A spatially localized ultracold matter wave is a long lived, coherent wave packet of high tunability, paving huge scope for emerging areas like quantum simulation and quantum sensing. Applications of ultracold atoms in quantum information science mostly rely on the external trap which can efficiently be engineered to a desired shape due to the unprecedented progress in the experimental front. However, investigating the dynamics of such system through exact theoretical approach becomes quite nontrivial due to its nonlinear nature and the presence of varying external trap upon engineering [1-5].
I will present exact theoretical methods to investigate BEC under various feasible engineered traps which include tunable optical lattices, ring-shaped trap and then use them for quantum information processing and quantum sensing [6,7]. We probe phase space interference structures of these systems which manifest Heisenberg limited measurements of observables like displacement, momentum and indirect measurements of temperature. Situations of negative absolute temperature is also explored [2,3], along with various nonlinear excitations.