Coupled fields in external background with application to non-thermal production of gravitinos

We provide the formalism for the quantization of systems of coupled bosonic and fermionic fields in a time dependent classical background. The occupat ion numbers of the particle eigenstates can be clearly defined and computed , through a generalization of the standard procedure valid for a single fie ld in which Bogolyubov coefficients are employed. We apply our formalism to the problem of non-thermal gravitino production in a two-fields model wher e supersymmetry is broken gravitationally in the vacuum. Our explicit calcu lations show that this production is strongly suppressed in the model consi dered, due to the weak coupling between the sector which drives inflation a nd the one responsible for supersymmetry breakdown.