Motivated by the recent realization of the Haldane model in shaking optical lattice, we investigate the effects of attractive interaction and 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 phase space for intermediate strength of the attractive interaction in the interaction-density plane. When inversion symmetry is broken, a new gapless 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.Introduction. Cold atoms in optical lattices can be used to simulate important models in condensed matter physics due to their high controllability and versatility [1,2]. This was demonstrated beautifully by the recent realizations of Haldane [3][4][5] and Hofstadter models using shaking lattice and Raman laser techniques, respectively. In these experiments, the existence of topological bands is verified using Bloch oscillations with either a Bose condensate [6] or free fermions [3]. With tunable interactions in the optical lattices, these advances open new avenues for the controlled study of interaction effects in topological system and pave the way to the possibility of realizing fractional Chern insulators.Perhaps by far the best studied interacting topological state is the fractional quantum Hall state, in which the strong Coulomb repulsion between electrons generate emergent fractional quasi-particles that obey abelian or non-abelian statistics [7][8][9][10][11][12][13]. Naturally, with the discovery of topological insulators [14,15], a great deal of efforts has been to investigate its interacting counterparts [16][17][18][19]. In the case of Haldane model, several recent studies have focused on the interplay between magnetic instabilities and the possible topological ground states in the case of repulsive interactions [20][21][22][23][24], as well as possible superconducting states with attractive interactions at half filling [25].In this Letter, we consider the analogous of BEC-BCS crossover [26] in the attractive Haldane-Hubbard model. In contrast to usual BEC-BCS crossover, we found that away from half filling, there are extended regions of parameter space (interaction-density) for which a topological superfluid is the ground state. With increasing breaking of inversion symmetry, a gapless topological state intervenes before the system enters a trivial superfluid. We also consider the fluctuation effects on the ground states and show that the usual Anderson-Bogoliubov mode becomes quadratic in the absence of time-reversal symme-

We investigate the dynamics of parity-and time-reversal (PT ) symmetric two-energy-level atoms in the presence of two optical and a radio-frequency (rf) fields. The strength and relative phase of fields can drive the system from unbroken to broken PT symmetric regions. Compared with the Hermitian model, Rabi-type oscillation is still observed, and the oscillation characteristics are also adjusted by the strength and relative phase in the region of unbroken PT symmetry. At exception point (EP), the oscillation breaks down. To better understand the underlying properties we study the effective Bloch dynamics and find the emergence of the z components of the fixed points is the feature of the PT symmetry breaking and the projections in x-y plane can be controlled with high flexibility compared with the standard two-level system with PT symmetry. It helps to study the dynamic behavior of the complex PT symmetric model.

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