# Measurement of Quasiparticle Diffusion in a Superconducting Transmon Qubit
## 3 Quasiparticle Probe  
In the high-frequency regime \(\hbar\omega_{ge}\gg E_{qp}-\Delta\), the down-transition rate induced by quasiparticles \(\Gamma_{\downarrow}^{qp}\) for a single-junction transmon qubit is [5, 7]:  
\[\Gamma_{\downarrow}^{qp}=\frac{\omega_{p}^{2}}{\omega_{ge}\pi}\sqrt{\frac{2 \Delta}{E_{ge}}}\chi_{qp}, \tag{1}\]  
where \(\omega_{p}=\sqrt{8E_{C}E_{J}}\) is the plasma frequency and \(\chi_{qp}\) is the quasiparticle density per Cooper pair.  
When the qubit is only excited thermally, the residual excited-state population is \(P_{e}^{r}=e^{-\frac{\hbar\omega_{pe}}{k_{B}T}}/(e^{-\frac{\hbar\omega_{pe}}{k_ {B}T}}+1)=1.2\times 10^{-5}\). The measured residual excited-state population is \(P_{e}^{r}=0.1208\), corresponding to an effective qubit temperature of 174 mK, which is much higher than that of the bath environment of T = 30 mK. This phenomenon indicates that the qubit is also partly excited by other sources, such as the \"hot\" quasiparticles. Hence, despite the uncharted energy distribution of \"hot\" quasiparticles, it is deemed that the qubit excitation rate can be written in the implicit function of the quasiparticle density \(\chi_{qp}\), that is, \(\Gamma_{\uparrow}(\chi_{qp})\).  
Consequently, we measure the energy relaxation time \(T_{1}\) and the residual excited-state population \(P_{e}^{r}\) to study the quasiparticles in the superconducting qubit. Since the longitudinal relaxation rate \(\Gamma_{l}=1/T_{1}\), we can obtain both the up- and down-transition rates \(\Gamma_{\uparrow}(\Gamma_{\downarrow})\) from the measured \(T_{1}\) and \(P_{e}^{r}\), which can be written as [25]:  
\[\begin{split} T_{1}&=\frac{1}{\Gamma_{\downarrow}- \Gamma_{\uparrow}}.\\ P_{e}^{r}&=\frac{\Gamma_{\uparrow}}{\Gamma_{ \uparrow}+\Gamma_{\downarrow}}.\end{split} \tag{2}\]  
In the T = 30 mK environment, the qubit excitation rate in thermal equilibrium can be neglected, since \(\Gamma_{\uparrow}=e^{-\hbar\omega_{ge}/(k_{B}T)}\Gamma_{\downarrow}\approx 10^{ -23}\Gamma_{\downarrow}\); as a result, the relaxation rate can simplified as \(\Gamma_{\downarrow}=1/T_{1}\).  
In this work, we measure \(T_{1}\) and \(P_{e}^{r}\) with different densities of quasiparticles by adjusting the parameters of the injection pulses. The power and the pulse length of the applied microwave correspond to the density of the quasiparticles, while the time gap between the injection and measurement denotes the diffusion time \(t_{diff}\). As shown in Figure 2, the energy relaxation time and the residual excited-state population vary according to different parameters of injection. Figure 2a,b show the experimental data of \(T_{1}\) and \(P_{e}^{r}\), respectively. We measured the Rabi oscillation between \(|1\rangle\) and \(|2\rangle\). As shown in Figure 2b, the reference Rabi oscillation, Rabi\({}_{ref}\), is measured after a \(\pi_{g-e}\) pulse is applied to the qubit in the ground state, while the signal one, Rabi\({}_{igg}\), is measured with no front operation. With both traces in Figure 2b, we can obtain the accurate \(P_{e}^{r}\)[11]. In Figure 2c,d, \(T_{1}\) and \(P_{e}^{r}\) are plotted as a function of diffusion time, from which we can capture some features of the quasiparticle dynamics. With the increase in the injected microwave power, the qubit is affected by more quasiparticles, resulting in a larger decrease in \(T_{1}\). After sufficient diffusion time, \(T_{1}\) converges to the original value, which means that the energy relaxation time is influenced by quasiparticles. The experimental data in Figure 2 indicate that \(T_{1}\) and \(P_{e}^{r}\) are sensitive to quasiparticles in the superconducting qubit and can be used to characterize the features of quasiparticles.