  We study the asymptotic dynamics of arbitrary linear quantum open systems
which are periodically driven while coupled with generic bosonic reservoirs. We
obtain exact results for the heat flowing into the network, which are valid
beyond the usual weak coupling or Markovian approximations. We prove the
validity of the dynamical third law of thermodynamics (Nernst unattainability
principle), showing that the ultimate limit for cooling is imposed by a
fundamental heating mechanism which becomes dominant at low temperatures: the
non resonant creation of pairs of excitations in the reservoirs induced by the
driving field. This quantum effect, which is missed in the weak coupling
approximation, restores the unattainability principle whose validity was
recently challenged.
