transitions among Landau levels as shown in Eq. (\[ll tr\]). In the non-relativistic theory, a charged particle in time-dependent magnetic fields similarly has a finite number of coupled oscillators, to which the quantum invariant method has been applied [@lewis-riesenfeld69; @fiore-gouba11] (for coupled, time-dependent oscillators, see also Refs. [@leach77; @ji-hong98; @dodonov00; @lee01]). The infinite degrees of freedom for the charged field necessarily involve renormalizing physical quantities such as the vacuum energy and the charge, in strong contrast to finite degrees of the freedom for the particle.

The quantum motion of the charged field may be classified according to the relative ratio $||H_{\rm O}|| / ||H_{\rm D}||$ for some appropriate measure. For instance, a measure has been introduced that the integrated rate of the change of Landau levels to the dynamical phase for any time interval [@kim14a] $$\begin{aligned}
{\cal R}_{\alpha} =  \frac{\int_{t_a}^{t_b} dt' |\Omega_{\alpha \beta} (t')|}{ \int_{t_a}^{t_b} dt' \omega_{\alpha} (t')},\end{aligned}$$ classifies the quantum motions (states) into the adiabatic change, the sudden change, and the nonadiabatic change. Now, these motions are classified by the relative ratio: (i) the adiabatic change when $||H_{\rm D}|| \gg  ||H_{\rm O}||$, static fields being an extreme limit of $||H_{\rm O}|| =0$, (ii) the sudden change when $||H_{\rm D}|| 