# Interference-Induced Phenomena in High-Order Harmonic Generation from Bulk Solids
## 2 Model and Method  
### Time-Dependent Hamiltonian  
In the following, we use single-electron approximation in order to show how the individual contribution of different initial states build up the time-dependent source of high harmonic radiation. To this end, we use a model that is closely related to well-developed, more complex approaches--see References [38; 39; 44]. The single-electron Hamiltonian in a crystal is given by:  
\[H(t)=\frac{1}{2m}\left(\mathbf{p}-\epsilon\mathbf{A}\right)^{2}+U(\mathbf{r}) +e\Phi, \tag{1}\]where \(e\) denotes the charge of the electron, and the lattice-periodic potential \(U(\mathbf{r})\) represents the solid. The external electromagnetic field is taken into account via the scalar and vector potentials, \(\Phi\) and \(\mathbf{A}\). For a given \(U\), one can look for the solution of the field-free problem in the form of Bloch states, \(\Psi_{n,\mathbf{k}}(\mathbf{r})=\exp(i\mathbf{k}\mathbf{r})u_{n,\mathbf{k}}( \mathbf{r})/\sqrt{\
u}\), where \(n\) denotes the band index, \(u_{n,\mathbf{k}}(\mathbf{r})\) are lattice-periodic functions, and \(\mathcal{V}\) is the crystal volume. The eigenvalue equation for these states reads:  
\[-\frac{\hbar^{2}}{2m}(\
abla+i\mathbf{k})^{2}u_{n,k}(x)+U(\mathbf{r})u_{n, \mathbf{k}}(\mathbf{r})=E_{n}(\mathbf{k})u_{n,\mathbf{k}}(\mathbf{r}). \tag{2}\]  
As we can see, these equations for different \(\mathbf{k}\) vectors are independent; thus, due to the periodicity of the functions \(u_{n,\mathbf{k}}(\mathbf{r})\), it is sufficient to determine the \(\mathbf{k}\) dependent eigenenergies \(E_{n}(\mathbf{k})\) in a single unit cell for all values of \(\mathbf{k}\) separately. Having obtained the Bloch states and the eigenenergies \(E_{n}(\mathbf{k})\) (i.e., the band scheme), we have the scene on which laser-induced dynamics take place. Let us note here that by using Bloch states, the band energies are always the same for \(\pm\mathbf{k}\), which means a certain limitation for the materials that can be described in this way. In more detail, this limitation certainly allows the description of direct bandgap semiconductors or dielectrics, where the extrema of the bands are at \(k=0\) and the momentum matrix elements also have the maximal magnitude around here (since, in this case, most of the significant transitions occur in a k-space range where the assumption of the symmetry \(E_{n}(\mathbf{k})=E_{n}(-\mathbf{k})\) holds). On the other hand, it is not expected that indirect bandgap materials (like Si) can be described in this way.  
In actual numerical calculations, one has to determine the electromagnetic gauge to be used. In the following, we consider the velocity gauge, where electromagnetic potentials can be expressed using the electric field \(\mathbf{F}(t)\) as \(\mathbf{A}(t)=-\int_{-\infty}^{t}\mathbf{F}(t^{\prime})dt^{\prime},\Phi=0\). Realistically, we assume that the characteristic wavelength of the exciting laser field is much larger than the unit cell; therefore, the spatial variation of the laser field can be neglected. Within this dipole approximation, the Hamiltonian (1) is diagonal in \(\mathbf{k}\), thus, the time evolutions of different Bloch states \(\Psi_{n,\mathbf{k}}\) and \(\Psi_{n,\mathbf{k}^{\prime}}\) (with \(\mathbf{k}\
eq\mathbf{k}^{\prime}\)) as initial states are independent. However, all these states interact with the _same_ HHG modes. This is the point we will analyze in Section 3.