A particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas

Abstract

The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number $ N$ of particles. This generalization yields a description of the Schrödinger picture field operators as the product of an annihilation operator $A$ for the total number of particles and the sum of a ``condensate wavefunction'' $ξ(x)$ and a phonon field operator $χ(x)$ in the form $ψ(x) \approx A\{ξ(x) + χ(x)/\sqrt{N}\}$ when the field operator acts on the N particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of $\sqrt{N}$, an thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different N.