## 4 Simulation
### Relaxation Time \(T_{1}\)  
Spontaneous relaxation of the qubit from the excited state to the ground state has a lifetime \(T_{1}\) to which contribute two main phenomena: losses in the dielectric media around the qubit, (corresponding to an intrinsic lifetime \(T_{int}\)), and losses via a coupling to the cavity (corresponding to a lifetime \(T_{purcell}\)):  
\[T_{1}^{-1}=T_{int}^{-1}+T_{purcell}^{-1} \tag{15}\]  
The losses over the cavity due to the Purcell effect are given by [18]:  
\[T_{purcell}=\frac{\Delta^{2}}{g_{01}^{2}\kappa} \tag{16}\]  
where \(\Delta=|\omega_{r}-\omega_{q}|\) is the difference in resonance frequency of the cavity and qubit and \(\kappa=\omega_{r}/2Q_{cav}\) is the decay rate of the resonator where \(Q_{cav}\) is the quality factor of the cavity. The dielectric losses can be described in terms of an intrinsic quality factor \(Q_{int}=\omega_{q}T_{1}^{int}\) of the transmon with contributions from different spatial regions \(i\) such as the silicon substrate and the layers of oxide on the surfaces [31]:  
\[Q_{int}^{-1}=\sum_{i}P_{i}tan(\delta_{i}). \tag{17}\]  
where \(P_{i}\) are the participation ratios and \(tan(\delta_{i})\) are the material-specific loss tangents. Due to a much larger loss tangent of the surface oxide layers compared to the silicon substrate and aluminium [31], only the surface layers were considered in our analysis. As  
Figure 8: Scheme of the capacitances and the charge density distribution of the transmon qubit. The two rectangles represent the upper pad (**up**) and the bottom pad (**down**) of the qubit. The charge of the pads is \(q_{up}=-q_{down}=q\) for the symmetry of the system, with a charge distribution \(\rho(\vec{r})=\rho_{up}(\vec{r})+\rho_{down}(\vec{r})\). The green rectangle represents an infinite ground plane. The capacitances between the pads and this plane are \(C_{\text{up}}\) and \(C_{\text{down}}\), while the capacitance between the two pads is \(C_{\text{pads}}\). they are only about \(5\,\mathrm{nm}\) thick, the electric field can be approximated to be constant over their thickness. The participating ratios are therefore given by:  
\[P_{i}=\frac{\epsilon_{0}\epsilon_{i}}{2W}\int_{A_{i}}d\vec{r}\,\tau_{i}|E(\vec{r })|^{2}=\frac{\epsilon_{0}\epsilon_{i}\tau_{i}C_{tot}}{q^{2}}\int_{A_{i}}d\vec{r }\,|E(\vec{r})|^{2} \tag{18}\]  
where \(\epsilon_{i}\) are the dielectric constants, \(\tau_{i}\) is the thickness of the layer, and \(A_{i}\) is the layer surface. The participation ratio is normalized by the total capacitor energy \(W=\frac{q^{2}}{2C_{tot}}\). The \(E(\vec{r})\) is the electric field on the surface \(A_{i}\) that is the sum of a parallel and a perpendicular component \(E(\vec{r})^{2}=E(\vec{r})_{\parallel}^{2}+E(\vec{r})_{\perp}^{2}\). The surface layers considered are the metal-air (MA), metal-substrate (MS), and substrate-air (SA). The following contributions can be derived [32] (Aluminium oxide: \(\epsilon_{\text{MS}}=\epsilon_{\text{MA}}=9.8\) Silicon dioxide: \(\epsilon_{\text{SA}}=3.8\) Silicon substrate: \(\epsilon_{\text{S}}=11.8\) Loss tangent: \(\delta_{i}=0.002\) Thickness: \(t_{i}=5\,\mathrm{nm}\). The value of the \(\epsilon_{i}\) are standard and taken from [31], the loss tangent is taken from [32], and the value of the thickness is an approximation):  
\[P_{\text{MS}} =\frac{\epsilon_{0}\epsilon_{S}^{2}}{\epsilon_{\text{MS}}}I_{ \text{MS}}\frac{C_{tot}}{q^{2}}\int_{MS}d\vec{r}\,|E_{\text{S}\perp}|^{2} \tag{19}\] \[P_{\text{MA}} =\frac{\epsilon_{0}}{\epsilon_{\text{MA}}}I_{\text{MA}}\frac{C_ {tot}}{q^{2}}\int_{MA}d\vec{r}\,|E_{0\perp}|^{2}\] (20) \[P_{\text{SA}} =\epsilon_{0}t_{\text{SA}}\frac{C_{tot}}{q^{2}}(\epsilon_{\text{ SA}}\int_{SA}d\vec{r}\,|E_{0\parallel}|^{2}+e_{\text{SA}}^{-1}\int_{SA}d\vec{r}\,|E_{0 \perp}|^{2}) \tag{21}\]  
where \(E_{0}\) is the electric field in the air and \(E_{S}\) is the electric field in the substrate while considering the dielectric constant of air as \(\epsilon=\epsilon_{0}\).