the transpose of the mode equation in the vector form (\[B-eq\]), $$\begin{aligned}
\ddot{U}_{(\pm)} + 2 \dot{U}_{(\pm)} \Omega + U_{(\pm)} \bigl({\bf \omega}^2 + \Omega^2 + \dot{\Omega} \bigr) = 0. \label{B-teq}\end{aligned}$$ However, the merit of the quantum invariant method does not lie in solving the classical field equation, but lies in constructing diverse quantum states. In the two-component first-order formalism, the Cauchy problem has been reduced to solving Eq. (\[B-eq\]), whose solutions may not be known for generic magnetic fields, and requires a perturbation method. In a similar manner, finding the exact invariants (\[q-inv2\]) is equivalent to solving Eq. (\[B-teq\]) or (\[B-eq\]). Therefore, we need a systematic method, such as perturbation theory, in order to find the invariants or directly to solve the time-dependent Schrödinger equation.

Adiabatic Change {#sec5-1}
----------------

The field $\phi$ is complex due to the longitudinal motion along the magnetic field in the four-dimensional spacetime, so we may quantize the time-dependent oscillators (\[diag\]) as for electric fields in Sec. \[sec4\]. Hence, the invariant operators are $$\begin{aligned}
\hat{\cal A}_{\alpha} (t) &=& \frac{i}{\sqrt{2}} \Bigl[ \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}_{\alpha} (t) + \hat{b}_{\alpha} (t)  \bigr), \nonumber\\
\hat{\cal A}^{\dagger}_{\alpha} (t) &=& - \frac{i}{\sqrt{2}} \Bigl[ 