Symmetries shape the current in ratchets induced by a biharmonic driving force

Abstract

Equations describing the evolution of particles, solitons, or localized structures, driven by a zero-average, periodic, external force, and invariant under time reversal and a half-period time shift, exhibit a ratchet current when the driving force breaks these symmetries. The biharmonic force f(t)=ϵ1 cos(qωt+ϕ1)+ϵ2 cos(pωt+ϕ2) does it for almost any choice of ϕ1 and ϕ2, provided p and q are two coprime integers such that p+q is odd. It has been widely observed, in experiments in semiconductors, in Josephson junctions, photonic crystals, etc., as well as in simulations, that the ratchet current induced by this force has the shape v∝ϵp1ϵq2 cos(pϕ1−qϕ2+θ0) for small amplitudes, where θ0 depends on the damping (θ0=π/2 if there is no damping, and θ0=0 for overdamped systems). We rigorously prove that this precise shape can be obtained solely from the broken symmetries of the system and is independent of the details of the equation describing the system.