  A phase space theory approach for treating dynamical behaviour of
Bose-Einstein condensates applicable to situations such as interferometry with
BEC in time-dependent double well potentials is presented. Time-dependent mode
functions are used, chosen so that one, two,.. highly occupied modes describe
well the physics of interacting condensate bosons in time dependent potentials
at well below the transition temperature. Time dependent mode annihilation,
creation operators are represented by time dependent phase variables, but time
independent total field annihilation, creation operators are represented by
time independent field functions. Two situations are treated, one (mode theory)
is where specific mode annihilation, creation operators and their related phase
variables and distribution functions are dealt with, the other (field theory)
is where only field creation, annihilation operators and their related field
functions and distribution functionals are involved. The paper focuses on the
hybrid approach, where the modes are divided up between condensate (highly
occupied) modes and non-condensate (sparsely occupied) modes. It is found that
there are extra terms in the Ito stochastic equations both for the stochastic
phases and stochastic fields, involving coupling coefficients defined via
overlap integrals between mode functions and their time derivatives. For the
hybrid approach both the Fokker-Planck and functional Fokker-Planck equations
differ from those derived via the correspondence rules, the drift vectors are
unchanged but the diffusion matrices contain additional terms involving the
coupling coefficients. Results are also presented for the combined approach
where all the modes are treated as one set.
