  Solid state materials hosting pseudospin-1 quasiparticles have attracted a
great deal of recent attention. In these materials, the energy band contains of
a pair of Dirac cones and a flat band through the connecting point of the
cones. As the "caging" of carriers with a zero group velocity, the flat band
itself has zero conductivity. However, in a non-equilibrium situation where a
constant electric field is suddenly switched on, the flat band can enhance the
resulting current in both the linear and nonlinear response regimes through
distinct physical mechanisms. Using the ($2+1$) dimensional pseudospin-$1$
Dirac-Weyl system as a concrete setting, we demonstrate that, in the weak field
regime, the interband current is about twice larger than that for
pseudospin-1/2 system due to the interplay between the flat band and the
negative band, with the scaling behavior determined by the Kubo formula. In the
strong field regime, the intraband current is $\sqrt{2}$ times larger than that
in the pseudospin-1/2 system, due to the additional contribution from particles
residing in the flat band. In this case, the current and field follows the
scaling law associated with Landau-Zener tunneling. These results provide a
better understanding of the role of the flat band in non-equilibrium transport
and are experimentally testable using electronic or photonic systems.
