  A phase of matter in which fermion quartets form a superconducting
condensate, rather than the paradigmatic Cooper pairs, is a recurrent subject
of experimental and theoretical studies. However, a comprehensive microscopic
understanding of charge-4$e$ superconductivity as a quantum phase is lacking.
Here, we propose and study a two-orbital tight-binding model with attractive
Hubbard-type interactions. Such a model naturally provides the Bose-Einstein
condensate as a limit for electron quartets and supports charge-4$e$
superconductivity, as we show by mapping it to a spin-1/2 chain in this
perturbative limit. Using both exact diagonalization and density matrix
renormalization group calculations for the one-dimensional case, we further
establish that the ground state is indeed a superfluid phase of 4$e$ charge
carriers and that this phase can be stabilized well beyond the perturbative
regime. Importantly, we demonstrate that 4$e$ condensation dominates over 2$e$
condensation even for nearly decoupled orbitals, a scenario suitable for
experiments with ultracold atoms in the form of almost decoupled chains. Our
model paves the way for both experimental and theoretical exploration of 4$e$
superconductivity and provides a natural starting point for future studies
beyond one dimension or more intricate 4$e$ states.
