  The response of a linear system to an external perturbation is governed by
the Fourier limit, with the inverse of the interaction time constituting a
lower limit for the system bandwidth. This does not hold for nonlinear systems,
which can thus exhibit sub-Fourier behavior. The present work identifies a
mechanism for sub-Fourier sensitivity in driven quantum systems, which relies
on avoided crossing between Floquet states. Features up to three orders of
magnitude finer than the Fourier limit are presented.
