  Optical Feshbach resonance is capable of inducing spatially varying
interactions in ultra-cold atoms. Its applications to pancake-shaped clouds of
bosons and fermions enable one to study several fresh phenomena. We examine
possibilities of inducing counter-intuitive structures such as creating a
superfluid enclave inside a Mott insulator for bosons and a normal-gas core
enclosed by a superfluid shell for fermions. We discuss feasible experimental
setups and signatures of those interesting structures, which can be very
different from common structures observed in experiments so far. While a
superfluid enclave in a Mott insulator can be useful for constructing atomic
devices for atomtronics, superconducting islands observed in scanning-tunneling
microscopy of heavily underdoped high-temperature superconductors may be
studied with cold Fermi gases with spatially varying attractions.
