Electron climbing a "devil's staircase" in wave-particle interaction.

Numerous nonlinear driven systems display spectacular responses to forcing, including chaos and complex phase-locking plateaus characterized by "devil's staircase," Arnold tongues, and Farey trees. In the universality class of Hamiltonian systems, a paradigm is the motion of a charged particle in two waves, which inspired a renormalization group method for its description. Here we report the observation of the underlying "devil's staircase" by recording the beam velocity distribution function at the outlet of a traveling wave tube versus the amplitude of two externally induced waves.