  We study the preparation (pump) and the detection (probe) of
far-from-equilibrium BCS superconductor dynamics in THz pump-probe experiments.
In a recent experiment [R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H.
Terai, Z. Wang, and R. Shimano, Phys. Rev. Lett. {\bf 111}, 057002 (2013)], an
intense monocycle THz pulse with center frequency $\omega \simeq \Delta$ was
injected into a superconductor with BCS gap $\Delta$; the subsequent post-pump
evolution was detected via the optical conductivity. It was argued that
nonlinear coupling of the pump to the Anderson pseudospins of the
superconductor induces coherent dynamics of the Higgs (amplitude) mode
$\Delta(t)$. We validate this picture in a two-dimensional BCS model with a
combination of exact numerics and the Lax reduction method, and we compute the
nonequilibrium phase diagram as a function of the pump intensity. The main
effect of the pump is to scramble the orientations of Anderson pseudospins
along the Fermi surface by twisting them in the $xy$-plane. We show that more
intense pump pulses can induce a far-from-equilibrium phase of gapless
superconductivity ("phase I"), originally predicted in the context of
interaction quenches in ultracold atoms. We show that the THz pump method can
reach phase I at much lower energy densities than an interaction quench, and we
demonstrate that Lax reduction (tied to the integrability of the BCS
Hamiltonian) provides a general quantitative tool for computing coherent BCS
dynamics. We also calculate the Mattis-Bardeen optical conductivity for the
nonequilibrium states discussed here.
