  We show that low-lying triplon excitations in a bilayer Heisenberg
antiferromagnet provide a promising avenue to realise magnetic analogs of
twisted superfluid and supersolid phases that were recently reported for
two-component ultracold atomic condensate in an optical lattice. Using a
cluster Gutzwiller mean field theory, we establish that Dzyaloshinskii-Moriya
interactions, that are common in many quantum magnets, stabilize these phases
in magnetic system, in contrast to pair hopping process that is necessary for
ultracold atoms. Our results provide useful guidance for the experimental
search of the twisted superfluid and supersolid phases of triplons in real
materials.
