  We study the interaction between elliptically polarized light and a
three-dimensional Luttinger semimetal with quadratic band touching using
Floquet theory. In the absence of light, the touching bands can have the same
or the opposite signs of the curvature; in each case, we show that simply
tuning the light parameters allows us to create a zoo of Weyl semimetallic
phases. In particular, we find that double and single Weyl points can coexist
at different energies, and they can be tuned to be type I or type II. We also
find an unusual phase transition, in which a pair of Weyl nodes form at finite
momentum and disappear off to infinity. Considering the broad tunability of
light and abundance of materials described by the Luttinger Hamiltonian, such
as certain pyrochlore iridates, half-Heuslers and zinc-blende semiconductors,
we believe this work can lay the foundation for creating Weyl semimetals in the
lab and dynamically tuning between them.
