  We consider the time evolution of the adiabatic particle number in both
time-dependent electric fields and in de Sitter spaces, and define a
super-adiabatic particle number in which the (divergent) adiabatic expansion is
truncated at optimal order. In this super-adiabatic basis, the particle number
evolves smoothly in time, according to Berry's universal adiabatic smoothing of
the Stokes phenomenon. This super-adiabatic basis also illustrates clearly the
quantum interference effects associated with particle production, in particular
for sequences of time-dependent electric field pulses, and in eternal de Sitter
space where there is constructive interference in even dimensions, and
destructive interference in odd dimensions.
