  We consider quantum field theoretic systems subject to a time-dependent
perturbation, and discuss the question of defining a time dependent particle
number not just at asymptotic early and late times, but also during the
perturbation. Naively, this is not a well-defined notion for such a
non-equilibrium process, as the particle number at intermediate times depends
on a basis choice of reference states with respect to which particles and
anti-particles are defined, even though the final late-time particle number is
independent of this basis choice. The basis choice is associated with a
particular truncation of the adiabatic expansion. The adiabatic expansion is
divergent, and we show that if this divergent expansion is truncated at its
optimal order, a universal time dependence is obtained, confirming a general
result of Dingle and Berry. This optimally truncated particle number provides a
clear picture of quantum interference effects for perturbations with
non-trivial temporal sub-structure. We illustrate these results using several
equivalent definitions of adiabatic particle number: the Bogoliubov, Riccati,
Spectral Function and Schrodinger picture approaches. In each approach, the
particle number may be expressed in terms of the tiny deviations between the
exact and adiabatic solutions of the Ermakov-Milne equation for the associated
time-dependent oscillators.
