# High-Precision Single-Photon Laser Time Transfer with Temperature Drift Post-Compensation
## 3 Experimental Data and Processing Methods  
With the above experiment setup, we finally achieved a single-shot precision of 45 ps expressed in terms of the root mean square (RMS) of the entire chain. The raw data are illustrated in Figure 2a, revealing in general the same trend of variation between the propagation delay and the temperature. According to the data collected at different temperatures, one could get the delay time by recognizing the peak position. Figure 2b shows a delay change of 240 ps as the temperature changes from 22 \({}^{\circ}\)C to  
Figure 1: Experimental setup of the laser time transfer based on the temperature drift post-compensation. GEN is a generator; L1, L2 and L3 are planoconvex lenses; A represents the aperture; M1 and M2 are high-reflection mirrors; ATT is an attenuator; SPAD is a single-photon detector based on Si; ET is an event timer (precision: 8 ps); and TR is a temperature recorder.  
34 \({}^{\circ}\)C. According to Figure 2, the whole temperature changing process can be divided into 7 sections. First comes a temperature stable period, which is followed by 6 temperature rising and stable periods. Each temperature rising period lasts 7-10 h, while the temperature stable period lasts 12 to 15 h. However, at the end of day 3, some errors occurred in the data recording system, so only 2 h of data for the temperature stable period were recorded that day.  
Figure 3 illustrates the trend of propagation delay along with the temperature variation. According to Figure 3, we assume the relationship of system propagation delay to temperature as a one-order linear change.  
Figure 3: Propagation delay vs. environment temperatures. (The black dots represent the data of the temperature stable period and the red dots represent the temperature rising period).  
Figure 2: (**a**) The propagation delay and room temperature over the time. (**b**) The propagation delay drift of environment temperatures from 22 \({}^{\circ}\)C to 34 \({}^{\circ}\)C.  
The raw data collected at different temperatures are represented by \(d_{i}\). One could get the delay time at different temperatures by recognizing the peak position or the mean value, which can be expressed by \(D_{j}\). The delay as a function of temperature can be plotted and linearly fitted as  
\[d=aT+b \tag{1}\]  
where \(d\) is the ideal delay time without compensation in unit of ps, \(T\) is the temperature in unit of \({}^{\circ}\)C and \(a\) is the slope of the fitting line curve. In principle, the variation of original propagation delay could be compensated to a value of temperature \(T_{0}\), which could be selected freely. The compensated delay time \(d_{ci}\) can be calculated by  
\[d_{ci}=d_{i}-(T_{i}-T_{0})\times a \tag{2}\]  
where \(T_{i}\) is the corresponding temperature of data \(d_{i}\). With the above experiment system and compensation method, we recorded all the data, took part of the data to get a fitting curve and the function and finally got the compensated data. Using the original and compensated data, we calculated and analyzed the time stability of the system before and after compensation.  
Though the system delay varies linearly with the temperature, we see some fluctuations when the temperatures begin to change. The most likely reason is that the temperature will take time to reach a steady state. This is why we separated the data into two groups corresponding to temperature stable and rising conditions for data analysis.