## 4 Simulations and Performance Estimation
### The Influence of the Pulse Width and Research on Repetition Frequency  
To match the accuracy requirement of polarization extinction ratio (PER) in the practical systems, we designed a simulation analysis for a relatively low polarization extinction ratio (PER) below 20 dB and conducted a multi-parameter joint optimization study under the condition of PER \(\leq\) 20 dB. According to the earlier works [12], the photon leakage between the pilot tone and quantum signal can be estimated as:  
\[\left\langle n_{pilot}^{e}\right\rangle=q\left\langle n_{pilot}\right\rangle \tag{24}\]  
where \(<n_{pilot}>\) is the average photon number of the pilot tone, \(<n_{pilot}^{e}>\) denotes the leaked photon number to the quantum signal mode, and q is the overlapping factor, which is related to the pulse extinction ratio. Assuming a Gaussian pulse shape, the overlapping factor q is given by:  
\[q=e^{-\frac{\Delta_{t}^{2}}{2\
u_{t}^{2}}} \tag{25}\]  
where \(\Delta_{t}\) is the time delay between the pilot and the quantum signal. In this scheme, it is related to the repetition frequency \(f_{rep}\), which can be written as:  
\[\Delta_{t}=\frac{1}{f_{rep}} \tag{26}\]  
Figure 5: Secure key rates versus different modulation variance VA and transmission distance curves corresponding to 20 dB PER. (The repetition frequency is 300 MHz, the pulse width is 1 ns, and the reverse reconciliation efficiency \(\beta\) is 0.95. s = 0.01, \(\
u_{dc}/N_{0}\) = 0.1, and the quantum efficiency \(\eta\) is 0.56. )\(\sigma_{t}\) is related to the full width at half maximum of the pulse \(\sigma_{FW}\),  
\[\sigma_{t}=\frac{\sigma_{FW}}{2\sqrt{\ln 2}} \tag{27}\]  
In the dual-polarization joint time and frequency division multiplexing CV-QKD system we proposed, the pulse width and repetition frequency will affect the leaked photons < \(n_{e1}\) >, < \(n_{e2}\) >, < \(n_{e3}\) >, < \(n_{e4}\) >, < \(n_{e5}\) >, and < \(n_{e6}\) > in Equations (18)-(23). In the following analysis, we take the 1 ns pulse width as an example to conduct research on repetition frequency under the optimal modulation variance that we concluded in Section 4.2. Here, we set the transmission distance as 25 km and the modulation variance VA as 1.6. As shown in Figure 6, the SKR increases with the repetition frequency for different PERs when the repetition frequency \(\leq\)302.5 MHz. In this case, the maximum SKR can be achieved during the repetition frequency of 292.5-302.5 MHz. Then, with increasing repetition frequency, the SKR goes into decline. This is mainly due to the fact that the pulse extinction ratio, or the overlapping factor q of pilot tones in the QS1 time-domain position, increased when the pulse repetition frequency increases to a relatively high level. From Figures 6 and 7, it can also be concluded that different PERs have different optimal repetition frequencies. As the dashed line shows, when the PER increases, the system has more redundancy for time-domain diffusion. At the bottom right of the dashed line, the repetition frequency becomes the main limiting factor for improving the SKRs. However, in the upper-left corner of the dashed line, PER is the main factor restricting the improvement of SKRs. First, we consider the ideal case in Figure 6, namely the PER is infinite. As the red solid line shows, the optimal repetition frequency can be higher than 302.5 MHz, and the maximum SKR is 11.74 Mbit/s. In this case, the time-domain attenuation factor \(q_{1}\) is 2.6337 \(\times\) 10\({}^{-7}\), equivalent to \(-\)65.8 dB, and \(q_{2}\) is equal to 4.8110 \(\times\) 10\({}^{-27}\) or \(-\)263.1 dB. When the PER deteriorates to 20 dB, the maximum SKR is about 5.863 Mbps under the 297.5 MHz optimal repetition frequency. In this situation, \(q_{1}=1.5760\times 10^{-7}\), or \(-\)68.0 dB; \(q_{2}=6.1684\times 10^{-28}\) or \(-\)272.1 dB. Then, when the PER deteriorates to 18 dB, an SKR of 2.933 Mbps@25 km can still be achieved based on the dual-polarization LLO CV-QKD system under the theoretical model we established in Section 3.