  We present a gauge-invariant density matrix description of non-equilibrium
superconductor (SC) states with spatial and temporal correlations driven by
intense terahertz (THz) lightwaves. We derive superconductor Bloch--Maxwell
equations of motion that extend Anderson pseudo-spin models to include the
Cooper pair center-of-mass motion and electromagnetic propagation effects. We
thus describe quantum control of dynamical phases, collective modes,
quasi-particle coherence, and high nonlinearities during cycles of carrier wave
oscillations, which relate to our recent experiments. Coherent photogeneration
of a nonlinear supercurrent with dc component via condensate acceleration by an
effective lightwave field dynamically breaks the equilibrium inversion
symmetry. Experimental signatures include high harmonic light emission at
equilibrium-symmetry-forbidden frequencies, Rabi--Higgs collective modes and
quasi-particle coherence, and non-equilibrium moving condensate states tuned by
few-cycle THz fields. We use such lightwaves as an oscillating accelerating
force that drives strong nonlinearities and anisotropic quasi-particle
populations to control and amplify different classes of collective modes, e.g.,
damped oscillations, persistent oscillations, and overdamped dynamics via Rabi
flopping. Recent phase-coherent nonlinear spectroscopy experiments can be
modeled by solving the full nonlinear quantum dynamics including
self-consistent light--matter coupling.
