Quantum Electrodynamics\nin High-Harmonic Generation:\nMultitrajectory Ehrenfest and Exact Quantum Analysis

High-harmonic generation (HHG) is a nonlinear process\nin which\na material sample is irradiated by intense laser pulses, causing the\nemission of high harmonics of incident light. HHG has historically\nbeen explained by theories employing a classical electromagnetic field,\nsuccessfully capturing its spectral and temporal characteristics.\nHowever, recent research indicates that quantum-optical effects naturally\nexist or can be artificially induced in HHG, such as entanglement\nbetween emitted harmonics. Even though the fundamental equations of\nmotion for quantum electrodynamics (QED) are well-known, a unifying\nframework for solving them to explore HHG is missing. So far, numerical\nsolutions have employed a wide range of basis-sets, methods, and untested\napproximations. Based on methods originally developed for cavity polaritonics,\nhere we formulate a numerically accurate QED model consisting of a\nsingle active electron and a single quantized photon mode. Our framework\ncan, in principle, be extended to higher electronic dimensions and\nmultiple photon modes to be employed in <i>ab initio</i> codes for realistic physical systems. We employ it as a model of\nan atom interacting with a photon mode and predict a characteristic\nminimum structure in the HHG yield vs phase-squeezing. We find that\nthis phenomenon, which can be used for novel ultrafast quantum spectroscopies,\nis partially captured by a multitrajectory Ehrenfest dynamics approach,\nwith the exact minima position sensitive to the level of theory. On\nthe one hand, this motivates using multitrajectory approaches as an\nalternative for costly exact calculations. On the other hand, it suggests\nan inherent limitation of the multitrajectory formalism, indicating\nthe presence of entanglement and true quantum effects (especially\nprominent for atomic and molecular resonances). Our work creates a\nroadmap for a universal formalism of QED-HHG that can be employed\nfor benchmarking approximate theories, predicting novel phenomena\nfor advancing quantum applications, and for the measurements of entanglement\nand entropy.