We have analytically explored the quantum phenomenon of particle scattering by harmonically trapped Bose and Fermi gases with the short ranged (Fermi-Huang δ 3 p [1]) interactions among the incident particle and the scatterers. We have predicted differential scattering cross-sections and their temperature dependence in this regard. Coherent scattering even by a single boson or fermion in the finite geometry gives rise to new tool of determining energy eigenstate of the scatterer. Our predictions on the differential scattering cross-sections, can be tested within the present day experimental setups, specially, for (i) 3-D harmonically trapped interacting Bose-Einstein condensate (BEC), (ii) BECs in a double well, and (iii) BECs in an optical lattice.

We report the appearance of multiple numbers of completely flat band states in an extended Lieb lattice model in two dimensions with five atomic sites per unit cell. We also show that this edge-centered square lattice can host intriguing topologically nontrivial phases when intrinsic spinorbit (ISO) coupling is introduced in the microscopic description of the corresponding tight-binding Hamiltonian of the system. This ISO coupling strength acts like a complex next-nearest-neighbor hopping term for this model and can be, in principle, tuned in a real-life experimental setup. In the presence of this ISO coupling, the band spectrum of the system gets gapped out, leading to nonzero integer values of the spin Chern number for different bands, indicating the nontrivial topological properties of the system. Furthermore, we show that for certain values of the ISO coupling, nearly flat bands with nonzero Chern numbers emerge in this lattice model. This opens up the possibility of realizing interesting fractional quantum spin Hall physics in this model when interaction is taken into account. This study might be very useful in an analogous optical lattice experimental setup. A possible application of our results can also be anticipated in the field of photonics using single-mode photonic waveguide networks. * ; Present address: Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel.

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