# Locating Two-Level Systems in a Superconducting Xmon Qubit
## 3 Time and Frequency Domain Analyses  
In this work, we conduct statistical analyses that are commonly used in frequency metrology [40]. We investigate both the Allan deviation and the power spectrum density of \(T_{1}\) and \(\Delta\omega/2\pi\) fluctuations in Figure 2c, respectively, and the results are displayed in Figure 3.  
The Allan deviation is a commonly used parameter in time-domain analysis, which helps to determine the nature of the stochastic processes responsible for data noise [41]. The slope of the Allan deviation, which corresponds to different kinds of noise, can be fitted to  
\[\sigma(\tau)=(\frac{h_{0}}{2})^{\frac{1}{2}}\tau^{-\frac{1}{2}}+(2 ln(2)h_{-1}) ^{\frac{1}{2}}+(\frac{4\pi^{2}}{6}h_{-2})^{\frac{1}{2}}\tau^{\frac{1}{2}}+ \frac{A\tau_{0}}{\tau}(4e^{-\frac{z}{z_{0}}}-e^{-\frac{2\pi}{z_{0}}}-3+2\frac {\tau}{\tau_{0}})^{\frac{1}{2}}, \tag{2}\]  
where the first three terms represent white noise, \(1/f\) noise, and random walk noise, respectively, and \(h_{i}\) (\(i=0,-1,-2\)) is the corresponding noise amplitude [42]. The last term stands for the exponentially correlated (Lorentzian) noise, which is the only noise process that can explain the single peak in the Allan analysis, where \(A\) is the noise amplitude and \(\tau_{0}\) is the characteristic timescale [31]. In Figure 3a, we model the \(T_{1}\) fluctuations by two  
Figure 2: (**a**) Time-domain measurement sequences: Simultaneous acquisition of \(T_{1}\) relaxation and \(T_{2}^{R}\) Ramsey measurements is achieved by interleaving single pulse sequences. The time \(\Delta t_{i}\) is the free evolution time. (**b**) Spectrally and temporally resolved \(T_{1}\). This dataset comprises 6480 \(T_{1}\) measurements, spanning 230 MHz and 21 h with 0.005 V and 16 min step sizes, respectively. (**c**) An interleaved series of 4700 \(T_{1}\) relaxation and \(T_{2}^{R}\) Ramsey measurements, spanning 85 h with a 65 s step size.  
Lorentzians with a white noise floor, and we obtain correlation times of approximately \(2\times 10^{3}\) s and \(3\times 10^{4}\) s. For the \(\Delta\omega/2\pi\) Allan deviation in Figure 3c, the correlation times are the same as the \(T_{1}\) fluctuations but with different amplitudes. Considering that the typical quasiparticle tunneling rate in transmons is in the range of 0.1 kHz to 30 kHz, we conclude that the two Lorentzians correspond to two TLSs.  
Another typical frequency domain analysis technique is the power spectral density (PSD) [43]. Similar to Equation (2), the noise PSD can also be represented by  
\[S(f)=p_{0}+\frac{p_{-1}}{f}+\frac{p_{-2}}{f^{2}}+\frac{2B}{\pi}\frac{\sigma}{4( f-f_{0})^{2}+\sigma^{2}}, \tag{3}\]  
where the four terms represent white noise, \(1/f\) noise, random telegraph noise, and exponentially correlated (Lorentzian) noise [42], with \(p_{i}\) (\(i=0,-1,-2\)) denoting the noise amplitude for the first three terms. \(\sigma\) is the full width at half of the maximum (FWHM) of the Lorentzian, and \(f_{0}\) is the characteristic frequency. Similar to the results of Allan variance, the PSD of the \(T_{1}\) and \(\Delta\omega/2\pi\) fluctuations can also be modeled by two Lorentzians of the same characteristic frequencies but different FWHMs, with \(1/f\) and a white noise floor. Again, it is confirmed that there are two TLSs affecting the qubit.