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We present a gauge-invariant density matrix description of non-equilibriumsuperconductor (SC) states with spatial and temporal correlations driven byintense terahertz (THz) lightwaves. We derive superconductor Bloch--Maxwellequations of motion that extend Anderson pseudo-spin models to include theCooper pair center-of-mass motion and electromagnetic propagation effects. Wethus describe quantum control of dynamical phases, collective modes,quasi-particle coherence, and high nonlinearities during cycles of carrier waveoscillations, which relate to our recent experiments. Coherent photogenerationof a nonlinear supercurrent with dc component via condensate acceleration by aneffective lightwave field dynamically breaks the equilibrium inversionsymmetry. Experimental signatures include high harmonic light emission atequilibrium-symmetry-forbidden frequencies, Rabi--Higgs collective modes andquasi-particle coherence, and non-equilibrium moving condensate states tuned byfew-cycle THz fields. We use such lightwaves as an oscillating acceleratingforce that drives strong nonlinearities and anisotropic quasi-particlepopulations to control and amplify different classes of collective modes, e.g.,damped oscillations, persistent oscillations, and overdamped dynamics via Rabiflopping. Recent phase-coherent nonlinear spectroscopy experiments can bemodeled by solving the full nonlinear quantum dynamics includingself-consistent light--matter coupling.

Lightwave Terahertz Quantum Manipulation of Non-equilibrium Superconductor Phases and their Collective Modes