# Locating Two-Level Systems in a Superconducting Xmon Qubit
## 4 TLS Location  
We can explain the fluctuations in Figure 2c using the interacting TLS model, where defects can not only interact with the qubit but also mutually interact with themselves [17], as shown in Figure 4c. If the transition energy of a TLS is below or near the thermal level \(\mathrm{k_{B}T}\), it undergoes stochastic state switching that is thermally activated. Longitudinal coupling \(g_{T}\) between the TLS with high transition energy near the qubit frequency (labeled as 'TLS1') and TLSs with energies below the thermal level causes telegraphic fluctuation  
Figure 3: (**a**,**c**) Allan deviations of \(T_{1}\) fluctuations and frequency shift (\(\Delta\omega/2\pi\)) fluctuations in Figure 2c, respectively. (**b**,**d**) PSDs of \(T_{1}\) fluctuations and frequency shift (\(\Delta\omega/2\pi\)) fluctuations in Figure 2c, respectively. All the plots feature a red line indicating the overall noise which comprises \(1/f\) noise (orange lines), two different Lorentzians (blue lines), and white noise (magenta lines). or spectral diffusion of the TLS1 frequency. This temporal frequency fluctuation of TLS1 results in the qubit parameters variations. In cases where the frequency of TLS1 is close to the qubit frequency, the coupling strength \(g\) becomes larger than \(g_{T}\), and we can simplify the model to two-qubit resonant coupling, where energy swap between the TLS1 and the qubit will happen. To locate the TLSs, we re-examine the \(T_{1}\) raw data. Typically, the qubit decays exponentially to the state \(|0\rangle\) due to spontaneous emission as the delay after the \(X_{\pi}\) pulse increases. However, we find two types of revival oscillations as shown in Figure 4a,b. All the other qubits in this device are tuned far away from the operation qubit, so the interaction between the qubits can be neglected. For quasiparticles, we discuss in the previous section that the quasiparticle tunneling rate is higher than the characteristic frequency of Allan deviation. The effect of quasiparticles on the qubit relaxation time usually remains in the form of exponential decay [13]. Both phase and flux qubits have also been found to exhibit these revivals [21,24], which were explained by coherently coupled TLSs residing in the qubit junctions. Accordingly, the revival oscillations are attributed to the energy exchange between the qubit and the two-level systems. The coupling strength \(g\) between a single TLS and the qubit can be extracted by fitting the oscillation to  
\[\langle\sigma_{Z}(t)\rangle=\langle\sigma_{Z}\rangle_{\infty}+a_{1}e^{-\Gamma_ {1}t}+a_{2}e^{-\Gamma_{2}t}+a_{\text{osc}}\cos{(2\pi ft+\phi)}e^{-\Gamma_{ \text{osc}}t}, \tag{4}\]  
Figure 4: (**a**) A dataset with a sinusoidal decay profile. (**b**) Another dataset with a much larger oscillation frequency. Resonant exchange with TLSs is the reason for the emergence of revivals. (**c**) Illustration of the interacting TLS model. The interaction between the TLS1 and TLSs with energies at or below \(\text{k}_{\text{B}}\text{T}\) (blue area) leads to frequency fluctuations of TLS1, which are then transformed into the qubit parameters fluctuations through coupling \(g\). (**d**) Electron microscope image of the qubit’s SQUID. The dimensions are indicated in the image. The red and green concentric circles have radii of 0.27 \(\upmu\)m and 1.2 \(\upmu\)m, respectively, indicating the locations of the TLSs. The insert depicts a schematic cross-sectional view of the junction and its native aluminum oxide layer that contains structural TLS (not depicted to scale). In addition to this, surface defects can also arise from fabrication residuals, atmospheric contaminants, and substrate surface amorphization resulting from circuit patterning.