the annihilation and creation operators for the qubit modes in the final frame by $\hat{\phi}_{i}=\sqrt{\frac{\hbar Z_{i}}{2}}(\hat{b}_{i}+\hat{b}_{i}^{\dagger})$ for $1\leq i\leq N$ and noting that the characteristic impedance $Z_{i}$ for the qubit mode $i$ is $Z_{i}=1/\omega_{i}$ in that frame

$$\begin{aligned}
J_{ij} & = & \frac{1}{2}\sqrt{Z_{i}Z_{j}}\left(\widetilde{\mathbf{M}}_{1}\right)_{ij}\nonumber \\
 & = & -\frac{1}{4}\sqrt{\frac{\omega_{i}\omega_{j}}{L_{i}L_{j}}}\mathrm{Im}\left[\frac{Z_{ij}\left(\omega_{i}\right)}{\omega_{i}}+\frac{Z_{ij}\left(\omega_{j}\right)}{\omega_{j}}\right]\end{aligned}$$

$J_{ij}$ in the above formula is in the units of radians per second. We note that this formula takes into account all the modes(possibly infinite) that might be present in the electromagnetic structure coupling the qubits.

Example 1: Two transmons coupled through a single mode $LC$ resonator bus
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![\[fig:2-Q-1-bus\]Example circuit of two transmons capacitively coupled through a single mode bus. Both transmons have the same shunting capacitance $C_{q}$ and the same coupling capacitances $C_{c}$ to the bus.](figs/2-tranmon-1-single-mode-bus)

In this section we will apply the formula in Eq. $\eqref{eq:The-impedance-formula-for-J}$ for the $J$-coupling rate derived in the previous section to the simple circuit of two transmons coupled through a lumped $LC$ resonator as shown in Fig. and compare it to the expression derived in [@Jay-Juelich]:

$$J=\frac{g_{1}g_{2}\left(\omega_{1}+\omega_{2}-2\omega_{r}\right)}{2\left(\omega_{1}-\omega_{r}\right)\left(\omega_{2}-\omega_{r}\right)}\label{eq:Jpert}$$

where $g_{1}$, $g_{2}$ are couplings of qubits $1,2$ to the bus, $\omega_{1}$, $\omega_{2}$ and $\omega_{r}$ are qubit and resonator frequencies; respectively.

The circuit in Fig. has the 