quadratic form $$\begin{aligned}
H (t) = \sum_{\alpha \beta} Z^{\dagger}_{\alpha} {\bf H}_{\alpha \beta} (t) Z_{\beta}, \label{q-ham}\end{aligned}$$ where the Hamiltonian matrix is $$\begin{aligned}
 {\bf H}_{\alpha \beta} (t) = \begin{pmatrix} \delta_{\alpha \beta} & 0\\ 0 & \omega_{\alpha}^2 \delta_{\alpha \beta} \end{pmatrix} \label{E-hmat}\end{aligned}$$ for the electric field, and it is $$\begin{aligned}
{\bf H}_{\alpha \beta} (t) = \begin{pmatrix} \delta_{\alpha \beta} & \Omega_{\alpha \beta} \\ -\Omega_{\alpha \beta} & \omega_{\alpha}^2 \delta_{\alpha \beta}\end{pmatrix} \label{B-hmat}\end{aligned}$$ for the magnetic field. Both matrices (\[E-hmat\]) and (\[B-hmat\]) are Hermitian, ${\bf H}^{\dagger} = {\bf H}$, and also make the Hamiltonian (\[q-ham\]) Hermitian.

The quantization of the Hamiltonian (\[q-ham\]) follows from the commutation relations $$\begin{aligned}
[\hat{\phi}_{\alpha}, \hat{\pi}_{\beta} ] = i \delta_{\alpha \beta}, \quad [\hat{\phi}^*_{\alpha}, \hat{\pi}^*_{\beta} ] = i \delta_{\alpha \beta},\end{aligned}$$ and all other commutators vanish. In other words, the commutation relations hold in the extended phase space such that $$\begin{aligned}
[\hat{Z}_{\alpha}, \hat{Z}^*_{\beta} ] = \begin{pmatrix} 0& -i \delta_{\alpha \beta} \\ i \delta_{\alpha \beta} & 0 \end{pmatrix} , \quad [\hat{Z}_{\alpha}, \hat{Z}_{\beta} ] = [\hat{Z}^*_{\alpha}, \hat{Z}^*_{\beta} ] = \begin{pmatrix} 0& 0 \\ 0 & 0 \end{pmatrix}.\end{aligned}$$ Then, the quantum evolution is governed by the time-dependent Schrödinger equation $$\begin{aligned}
i \frac{\partial}{\partial t} \vert \Psi (t) \rangle = \sum_{\alpha \beta}  \hat{Z}^{\dagger}_{\alpha} {\bf H}_{\alpha \beta} (t) \hat{Z}_{\beta} \vert \Psi (t) 