Number-conserving approaches to $n$-component Bose-Einstein condensates

We develop the number-conserving approach, which has previously been used in a single-component Bose-Einstein condensed dilute atomic gas, to describe consistent coupled condensate and noncondensate number dynamics, to an $n$-component condensate. The resulting system of equations is comprised, for each component, of a generalized Gross-Pitaevskii equation coupled to modified Bogoliubov–de Gennes equations. Lower order approximations yield general formulations for multicomponent Gross-Pitaevskii equations, and systems of multicomponent Gross-Pitaevskii equations coupled to multicomponent modified number-conserving Bogoliubov–de Gennes equations. The analysis is left general, such that, in the $n$-component condensate, there may or may not be mutually coherent components. An expansion in powers of the ratio of noncondensate-to-condensate particle numbers for each coherent set is used to derive the self-consistent, second-order, dynamical equations of motion. The advantage of the analysis developed in this article is in its applications to dynamical instabilities that appear when two (or more) components are in conflict and where a significant noncondensed fraction of atoms is expected to appear.