 we expand the exponential and Laguerre in Eq. (\[auxf1\]) up to second order in $\eta$ to find $$\begin{aligned}
                                                                         \label{fauxap}
\left|{f_n^m}\right|^{2} &\approx& \frac{(n+m)!}{n!m!^2} 
\left[1 - \frac{2n+m+1}{m+1} \eta^2 \right]{\eta^{2m}}.         \end{aligned}$$ To obtain this expression we used $d L_n^m(x)/d x = - L_{n-1}^{m+1} (x)$ and $L_n^m(0) = (n+m)!/(n!m!)$ [@gradshteyn]. Now, by keeping just terms up to $\eta^2$ in $\left|f_n^m\right|^{2}$, one gets $$\label{fauxap2}
\left|f_n^m\right|^{2} \approx [1 - (2n+1) \eta^2]  \delta_{m 0 } + 
(n+1) \eta^2 \delta_{m 1}.$$ Terms with $m\ge 2$ appear only in higher powers of $\eta$. Notice that for $m = 1$, $\left|f_n^1\right|^{2} \to 0$ and $\mathcal{Z}_{\pm}(\lambda_f) \to \mathcal{Z}(\lambda_i)$ as $\eta \to 0$, whhich makes $\mathcal {L} \to 0$. On the other hand, $\left|f_n^0\right|^{2}$ in Eq. (\[fauxap2\]) is a concave function of $\eta$ with $\lim_{\eta\to 0}\left |f_n^0\right|^{2}=1$, $\forall n$. Consequently, $\mathcal{L}\neq 0$ as $\eta\to 0$. All these features can be seen from Fig. \[fig1L1\]. For $m > 1$, only higher order terms in $\eta$ contribute to $|f_n^m|$, forcing $\mathcal Z_{\pm}(\lambda_f) \to \mathcal Z(\lambda_i)$ as $\eta \to 0$, just like what happens when $m=1$. The physical explanation for the distinct behavior found in the carrier transition $m=0$ lies in the system Hamiltonian after and before laser application. From Eq. (\[Omegaux1\]), one can see that $$\label{etalim}
\lim_{\eta \to 0} \hat{\Omega}_{m}^\pm = \frac{\Omega}{2} \delta_{m 0}{{\sf 1 \hspace{-0.3ex} \rule{0.1ex}{1.52ex}\rule[-.01ex]{0.3ex}{0.1ex}}},$$ where ${{\sf 1 \hspace{-0.3ex} \rule{0.1ex}{1.52ex}\rule[-.01ex]{0.3ex}{0.1ex}}}$ is the identity operator for the center-of-mass motion. By taking Eqs. (\[hamct\]) and (\[etalim\]) into account, it follows that, when $m=0$, the laser is able to drive transitions between the two electronic states, even when $\eta = 0$. In other words, the pre-