Cubic-quintic nonlinearity in superfluid Bose-Bose mixtures in optical lattices: Heavy solitary waves, barrier-induced criticality, and current-phase relations

We study superfluid (SF) states of strongly interacting Bose-Bose mixtures with equal mass and intracomponent interaction in optical lattices both in the presence and absence of a barrier potential. We show that the SF order parameters obey the two-component nonlinear Schrödinger equation (NLSE) with not only cubic but also quintic nonlinearity in the vicinity of the first-order transitions to the Mott insulators with even fillings. In the case of no barrier potential, we analyze solitary-wave solutions of the cubic-quintic NLSE. When the SF state changes from a ground state to a metastable one, a standard dark solitary wave turns into a bubblelike dark solitary wave, which has a nonvanishing density dip and no π-phase kink even in the case of a standing solitary wave. It is shown that the former and latter solitary waves are dynamically unstable against an out-of-phase fluctuation and an in-phase fluctuation, respectively, and the dynamical instabilities are weakened when one approaches the transition point. We find that the size and the inertial mass of the solitary waves diverge at the first-order transition point. We suggest that the divergence of the inertial mass may be detected through measurement of the relation between the velocity and the phase jump of the solitary wave. In the presence of a barrier potential, we reveal that when the barrier strength exceeds a certain critical value, the SF state that was metastable without the barrier is destabilized towards complete disjunction of the SF. The presence of the critical barrier strength indicates that the strong barrier potential qualitatively changes the criticality near the metastability limit of the SF state. We derive critical behaviors of the density, the compressibility, and the critical current near the metastability limit induced by the barrier. It is also found that the relation between the supercurrent and the phase jump across the barrier exhibits a peculiar behavior, owing to the nontopological nature of the bubblelike solitary wave.