# Giant Dipole Multi-Resonances Excited by High-Frequency Laser Pulses
## 2 Neutron-Proton Mixtures in an External Electromagnetic Field  
In what follows, we extend the traditional framework of nuclear hydrodynamics, as presented, for example, in Refs. [17, 22], to a proton-mixture interacting via Skyrme-type forces [28, 30]. Let us introduce the constituent velocities \(\chi^{\prime}_{p,n}\) via the definition of the corresponding mass-current densities (we employ the term mass-current density, present in the Skyrme energy density \(\mathcal{H}_{\rm Sky}\), to differentiate from the charge-current density.) \[\dot{J}_{q}=\frac{m}{\hbar}\rho_{q}\;\chi^{\prime}_{q}\;\;.\;\;\;\;(q=p,n) \tag{1}\]  
The kinetic energy of the fluid mixture then reads  
\[T=\frac{1}{2}m\int d\mathbf{r}(\rho_{p}\;\chi^{\prime 2}_{p}+\rho_{n}\;\chi^{ \prime 2}_{n}). \tag{2}\]  
The internal energy originating from nuclear interactions is expressed by means of Skyrme parametrization (encoded in the \(B\)-coefficients, as seen below) of the energy density  
\[\int d\mathcal{H}_{\rm Sky} \tag{3}\]  
where \(\mathcal{H}_{\rm Sky}\), neglecting the spin-orbit contribution, justified for spin-saturated nuclei, assumes the compact form in terms of the neutron and proton (\(q=n,p\)) local \(\rho_{q}\), kinetic energy \(\tau_{q}\) and the mass-current \(\dot{J}_{q}\) densities [31]  
\[\mathcal{H}_{\rm Sky} =\frac{\hbar^{2}}{2m}(\tau_{p}+\tau_{n})+B_{1}\rho^{2}+B_{2}(\rho _{p}^{2}+\rho_{n}^{2}) \tag{4}\] \[+B_{3}(\rho\tau-\dot{J}^{2})+B_{4}(\rho_{p}\tau_{p}-\dot{j}_{p}^{ 2}+\rho_{n}\tau_{n}-\dot{j}_{n}^{2})\] \[-B_{5}(\
abla\rho)^{2}-B_{6}\Big{[}(\
abla\rho_{p})^{2}+(\
abla \rho_{n})^{2}\Big{]}+\Big{[}B_{7}\rho^{2}+B_{8}(\rho_{p}^{2}+\rho_{n}^{2}) \Big{]}\rho^{\alpha}.\]  
The corresponding central one-body potential \(\mathcal{U}_{q}\) is defined as the functional derivative of the energy density with respect to the \(q\)-th constituent density  
\[\mathcal{U}_{q}\equiv\frac{\delta\mathcal{H}_{\rm Sky}}{\delta\rho_{q}(\mathbf{r} )}=\frac{\partial\mathcal{H}_{\rm Sky}}{\partial\rho_{q}}-\
abla\cdot\frac{ \partial\mathcal{H}_{\rm Sky}}{\partial\
abla\rho_{q}}+\Delta\frac{\partial \mathcal{H}_{\rm Sky}}{\partial\Delta\rho_{q}} \tag{5}\]  
In a finite system to the above form of the internal energy density, the following contribution has to be added:  
\[U_{\rm C}(\mathbf{r})=\frac{1}{2}\epsilon\rho_{p}(\mathbf{r})\Phi(\mathbf{r})=\frac{e^{2} }{2}\rho_{p}(\mathbf{r})\int d\mathbf{r}\frac{\rho_{p}(\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{ r}^{\prime}|}. \tag{6}\]  
where \(\Phi(\mathbf{r})\) is the Coulomb potential. Thus, the internal energy can be written as  
\[U=\int d\mathbf{r}\Big{(}\mathcal{H}_{\rm Sky}(\mathbf{r})+U_{\rm C}(\mathbf{r})\Big{)} \tag{7}\]  
In this paper, we consider a neutron-proton \"plasma\" placed in an external electromagnetic field (laser) of frequency \(\omega\) and electric and magnetic strengths  
\[\mathbf{E}_{0}(t)=\hat{\mathbf{\varepsilon}}\mathcal{E}_{0}\cos\omega t,\;\;\;\mathbf{B}_{ 0}(t)=(\hat{\mathbf{k}}\times\hat{\mathbf{\varepsilon}})\frac{\mathcal{E}_{0}}{c} \sin\omega t, \tag{8}\]where \(\hat{e}\) is the unit vector along the laser polarization, and \(\hat{\mathbf{k}}\) is the unit vector along the propagation direction such that \(\hat{\mathbf{k}}\perp\hat{\mathbf{e}}\). The interaction of the charged component of the fluid mixture with the external electromagnetic field of strength \((\mathbf{E},\mathbf{B})\) reads [32]  
\[T=\frac{1}{2}m\int d\mathbf{r}(\rho_{p}\;\chi^{\prime 2}_{p}+\rho_{n}\;\chi^{ \prime 2}_{n}). \tag{2}\]0  
where \(\mathbf{\chi}_{p}\) is the proton fluid displacement field, trivially related to the above introduced proton fluid velocity field  
\[T=\frac{1}{2}m\int d\mathbf{r}(\rho_{p}\;\chi^{\prime 2}_{p}+\rho_{n}\;\chi^{ \prime 2}_{n}). \tag{2}\]1  
Since the coherent motion tends to be distributed in the nuclear volume mainly by two-body scattering, a damping mechanism occurs. This can be viewed from a microscopic point of view as the friction between the proton and neutron fluids.