  Quasicrystal is a class of ordered structures defying conventional
classification of solid crystals and may carry classically forbidden (e.g.,
5-fold) rotational symmetries. In view of long-sought supersolids, a natural
question is whether a superfluid can spontaneously form quasicrystalline order
that is not possessed by the underlying Hamiltonian, forming
"superfluid-quasicrystals". Here we show that a superfluid-quasicrystal stripe
state with the minimal 5-fold rotational symmetry can be realized as the ground
state of a Bose-Einstein condensate within a practical experimental scheme.
There exists a rich phase diagram consisting of various
superfluid-quasicrystal, supersolid, and plane-wave phases. Our scheme can be
generalized for generating other higher-order (e.g., 7-fold) quasicrystal
states, and provides a platform for investigating such new exotic quantum
matter.
