# Measurement of Quasiparticle Diffusion in a Superconducting Transmon Qubit
## 4 Quasiparticle Diffusion in the Superconducting Qubit  
The transition rates \(\Gamma_{\uparrow}\) and \(\Gamma_{\downarrow}\) presented in Figure 3a were calculated with Equation (2). Note that \(\Gamma_{\uparrow}\) are an order of magnitude smaller than \(\Gamma_{\downarrow}\), which also almost overlap, indicating that the density of \"hot\" quasiparticles is independent of the injection power. Meanwhile, there is a positive correlation between \(\Gamma_{\downarrow}\) and the injection power, demonstrating that most of the generated quasiparticles are \"cold\" in this injection protocol. Now, we study the phenomenological energy distribution of the quasiparticles injected into the superconducting qubit. To simplify the calculation, the diffusion, which is clearly related to the energy of the quasiparticles, is taken into account to model the quasiparticles near the junction [21]  
\[\frac{\partial\chi_{qp}}{\partial t_{diff}}=D\
abla^{2}\chi_{qp}. \tag{3}\]  
Figure 2: Experimental impacts of injected quasiparticles. (**a**) State evolution of the energy relaxation time \(T_{1}\) measured by injecting the power of 7 dB. The green data were measured with a diffusion time of 50 \(\upmu\)s, while the red data were measured when \(t_{diff}=\)200 \(\upmu\)s. (**b**) Rabi oscillations observed with 7 dB of injection power and 50 \(\upmu\)s of diffusion time. The green line represents the reference Rabi oscillation, while the red line represents the signal Rabi oscillation. From the amplitudes of the Rabi oscillation, we can calculate the residual excited-state population [11]. (**c**) The energy relaxation time \(T_{1}\) and (**d**) residual excited-state population \(P_{\epsilon}^{r}\) were measured as a function of \(t_{diff}\). The four curves in each figure represent, respectively, the responses of the qubit induced by quasiparticle injection with injection powers of 4, 7, 10, and 13 dB. The yellow line represents the data without the quasiparticle injection. The data were measured with an injection duration of 3 \(\upmu\)s.  
In Equation (4), \(D=60v_{qp}\) cm\({}^{2}\)/s is the diffusion constant in aluminum, where the normalized velocity \(v_{qp}=(1-\Delta^{2}/E_{qp}^{2})^{1/2}\) depends on the quasiparticle's energy \(E_{qp}\)[5; 23]. The Gaussian solution of this diffusion equation can be written as  
\[\chi_{qp}(x,t_{diff})=\frac{C_{0}}{\sqrt{\pi Dt_{diff}}}exp(\frac{-x^{2}}{4D_{ diff}}), \tag{4}\]  
where \(C_{0}\) is the initial value of \(\chi_{qp}\) and \(t_{diff}\) represents the diffusion time. \(x\) is the distance over which the created quasiparticles diffuse from the outer side of the electrode to the junction [23]. For simplicity, diffusion is considered in only one dimension, and \(x=0.5\) mm in our sample. With Equations (1) and (4), we obtain the relationship between the diffusion time and \(\Gamma_{\downarrow}\). As shown in Figure 3a, the transition rates of the qubit extracted from the experiment change with the diffusion time. The four colored curves point to the four injection powers of 4, 7, 10, and 13 dB, respectively, and the maximum values, which are denoted as \(\Gamma_{\downarrow}^{max}\), corresponding to four injection powers are \(2.586\times 10^{-4}\), \(3.454\times 10^{-4}\), \(4.468\times 10^{-4}\), and \(5.22\times 10^{-4}\). With the increase in the injection power, \(\Gamma_{\downarrow}\) of the quasiparticle takes more time to reach \(\Gamma_{\downarrow}^{max}\). This contradicts the previous theoretical model, in which only the \"cold\" quasiparticles are considered. In this original model, the power is only related to the  
Figure 3: Quasiparticle diffusion in the superconducting qubit. (**a**) Transition rate as a function of diffusion time. The solid circles on the dotted (solid) line represent the up- and down-transition rates \(\Gamma_{\uparrow}\) (\(\Gamma_{\downarrow}\)) extracted from \(T_{1}\) and \(P_{e}^{r}\). The four colored curves point to the four injection powers. These data were measured with an injection duration of 3 μs.