  For decades, most research on high harmonic generation (HHG) considered
matter as quantum but light as classical, leaving the quantum-optical nature of
the harmonics an open question. Here we explore the quantum properties of high
harmonics. We derive a formula for the quantum state of the high harmonics,
when driven by arbitrary quantum light states, and then explore specific cases
of experimental relevance. Specifically, for a moderately squeezed pump, HHG
driven by squeezed coherent light results in squeezed high harmonics. Harmonic
squeezing is optimized by syncing ionization times with the pump's squeezing
phase. Beyond this regime, as pump squeezing is increased, the harmonics
initially acquire squeezed thermal photon statistics, and then occupy an
intricate quantum state which strongly depends on the semi-classical nonlinear
response function of the interacting system. Our results pave the way for the
generation of squeezed extreme-ultraviolet ultrashort pulses, and, more
generally, quantum frequency conversion into previously inaccessible spectral
ranges, which may enable ultrasensitive attosecond metrology.
