by Eq.

$$m_{id}=C_{p(d)}\mathrm{Im}\left[Z_{i,p(d)}\left(\omega_{i}\right)\right]/\sqrt{L_{i}}\label{eq:mij-final-frame}$$

where $m_{id}$ is the coupling of the bath due to the $d$-th drive line to the qubit mode $i$.

We need to now compute the spectral densities of the baths corresponding to the transmission lines. $\bar{\mathbf{C}}_{Z}\left(\omega\right)$ matrix defined in Eq. is also a scalar in the case of a single bath corresponding to the $d$-th drive line and is given by

$$\bar{C}_{Z,d}\left(\omega\right)=-\frac{i\omega Z_{0}}{1+i\omega C_{p(d)}Z_{0}}$$

Kernel of the bath due to the $d$-th drive line is given in Eq. (35) of [@Burkard] as

$$K_{d}\left(\omega\right)=\frac{\bar{C}_{Z,d}\left(\omega\right)}{1+\bar{\mathbf{m}}_{d}^{T}\mathbf{C}^{-1}\bar{\mathbf{m}}_{d}\bar{C}_{Z,d}\left(\omega\right)}$$

The term $\bar{\mathbf{m}}_{d}^{T}\mathbf{C}^{-1}\bar{\mathbf{m}}_{d}$ can be evaluated in the final frame using Eq. and noting that $\mathbf{C=1}$ in the final frame. Hence

$$\bar{\mathbf{m}}_{d}^{T}\mathbf{C}^{-1}\bar{\mathbf{m}}_{d}=C_{p(d)}^{2}\underset{i}{\sum}\left(\mathrm{Im}\left[Z_{i,p(d)}\left(\omega_{i}\right)\right]\right)^{2}/L_{i}$$

The spectrum of the bath is given by

$$\begin{aligned}
J_{d}\left(\omega\right) & = & -\mathrm{Im}\left[K_{d}\left(\omega\right)\right]\nonumber \\
 & = & \frac{\omega Z_{0}}{1+\omega^{2}Z_{0}^{2}\left(C_{p(d)}+\bar{\mathbf{m}}_{d}^{T}\mathbf{C}^{-1}\bar{\mathbf{m}}_{d}\right)^{2}}\nonumber \\
 & \simeq & \frac{\omega Z_{0}}{1+\omega^{2}Z_{0}^{2}C_{p(d)}^{2}}\end{aligned}$$

assuming $\left(\bar{\mathbf{m}}_{j}^{T}\mathbf{C}^{-1}\bar{\mathbf{m}}_{j}\right)/C_{p(d)}\ll1$ which holds for typical parameter values and frequencies.

Finally $T_{1}$ rate of the qubit mode $i$ due to the $d$-th drive line can be calculated using Eq. (44) of [@Burkard]

$$\begin{aligned}
\frac{1}{T_{1}^{i,d}} & = & \frac{4}{\hbar}\left|\left\langle 0\right|m_{id}\mathbf{\hat{Q}}_{i}\left|1_{i}\right\rangle \right|^{2}J_{d}\left(\omega_{i}\right)\coth(\frac{\hbar\omega_{i}}{2k_{B}T})\end{aligned}$$

which can be simplified assuming $\coth(\frac{\hbar\omega_{i}}{2k_{B}T})\simeq1$ for the typical chip temperatures as

$$\begin{aligned}
\frac{1}{T_{1}^{i,d}} & =\frac{4}{\hbar}\left|\left\langle 0\right|m_{id}\mathbf{\hat{Q}}_{i}\left|1_{i}\right\rangle \right|^{2}J_{d}\left(\omega_{i}\right)\nonumber \\
 & =\frac{2}{L_{i}}\mathrm{Im}\left[Z_{i,p(d)}\left(\omega_{i}\right)\right]^{2}\frac{\omega_{i}^{2}Z_{0}C_{p(d)}^{2}}{1+\omega_{i}^{2}Z_{0}^{2}C_{p(d)}^{2}}\label{eq:T1-exp-long-1}\end{aligned}$$

![\[fig:Transmon-readout-line\]Example circuit of a Transmon 