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Based on the Cornwall-Jackiw-Tomboulis (CJT) effective action approach and the requirement to preserve the Goldstone theorem within two-loop approximation, we study the universal quantum effect, which is related to the two-body correlation, in a homogeneous weakly interacting Bose gas at zero temperature. We find the state equations that provide the condensate fraction, energy density, chemical potential, and sound speed. They are the series expansions of the gas parameter.

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The density of condensates of a binary mixture of Bose gases restricted by two parallel planar walls is investigated within the framework of Cornwal-Jackiw-Tomboulis effective action approach in improved Hartree-Fock approximation. It results that the density of condensates strongly depend not only on the distance between two walls but also on the interspecies interaction strength and they are equal their expectation values of the field operators after adding a term associated with the quantum fluctuations.

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