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Motivated by recent experiments in low-dimensional trapped fermionicsuperfluids, we study a quasi-one-dimensional (quasi-1D) superfluid with apopulation imbalance between two hyperfine states using an exact mean-fieldsolution for the order parameter. When an effective "magnetic field" exceeds acritical value, the superfluid order parameter develops spatial inhomogeneityin the form of a soliton lattice. The soliton lattice generates a band ofquasiparticle states inside the energy gap, which originate from the Andreevbound states localized at the solitons. Emergence of the soliton lattice isaccompanied by formation of a spin-density wave, with the majority fermionsresiding at the points in space where the Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) order parameter vanishes. We discuss possibilities for experimentaldetection of the quasi-1D FFLO state using elastic and inelastic optical Braggscattering and radio-frequency spectroscopy. We show that these measurementscan provide necessary information for unambiguous identification of thespatially-inhomogeneous quasi-1D FFLO state and the soliton lattice formation.

Spectroscopy of the soliton lattice formation in quasi-one-dimensional fermionic superfluids with population imbalance