High-order harmonic generation in a crystal driven by quantum light

We study intraband high-harmonic generation (HHG) in a crystal driven by quantum light. Previous theoretical studies have developed a framework based on coherent state expansions in terms of $P$ distributions to consider nonclassical driving fields for HHG in atoms. Here we adapt this framework to the context of solids and consider an intraband model of ZnO. We investigate the effect of the quantum optical nature of the driving field on the harmonic spectra including the cutoff and the intensity scaling of the harmonics with driving field intensity. Based on analytical calculations in the Floquet limit, we explain why driving with thermal light or bright-squeezed vacuum (BSV) produces a much higher cutoff than when driving with fields described by coherent or Fock states. Further, we derive an expression for the generated time-dependent electric field and its fluctuations and find that it inherits characteristics of the driving field. Finally, we discuss the limitations of an approximative positive $P$ representation, which is introduced to be able to reduce the numerical complexity for Fock and BSV driving fields.