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Experimental realizations of topological quantum systems and detections of topological invariants in ultracold atomic systems have been a greatly attractive topic. In this work, we propose a scheme to realize topologically different phases in a bichromatic optical lattice subjected to a periodically driven tilt harmonic oscillation, which can be effectively described by a superlattice model with tunable long-range hopping processes. By tuning the ratio of nearest-neighbor (NN) and next-nearest-neighbor (NNN) hopping amplitudes, the system undergoes a topological phase transition accompanied by the change of topological numbers of the lowest band from -1 to 2. Using a slowly time-periodic modulation, the system emerges distinct quantized topological pumped charges (TPCs) of atoms in the filled band for different topological phases. Our scheme is realizable in current cold atomic technique. [ABSTRACT FROM AUTHOR]

Motivated by the recent realization of the Haldane model in a shaking optical lattice, we investigate the effects of attractive interaction and the BEC-BCS crossover in this model at and away from half-filling. We show that, contrary to the usual s-wave BEC-BCS crossover in the lattice, a topological superfluid with Chern number C = 2 appears in an extended region of the phase space for intermediate strength of the attractive interaction on the interaction-density plane. When inversion symmetry is broken, a gapless weak topological state is realized. We also investigate the fluctuations in these superfluid phases and show that the Anderson-Bogoliubov mode is quadratic due to time-reversal symmetry breaking and the existence of an undamped Leggett mode in the strong-coupling limit. Near the topological phase transition, the damping of the Leggett mode reaches its maximum. [ABSTRACT FROM AUTHOR]