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Abstract
The two-way quantum clock synchronization has been shown to provide femtosecond-level synchronization capability and security against symmetric delay attacks, thus becoming a prospective method to compare and synchronize distant clocks with enhanced precision and safety. In this letter, a field test of two-way quantum synchronization between a H-maser and a Rb clock linked by a 7 km-long deployed fiber is implemented by using time-energy entangled photon-pair sources. Limited by the intrinsic frequency stability of the Rb clock, the achieved time stability at 30 s is measured as 32 ps. By applying a fiber-optic microwave frequency transfer technology to build frequency syntonization between the separated clocks, the limit set by the intrinsic frequency stability of the Rb clock is overcome. A significantly improved time stability of 1.9 ps at 30 s is achieved, which is mainly restrained by the low number of acquired photon pairs due to the low sampling rate of the utilized coincidence measurement system. Such implementation demonstrates the high practicability of the two-way quantum clock synchronization method for promoting field applications.
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1. Introduction
Precise synchronization between distant clocks is of great significance due to its essential role in almost every type of precision measurement. The two-way time transfer is a popular way to compare and synchronize distant time scales precisely. Among different two-way time transfer (TWTT) methods, the widely utilized satellite-based TWTT has reached time stability of 100ps [1,2]. With a much higher frequency and bandwidth than radio radiations, the optical TWTT allows realizing time stability of a few picoseconds and accuracy better than 100 ps [3–6]. By utilizing optical frequency comb as the light source, femtoseconds of time transfer stability in free-space have been reported [7,8], at the cost of a complex configuration and strict requirement for the frequency comb propagation with high-fidelity. Benefitting from the low loss, high reliability, and high stability of optical fibers, the fiber-optic time and frequency transfer offer an alternative method to the free-space transfer with potentially superior performance [9–11]. However, the best time stability of the classical two-way transfer approach based on the traditional modulation remains at subpicosecond level under the common clock Ref. [12]. On the other hand, the fiber dispersion makes the utilization of optical frequency comb for time transfer over fiber links encounter insuperable difficulties, and demonstration has only been realized over dispersion-compensated kilometer-scale optical fibers [13]. Furthermore, secure time transfer is critical to widespread technologies and infrastructures [14,15]. As classical techniques are susceptible by nature to interference by malicious parties [16], the need for introducing the quantum technology to ensure security becomes compelling.
By virtue of the strong temporal correlation characteristics of time-energy entangled photon pair sources, fiber-based two-way quantum clock synchronization protocol was proposed to compare and synchronize the distant clocks [17]. Benefitted from this highly time-correlated creation of the photon pairs, a narrow correlation peak in their arrival time at remote sites can be achieved. By using the correlation peak location, the time and frequency differences between remote clocks can be easily extracted. Compared with the classical method, the requisite modulation and demodulation are avoided, simplifying the time transfer process and significantly improving single-shot measurement accuracy. Moreover, the Rayleigh backscattering noise [18] inherent in the traditional fiber-optic time transfer system can be easily avoided as the Rayleigh backscattered photons cannot contribute to the effective two-photon coincidence for the time difference measurement. In addition, the nonlocal dispersion cancellation possessed by the time-energy entangled photon-pair sources has been shown capable of resolving the entanglement loss and extending the transmission to a farther distance [19]. With the common clock reference, the synchronization performance achievable by the two-way quantum synchronization technique has been demonstrated and reported on a 20km spooled fiber link with a few tens of femtoseconds synchronization stability and an accuracy of 2.46 ps in our previous work [20]. With the development of quantum repeater techniques, high-precision quantum clock synchronization over fiber distance reaching hundreds of kilometers can be expected. Furthermore, the security of the clock synchronization can be guaranteed by the complementarity principle of quantum mechanics, and the requisite security test can be reliably implemented based on quantum techniques such as the unique nonlocality of the quantum entanglement [21–23], quantum key distribution [24], and so on.
As the two-way quantum clock synchronization technique has been demonstrated in the laboratory with apparent potential, exploring its possible advancement in tackling the practical synchronization of distant clocks becomes imperative. In this paper, we carry out a field test of the two-way quantum clock synchronization between a H-maser located at the campus of National Time Service Center (NTSC) and a Rb clock at the Lishan Observatory (LSO) of NTSC. Through the deployed telecommunication fiber with a distance of 7 km, the two sites are physically connected. The time stability of 32 ps at 30 s is obtained, which is restrained by the stability of Rb clock at 30 s. By introducing the fiber-based microwave frequency transfer into the experiment, the stability is improved by more than one order of magnitude to 1.9 ps with only 1440 coincidence counts in 30 s, which is due to the low sampling rates of the utilized event timers (ET, ∼6 kHz) for coincidence measurement. Further improvement of the time stability is highly expected by utilizing new ETs with MHz-level sampling rates [25]. This experimental demonstration lays a solid foundation for the application of two-way quantum clock synchronization in practical systems for potential sub-picosecond precision. Benefitted from it, the time-bin-encoded quantum key distribution can also be implemented without an additional dedicated synchronization procedure [26,27].
2. Experimental setup
The schematic diagram of the field two-way quantum clock synchronization setup is shown in Fig. 1. The H-maser sited at the NTSC lab and the Rb clock (PRS10, SRS. Inc) at the LSO lab are connected by a 7 km-long deployed fiber link, with a transmission loss of ∼2.5 dB. At each site, there is a time-energy entangled photon-pair source, a pair of single photon detectors (SPDs) and an ET referenced to the local clock. The photon pair sources are generated by using a 780 nm DBR laser (Photodigm Inc) to pump a 10 mm-long periodically poled Lithium Niobate (PPLN, type II) waveguide with the poling period of ∼8.4 µm. Via the spontaneous parametric down conversion (SPDC), degenerate photon pairs at 1560 nm (denoted as signal and idler photons) are obtained with the spectral bandwidth being 2.4 nm [28]. The four SPDs are superconductive nanowire single photon detectors with efficiency of 65% (SNSPD, Photec Ltd.) [29,30], denoted as D1-D4. Two commercial ETs (A033-ET, Eventech Ltd, ET1& ET2), each having two input ports (A and B ports) and a sampling rate of 6 kHz for one port, are utilized as the time tagging devices. At NTSC, the signal photon (s1) is kept for local detection by D1, and the idler photon (i1) is transmitted forward through the 7 km fiber to LSO to be detected by D2. Similarly, s2 is maintained locally at LSO and detected by D4, while i2 is transmitted backward to NTSC and detected by D3. The optical circulators (OC1 and OC2) ensure bidirectional transmission through the same fiber. In front of each SNSPD, there is a fiber-based bandpass filter of 1560nm±5nm to filter out the spurious background noise from the transmission link.
Fig. 1. (a) Aerial and (b) schematic view of the experimental setup of the field two-way quantum clock synchronization between the H-maser at NTSC and the Rb clock at LSO.
The photons detected by D1, D2, D3, D4 are subsequently tagged by the individual ET ports ET2A, ET1A, ET1B, ET2B (abbreviated to be 2A, 1A, 1B, 2B) as $\{ \textrm{t}_1^j\} $, $\{ \textrm{t}_2^j\} $, $\{ \textrm{t}_3^j\} $, $\{ \textrm{t}_4^j\} $, where the superscript j denotes the j-th tagged photon. The time tags $\{ \textrm{t}_2^j\} $, $\{ \textrm{t}_3^j\} $, at LSO are sent to NTSC via classical communication channel for acquiring the time differences t2-t1 and t3-t4 by nonlocal coincidence identification algorithm [31]. According to the bidirectional symmetry, the time offset Δt0 between the two clocks can be extracted by ((t2-t1)-(t3-t4))/2 [20].
3. Experimental results
3.1 Evaluation of synchronization stability dependent on the reference clocks
The influence of the reference clocks on the synchronization stability is firstly analyzed by setting the two-way quantum clock synchronization setup locally in the NTSC lab and using a 3-m fiber to link the two sites. Each reference clock offered a 10 MHz and a 1 PPS signal to the ET. The 10 MHz is used to syntonize the internal clock of the ET, which dominates the stability of the single-photon arrival time measurement. The 1 PPS signal is used to synchronize the rising edge of the internal clock of the ET to its rising edge and define the starting time of each measurement, which determines the accuracy of the arrival time measurement. In order to test the impact of the references, short electronic cables of 10 m length are symmetrically used in the system and kept unchanged during the evaluation measurement. To clarify the roles of the 1 PPS and 10 MHz references, four cases of settings are tested: case 1) both ETs are referenced to the same 10 MHz and 1 PPS from the H-maser, case 2) both ETs are referenced to the same 10 MHz from the H-maser, but the 1 PPS signals for them are individually from the H-maser and the Rb clock, case 3) both ETs are referenced to the same 1 PPS from H-maser, but the 10 MHz signals are individually from H-maser and Rb clock, and case 4) the two ETs are respectively referenced to H-maser and Rb clock. The corresponding time offset results versus the elapsed time of about 190 s are presented in Fig. 2. It can be seen that, with the same 10 MHz reference frequency, the measured time offsets remain independent on the elapsed time (in wine stars and blue up-triangles) even when the referenced 1 PPS signals are different; while for different reference frequencies, the measured time offsets experience significant drift with respect to the elapsed time (in black squares and magenta down-triangles). Thus it makes sense to syntonize 10 MHz frequency reference to improve the synchronization stability.
Fig. 2. The time offset results acquired under the cases of 1) the same reference clock (in wine stars), 2) the same 10 MHz reference (in blue up-triangles), 3) different 10 MHz frequency references (in magenta down-triangles), and 4) different reference clocks (in black squares).
For the case with independent reference clocks, a long-term time offset measurement is implemented and Fig. 3(a) depicts the 2-hour results in black squares. By applying a quadratic polynomial fitting to the result [32,33], the relative frequency accuracy of the Rb clock to the H-maser is given as ∼1×10−11, which coincides nicely with the nominal relative frequency accuracy (<5×10−11) of the utilized Rb clock. The relative frequency drift is given as 8×10−17/s, representing the Rb clock's aging. After subtracting the quadratic polynomial fitting, the residuals of the measured time offsets are plotted in Fig. 3(b), which shows a fluctuation of about 162.6 ps in standard deviation. Note that, when the relative frequency accuracy contribution is removed from the measured time offsets, it resembles the disciplining of the Rb clock to the H-maser. Thus, one can consider the Rb clock has been synchronized to the H-maser by software. The attainable stability between the two clocks by evaluating the residuals of the time offsets in terms of Allan deviation (ADEV) is given by black squares in Fig. 3(c), which reaches 2.2×10−12 at an averaging time of 30 s. For comparison, the relative frequency stability of the Rb clock to the H-maser is measured by a phase noise analyzer (Symmetricom 5125A) and plotted in Fig. 3(c) by blue diamonds. The consistency between the two curves within the averaging times of 30-400 s implies that the short-term synchronization performance is determined by the intrinsic stability of the Rb clock. The upward trend of the blue-diamond curve beyond 500 s is from the relative frequency drift of the Rb clock. In comparison, the ADEV curve given by black squares remains the downward trend due to the disciplining effect of the H-maser on the Rb clock.
Fig. 3. Evaluated results of the synchronization performance under independent reference clocks. (a) The measured time offsets versus the elapsed time in two hours (black squares) and the corresponding polynomial fit (red line). (b) The residuals of the time offset results as a function of the elapsed time. (c) The attainable ADEV synchronization stability (black squares), and the relative frequency stability of Rb clock to H-maser (blue diamonds).
3.2 Evaluation of the quantum clock synchronization accuracy
Just as the accuracy evaluation in the classical time transfer system, we evaluate the time bias contributions of our two-way quantum clock synchronization system, which include the propagation asymmetry caused by the chromatic dispersion (τdisp), the systematic biases introduced by the two 1.245 km-long DCFs for dispersion compensation (τDCF), the superconductive single photon detectors (τSNSPD), the event timers (τET), and the short fiber pigtails and electronic cables for interconnections between instruments(τCABLE). For the evaluation, we implement the measurement at the NTSC lab using a common clock reference and a 7 km spooled fiber as the two-way transfer link. The time offset between the fiber interconnected sites of A and B is measured to be τB-A = 65978 ± 2 ps. After correction for the above systematic biases, the calibrated accuracy of the two-way quantum clock synchronization system can be evaluated, i.e., τcal=τB-A-(τdisp+τDCF+τSNSPD+τET+τCABLE).
Take the evaluation of the time tagging performance of the two utilized ETs as an example, the H-maser is used as the common frequency reference for both of them. The signal generator (AFG31052, Tektronix. Inc), also referenced to the H-maser, generates two 1 PPS signals with a preset relative delay τ between them. By individually sending them to the two input ports (denoted by A, B) of ET1 and ET2, the measured time offsets as a function of the preset τ for both ETs are shown in Fig. 4(a) in black squares and red circles, respectively. The consistency of the two curves and their preeminent linearity with the preset time value τ can be seen in the whole range of measured time offsets. By zooming in the plots, there is a fixed difference of 19.7 ± 4.8 ps between the measured time offsets of ET1 and that of ET2, which can be explained as the slight inhomogeneity between the input ports of individual ET.
Fig. 4. (a) Measured time offsets of ET1 (black squares) and ET2 (red circles) as a function of the varying time delay between the two input 1PPS signals. (b) Under the fixed time delay of 100.147 ns, measured time offset results with time under different input configurations of 1A2A (in black squares), 1A2B (in red circles), 1B2A (in blue up-triangles), 1B2B (in purple down-triangles), 1A1B (in olive diamonds), and 2A2B (in navy hexagons).
Subsequently, the homogeneity between the two ETs is also evaluated by sending the two 1 PPS signals with a fixed delay of 100.147 ns into either two of the input ports. Therefore, six cases are considered: (1) input A of ET1 as the start, input A of ET2 as the stop (1A2A), (2) input A of ET1 as the start, input B of ET2 as the stop (1A2B), (3) input B of ET1 as the start, input A of ET2 as the stop (1B2A), (4) input B of ET1 as the start, input B of ET2 as the stop (1B2B), (5) input A of ET1 as the start, input B of ET1 as the stop (1A1B), and (6) input A of ET2 as the start, input B of ET2 as the stop (2A2B). The measured time offsets are shown in Fig. 4(b) by black squares, red circles, blue up-triangles, purple down-triangles, olive diamonds, and navy hexagons, respectively. The measured time delays are then evaluated to be 101.034 ± 0.009 ns for 1A2A, 101.097 ± 0.007 ns for 1A2B, 100.984 ± 0.008 ns for 1B2A, 101.045 ± 0.008 ns for 1B2B, 100.196 ± 0.007 ns for 1A1B, and 100.211 ± 0.007 ns for 2A2B. According to the experimental setup, the time offset measurement is based on the time-tagging data of 2A-1A and 1B-2B. From the above analysis, the ETs introduce a time bias of τET=−893.0 ± 9.9 ps. By restarting the ETs several times, the measured asymmetric bias fluctuates with a standard deviation of 2 ps, which is within the variation range of 9.9 ps, verifying the reproducibility of ET measurements.
The time bias from the propagation asymmetry can be evaluated by τdisp = 1/2(LDΔλ), where L is the fiber length, D is the dispersion of the fiber and Δλ is the center wavelength difference of the idler photons of the two entangled sources [34]. Given Δλ=0.37 ± 0.06 nm, D = 17 ps/nm/km and L = 7 km, the asymmetric propagation bias is estimated to be τdisp = 22 ± 3.6 ps. τDCF = 66871.6 ± 3.6 ps, τSNSPD = 3.0 ± 1.1 ps, and τCABLE=−18.00 ± 1.40 ps have also been deduced. Thus, the overall asymmetric delay is evaluated to be 65985.6 ± 11.3 ps. Compared with the measured time offset of τB-A = 65978 ps, the calibrated accuracy can be given by τCAL= 7.6 ps, which is within the uncertainty budget of 11.3 ps. The above accuracy calibration verified the feasibility of the two-way quantum clock synchronization protocol. The time bias contributions for the synchronization accuracy analysis are also listed in Table 1.
Table 1. The evaluated time bias contributions
One should note that, the influences due to the Sagnac effect (0.05ps/km [11]) and polarization mode dispersion (PMD, 0.05 ps/km1/2) is neglected for our 7 km fiber link, as their contributions are much less than 1 ps (about 0.48 ps) in total.
3.3 Field test result of the quantum clock synchronization setup
Then the two-way quantum clock synchronization between the H-maser in the NTSC lab and the Rb clock in the LSO lab are performed. Based on the experimental system, the coincidence widths of the time differences t2-t1 and t3-t4, are measured to be around 285 ps and 286 ps respectively. According to Ref. [35], the detected coincidence width of the entangled photons can be derived by
where σdisp is the dispersion broadening width of the entangled photons, σjitter is the detector jitter (about 51 ps for our SNSPDs), and σclock =Δu*t is the broadening due to the clock frequency difference (Δu is the relative frequency accuracy, and t is data acquisition time instead of time interval). Based on the experimental parameters, the dispersion broadening can be estimated as σdisp = 17ps/nm/km×7km×2.4 nm/(2ln2)1/2≈247 ps. With the H-maser and Rb clock as the reference, Δu =1×10−11 and t = 10 s give rise to a value of σclock =100 ps. Combined with σjitter of 51 ps, the coincidence width is simulated to be about 267 ps, which shows good agreement with the measured results. The extracted time offset t0 between the two clocks as a function of the elapsed time is plotted in Fig. 5(a) by black squares. By applying a quadratic polynomial fitting (red curve), the relative frequency accuracy and frequency drift of the Rb clock versus the H-maser are evaluated as 7.1×10−11 and 7×10−17/s, respectively. It can be readily seen that, the relative frequency drift of 7×10−17/s evaluated via the field fiber link is very close to that derived from the direct comparison between the Rb clock and the H-maser, which is 8×10−17/s. The relative frequency accuracy is deduced from the time offset measurement, and thus is proportional to the measured coincidence width of the entangled photons. Given the other parameters remaining constant, the measured coincidence width of the entangled photons has been broadened by nearly six times due to the 7 km fiber transmission. Therefore, the deviation of the evaluated relative frequency accuracy of 7.1×10−11 from the previous result of 1×10−11 should be mainly attributed to the field 7 km fiber link. The residuals after the fit are depicted in Fig. 5(b) and a breathing fluctuation is shown, which can be attributed to the independent ambient condition at the two sites as their temperature variations feature different periodic behavior.Fig. 5. (a) Measured time offsets between the two remote clocks and (b) the residual of the polynomial fit as a function of the elapsed time.
The synchronization stability between the two spatially separated clocks in time deviation (TDEV) is also evaluated based on the residuals of the fitted time offset results and plotted in Fig. 6 by black squares, in which a value of 38.1 ps at 30 s is obtained. In order to eliminate the dispersion impact in transmission paths, one dispersion compensation fiber (DCF) with a length of 1.245 km is inserted in each local arm. Without considering the contribution of σclock, the narrowed coincidence widths of about 120 ps are expected due to the nonlocal dispersion cancellation (NDC) [19]. Taking σclock into consideration, the coincidence width is simulated to be about 156 ps, which agrees with the measured result of 177–180 ps as well. Under NDC configuration, the achieved time stability is also shown in Fig. 6 (red circles) and gives a TDEV of 32 ps at 30 s. We can see that the NDC has a negligible impact on the achieved stability with independent Rb clock and H-maser as the references. Comparing the long-term time stabilities between the cases with and without NDC in the setup, the almost same results of 19.3 ps at 7680 s further indicate that the NDC has negligible impact on the achieved stability with independent Rb clock and H-maser as the references. For clarity, the TDEV result of the two-way quantum clock synchronization setup co-located in the NTSC lab is also given in Fig. 6 by blue down-triangles. It can be seen that, all the TDEV curves first rise with the increment of averaging time and then behave drop-trend. The equivalence of the three TDEV curves before the averaging time reaching 1000s shows that the attainable short-term synchronization stability in the field test is also limited by the relative frequency stability of the Rb clock. With the increase of averaging time, the synchronization stability starts to depend more on the ambient fluctuations. In comparison with the case with the two reference clocks at the same lab (blue down-triangles), whose inflexion arrives at 1000 s, the inflexion points with the two clocks located at different labs move to around 2000 s. This movement reflects the independent ambient influence introduced by individual laboratory.
Fig. 6. Synchronization stability between two independent clocks separated by 7 km deployed fibers without DCF in fiber link (black squares), with DCF in fiber link (red circles), and without 7 km deployed fibers (blue down-triangles); two-way quantum clock synchronization stability with transferring the reference frequency of the H-maser to LSO laboratory (olive diamonds).
3.4 Synchronization stability improvement with the microwave frequency transfer
As it has been shown that the relative frequency stability of the reference clocks dominates the clock synchronization performance, a distinct improvement can be expected by applying stabilized 10 MHz reference frequency between the two distant sites. In view of that, utilizing a fiber-based microwave frequency transfer technology, the 10 MHz frequency reference from the H-maser at NTSC is stably transferred with a stability of 3.7×10−15@30 s and used as the ET frequency reference at LSO. Detailed frequency transfer experimental configuration and results are shown in Ref. [36]. Under this accomplishment, the two-way quantum clock synchronization is performed and the resultant time stability is presented in Fig. 6 by olive diamonds. The short-term stability result gives a TDEV of 1.9 ps and a corresponding ADEV of 1.1×10−13 at 30 s, which shows a distinguished improvement by more than one order of magnitude. To compare with the above achieved frequency transfer stability, there is a gap of almost two orders of magnitude to be analyzed. According to the quantum theoretical model, the measured standard deviation (SD) of the time offset Δt0 can be given by the detected coincidence width σ of the entangled photons and the number of photon pairs N within a certain measurement time [20]:
By applying frequency transfer to build the common frequency reference, σclock = 0 can be achieved. In this case, the coincidence width is estimated to be 120 ps, which is slightly smaller than the measured results of about 132 ps for both t2-t1 and t3-t4. Due to the low sampling rate of the utilized event timers (∼6 kHz for each port), the acquired number of photon pairs N in 30 s is only around 1440 cps. By substituting these two parameters into Eqn. (2), theoretical estimation of 1.7 ps is derived and consistent with the experimental result. From the above analysis, the improvement of Δt0 for two orders of magnitude can be expected by narrowing the coincidence width with NDC optimization [37] and utilizing new ETs with MHz-level sampling rate.
With the increasing of the average time, the TDEV curve reaches the inflexion at about 1000 s instead of 2000 s, indicating the effect of the frequency transfer technology on suppressing the influence of the independent circumstance fluctuations. The long-term time stability approaches 5.2 ps at 7680 s, which shows an improvement by four times. The dropping trend with the measurement time promises better time stability at a longer averaging time beyond 10000 s.
4. Conclusions
In summary, we have demonstrated a two-way quantum clock synchronization experiment between spatially separated clocks. Over a 7 km-long deployed telecommunication fiber link, the H-maser at NTSC and the Rb clock at LSO are compared and synchronized via the two-way quantum clock synchronization method. The short-term synchronization stability reached the intrinsic frequency stability of the Rb clock to the H-maser, which was measured as 32 ps at 30 s. The long-term synchronization stability was achieved to be 19.3 ps at 7680 s. Further applying the fiber-based microwave frequency-transfer to the experiment, a significantly improved time stability was achieved with a short-term TDEV of 1.9 ps at 30 s and a long-term TDEV of 5.2 ps at 7680 s. The nice agreement between the short-term TDEV result and theoretical analysis proves that, both the low sampling rates of the utilized event timers (ET, ∼6 kHz) for coincidence measurement and imperfect NDC limit the achievable stability. Utilizing new ETs with MHz sampling rate and optimizing the NDC effect, a magnificent advancement of the stability to the level set by the transferred frequency stability can be achieved. This experiment proves the high potential of the two-way quantum clock synchronization in field applications to promote the synchronization performance. For example, our method can be easily applied to the time-bin encoded quantum key distribution (QKD), to improve the synchronization required for much enhanced key rate.
Funding
National Natural Science Foundation of China (12033007, 61801458, 61875205, 91836301); Key Research Program of Frontier Science, Chinese Academy of Sciences (QYZDB-SW-SLH007); Strategic Priority Research Program of CAS (XDC07020200); Chinese Academy of Sciences Key Project (ZDRW-KT-2019-1-0103); Western Young Scholar Project of CAS (XAB2019B15, XAB2019B17); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021408); West Light Foundation of the Chinese Academy of Sciences (29202082).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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