  We theoretically propose an idea based on reservoir engineering to process
the structure of a Fermi edge to split into multiple Fermi edges, so as to be
suitable for the state which we want to realize. When one appropriately tunes
the chemical-potential difference between two reservoirs being coupled with the
system, the system is shown to be in the non-equilibrium steady state with the
momentum distribution having a two-edge structure. We argue that these edges
play similar roles to two Fermi surfaces, which can be designed to realize
exotic quantum many-body states. To demonstrate this, we consider a model
driven-dissipative two-component Fermi gas with an attractive interaction as a
paradigmatic example and show that it exhibits an unconventional Fermi
superfluid. While the superfluid order parameter of this state has the same
form as that in the Fulde-Ferrell state discussed in metallic superconductivity
under an external magnetic field, the former non-equilibrium pairing state is
not accompanied by any spin imbalance. Our proposed reservoir engineering to
process the Fermi momentum distribution would provide further possibilities of
many-body quantum phenomena beyond the thermal equilibrium case.
