# Giant Dipole Multi-Resonances Excited by High-Frequency Laser Pulses
## 2 Neutron-Proton Mixtures in an External Electromagnetic Field
From the same boundary condition, we are left with non-vanishing components only in the \(n=\pm 1\) channels of the density fluctuation (48),  
\[\mathcal{U}_{q}\equiv\frac{\delta\mathcal{H}_{\rm Sky}}{\delta\rho_{q}(\mathbf{r} )}=\frac{\partial\mathcal{H}_{\rm Sky}}{\partial\rho_{q}}-\
abla\cdot\frac{ \partial\mathcal{H}_{\rm Sky}}{\partial\
abla\rho_{q}}+\Delta\frac{\partial \mathcal{H}_{\rm Sky}}{\partial\Delta\rho_{q}} \tag{5}\]6  
The electric field component of the laser pulse, oriented along the \(z\)-axis, produces a dynamic polarization of the \(p-n\) \"plasma\", and therefore induces a non-vanishing dynamic dipole moment only along this direction  
\[\mathcal{U}_{q}\equiv\frac{\delta\mathcal{H}_{\rm Sky}}{\delta\rho_{q}(\mathbf{r} )}=\frac{\partial\mathcal{H}_{\rm Sky}}{\partial\rho_{q}}-\
abla\cdot\frac{ \partial\mathcal{H}_{\rm Sky}}{\partial\
abla\rho_{q}}+\Delta\frac{\partial \mathcal{H}_{\rm Sky}}{\partial\Delta\rho_{q}} \tag{5}\]7  
Next, to calculate the classical absorption cross-section, we need the time-derivative of (64). Then, we take the average energy absorbed per unit time,  
\[\mathcal{U}_{q}\equiv\frac{\delta\mathcal{H}_{\rm Sky}}{\delta\rho_{q}(\mathbf{r} )}=\frac{\partial\mathcal{H}_{\rm Sky}}{\partial\rho_{q}}-\
abla\cdot\frac{ \partial\mathcal{H}_{\rm Sky}}{\partial\
abla\rho_{q}}+\Delta\frac{\partial \mathcal{H}_{\rm Sky}}{\partial\Delta\rho_{q}} \tag{5}\]8 \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Skyrme** & \multirow{2}{*}{**\(\hbar\omega_{p}\) (MeV)**} & \multirow{2}{*}{**\(\hbar\xi_{n}\) (MeV)**} & \multirow{2}{*}{**\(\hbar\Omega\) (MeV)**} & \multirow{2}{*}{**\(\hbar\Omega\times A^{\text{1/3}}\)**} & \multirow{2}{*}{**\(\hbar\Omega_{\text{exp}}\times A^{\text{1/3}}\)**} \\
**Interaction** & & & & & **(MeV)** & **(MeV)** \\ \hline
**SIII** & 8.4 & 10.9 & 10.8 & 63.9 & \\
**SkM** & 8.8 & 14.0 & 13.9 & 82.5 & \\
**SkA** & 8.7 & 14.2 & 14.2 & 84.0 & \\
**SkP** & 8.9 & 14.7 & 14.6 & 86.7 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Giant dipole fundamental mode, plasma frequency, and energy of the GDR fundamental overtone for various Skyrme parametrizations. The derivation of the final form of \(\sigma_{\rm abs}\) amounts to generalizing the derivation in the field-free case, as detailed in Ref. [17],  
\[\sigma_{\rm abs}=\frac{Ze^{2}}{m\epsilon_{0}}\Bigg{(}1+\frac{c_{np}^{2}-c_{pn}^{ 2}}{c_{pp}^{2}-c_{pn}^{2}}\Bigg{)}\sum_{n=\pm 1}\frac{\Gamma/c}{{\zeta_{n}}^{2}-2} \Bigg{\{}\left[\Omega+n\omega+\frac{\omega_{p}^{2}-\zeta_{n}^{2}}{\Omega+n \omega}\right]^{2}+\Gamma^{2}\Bigg{\}}^{-1}. \tag{66}\]  
and results in a two-peaked Lorentz function when \(\omega\
eq 0\). The action of a high-frequency laser field will split the one-peaked distribution in a manner analogous to the dipole strength in deformed nuclei [17]. It can be remarked from Figure 1 that the splitting of the GDR centroid of \(\hbar\Omega_{0}=13.9\) MeV for the spherical heavy nucleus \({}^{208}\)Pb is visible for frequencies from the \(\gamma\)-ray domain. Within a quantum treatment of the problem, we expect the coming into play of higher terms (\(|n|\geq 2\)) in the sum over the number of exchanged photons, \(\sigma_{\rm abs}=\sum_{n}\sigma_{n}\). Depending on the frequency and the strength of the laser field, the broad resonance centered on the GDR fundamental mode \(\Omega_{0}\) will display a series of maxima corresponding to the satellites \(\Omega_{0}\pm\omega\), \(\Omega_{0}\pm 2\omega\),.... More precisely, the effect of a laser pulse in a continuous wave form produces a multiresonance shape of the absorption cross-section.