  QCD-like theories possess a positively definite fermion determinant at finite
baryon chemical potential $\mu_{\text B}$ and the lattice simulation can be
successfully performed. While the chiral perturbation theories are sufficient
to describe the Bose condensate at low density, to describe the crossover from
Bose-Einstein condensation (BEC) to BCS superfluidity at moderate density we
should use some fermionic effective model of QCD, such as the
Nambu-Jona-Lasinio model. In this paper, using two-color two-flavor QCD as an
example, we examine how the Nambu-Jona-Lasinio model describes the weakly
interacting Bose condensate at low density and the BEC-BCS crossover at
moderate density. Near the quantum phase transition point $\mu_{\text B}=m_\pi$
($m_\pi$ is the mass of pion/diquark multiplet), the Ginzburg-Landau free
energy at the mean-field level can be reduced to the Gross-Pitaevskii free
energy describing a weakly repulsive Bose condensate with a diquark-diquark
scattering length identical to that predicted by the chiral perturbation
theories. The Goldstone mode recovers the Bogoliubov excitation in weakly
interacting Bose condensates. The results of in-medium chiral and diquark
condensates predicted by chiral perturbation theories are analytically
recovered. The BEC-BCS crossover and meson Mott transition at moderate baryon
chemical potential as well as the beyond-mean-field corrections are studied.
Part of our results can also be applied to real QCD at finite baryon or isospin
chemical potential.
