\varphi_{\alpha} (\hat{\pi}^*_{\alpha} + \hat{\pi}_{\alpha}) - \dot{\varphi}_{\alpha} (\hat{\phi}^*_{\alpha} + \hat{\phi}_{\alpha}) \Bigr] = \frac{1}{\sqrt{2}} \bigl( \hat{a}^{\dagger}_{\alpha} (t) + \hat{b}^{\dagger}_{\alpha} (t)  \bigr). \label{B-in}\end{aligned}$$ Here, the auxiliary field $\varphi_{\alpha}$ for $\alpha = (n, k_z)$ is a complex solution to the mode equation $$\begin{aligned}
\ddot{\varphi}_{\alpha} (t) + \omega^2_{\alpha} (t) \varphi_{\alpha} (t) = 0 \label{aux sol2}\end{aligned}$$ with $\omega^2_{\alpha} = |qB| (2n+1) + m^2 + k^2_z$ and ${\rm Wr} [\varphi_{\alpha}, \varphi^*_{\alpha} ] = i$. Then, the diagonal Hamiltonian becomes $$\begin{aligned}
H_{\rm D} = \sum_{\alpha} \Bigl[ \bigl( \dot{\varphi}^*_{\alpha} \dot{\varphi}_{\alpha} + \omega^2_{\alpha} \varphi^*_{\alpha} \varphi_{\alpha} \bigr) \bigl(\hat{a}^{\dagger}_{\alpha} \hat{a}_{\alpha} + \hat{b}_{\alpha}\hat{b}^{\dagger}_{\alpha} \bigr) + \bigl( \dot{\varphi}^{*2}_{\alpha} + \omega^2_{\alpha} \varphi^{*2}_{\alpha} \bigr) \hat{a}^{\dagger}_{\alpha} \hat{b}^{\dagger}_{\alpha} + \bigl( \dot{\varphi}^{2}_{\alpha} + \omega^2_{\alpha} \varphi^{2}_{\alpha} \bigr) \hat{a}_{\alpha} \hat{b}_{\alpha} \Bigr], \label{diag2}\end{aligned}$$ and the off-diagonal Hamiltonian takes the form $$\begin{aligned}
H_{\rm O} = \sum_{\alpha \beta} \Bigl[ \bigl(  \dot{\varphi}^*_{\alpha} \varphi_{\beta} -  \varphi^*_{\alpha} \dot{\varphi}_{\beta} \bigr) \Omega_{\alpha \beta} \bigl( \hat{a}^{\dagger}_{\alpha} \hat{a}_{\alpha} + \hat{b}_{\alpha}\hat{b}^{\dagger}_{\alpha} \bigr)
+ \bigl( \dot{\varphi}^*_{\alpha} \varphi^*_{\beta} - \varphi^*_{\alpha} \dot{\varphi}^*_{\beta} \bigr) \Omega_{\alpha \beta} \hat{a}^{\dagger}_{\alpha} \hat{b}^{\dagger}_{\beta}
+ \bigl( \dot{\varphi}_{\alpha} \varphi_{\beta} - \varphi_{\alpha} \dot{\varphi}_{\beta} \bigr) \Omega_{\alpha \beta} \hat{a}_{\alpha} \hat{b}_{\beta} \Bigr]. \label{off2}\end{aligned}$$ The invariant operators (\[B-in\]) satisfy Eq. (\[ln eq\]) for the diagonal Hamiltonian (\[diag2\]) when $\varphi_{\alpha}$ is the solution to Eq. (\[aux sol2\]).

In the adiabatic change, we find the quantum states for the unperturbed 