Fully quantum theory of strong-field driven tunable entangled multi-photon states in HHG

Quantum high-harmonic generation (HHG) is a growing field of research with capabilities of providing high photon-number entangled states of light. However, there is an open debate regarding the theory level required for correctly describing the quantum aspects of HHG emission, such as squeezing or entanglement. Previous approaches have employed non-interacting classical ensembles of trajectories, or perturbation theory utilizing the classical trajectories as a starting point, missing out key entanglement features. In this Letter, we develop a full quantum theory for entanglement measures in HHG solving exactly the light-matter interaction Hamiltonian and employ it for evaluating the entanglement between emitted photons of different harmonics. For the first time, we reach qualitative agreement of theory with recent experiments showing that the R entanglement parameter decreases with increasing laser power for below-threshold harmonics. Our results indicate that fine-tuning the laser power could enhance HHG entanglement features, which are observed to oscillate with the driving power and exhibit local non-classical maxima structures. Similarly, our theory predicts that the oscillatory behavior of entanglement observed for below-threshold harmonics also appears for entanglement involving above-threshold harmonics. We also show that the long-range behavior of driven electronic trajectories can qualitatively change the resulting entanglement. Lastly, we show that focal averaging over classical degrees of freedom, which has thus far been ignored in quantum HHG theories, plays a key role in entanglement measures and can change the qualitative behavior of observables. Our work establishes the state-of-the art in exploring entanglement features in HHG, and paves way for analysis and engineering of 'truly-quantum' multi-photon states in the XUV and ultrafast regime for more complex matter systems.