  Recent experimental progress in the fields of cold quantum gases and
ultrafast optical spectroscopy of quantum materials allows to controllably
induce and probe non-adiabatic dynamics of superconductors and superfluids. The
time-evolution of the gap function before relaxation with the lattice is
determined by the superposition of coherently evolving individual Cooper pairs
within the manifold of the Bardeen-Cooper-Schrieffer (BCS) wavefunction. While
dynamics following an abrupt quench of the pairing interaction strength in the
single-band BCS model has been exactly solved due to the integrability of the
model, the dynamics of post-quench multi-band superconductors remain under
scrutiny. Here, we develop a generalization of the Volkov-Kogan Laplace-space
perturbative method that allows us to determine the non-adiabatic gap dynamics
of two-band fully gapped superconductors for a wide range of quench amplitudes.
Our approach expands the long-time dynamics around the steady-state asymptotic
value of the gap, which is self-consistently determined, rather than around the
equilibrium value of the gap. We explicitly demonstrate that this method
recovers the exact solution of the long-time gap dynamics in the single-band
case and perfectly agrees with a numerical solution of the two-band model. We
discover that dephasing of Cooper pairs from different bands leads to faster
collisionless relaxation of the gap oscillation with a power-law of $t^{-3/2}$
instead of the well-known $t^{-1/2}$ behavior found in the single-band case.
Furthermore, the gap oscillations display beating patterns arising from the
existence of two different asymptotic gap values. Our results have important
implications to a variety of two-band superconductors driven out of
equilibrium, such as iron-based superconductors, MgB$_{2}$, and SrTiO$_{3}$.
