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We consider a dilute and ultracold bosonic gas of weakly-interacting atoms.Within the framework of quantum field theory we derive a zero-temperaturemodified Gross-Pitaevskii equation with beyond-mean-field corrections due toquantum depletion and anomalous density. This result is obtained from thestationary equation of the Bose-Einstein order parameter coupled to theBogoliubov-de Gennes equations of the out-of-condensate field operator. We showthat, in the presence of a generic external trapping potential, the key stepsto get the modified Gross-Pitaevskii equation are the semiclassicalapproximation for the Bogoliubov-de Gennes equations, a slowly-varying orderparameter, and a small quantum depletion. In the uniform case, from themodified Gross-Pitaevskii equation we get the familiar equation of state withLee-Huang-Yang correction.

Self-consistent derivation of the modified Gross-Pitaevskii equation with Lee-Huang-Yang correction