Symmetries shape the current in ratchets induced by a bi-harmonic force

TL;DR

It is rigorously proved that the ratchet current induced by the biharmonic force has the shape v proportional, variant1p2q cos(pphi1-qphi2+theta0) for small amplitudes, where theta0 depends on the damping.

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 bi-harmonic force $f(t)=ε_1\cos(q ωt+φ_1)+ε_2\cos(pωt+φ_2)$ does it for almost any choice of $φ_{1}$ and $φ_{2}$, provided $p$ and $q$ are two co-prime integers such that $p+q$ is odd. It has been widely observed, in experiments in Josephson-junctions, photonic crystals, etc., as well as in simulations, that the ratchet current induced by this force has the shape $v\proptoε_1^pε_2^q\cos(p φ_{1} - q φ_{2} + θ_0)$ for small amplitudes, where $θ_0$ depends on the damping ($θ_0=π/2$ if there is no damping, and $θ_0=0$ 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.