A number-conserving approach to a minimal self-consistent treatment of   condensate and non-condensate dynamics in a degenerate Bose gas
  We describe a number conserving approach to the dynamics of Bose-Einstein condensed dliute atomic gases. This builds upon the works of Gardiner [C. W. Gardiner, Phys. Rev. A 56, 1414 (1997)], and Castin and Dum [Y. Castin and R. Dum, Phys. Rev. A 57, 3008 (1998)]. We consider what is effectively an expansion in powers of the ratio of non-condensate to condensate particle numbers, rather than inverse powers of the total number of particles. This requires the number of condensate particles to be a majority, but not necessarily almost equal to the total number of particles in the system. We argue that a second-order treatment of the relevant dynamical equations of motion is the minimum order necessary to provide consistent coupled condensate and non-condensate number dynamics for a finite total number of particles, and show that such a second-order treatment is provided by a suitably generalized Gross-Pitaevskii equation, coupled to the Castin-Dum number-conserving formulation of the Bogoliubov-de Gennes equations. The necessary equations of motion can be generated from an approximate third-order Hamiltonian, which effectively reduces to second order in the steady state. Such a treatment as described here is suitable for dynamics occurring at finite temperature, where there is a significant non-condensate fraction from the outset, or dynamics leading to dynamical instabilities, where depletion of the condensate can also lead to a significant non-condensate fraction, even if the non-condensate fraction is initially negligible. 
S. A. Gardiner and S.A. Morgan