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 wave function” $\xi \left(\mathbf{x}\right)$ and a phonon field operator $\chi \left(\mathbf{x}\right)$ in the form $\psi \left(\mathbf{x}\right)\approx A\{\xi \left(\mathbf{x}\right)+\chi \left(\mathbf{x}\right)/\sqrt{N}\}$ when the field operator acts on the $N$ particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of $\sqrt{N}$, and 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 $N$. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if $\xi \left(\mathbf{x}\right)$ is a solution of the time-independent Gross-Pitaevskii equation for $N$ particles and any chemical potential $\mu$, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed $N$ is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of $\sqrt{N}$ for the equations of motion. In this expansion the condensate wave function has the time-dependent form $\xi (\mathbf{x},t)$, and the condition for the validity of the expansion is that $\xi (\mathbf{x},t)$ satisfies the time-dependent Gross-Pitaevskii equation $\partial \xi /\partial t=-({ħ}^2/2m){\nabla }^2\xi +V\xi +Nu|\xi {|}^2\xi$. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as $\chi (\mathbf{x},t)={\sum }_k{\xi }_k(\mathbf{x},t){\alpha }_k,$ where the operators ${\alpha }_k$ are time-independent annihilation operators, and the state vector has a time evolution described by a Schrödinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function $\xi (\mathbf{x},t)$.

• Received 13 December 1996

DOI:https://doi.org/10.1103/PhysRevA.56.1414