  We explore the roles of electronic band structure and Coulomb interactions in
solid-state HHG by studying the optical response of linear atomic chains and
carbon nanotubes to intense ultrashort pulses. Specifically, we simulate
electron dynamics by solving the single-particle density matrix equation of
motion in the presence of intense ultrafast optical fields, incorporating
tight-binding electronic states and a self-consistent electron-electron
interaction. While linear atomic chains constitute an idealized system, our
realistic 1D model readily provides insight on the temporal evolution of
electronic states in reciprocal space, both in the absence or presence of
electron interactions, which we demonstrate to play an important role in the
HHG yield. This model further predicts that doped semiconductors generate high
harmonics more efficiently than their metallic and undoped counterparts. To
complement this idealized system we also show results for HHG in more realistic
quasi-1D structures such as carbon nanotubes, the behavior of which is found to
be in good qualitative agreement with the atomic chains. Our findings apply
directly to extreme nonlinear optical phenomena in atoms on surfaces,
carbon-based structures, linear arrays of dopant atoms in semiconductors, and
linear molecules, such as polycyclic aromatic hydrocarbon chains, and can be
straightforwardly extended to optimize existing platforms for HHG or identify
new solid-state alternatives in the context of nonlinear plasmonics.
