# Clock synchronization with pulsed single photon sources

Christopher Spiess<EMAIL_ADDRESS>Friedrich Schiller
University Jena, Fürstengraben 1, 07743 Jena  Fraunhofer Institute for Applied
Optics and Precision Engineering, Albert-Einstein-Strasse 7, 07745 Jena
Fabian Steinlechner<EMAIL_ADDRESS>Fraunhofer
Institute for Applied Optics and Precision Engineering, Albert-Einstein-
Strasse 7, 07745 Jena  Friedrich Schiller University Jena, Abbe Center of
Photonics, Albert-Einstein-Strasse 6, 07745 Jena

###### Abstract

Photonic quantum technology requires precise, time-resolved identification of
photodetection events. In distributed quantum networks with spatially
separated and drifting time references, achieving high precision is
particularly challenging. Here we build on recent advances of using single-
photons for time transfer and employ and quantify a fast postprocessing scheme
designed to pulsed single-photon sources. We achieve an average root mean
square synchronization jitter of 3.0 ps. The stability is comparable to
systems with Rb vapor cell clocks with 19 ps at 1 second integration time, in
terms of Allan time deviation. Remarkably, our stability is even better than
classical high-precision time transfer, like the White Rabbit protocol,
although we use significantly less signal (single-photon level). Our
algorithms allow local processing of the data and do not affect the secure key
rate. It compensates substantial clock imperfections from crystal oscillators
and we foresee great potential for low signal scenarios. The findings are
naturally suited to quantum communication networks and provide simultaneous
time transfer without adding hardware or modifying the single-photon sources.

Quantum Cryptography, Quantum Key Distribution, Entanglement, Clock
Synchronization, Time Transfer

††preprint: APS/123-QED††preprint: AIP/123-QED

##  I Introduction

The synchronization of remote clocks plays a crucial role in communication
networks [1, 2, 3]. To this end, the network time protocol [4] is widely used
for the synchronization of computers in communication networks, with a typical
frequency stability of the order of 1E-7. This can be further enhanced on the
order of 1E-19 using optical timing transfer technologies [5, 6]. Such levels
of stability are crucial in emerging quantum communication networks. In this
context, clock synchronization methods that use single photons to transfer
timing information have been proposed and implemented [7, 8]. The critical key
benefit of these approaches is not necessarily that they improve performance
over their classical counterparts. Instead, they avoid the need for additional
hardware, such as rubidium clocks [9, 10, 11], a receiver for the global
navigation satellite system [12, 13], or auxiliary synchronization lasers [14,
15, 16, 17, 18, 19] in existing quantum communication networks. A crucial
aspect for the synchronization method is the photon source operation mode that
can be continuous wave / asynchronous or pulsed / clocked [20, 21]. A notable
example is a spontaneous parametric down-conversion (SPDC) source driven by a
continuous-wave pump [22, 17]. Such sources can be realized with low effort
using commercial off-the-shelf components and achieve very high quantum-state
transmission rates. However, the random emission and arrival times of the
photons complicate the time synchronization [23].

In contrast, pulsed single-photon sources confine the photon arrival time to
an envelope given by the pump, by providing an additional time reference. This
greatly simplifies synchronization, as its shown impressively for entanglement
swapping [24] or quantum teleportation [25, 26, 27] that require high-quality
interference. In this paper, we focus on (clocked) weak coherent pulse sources
[19, 28, 29, 30]. The pulse train acts as a structure to transfer clock
frequencies from the source to the receiver. A widlely employed approach for
matching the clock frequency of source and receiver is to perform an analysis
of the arrival time of photons in the frequency domain [31, 32]. On an optical
link with constant losses, a higher frequency of arriving photons corresponds
to a faster clock of the sender. It is easy to match the clock frequencies. On
the other hand, when there are high losses on the link, the received photon
detection rate is much lower than the clock rate of the sender. In such cases,
it is difficult to extrapolate a much higher sender clock frequency from a
low-frequency receiver signal. In other words, the signal-to-noise ratio
reduces with strong link losses and limits this approach to short-distance
links with small losses. A possible solution is to use dedicated
synchronization patterns to improve the signal-to-noise ratio [31, 33, 34], as
shown in [35, 32]. The authors show a remarkably low standard deviation
(1$\sigma$) of the timing offset variation of approximately 500 ps (estimated
from maximum deviation 3$\sigma$ = 1.5 ns, integration time 1 second).
However, since these patterns are part of the single-photon stream, this
method requires modifications to the source software. The synchronization
pattern reserves a selected period of time that could be used to transfer data
bits otherwise. In consequence, the effective data rate reduces.

In this work, we extend upon initial demonstrations of correlation-based clock
synchronization [8] that require neither hardware (sender and receiver) nor
software modifications of the single-photon source [36, 37]. Our overarching
goal is to reduce the need for additional synchronization hardware in quantum
communication scenarios and show useful ideas for classical time transfer. In
particular, we provide a comprehensive summary of a clock frequency tracking
algorithm, analyze its performance, and compare it to other synchronization
techniques. The clock frequencies are matched by comparing the sender source
clock rate with the received photon arrival time. Specifically, we propose a
dedicated algorithm that is uniquely suited to pulsed sources. The algorithm
results in a substantial reduction in computation effort and allows the
feasibility of low-signal scenarios, such as long-distance links [13]. In this
work, we derive the absolute timing offset by comparing the actual transmitted
and received data streams once during the initialization of the
synchronization in the BB84 quantum key distribution protocol [38]. After
initialization, this data exchange is no longer required in normal operation
mode, as we just analyze the arrival time of the photons within a given
window. As a consequence, unlike synchronization strings, this does not
compromise the effective data rate.

We implement the synchronization protocol using weak coherent pulses as an
exemplary single-photon source. In doing so, we ensure average synchronization
timing jitters of 3.0 ps in a 5-min communication session by actively tracking
the arrival time statistics, similar to the correlation peaks of photon pairs
[7, 8]. The time deviation of the residual timing offset variation is only 19
ps at 1 second integration time which is more than one order of magnitude
lower than previously reported [32] and comparable to systems employing Rb
vapor cell clocks [39, 40]. Even remarkably, although we just use single
photons, the time deviation is better than with the White Rabbit protocol that
uses standard strong optical signal [41]. Our post-processing scheme
represents a simple addition to arbitrary single photon sources, which
increases the scaling ability of large quantum communication networks [42] and
has potential application in secure time transfer and synchronization [43].

We structure this work by first comparing our algorithm for pulsed single-
photon sources to continuous-wave driven single-photon sources as motivation
(Section II.1). Second, we describe the experimental setup (Section II.2) and
the algorithms for initialization of the synchronization (Section II.3). In
the following, we explain how to keep the time reference locked during the
communication session (Section II.4). Finally, we evaluate and discuss the
performance parameters, synchronization timing jitter, and timing stability in
terms of the Allan time deviation.

##  II Results

As a summary of our results, we characterize the timing stability of our
algorithms in terms of the modified Allan time deviation (MDEV). From the
residual variations of the timing offset, we determine the Allan timing
stability (TDEV) to 2.8 (3.2) ps in a 30 (100) second integration time $\tau$,
via TDEV$=(\tau/\sqrt{3})\,$MDEV [44]. It is much lower than recently
published values using ultra-stable clocks: 38.1 ps at 30 s integration time
with a Rb vapor cell clock at site A and H-maser at site B [40] and 88 ps at
100 s integration time with two Rb vapor cell clocks [39]. We show feasibility
with standard crystal oscillators that have 8 orders of magnitude worse
performance than atomic references [45, 46] and still achieve timing jitters
comparable to links deployed with GPS-referenced systems [13, 10, 47].

###  II.1 Synchronization schemes with single photon sources

Strong coherent pulses are very straightforward to use for timing applications
(Fig. 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 (Fig. 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.

Figure 1: Application of coherent and weak coherent pulses for time transfer.
(a) It is easy to extract the rising edge from coherent pulses with high
accuracy, due to the large number of photons. The photons are distributed
according to the pulse envelope. (b) Weak coherent pulses are attenuated
strongly until they have a mean photon number $<1$. The arrival time of the
photon is described by the pulse envelope that serves as a probability
distribution. The precise time is uncertain by the pulse width.

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 can not 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 (Fig. 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.

Figure 2: Types of clock synchronization schemes with single photons in a
quantum communication environment. (a) Continuous-wave (CW) driven and passive
entangled photon sources (EPS) act as mediator. The two receivers (A, B)
communicate the arrival time of photons to find correlation events. In the
following, they first reduce their frequency difference
$\mathrm{f}_{\mathrm{A}}-\mathrm{f}_{\mathrm{B}}$ and second minimize their
absolute time difference $\mathrm{t}_{\mathrm{A}}-\mathrm{t}_{\mathrm{B}}$.
(b) When pulsing the pump, the receivers have the option to synchronize their
individual clock frequencies to the sender’s clock with frequency
$\mathrm{f}_{\mathrm{0}}$ through the source clock rate. This procedure does
not require any communication between the receiver. They just locally process
the arrival times of the photons. In a second step, the receivers communicate
their arrival times to find the absolute timing offset. (c) The
synchronization scheme for sources that base on weak coherent pulses (WCS) is
almost identical to (b). The difference is that the clock frequency and timing
offset of the receiver is matched to the master clock.

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 (Fig. 2(b)). The procedure has two important consequences:

  1. 1.

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.

  2. 2.

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 (Fig. 2(c)). The upcoming sections describe the
underlying methods in more detail with the example of weak coherent sources.

###  II.2 Experimental setup

Figure 3: Synchronization of clocks with single photons from pulsed sources in an exemplary quantum key distribution system. (a) Typical example of pulsed single photon sources are setups based on weak coherent pulses. The amplitude modulator generates optical pulses in bins called early, late or both. Single photon detectors (SPD) measure the arrival time after attenuation to single photon level. Upon the arrival of a photon, they create sharp electric pulses that allow triggering of time-to digital converters. When triggered, they generate time stamps according to its internal clock for further processing on a computer. The clocks of the sender and receiver can be either rubidium clocks (1) or quartz oscillators (2). The reference measurement is established via an RF cable (3) that transfers a 10 MHz clock signal for classical synchronization. (b) Matching clock frequencies between sender and receiver results in highest signal-to-noise ratios of the arrival time statistics in the source clock rate window. (c) The peaks reduce in strength for not compensated clock frequency difference. Note that the clock frequency difference of 5 ns/s results in a full-width correlation peak spread of approximately 500 ps over 100 ms integration time and overlaps with the neighboring correlation peak. The relative timing offset within the source clock rate window is derived as well. (d) Finding the absolute timing offset with pulsed single photon sources. The cross-correlation between received and sent data indicates a correlation dip in the error rate. When received patterns are equal to the sent patterns, the error is smallest. Analog-to-digital converter |  Single-photon source |  Single-photon detector |  Total
---|---|---|---
3 ps |  37 ps |  13 ps |  39 ps
Table 1: RMS timing jitter budget of the experimental setup.

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 (Fig. 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 $\ket{\mathrm{Early}}$/$\ket{\mathrm{Late}}$ and in
the superposition basis $\ket{+}$ (the phase difference is zero) and
$\ket{\mathrm{-}}$ (the phase difference is $\pi$). 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 V.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

  1. 1.

low precision, because of the considerable frequency differences that
complicate the initialization of the synchronization (addressed in Section
II.3) and,

  2. 2.

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 II.4).

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.

###  II.3 Initialization of the quantum communication session

Figure 4: Workflow for tracking the sender clock frequency (a) The timing
offset is stored continuously in a buffer. When reaching the clock frequency
update time, the software reads out the buffer and updates the clock
frequency. We call the timing jitter before and after such a clock frequency
update the a priori and a posteriori timing jitter, respectively. Note that
the a priori cross-correlation is seeded by the clock-frequency from the
previous data set. The a posteriori cross-correlation function receives the
clock frequency of the current data set. (b) Any deviation of the predicted
timing offset from the measured value serves as error signal (c). This
provides an updated correction value for the clock frequency at the receiver
(clock frequency difference) (d).

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 $N^{\mathrm{th}}$ transmitted photon
is compared with all $M^{\mathrm{th}}$ 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 $1/f_{c}$, given by the source clock
rate $f_{c}$. 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

$T_{i}=\bmod_{1/f_{c}}(t_{i}).$ (1)

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

$c(\tau)=\mathrm{Histogram}\left[T_{i},\tau\right].$ (2)

The nature of this operation provides only a relative timing offset within the
source clock rate window (0 … $1/f_{c}$). 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 V.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 [37, 23]. 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
(Fig. 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

  1. 1.

identifying the relative timing offset (0 … $1/f_{c}$) via the modulo operator
(in initialization and during the communication session), and

  2. 2.

measuring the absolute timing offset by comparing the sent bit sequence with
the received sequence (multiple integers of $1/f_{c}$, in initialization
only).

The relative timing offset is a by-product of the frequency search (see Fig.
3(b,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 Fig. 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].

###  II.4 Live tracking during the quantum communication session

Figure 5: Tracking of the sender clock frequency to enable lowest timing
jitters with quartz oscillator at sender and receiver. (a) Correlation peak
live tracking with clock frequency update times of 150 ms. The total timing
jitter root-mean squared during a 5-min measurement session is depicted with
its corresponding relative occurrence probability in the inset. (b) The
tracked and compensated clock frequency difference quickly accumulates to more
than 40 ns/s. Although the frequency difference changes a lot over the first
minute, our algorithm applies compensation values reliably and returns low
timing jitters. (c) We compare the measured mean root-mean-squared (RMS)
timing jitter to the reference of an ideal clock frequency transfer through an
RF cable between source and receiver. The measured synchronization jitter
(Equation 4) increases with the clock frequency update time almost linearly
(see dotted lines for guidance). The a priori correction timing jitters refer
to applied clocks frequency differences of the previous data set, i.e. the
last update (see Fig. 4). A posteriori correction refers to timing jitters
after application of an updated clock frequency from the current data set.
Every data point shows the mean timing jitter in every 5-min measurement
session. The clock rate of the sender is 500 MHz. The receiver detects signal
with count rates of $270\pm 20$ kcps and correlation rate of $160\,\pm 20$
kcps with correlation window of 39 ps (equal to average RMS timing jitter). We
achieve RMS synchronization jitters of as small as 3.0 ps with an update time
of approximately 150 ms. The acquisition time amounts to 100 ms.

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,

$\Delta u=\frac{\delta T}{T_{\mathrm{f}}}.$ (3)

Note that $\delta T\leq 1/f_{c}$ should be fulfilled when using the modulo
operator, since the timing offset observation window is limited to the current
symbol with a range of $0\dots 1/f_{c}$. 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.

Figure 6: Timing stability measurements. (a) Timing offset change for
different experimental setups. (1.) Quartz oscillators at source and receiver
without updating the clock frequencies (linear component of 15.8 ns/s
subtracted after fit). (2.) Quartz oscillators at source and receiver with
tracking of the clock frequencies, denoted with (+). (3.) Free-running
rubidium (Rb) clocks at source and receiver with stability specifications
(MDEV) of each being 3E-12 at 1 s, 3E-12 at 10 s and 1E-12 at 100 s. (4.) An
RF-cable provides the 10 MHz reference output of the source to the receiver.
(b) Modified Allan deviation [44], for varying averaging times. For reference,
we included the Allan time deviation (TDEV) in the figure as dotted grey lines
(see Table 2 for the TDEV of case (2.) that is derived from the MDEV). The
measurements are fit by a linear function to find the noise source that each
follow a different power law ($\tau^{0}$ Flicker FM, $\tau^{-1}$ Flicker PM,
$\tau^{-3/2}$ White PM). Note that this stability describes the timing offset
uncertainty of the upcoming data set after wrong prediction of the clock
frequency (a priori correction). It is directly proportional to the clock
instability during the clock update time. TDEV from literature: Lee (two Rb
clocks) [39] and Quan ((i) Rb clock and H-maser only, (ii) additional
microwave frequency transfer for increased stability) [40] and the White
Rabbit protocol [41]. The TDEV and overlapping Allan deviation is graphically
represented in Appendix V.3.

The frequency update algorithm records timing offsets and updates the receiver
clock frequency after a given feedback loop time (Fig. 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 $\delta
T$ (Fig. 4 (b-c)). This difference between predicted and actual values
produces an updated clock frequency, as described in Equation 3 (Fig. 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 (Fig. 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.

$\tau$ (s) | 0.15 | 0.2 | 0.4 | 1 | 2 | 4 | 10 | 20 | 30 | 40 | 100
---|---|---|---|---|---|---|---|---|---|---|---
TDEV (ps) | 54 | 46 | 32 | 19 | 13 | 9.5 | 6.0 | 3.5 | 2.8 | 1.9 | 3.2
MDEV | 630 | 403 | 138 | 34 | 11 | 4.1 | 1.0 | 0.31 | 0.16 | 0.08 | 0.06
Table 2: Time deviation (TDEV) and modified Allan deviation (MDEV
$\times$1E-12) [44] for varying averaging times $\tau$ with frequency tracked
quartz crystal oscillators. The values are close to what is reported in the
literature with highly stable clocks - 88 ps at 100 s averaging time [39] (two
rubidium clocks), 38.1 ps at 30 s averaging time [40] (rubidium clock and
H-maser). When comparing our results to classical synchronization schemes, the
overall time deviation is identical to the White Rabbit protocol (WR: 11 ps at
1 s and 2 ps at 30 s [50]) or even smaller (WR: 45 ps at 1 s, 15 ps at 10 s
and 6 ps at 100 s [41]). The TDEV and overlapping Allan deviation is
graphically represented in Appendix V.3.

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 (Fig. 5(b)). The
synchronization timing jitter $\sigma_{\mathrm{sync}}$ is derived from the
measured mean timing jitter $\sigma$ and the reference timing jitter
$\sigma_{0}$ as

$\sigma_{\mathrm{sync}}=\sqrt{\sigma^{2}-\sigma_{0}^{2}}.$ (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 $270\pm 20\,$kcps (correlation rate $160\,\pm 20$
kcps). This corresponds to a mean photon number of $5.4\times 10^{-4}$ 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. 1.

internal quartz oscillators without tracking of the clock frequencies,

  2. 2.

internal quartz oscillators with tracking of the clock frequencies,

  3. 3.

free-running rubidium clocks at the source and receiver, and

  4. 4.

the reference with an RF cable for clock transfer between the two quartz
oscillators at the source and receiver.

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 (Fig. 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.

##  III Discussion

Our cross-correlation approach for pulsed single-photon sources is superior
for synchronizing distant clocks compared to continuous-wave sources. The
above becomes computationally more efficient than Fourier-based methods,
because of the following reasons:

  1. 1.

The modulo operation is intrinsically more computational power saving. The
complexity depends only on the number of time stamps at a single receiver and
the observation window is only $0\dots 1/f_{c}$, instead of over the full
integration time.

  2. 2.

The cross-correlation time is independent of the desired frequency resolution.
It depends only on the number of time stamps to be processed and uses the
provided digital resolution of the analog-to-digital converter of 1 ps. In
continuous-wave sources, the resolution is reduced by binning with $N$ number
of bins. This is related to the frequency resolution as $1/N$ [23, 7] .

  3. 3.

All processing is performed locally after finding the absolute timing offset
once during initialization. This reduces the time to update the frequency in
the live session.

After obtaining the relative timing offset with the modulo operator, the
absolute timing offset is later derived by matching the sent sequences with
the received sequences. Our live-tracking algorithms continuously align the
clock frequencies as they strongly drift during long-term sessions. Here, we
show synchronization timing jitters of 3.0 ps with 150 ms frequency update
times that are smaller than previously reported 30-50 ps from free-running
rubidium clock-based setups without time transfer with single photons [13,
10].

Low detector timing jitters, high signal-to-noise ratio, and pulsed operation
are beneficial for any synchronization scheme. Here, we take advantage of
nanowire single-photon detectors with low root mean squared timing jitters of
$13\pm 2$ ps and 160 kcps correlation events. The main source of timing jitter
during the initialization of the synchronization is the difference in the
clock frequency. Hence, small single photon detector timing jitters improve
the situation only slightly. However, they produce a lower synchronization
jitter during the communication session. Here, the peak correlation time
shifts are derived with higher resolution, and the subsequent clock frequency
is accurately estimated. On the other hand, low signal rates increase the
probability of failure during both the initialization of the synchronization
and the subsequent frequency update algorithm. By pulsing the quantum source,
we reduce this probability, increase the robustness, and allow for the
application to arbitrary clock and link setups.

The frequency sweeps during the initialization of the synchronization require
repeated cross-correlations. We implement it as serial for loops that are
known for their high computation times. In the next step, the calculations
could be parallelized and processed on a graphical processing unit for higher
efficiency. Furthermore, if we knew the clock frequency differences from
previous measurements, the number of loops would be drastically reduced. This
option is especially applicable in systems with a higher duty cycle. Unlike
previous work [31, 32, 7], this method is less dependent of the signal
strength. We find that pulsed single-photon sources allow for synchronization
at loss levels that would preclude continuous-wave correlation-based
approaches previously demonstrated [23], giving rise to synchronization
feasibility under low signal conditions, such as long-distance free space
links [13] with only crystal oscillators (see Appendix V.1).

Quantum communication networks that employ pulsed correlated photon pair
sources benefit from a significantly enhanced signal-to-noise ratio for
synchronization. Each individual receiver can locally process the arrival time
of photons to match their clock frequency with the common master clock of the
source. As this does not require correlation events between the receivers, the
signal-to-noise ratio is drastically enhanced and opens the application to
low-signal scenarios. Furthermore, the amount of data to be transferred for
synchronization is significantly reduced. In turn, the update time of the
clock frequency reduces, which follows in a smaller time deviation.

Although we think our scheme applies very well to quantum communication
scenarios by reducing additional hardware, there are some important downsides
in comparison to classical synchronization methods. They mainly relate to the
limited signal strength and the intrinsic timing uncertainty of the photons in
weak coherent sources. The maximum signal is bounded by the single-photon
detectors. Single-photon count rates exceeding GHz are only possible through
multi-pixel detection units, but at the expense of the detection efficiency
[51]. Such a count rate allows us to barely detect a standard 10-MHz clock
signal as used in classical communication. However, this becomes difficult in
lossy scenarios as the signal can not be amplified in quantum communication.
In contrast, its straightforward with classical strong laser light due to the
abundant number of photons and the option for optical amplification.

Compared to the White Rabbit protocol, the single photons are used not only
for synchronization but also as quantum data bits. This has the consequence
that no two-way time transfer is necessary to isolate the run-time
fluctuations of the link. Our one-way time transfer scheme simultaneously
compensates for clock drifts and run-time fluctuations. Thus, it provides a
reliable reference frame of the sent and received quantum bits. This is a
significant reason for our results to be comparable to the White Rabbit
protocol, although our feedback cycle (7 Hz) is much slower than the White
Rabbit PLL (30 Hz) [52].

##  IV Conclusions

Our frequency tracking protocol universally applies to any single-photon
source, regardless of its operating mode. We enable synchronization in post-
processing without modifying the quantum source on the hardware side and
without synchronization strings. These algorithms can be seamlessly integrated
into state-of-the-art quantum communication schemes by using correlation
events to find the initial absolute timing offset. Processing of the arrival-
time statistics during the session helps to keep the clock frequency locked
without sharing any secure data bits. We show that the requirements on the
clock performance can be more flexible as compensation mechanisms counteract
and balance strong drift and clock frequency differences. This feature enables
standard computers to be connected to future up-scaled quantum communication
networks [42], as ultra-precise clocks become redundant. Here, our methods
represent important advances for secure clock synchronization [43] and global
precision time distribution [53] and show how single photons become both
information and timing carriers.

## Acknowledgments

This research was carried out within the scope of the QuNET project, funded by
the German Federal Ministry of Education and Research (BMBF) in the context of
the federal government’s research framework in IT security ’Digital. Secure.
Sovereign.’ Further funding was received under the FastPhoton project from the
European Union, the European Social Funds and the Federal State of Thuringia
(2019FGR0101). Christopher Spiess is a member of the Max Planck School of
Photonics, supported by the BMBF, the Max Planck Society, and the Fraunhofer
Society. The authors acknowledge the support of Sebastian Töpfer, Meritxell
Cabrejo Ponce, Karin Burger and Carlos Andres Melo Luna for the internal
review.

## Author Contributions

C.S. designed and performed the experiments. F.S. proposed and directed the
research. The first draft of the manuscript was written by C.S. and F.S. All
authors discussed the results and reviewed the manuscript.

## Competing Interests statement

The authors declare that they have no competing interests.

##  V Appendix

###  V.1 Computation time and feasibility for low-signal scenarios

Figure 7: Computation time of modulo-based cross-correlations and comparison
with literature. (a) The computation time increases linearly with the number
of time tags. (b) The figure depicts the total computation time for varying
frequency resolutions $\delta u$. The total time is $T_{\mathrm{tot}}=T(\delta
u)\Delta U/\delta u$ with a clock frequency difference accuracy $\delta u$ for
a range of $\Delta U=40\,$ppm ($-20\,\mathrm{ppm}\dots+20\,\mathrm{ppm}$, as
given by crystal oscillators [45]) and $T(\delta u)=\mathrm{const}=37\,$ms
(for $N=10^{7}$ time tags). With a maximum computation time of 2 hours [23],
the clock frequency resolution can be as low as 0.2 ns/s. (c) Figure modified
from [23]. The significance of correlation peaks indicates its visibility and
its likelihood of detection. As example, a significance of 3 refers to a
confidence of 99.7 % that it is not just noise. Smaller residual clock
frequency difference increase this probability and small signal-to-noise ratio
reduce it. By the clock frequency compensation during initialization, we open
feasibility of clock synchronization to regimes of very low signal noise ratio
ratios, as in Ecker2021 [13] (signal to noise ratio of 5, 143 km free-space
quantum key distirbution link) or in the low signal regime of Spiess2021-low
[23]. Other references are Steinlechner2017 [10] and Ho2009 [7]. The
specifications of our computer are described in Table 3.

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 $10^{7}$ time tags at the receiver (Fig.
7 (a)). This enables high-resolution clock-frequency sweeps during
initialization of the synchronization procedure down to smaller than 0.2 ns/s
(Fig. 7 (a)) at 2 hours of computation time. In comparison, Fast-Fourier
transformations require more than 30 seconds for the same array size and limit
the clock frequency resolution to 140 ns/s 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 $<5$ [13] (Fig. 7 (c)).

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 correaltion peak and not just noise. The significance has the following
relation [7],

$S=\sqrt{\frac{r_{C}^{2}}{r_{A}r_{B}\Delta u}},$ (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 $\Delta u$.

Parameter | Value
---|---
Processor | Intel(R) Core(TM) i5-8250U CPU
Speed | 1.60 GHz (1.80 GHz)
RAM | 16 GB
System | Windows 10, 64-bit based processor
Environment | Python 3.7.0 64 bit, NumPy 1.19.4
Table 3: Specifications of the computer.

###  V.2 Timing uncertainty from fiber length

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 $\lambda$ [54],

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

The thermo-optic coefficient $\epsilon=\partial n/\partial T=11\times
10^{-6}\,\mathrm{K}^{-1}$ is at least one order of magnitude higher than the
thermal expansion coefficient $\alpha=\partial L/\partial T=0.55\times
10^{-6}\,\mathrm{K}^{-1}$ 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 ($>$10 s) when it is not compensated.

###  V.3 Allan time deviation

Ref. [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.

Figure 8: Representation of the stability of our algorithm in terms of
overlapping Allan deviation (a) and Allan time deviation (TDEV) in (b). The
underlying data set is the same as in Fig. 6.(1.) Quartz oscillators at source
and receiver without updating the clock frequencies. (2.) Quartz oscillators
at source and receiver with tracking of the clock frequencies, denoted with
(+). (3.) Free-running rubidium (Rb) clocks at source and receiver with
stability specifications (MDEV) of each being 3E-12 at 1 s, 3E-12 at 10 s and
1E-12 at 100 s. (4.) An RF-cable provides the 10 MHz reference output of the
source to the receiver.

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