Topological Superfluids with Finite Momentum Pairing and Majorana Fermions

Majorana fermions, quantum particles that are their own anti-particles, are not only of fundamental importance in elementary particle physics and dark matter, but also building blocks for fault-tolerant quantum computation. Recently Majorana fermions have been intensively studied in solid state and cold atomic systems. These studies are generally based on superconducting pairing between two Fermions with opposite momenta (\textit{% i.e.}, zero total momentum). On the other hand, finite total momentum Cooper pairings, known as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, were predicted 50 years ago and then widely studied in many branches of physics. However, whether FFLO superconductors can also support Majorana fermions has not been explored. Here we show that Majorana fermions can exist in certain types of gapped FFLO states, yielding a new topological quantum matter: topological FFLO superfluids/superconductors. We demonstrate the existence of such topological FFLO superfluids and the associated Majorana fermions using spin-orbit coupled degenerate Fermi gases and derive their physical parameter regions. The potential implementation of topological FFLO superconductors in semiconductor/superconductor heterostructures are also discussed.