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We have studied the effect of time-independent repulsive random impurity potential on the quantum droplets of Bose-Einstein condensation of two different species of Bose atoms in two dimensions. We have solved the Gross-Pitaevskii equation to get the density profile and the energy of the condensate. In our study, we found that quantum droplets are incredibly robust against the non-chemical reactive impurity.

We have studied the collective excitation of Bose-Einstein condensate of short range weak interacting atoms with spin-orbit coupling (SOC) in two dimension in liquid phase. In our study, we have included Rashba, Dresselhaus SOC as well as Raman SOC. The study of Bogoliubov excitation shows that in small interaction case only phonon modes are present and for higher interactions roton modes start to appear. Energy spectra contain two roton modes in the system relatively strong interacting atomic system.

We have studied the collective excitation of fractional quantum Hall effect (FQHE) in the rotating Bose-Einstein condensate (BEC) using CF theory at the filling fraction $\nu=1/2$. The roton type of excitation in the FQHE of electron system is established over the years for all the filling fraction, whereas the collective excitation at $\nu=1/2$ filling fraction in the rotating BEC shows no roton minimum. We have investigated this using composite fermion theory with the P$\ddot{o}$schl-Teller interaction potential between the particles. We have seen that the long range interaction will give the roton, whereas short range interaction gives no roton minimum.

In this work, we would like to present the Bose-Einstein Condensation in such a system where rotation is decreasing radially from the centre of the condensate. That non-uniform rotation is defined by a rotating parameter called $\lambda$. The system is defined by a modified Gross-Pitaevskii equation. The result shows very different behaviour from uniformly rotating condensate. In a uniformly rotating case there is formation triangular vortex lattice but in our case, we are watching that vortices are formed in circular ring shape above a specific amount rotation defined by $\lambda_c$. Below $\lambda_c$ there is a distortion i.e. there is neither circular ring shape nor triangular symmetry among the vortices. We have studied the energy and chemical potential of the system. We have seen a sharp change in the energy and chemical potential of the systems at the point of $\lambda_c$. In this rich complex phase, we have drawn a phase diagram associated with the phase transition from disordered to circular ring shape pattern.