  High harmonic generation (HHG) is an extremely nonlinear effect, where a
medium is driven by a strong laser field, generating coherent broadband
radiation with photon energies ranging up to the X-ray and pulse durations
reaching attosecond timescales. Conventional models of HHG treat the medium
quantum mechanically, while the driving and emitted fields are treated
classically. Such models are usually very successful, but inherently cannot
capture the quantum-optical nature of the process. Despite prior works
considering quantum HHG, it is still not known in what circumstances the
spectral and statistical properties of the radiation considerably depart from
the known phenomenology of HHG. Finding such regimes in HHG could lead to novel
sources of attosecond light with intrinsically quantum statistics such as
squeezing and entanglement. In this work, we present a fully quantum
electrodynamical theory of extreme nonlinear optics with which we predict new
quantum effects that alter both spectral and statistical properties of HHG. We
predict the emission of shifted frequency combs resulting from transitions
between perturbed states of the driven atom, and identify new spectral features
that arise from the breakdown of the dipole approximation for the emission.
Moreover, we find that each HHG emitted photon is a superposition of all
frequencies in the spectrum - e.g., HHG creates single photon combs. We also
describe how the HHG process changes in the single atom regime, and discuss how
our various predictions can be tested experimentally. Our approach is also
applicable to a wide range of nonlinear optical processes, paving the way
toward novel quantum information techniques in the EUV and X-ray, and in
ultrafast quantum optics.
