequency sweep, several correlation calculations are required, which increases the computation time. For example, continuous wave sources need computationally heavy Fourier-based methods [23], which makes initialization under low signal conditions infeasible. On the other hand, we propose to use the modulo operator in pulsed sources to allow for much reduced total computation time, making pulsed sources superior in low-signal scenarios.

Finding the absolute timing offset between the sender and receiver is the result of a two-stage process. It comprises

• (i)  
  identifying the relative timing offset ($0 \ldots 1/f_\textrm c$) via the modulo operator (in initialization and during the communication session), and
• (ii)  
  measuring the absolute timing offset by comparing the sent bit sequence with the received sequence (multiple integers of $1/f_\textrm c$, in initialization only).

The relative timing offset is a by-product of the frequency search (see figures 3(b) and (c))—the absolute timing offset is obtained by comparing the sent and received symbols (data bits). Note that the data bits act as a synchronization string during initialization, similar to publication [31]. In contrast, we use the actual data bits for quantum key distribution a single time instead of adding an additional sequence. In our emulated quantum key distribution test environment, we encode 1000 symbols in the arbitrary waveform generator with a clock rate of 500 MHz. This sequence of 1000 data bits repeats periodically. We shift the received symbols by multiples of the source clock rate until they match the sent bit sequence. At this point, the quantum bit error rate is lowest with its corresponding absolute timing offset (see figure 3(d)).

The method of finding the absolute timing offset neither poses a security risk nor compromises the effective key rate. The absolute value of the timing offset is determined in a single step during the initialization. The bits revealed to access the error rate in this step are not safe to use for the secure key. However, no single bit of information is shared after the initialization is performed, as all processing to find the relative timing offset is performed locally. As a consequence, it does not affect the secure key rate during the communication session. The correctness of the absolute timing offset is continuously verified by a low quantum bit error rate, which is already part of the parameter estimation in the protocol for error correction [49].

The first section describes how to determine the absolute timing offset and how to match the clock frequencies between senders and receivers. As the clock frequency drifts in the following communication session, the timing uncertainty would increase without any compensation applied. We avoid this by introducing our frequency update algorithm. A useful indicator of current frequency differences $\Delta u$ is the change of timing offset $\delta T$ after one feedback loop of duration $T_\mathrm{f}$ in the tracking algorithm,

$$\begin{equation} \Delta u = \frac{\delta T}{T_\mathrm{f}}.\end{equation} \tag{ 3 }$$

Note that $\delta T \unicode{x2A7D} 1/f_\textrm c$ should be fulfilled when using the modulo operator, since the timing offset observation window is limited to the current symbol with a range of $0 \dots 1/f_\textrm c$. For visualization, let us take our example with a symbol range of $0 \dots 2\,$ns (500 MHz). When the timing offset changes by 2.5 ns, the correlation peak slips into the neighboring symbol range, but the modulo operator would just provide an incorrect (relative) offset of 0.5 ns. As a consequence, the error rate increases to 50%. In such cases, we recommend searching in the ranges of neighboring symbols to recover synchronization. This is done by changing the timing offset by, for example, ${\pm}1$ or ${\pm} 2$ of the repetition rate and checking the corresponding error rate. In our example, adding 2 ns works for recovery. To avoid all this, it is useful to reduce the data acquisition time or to choose shorter feedback loops for frequency adjustments to avoid the correlation features from leaving the observation window.

The frequency update algorithm records timing offsets and updates the receiver clock frequency after a given feedback loop time (figure 4(a)). Upon receiving new data, the software calculates preliminary arrival-time statistics over the modulo operator (i.e. performs a cross-correlation). The calculation includes the last clock frequency and returns an a priori timing jitter (from a Gaussian fit), estimated from all previous clock frequencies, but without including the current measurement. On the contrary, the a posteriori timing jitter is an estimate of all previous clock frequencies, including the current measurement. As the clock frequency changes, the predicted timing offset (estimated with the clock frequency of the last data sets) deviates by $\delta T$ (figures 4(b) and (c)). This difference between predicted and actual values produces an updated clock frequency, as described in equation (3) (figure 4(d)). With the updated clock frequency, the algorithm calculates an a posteriori cross-correlation that results in a smaller timing jitter (a posteriori timing jitter) as the clock frequencies between sender and receiver match more closely (figure 5). The applied clock frequency for the a priori cross-correlation is always outdated by the update time, as it originates from the last data set. This time delay by the update time becomes apparent in the difference in synchronization jitter between the a priori and a posteriori correlations at fast update times. The correction function enables extremely small a posteriori timing jitters, as the clock frequency of the current data set is applied. We determine the stability from the cumulative timing offset deviation that updates the clock frequency. Hence, the stability refers to an outdated clock frequency from the last data sets. This means that our estimated timing stability is related to an upper bound. In contrast, the a posteriori timing jitter includes the current clock frequency and may reach near-ideal performance. It is very suitable for quantum communication, which aims to obtain the lowest timing jitters for noise reduction and high data rates.

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The timing jitter due to poor synchronization strongly depends on the update time of the receiver clock frequency. The residual synchronization timing jitter is caused by a weak frequency stability that accelerates the clock. In a feedback algorithm, we track the instantaneous clock frequency difference and compensate for it at the receiver. We measure the average clock stability in several 5 min sessions. Here, we find an almost linear relationship between the synchronization jitter and feedback loop time (figure 5(b)). The synchronization timing jitter $\sigma_\mathrm{sync}$ is derived from the measured mean timing jitter σ and the reference timing jitter σ₀ as

$$\begin{equation} \sigma_\mathrm{sync} = \sqrt{\sigma^2 - \sigma_0^2}.\end{equation} \tag{ 4 }$$

We establish the reference (ground truth) for the timing jitter through an RF cable between the source and the receiver that transfers a 10 MHz clock signal. We take this reference measurement just a few minutes before the actual measurement to calibrate the system. The single-photon count rate is constant and amounts to $270\pm20\,$kcps (correlation rate $160\,\pm20$ kcps). This corresponds to a mean photon number of $5.4\times10^{-4}$ per pulse. Taking into account the fastest clock frequency update time of 150 ms, the a posteriori correction timing jitter increases by only 110 fs, indicating average synchronization timing jitters of 3.0 ps.

The stability criterion of the algorithm is the time offset deviation that describes the change in clock frequency over the update time. The a posteriori timing jitter gives much better stability measures, because the current clock frequency is from the immediate data set. However, here we use the timing offset as an indicator for a better comparison with the literature. Figure 6(a) shows the variation in timing offset for different types of experimental setups:

• 1.  
  Internal quartz oscillators without tracking of the clock frequencies,
• 2.  
  Internal quartz oscillators with tracking of the clock frequencies,
• 3.  
  Free-running rubidium clocks at the source and receiver, and
• 4.  
  The reference with an RF cable for clock transfer between the two quartz oscillators at the source and receiver.

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The high stability of rubidium oscillators results in particular small timing offset variations, in contrast to quartz oscillators without our applied frequency update algorithm. We summarize the timing offset variations in terms of the Allan time deviation [44] which describes the average standard deviation of the timing offset for varying the averaging times (figure 6(b)). Note that tracking the frequency of the quartz oscillators increases the stability by more than two orders of magnitude at averaging times of 100 s and approaches the stability of rubidium clocks. The timing stability with quartz oscillators is shown in table 2. In conclusion, fast update times for clock frequency adjustments reduce timing jitter caused by poor-performing clocks and improve overall synchronization stability.

The computation time on our computer (table 3) is very low with cross-correlations based on the modulo operator (equation (1)). It takes about 37 ms to calculate a cross-correlation with 10⁷ time tags at the receiver (figure 7(a)). This enables high-resolution clock-frequency sweeps during initialization of the synchronization procedure down to smaller than 0.2 ns s⁻¹ (figure 7(a)) at 2 h of computation time. In comparison, fast-Fourier transformations require more than 30 s for the same array size and limit the clock frequency resolution to 140 ns s⁻¹ for continuous wave sources [23]. Strengthening a poor signal from large clock frequency differences is the basis of noise resistance in pulsed single-photon sources and provides the feasibility for synchronization even in very low signal regimes with coincidence-to-accidental ratios $\lt$5 [13] (figure 7(c)).

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The significance of the correlation peak S describes the ratio of the signal (i.e. the number of correlations) to the standard deviation of the noisy background. As an example, let us imagine a significance of 3 for the correlation peak. This refers to a confidence of 99.7% that this is really the correlation peak and not just noise. The significance has the following relation [7],

$$\begin{equation} S = \sqrt{\frac{r_C^2}{r_A r_B \Delta u}}, \end{equation} \tag{ 5 }$$

with the single rate of receiver A (or sender) and receiver B, $r_A, r_B$, correlation rate r_C and residual clock frequency difference $\Delta u$.

The optical fiber changes its refractive index and geometry due to temperature. The phase change can be described by a geometric change in the fiber length L and its refraction index n with temperature T for the wavelength λ [54],

$$\begin{equation} \Delta \phi = \frac{2\pi}{\lambda} \left( n \frac{\partial L}{\partial T} + L \frac{\partial n}{\partial T}\right) \Delta T.\end{equation} \tag{ 6 }$$

The thermo-optic coefficient $\epsilon = \partial n / \partial T = 11\times10^{-6}\,\mathrm{K}^{-1}$ is at least one order of magnitude higher than the thermal expansion coefficient $\alpha = \partial L / \partial T = 0.55\times10^{-6}\,\mathrm{K}^{-1}$ for the core of silica glass [55, 56]. When applying the equation (6) to 10 m of fiber, a change of temperature by $\Delta T = 1\,$K gives rise to a time delay of 0.37 ps at 1550 nm. As the temperature changes slowly in our air-conditioned lab, it affects the timing offset stability slightly at longer averaging times ($\gt$10 s) when it is not compensated.

Allan and Barnes [57] describes that the modified Allan deviation (MDEV) is useful for characterizing noise sources in oscillators or time transfer systems. However, the weighting in the MDEV gives lower stability outcomes that might be misleading [58]. For this reason, we also give a representation of our results in the overlapping Allan deviation and the Allan time deviation (TDEV) in figure 8.

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