  We study the ground-state properties of a two-component fermionic mixture
effectively confined in a one-dimensional harmonic trap. We consider scenarios
when numbers of particles in components are the same but particles have
different masses. We examine whether it is possible to detect signatures of an
unconventional pairing between opposite-spin fermions in the presence of
attractive interactions. For this purpose, we perform the exact diagonalization
of the many-body Hamiltonian and study the two-particle reduced density matrix.
In agreement with expectations, we confirm that the many-body ground state is
dominated by conventional pairs with a negligible total momentum for a small
mass imbalance. Furthermore, we show that for sufficiently large mass ratios
the domination of fundamentally different pairs is established and the
Fulde-Ferrell-Larkin-Ovchinnikov phase is supported. Finally, we argue that the
two mechanisms can coexist in the regime of moderate mass ratios. Due to the
current experimental progress in obtaining ultra-cold fermionic systems in a
few-body regime, our predictions may have some importance for the upcoming
experiments.
