Beyond the Gross-Pitaevskii Equation: A Perturbative Approach

Abstract: We propose and implement a beyond Gross-Pitaevskii approach based on Many-Body Perturbation theory (MBPT) for the study of Bose-Einstein Condensate (BECs) ground states. Two partitions of the system Hamiltonian are considered. Firstly, a bosonic analogue to Møller-Plesset simply dubbed RSPT, and secondly an approach based on the Epstein-Nesbet partitioning scheme labeled ENPT. We consider one-dimensional BECs and work in the Harmonic Oscillator (HO) basis. Both third order RSPT and ENPT show overall good agreement with Full Configuration Interaction (FCI) in the low-particle number regime for a harmonically trapped BEC. For the same system, fast convergence is also seen towards the GP energy in the mean-field limit, as expected. Moreover, third order ENPT is seen to consistently produce lower ground state energies with better accuracy, compared to RSPT. Finally, we explore more complicated systems. Firstly, a BEC trapped in a double-well potential, where the mean-field ground state exhibits symmetry breaking. For low enough particle counts, when far away from the mean-field limit, we found that third order ENPT applied to a symmetric mean-field state results in lower energies compared to starting from the asymmetric mean-field ground state. Further studies examining the perturbed wave functions are necessary to determine whether or not this ground state is symmetric. Lastly, we study self-bound BEC droplet states, and although our methods are not able to correctly reproduce the characteristic energy minima they do hold promise in the study of self-bound states. Future studies exploring other starting points such as the extended GP equation are proposed.