In the context of a gas of ultracold atoms with effective spin $S=3/2$ confined to an elongated trap, we study the one-dimensional Fermi gas interacting via an attractive $\delta$-function potential within the grand-canonical ensemble. The particles can be either unbound or clustered in bound states of two, three, and four fermions. The rich $\mu$ versus $H$ ground-state phase diagram ($\mu$ is the chemical potential and $H$ the external magnetic field) consists of the four basic states and the various possible mixed phases in which some these states coexist. Extending the analysis of K. Yang [Phys. Rev. B 63, 140511(R) (2001)] for $S=1/2$, we study the correlation functions of the generalized Cooper clusters of bound states of two, three, and four particles using conformal field theory and the exact Bethe Ansatz solution. The correlation functions consist of a power law with distance times a sinusoidal term oscillating with distance. In an array of tubes with weak Josephson tunneling, the type of superfluid order is determined by these correlation functions. The wavelength of the oscillations is related to the periodicity of a generalized Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state for higher spin particles. All the relevant states are analyzed for $S=3/2$.

• Received 23 December 2011

DOI:https://doi.org/10.1103/PhysRevB.85.205129