Ultracold atomic systems provide a versatile platform for exploring quantum phenomena, offering tunable interactions and diverse trapping geometries. In this study, we investigate a one-dimensional system of trapped fermionic atoms using the composite boson formalism, which describes pairs of opposite-spin fermions as cobosons (short for composite bosons). We solve the Schrödinger equation for a $2\uparrow +2\downarrow$ fermionic system across a broad range of attractive and repulsive interactions by constructing the system's wave function as a superposition of two-coboson states. We determine key observables such as particle density profiles and two-body correlations. The density profiles calculated agree with both the Tonks-Girardeau and Lieb-Liniger limits and provide a portrait of the transition between those regimes. In the strong repulsive regime, the ansatz captures the Fridel-Wigner transition characterized by a doubling of the peaks in the density profiles. Additionally, we compute the low-lying energy spectrum and estimate the pairing gap. Our results highlight the usefulness of the coboson superposition ansatz for exploring quantum phenomena in strongly correlated few-body systems.

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• Composite bosons
• Fermi gases
• First-principles calculations
• Schroedinger equation