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 f͑t͒ = ⑀ 1 cos͑qt + 1 ͒ + ⑀ 2 cos͑pt + 2 ͒ does it for almost any choice of 1 and 2 , 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 ϰ ⑀ 1 p ⑀ 2 q cos͑p 1 − q 2 + 0 ͒ for small amplitudes, where 0 depends on the damping ͑ 0 = / 2 if there is no damping, and 0 = 0 for overdamped systems͒. 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. Ratchetlike transport phenomena, where a net motion of particles or solitons is induced by zero-average forces, can be observed in many physical systems. Such is, for instance, the dc current in semiconductors ͓1-3͔, the net motion of fluxons in long Josephson junctions ͑JJs͒ ͓4,5͔, of vortices in superconductors ͓6͔, of cold atoms in optical lattices ͓7,8͔, or the rectification of Brownian motion ͓9-11͔. In some of these systems, the ratchetlike motion is induced by means of spatial asymmetries ͓12,13͔. In the others the transport can also appear if some temporal symmetries are broken by timedependent forces, e.g., ͓13-18͔. This latter case has two advantages: it is generally easier to analyze theoretically, and it is more amenable to experimental observation, e.g., in semiconductors ͓2͔, in JJs ͓4,5͔, or in optical lattices ͓8,19͔.A large number of simulations and experiments have show ͓2,4,5,7,8,11,20-27͔ that in many different systems the behavior of the ratchet velocity v driven by the T-periodic biharmonic forcewhere T =2 / , 1 and 2 are the phases, p and q are coprimes with p + q odd, and the amplitudes ⑀ 1 and ⑀ 2 are small, is given by the expressionwhere B and 0 depend on the parameters of the model and on but neither on the amplitudes nor on the phases ͓4,8,17,19,22,23,26͔. It has also been shown for specific systems that nondissipative dynamics have 0 = / 2 ͓7,15͔, whereas overdamped ones have 0 =0 ͓5,21,27͔. The aim of this Rapid Communication is to show that symmetry considerations alone are enough to predict behavior ͑2͒. This is a strong result because it is valid for any equation that describes the system, no matter the type of nonlinear terms it may contain, as long as it shows invariance under certain symmetry transformations-which will state precisely below. Attempts at determining the shape of current ͑2͒ can be found even in the pioneering works ͓2,20͔, aimed at developing a sensitive method of measuring deviations from Ohm's law. Their analysis, however, relies on an expansion of v in odd moments of f͑t͒, justified by the...

Universal asymptotic behavior in nonlinear systems driven by a two-frequency forcing

“…As shown in Ref. [11], the functional form in Eq. (14) is a consequence solely of the system symmetries, being independent of the interaction details.…”

“…(12), and satisfies the conditions imposed by Eqs. (10) and (11). It is to be noted that, in this figure,V ω 2,0 (0) =V (ω 2,0 ,0) = λ/T 0 , where T 0 = 2πq 0 /ω 1 = 6π/ω 1 is the period of the biharmonic force (see Ref.…”

Current reversals in a rocking ratchet: The frequency domain

“…For no dissipation (Hamiltonian case), the system is symmetric under time reversal for φ = nπ with n integer, and therefore for these values directed motion cannot be produced. It can be shown [9,23] that the dependence of the average velocity on the phase φ is, in leading order, v = A sin φ. Consider now the case of nonzero dissipation.…”

“…Thus, a current can be generated also for these values of the phase φ. For weak dissipation, the dependence of the average velocity on the phase φ is, in leading order, v = A sin(φ − φ 0 ), where φ 0 is a dissipation-induced symmetry-breaking phase lag which vanishes in the Hamiltonian limit [10,23,24]. Finally, in the overdamped regime, the system is invariant under the so-called supersymmetry [11] (x,p,t) → (x + λ/2, − p, − t), with λ the spatial period of the potential, for φ = π/2 + nπ , with n integer.…”

Dynamical Systems with Spatiotemporal Periodicities through the Symmetries

“…Time or space periodicity is reflected in the properties of these systems through a dependence on the parameters of their periodic terms [1,2]. In this talk, it is shown that simple symmetry considerations determine how their properties depend functionally on the amplitudes and the phases of the periodic terms, regardless of whether they are classical or quantum, stochastic or deterministic, dissipative or nondissipative [3].…”