  We study the possible ground state configurations of two strongly coupled
chains of charge neutral spin-3/2 fermionic atoms interacting via short range
van der Waals interaction. The coupling between the two chains is realized by
relatively large hopping amplitude. Exploiting that such a ladder configuration
can be mapped to an effective one-band model we analyze the emerging ground
states of the system. We show that various spatially inhomogeneous states,
valence bond states, plaquette states compete depending on the filling and the
ratio of the interaction strengths in the singlet and quintet scattering
channel. We find that a Luttinger liquid state is the ground state of the
strongly coupled ladder in an extended region of the parameter space, and we
also show that a topologically nontrivial charge Haldane state can emerge in
the strongly coupled ladder at quarter and three-quarter fillings.
