  We develop a theory to describe the dynamics of a driven-dissipative
many-body Fermi system, to pursue our proposal to realize exotic quantum states
based on reservoir engineering. Our idea is to design the shape of a Fermi
surface so as to have multiple Fermi edges, by properly attaching multiple
reservoirs with different chemical potentials to a fermionic system. These
emerged edges give rise to additional scattering channels that can destabilize
the system into unconventional states, which is exemplified in this work by
considering a driven-dissipative attractively interacting Fermi gas. By
formulating a quantum kinetic equation using the Nambu-Keldysh Green's function
technique, we explore nonequilibrium steady states in this system and assess
their stability. We find that, in addition to the BCS-type isotropic pairing
state, a Fulde-Ferrell-type anisotropic superfluid state being accompanied by
Cooper pairs with non-zero center-of-mass momentum exists as a stable solution,
even in the absence of a magnetic Zeeman field. Our result implies a great
potential of realizing quantum matter beyond the equilibrium paradigm, by
engineering the shape and topology of Fermi surfaces in both electronic and
atomic systems.
