Quantum optics of strongly laser--driven atoms and generation of high photon number optical cat states.

Recently we have demonstrated the quantum nature of light in strongly laser driven atoms, and we have shown how the process of high harmonic generation can be used for the creation of highly non-classical light states, in particular superpositions of two coherent states, i.e., optical Schr\"{o}dinger "cat" states [M. Lewenstein et al., Generation of optical Schr\"odinger cat states in intense laser-matter interactions, Nat. Phys. (2021)]. Here, we investigate the quantum optical description of the interaction, incorporating many atoms and the back-action of the high harmonic generation or above threshold ionization processes on the coherent state of the driving laser field. We show how the conditioning on high harmonic generation and above threshold ionization can lead to non-classical light states, and we discuss the key parameters that can be used for controlling and characterizing the non-classical states obtained after high harmonic generation. The theoretical results have been experimentally confirmed by demonstrating how the obtained coherent state superposition transitions from an optical "cat" to "kitten" state by changing the number of atoms participating in the high harmonic generation process. Additionally, we experimentally verify the applicability of the approach for generating high photon number non-classical light states. This is shown by demonstrating the generation of a 9--photon optical "cat" state which, to our knowledge, is the highest photon number optical "cat" state experimentally reported. Our findings anticipate the development of new methods that naturally lead to the creation of high photon number controllable coherent state superpositions, advancing investigations in quantum technology.