# Interference-Induced Phenomena in High-Order Harmonic Generation from Bulk Solids
## 3 Results
### Even and Odd Harmonics  
As we have seen, it is possible to calculate spectra that result from any subset of \(k\) values chosen from the first Brillouin zone. In other words, we can artificially restrict the summation over \(k\) in Equation (3), and investigate how the interference of the contributions of these states as sources to the complete current density result in the net HHG signal. Focusing on the practical absence of the even-order harmonics from the spectra, Figure 5 provides an explanation. As we can see, HHG signals corresponding to a given value of \(k\) contain all harmonics with approximately the same weights. However, as the example of Figure 5 shows, when we combine the current densities that result from the time evolution of a given \(k\) and its opposite, the even harmonics tend to disappear. That is, the fields emerging from the time evolution of states that--without the external field--have phase velocities with the same magnitudes but opposite signs, interfere constructively for the odd harmonics, and destructively for the even ones. Additionally, this result does not depend on the particular choice of the model potential \(U\), since it is visible for both \(U^{(a)}\) and \(U^{(b)}\). Now, it is important to emphasize that the question of whether even-order harmonics will appear or not, can be decided using very fundamental, symmetry based considerations. Besides inversion symmetry [51], the periodicity of the excitation is also crucial (we thank one of our referees for drawing our attention to this point), since destructive interferences appear in consecutive half-cycles. In this way, the inversion symmetry of the material, together with the repetition of the field with the opposite sign in the next half-cycle, is responsive for the suppression of even-order harmonics. However, when no real-space symmetry can be associated to the emitters (like in the case of abstract two-level systems [52]), generally both odd and even order harmonics appear. As we see here, when a system of few-level quantum systems can be related to a physical system in real space, this situation changes, but in this case, the interference of fields emitted by all individual few-level systems is needed. The trivial method of discovering the role of inversion in our treatment would be comparing the spectra that correspond to potentials with and without inversion symmetry. However, using different pairs of potentials (not only \(U^{(a)}\) and \(U^{(b)}\)), we observed no qualitative difference, and the even-order harmonics were practically always missing. This effect can originate from the dimensionality of the model, since working in one dimension means assuming that only a very narrow set of k vectorsfrom the orthogonal plane plays a relevant role in the time evolution, and this may not be the case for real materials. On the other hand, since the appearance of the even-order harmonics is not determined by the crystal structure alone, the symmetry of the exciting field is equally important. The \"half-cycle by half-cycle sign change symmetry\" of a long, many-cycle infrared pulse can be broken by adding a slowly varying, few-cycle electromagnetic pulse, such as in the THz regime. The result of this kind of \"THz-assisted\" HHG is shown by Figure 6. As we can see, in this case, the even-order harmonics appear--practically as strongly as the odd-order ones--already in a 1D model. In order to see this in more detail, we investigated the role of the delay between the THz and the infrared pulses. (Note that this delay was zero for Figure 6.) As we can see in Figure 7, when there is no overlap between the pulses, the strength of the harmonics are practically the same as without the THz radiation, regardless of the sign of the delay. This underlines the fact that it is the infrared radiation that produces high harmonics, and the THz pulse itself is not intense enough in this sense.