Formulation for zero mode of Bose-Einstein condensate beyond Bogoliubov approximation

It is shown for the Bose-Einstein condensate of cold atomic system that the new unperturbed Hamiltonian, which includes not only the first and second powers of the zero mode operators but also the higher ones, determines a unique and stationary vacuum at zero temperature. From the standpoint of quantum field theory, it is done in a consistent manner that the canonical commutation relation of the field operator is kept. In this formulation, the condensate phase does not diffuse and is robust against the quantum fluctuation of the zero mode. The standard deviation for the phase operator depends on the condensed atom number with the exponent of $-1/3$, which is universal for both homogeneous and inhomogeneous systems.