Theory of High Harmonic Generation with Quantum Light

High harmonic generation (HHG) driven by intense quantum light has recently emerged as a new frontier in strong-field physics. Yet the mechanism underlying HHG with nonclassical fields such as bright squeezed vacuum (BSV) light --- characterized by a vanishing mean electric field --- remains elusive. In this work, we develop a fully quantum theory for HHG under arbitrary quantum driving fields. We show that the fluctuation component of the dipole correlation operator plays a central role in HHG with quantum light, and is essential for BSV-driven HHG. Furthermore, our framework reveals the photon statistics distribution within individual harmonics and across different harmonics, demonstrating that the statistics inherit the statistical properties of the driving field. This theory overcomes key limitations of existing approaches and uncovers new quantum features of HHG in the presence of nonclassical light.