## 2 Adiabatic Evolution of the Fock States  
The adiabatic (quasiclassical) approximate complex solution to Equation (7) with \(\omega(t)>0\) can be written in the form  
\[\varepsilon(t)\approx[\omega(t)]^{-1/2}e^{i\hat{\varphi}(t)},\quad\hat{ \varepsilon}(t)\approx i[\omega(t)]^{1/2}e^{i\hat{\varphi}(t)},\qquad\tilde{ \varphi}(t)=\int_{t_{i}}^{t}\omega(z)dz. \tag{11}\]  
In this approximation, the solution (6) assumes the form (3) with the instantaneous value of function \(\omega(t)\). This result agrees with the famous Born-Fock adiabatic theorem [15]. The mean energy evolves according to Equation (5) with \(\omega=\omega(t)\). In particular, all states with \(m\geq 0\) have the same mean energy \(\hbar\omega(t)(1+2n_{r})\) if \(\omega(t)>0\). The magnetic moment operator has the form [24; 25; 26; 27; 28; 29; 30; 31; 32]  
\[\hat{\mathcal{M}}=\frac{e}{2c}\big{(}\hat{x}\hat{\pi}_{y}-\hat{y}\hat{\pi}_{x }\big{)},\qquad\hat{\pi}_{j}\equiv\hat{p}_{j}-eA_{j}(\hat{x},\hat{y}). \tag{12}\]  
For the \"circular\" gauge of the vector potential, \(\mathbf{A}=B(-y,x)/2\), we can write  
\[\hat{\mathcal{M}}=\Big{[}\hat{x}\hat{p}_{y}-\hat{y}\hat{p}_{x}-\omega\Big{(} \hat{x}^{2}+\hat{y}^{2}\Big{)}\Big{]}e/(2c). \tag{13}\]  
The magnetic moment operator does not commute with Hamiltonian (2):  
\[\big{[}\hat{H},\hat{M}\big{]}=i\omega\mu_{B}\big{(}\hat{x}\hat{\pi}_{x}+\hat{ \pi}_{x}\hat{x}+\hat{y}\hat{\pi}_{y}+\hat{\pi}_{y}\hat{g}\big{)}, \tag{14}\]  
where \(\mu_{B}=e\hbar/(2c)\) is the Bohr magneton (for the unit mass chosen in this paper). The mean value \(\langle r^{2}\rangle\equiv\langle x^{2}+y^{2}\rangle\) in the state (6) is given by a simple formula (see Appendix A)  
\[\langle r^{2}\rangle_{n_{r},m}=[K(t)]^{-1}(2n_{r}+|m|+1). \tag{15}\]Consequently, the mean value of the magnetic moment in the Fock state (6) equals  
\[\langle\mathcal{M}\rangle_{n_{r},m}=\mu_{B}\Big{[}m-\omega(t)|\varepsilon(t)|^{2} (2n_{r}+|m|+1)\Big{]}. \tag{16}\]  
This is not an eigenvalue of the magnetic moment operator, since the spectrum of this operator is continuous [33]. Formula (16) is interesting because it shows that the _mean value_ of the magnetic moment in the energy eigenstate (when \(\omega=const>0\) and \(\omega|\varepsilon|^{2}=1\)) is proportional to the energy eigenvalue, although non-commuting operators \(\hat{H}\) (2) and \(\hat{M}\) (13) have quite different structures and spectra. Fluctuations of the magnetic moment, characterized by the variance \(\sigma_{M}=\langle\hat{\mathcal{M}}^{2}\rangle-\langle\hat{\mathcal{M}} \rangle^{2}\), can be calculated with the aid of formula (see Appendix A)  
\[\langle r^{4}\rangle_{n,m}=[K(t)]^{-2}[6n_{r}(n_{r}+|m|+1)+(|m|+1)(|m|+2)]. \tag{17}\]  
The result is  
\[\sigma_{M}^{(n,m)}=\Big{[}\mu_{B}\omega(t)|\varepsilon(t)|^{2}\Big{]}^{2}[2n _{r}(n_{r}+|m|+1)+|m|+1]. \tag{18}\]  
We see that fluctuations of the magnetic moment can be strong in the initial energy eigenstates (when \(\omega|\varepsilon|^{2}=1\)) with large values of the quantum numbers \(n_{r}\) and \(|m|\). However, they are strong in the ground state (\(n_{r}=m=0\)) as well: \(\sigma_{M}^{(0,0)}=[\langle\mathcal{M}\rangle_{0,0}]^{2}\). In the adiabatic regime (11), the mean magnetic moment does not depend on time, providing an example of adiabatic invariants:  
\[\langle\mathcal{M}\rangle_{n_{r},m}=\mu_{B}[m-(2n_{r}+|m|+1)]=-E_{n_{r},m}(t)/ [\hbar\omega(t)]. \tag{19}\]  
However, the solution (11) holds under the condition  
\[|\omega|/\omega^{2}(t)\ll 1, \tag{20}\]  
which is clearly broken when \(\omega(t)=0\). Nonetheless, when the frequency slowly passes through zero value and slowly becomes not too small again, the conditions of the quasiclassical approximation are reestablished. We suppose that \(\omega=0\) at some instant \(t_{*}>t_{i}\).