  We consider the adiabatic limit of Hadamard states for free quantum
Klein-Gordon fields, when the background metric and the field mass are slowly
varied from their initial to final values. If the Klein-Gordon field stays
massive, we prove that the adiabatic limit of the initial vacuum state is the
(final) vacuum state, by extending to the symplectic framework the adiabatic
theorem of Avron-Seiler-Yaffe. In cases when only the field mass is varied,
using a abstract version of the mode decomposition method we can also consider
the case when the initial or final mass vanishes, and the initial state is
either a thermal state or a more general Hadamard state.
