  The present paper outlines a basic theoretical treatment of decoherence and
dephasing effects in interferometry based on single component BEC in double
potential wells, where two condensate modes may be involved. Results for both
two mode condensates and the simpler single mode condensate case are presented.
A hybrid phase space distribution functional method is used where the
condensate modes are described via a truncated Wigner representation, and the
basically unoccupied non-condensate modes are described via a positive P
representation. The Hamiltonian for the system is described in terms of quantum
field operators for the condensate and non-condensate modes. The functional
Fokker-Planck equation for the double phase space distribution functional is
derived. Equivalent Ito stochastic equations for the condensate and
non-condensate fields that replace the field operators are obtained, and
stochastic averages of products of these fields give the quantum correlation
functions used to interpret interferometry experiments. The stochastic field
equations are the sum of a deterministic term obtained from the drift vector in
the functional Fokker-Planck equation, and a noise field whose stochastic
properties are determined from the diffusion matrix in the functional
Fokker-Planck equation. The noise field stochastic properties are similar to
those for Gaussian-Markov processes in that the stochastic averages of odd
numbers of noise fields are zero and those for even numbers of noise field
terms are sums of products of stochastic averages associated with pairs of
noise fields. However each pair is represented by an element of the diffusion
matrix rather than products of the noise fields themselves. The treatment
starts from a generalised mean field theory for two condensate mode. The
generalized mean field theory solutions are needed for calculations using the
Ito stochastic field equations.
