# Measurement of Quasiparticle Diffusion in a Superconducting Transmon Qubit
## 4 Quasiparticle Diffusion in the Superconducting Qubit
(**b**) \(\Gamma_{\downarrow}\) as a function of diffusion time fitted by the modified model with different injection powers. (**c**) The diffusion time corresponding to the fitted \(\Gamma_{\downarrow}^{max}\) with (without) the existence of “hot” quasiparticles. (**d**) \(\Gamma_{\downarrow}^{max}\) corresponding to different injection powers when measured with injection durations of 3 and 4.5 μs. The experimental data and simulated results are represented by circles and dotted lines, respectively. density of the quasiparticles. With the bath temperature of T = 30 mK, quasiparticles with the average energy \(E_{qp}=\Delta+kT\) have the same velocity, resulting in the same diffusion feature. Therefore, the diffusion time corresponding to \(\Gamma^{max}_{\downarrow}\) should be the same with different injection powers (see the blue line in Figure 3c). Therefore, we modify the original model by introducing the \"hot\" quasiparticles with the average energy of \(E_{qp}=\Delta+E_{ge}\). Considering the phenomenon in which \(\Gamma_{\uparrow}\) is independent of the injection power, the density of quasiparticles can be phenomenologically written as:  
\[\chi_{qp}(p,t_{diff})=\chi^{c}_{qp}(p,t_{diff})e^{-\Gamma^{c}_{d}t_{diff}}+\chi ^{h}_{qp}(t_{diff})e^{-\Gamma^{h}_{d}t_{diff}}+\chi^{0}_{qp}. \tag{5}\]  
where p represents the injection power and \(\chi^{0}_{qp}\) represents the quasiparticles that are present when no quasiparticles are injected, and the system is in thermal equilibrium. We assume that the \"hot\" quasiparticles are independent of the injection power. With the presence of \"hot\" quasiparticles, the fitted diffusion time corresponding to the \(\Gamma^{max}_{\downarrow}\) changes with different injection powers (see the gray line in Figure 3c), which is consistent with the experimental data in Figure 3a. Furthermore, the decay rates of the quasiparticles \(\Gamma_{d}\) are also introduced to fit the sharply descending trace on the right side of \(\Gamma^{max}_{\downarrow}\). Here, we assume that recombination and vortex trapping [26; 27] are important channels for the decay of quasiparticles. From our model, with the measurement errors taken into account, we extracted the decay rates of the \"cold\" and \"hot\" quasiparticles, which were \(\Gamma^{c}_{d}=1/140\) and \(\Gamma^{h}_{d}=1/350\) us, respectively. With the injection powers of 4, 7, 10, and 13 dB, the fitting \(C^{c}_{0}\) for \"cold\" quasiparticles in Equation (4) was \(0.75\times C_{ini},1.8\times C_{ini},2.9\times C_{ini},and4.1\times C_{ini}\), where \(C_{ini}=[2.8\times 10^{-3},2.95\times 10^{-3}]\) for the injection durations of 3 and 4.5 us, and \(C^{h}_{0}\) of the \"hot\" quasiparticles is \(2\times 10^{-3}\). Figure 3d demonstrates that the fitting results are in good agreement at different injection powers. The deviation between the experimental data and the theoretical fitting could be caused by the following reasons: (1) the transitions between \"hot\" and \"cold\" quasiparticles during diffusion; (2) the use of the quasiparticles\" average energies instead of the integral of the energies when calculating the normalized quasiparticle diffusion velocities; (3) recombination and vortex trapping were not taken into account in our model in order to simplify the dynamic features of quasiparticles.