  We study the coherent dynamics of a quantum many-body system subject to a
time-periodic driving. We argue that in many cases, destructive interference in
time makes most of the quantum averages time-periodic, after an initial
transient. We discuss in detail the case of a quantum Ising chain periodically
driven across the critical point, finding that, as a result of quantum
coherence, the system never reaches an infinite temperature state. Floquet
resonance effects are moreover observed in the frequency dependence of the
various observables, which display a sequence of well-defined peaks or dips.
Extensions to non-integrable systems are discussed.
