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

Symmetries shape the current in ratchets induced by a biharmonic driving force Niurka R. Quintero,1,*José A. Cuesta,2,†and Renato Alvarez-Nodarse3,‡ 1Departamento de Física Aplicada I, E.U.P., Universidad de Sevilla, Virgen de África 7, E-41011 Sevilla, Spain 2Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, Leganés, E-28911 Madrid, Spain 3Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, E-41080 Sevilla, Spain 共Received 28 June 2009; revised manuscript received 15 December 2009; published 23 March 2010兲 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共t兲= ⑀ 1cos共q ␻ t+ ␾ 1兲+ ⑀ 2cos共p ␻ t + ␾ 2兲does it for almost any choice of ␾ 1and ␾ 2, provided pand qare two coprime integers such that p+qis 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⬀ ⑀ 1 p ⑀ 2 qcos共p ␾ 1 −q ␾ 2+ ␪ 0兲for small amplitudes, where ␪ 0depends on the damping 共 ␪ 0= ␲ /2 if there is no damping, and ␪ 0 =0 for overdamped systems兲. 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. DOI: 10.1103/PhysRevE.81.030102 PACS number共s兲: 05.60.Cd, 05.45.Yv 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 关1–3兴, the net motion of fluxons in long Josephson junctions 共JJs兲关4,5兴, of vortices in superconductors 关6兴, of cold atoms in optical lattices 关7,8兴,orthe rectification of Brownian motion 关9–11兴. In some of these systems, the ratchetlike motion is induced by means of spatial asymmetries 关12,13兴. In the others the transport can also appear if some temporal symmetries are broken by timedependent forces, e.g., 关13–18兴. 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 关2兴,inJJs关4,5兴, or in optical lattices 关8,19兴. A large number of simulations and experiments have show 关2,4,5,7,8,11,20–27兴that in many different systems the behavior of the ratchet velocity vdriven by the T-periodic biharmonic force f共t兲= ⑀ 1cos共q ␻ t+ ␾ 1兲+ ⑀ 2cos共p ␻ t+ ␾ 2兲,共1兲 where T=2 ␲ / ␻ , ␾ 1and ␾ 2are the phases, pand qare coprimes with p+qodd, and the amplitudes ⑀ 1and ⑀ 2are small, is given by the expression v=B ⑀ 1 p ⑀ 2 qcos共p ␾ 1−q ␾ 2+ ␪ 0兲,共2兲 where Band ␪ 0depend on the parameters of the model and on ␻ but neither on the amplitudes nor on the phases 关4,8,17,19,22,23,26兴. It has also been shown for specific systems that nondissipative dynamics have ␪ 0= ␲ /2关7,15兴, whereas overdamped ones have ␪ 0=0 关5,21,27兴. The aim of this Rapid Communication is to show that symmetry considerations alone are enough to predict behavior 共2兲. 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 共2兲can be found even in the pioneering works 关2,20兴, aimed at developing a sensitive method of measuring deviations from Ohm’s law. Their analysis, however, relies on an expansion of vin odd moments of f共t兲, justified by the adiabatic response of the system to an applied field 共see also 关28兴兲. While it cannot be ruled out that such an expansion holds for some systems, or in this adiabatic limit, it is certainly not valid in general. In fact, if one applies that expansion to related dissipative systems, like those of Refs. 关2,4,7,8,15,17,22,23,26兴, the value ␪ 0=0 is always obtained, whereas ␪ 0⫽0 in general—it can even be ␪ 0= ␲ /2 when dissipation vanishes. We illustrate this fact by analyzing in the supplementary material 关38兴an exactly solvable example. Therefore one can readily see that the moment expansion is in general an incorrect assumption; only in the overdamped or the adiabatic limits could this expansion become correct, but we do not know of any proof that this holds for systems other than this specific example. Let E关x共t兲,f共t兲兴=0 denote a functional equation 共which can represent an ordinary or partial differential equation, an integral equation, etc.兲describing the evolution of a particle, soliton, or localized structure whose position is given by x共t兲, under the driving of a zero-average, external, periodic force f共t兲=f共t+T兲,T⬎0. One such system is said to have ratchetlike behavior if the average velocity, defined as 关13兴 v= lim t→⬁ 1 t 冕 0 t x ˙共 ␶ 兲d ␶ = lim t→⬁ x共t兲 t,共3兲 independent of the initial conditions 关29兴, is nonzero. Consider two temporal transformations: time reversal 共R:t哫 −t兲and time shift 共S:t哫t+T/2兲, and suppose that their action on the force f共t兲is given by *[email protected] †[email protected] ‡[email protected] PHYSICAL REVIEW E 81, 030102共R兲共2010兲 RAPID COMMUNICATIONS 1539-3755/2010/81共3兲/030102共4兲©2010 The American Physical Society030102-1 共Rf兲共t兲=f共−t兲=f共t兲,共4兲 共Sf兲共t兲=f共t+T/2兲=−f共t兲.共5兲 Suppose further that any of these transformations—with the appropriate transformation of x共t兲—leaves E关x共t兲,f共t兲兴=0 invariant. Nondissipative systems provide typical examples of this kind of behavior such as the equation of motions of cold atoms in optical lattices 关7兴, the dynamic of a particle in a symmetric potential 关15兴, and the soliton ratchets in the extended systems 关17,25兴. For these kind of systems and forces which satisfy either Eq. 共4兲or 共5兲—or both—there can be no ratchet effect because any of the two transformations changes the sign of v 共Rbecause time goes backward, and S, because fchanges sign兲. As a matter of fact, this is a nice illustration of Curie’s principle 关30兴. In some other cases, time reversal leaves the equation invariant provided the force transforms itself as 共Rf兲共t兲=f共−t兲=−f共t兲共6兲 instead of Eq. 共4兲. The most prominent examples of this are equations describing overdamped systems such as the vortex motion in JJs 关5兴, the overdamped Brownian motion 关16兴, and the ratchet dynamics of breathers in the discrete Schrödinger equation 关27兴. Again, and for the same reason, no ratchet effect can appear if the driving force fulfills Eq. 共6兲. In this case, however, in general breaking symmetries 共5兲and 共6兲is not enough to induce a ratchet current, some additive noise is necessary as well 关21兴. Whichever the case, a biharmonic force such as Eq. 共1兲is able to break both Eq. 共4兲关or Eq. 共6兲兴 and Eq. 共5兲and induce a ratchet current. In what follows we will prove that, provided a ratchet current is produced, the symmetries impose that it be of form 共2兲. Let us begin by noticing that vmust be a functional of f共t兲, which we can expand as v关f兴=v0+兺 n=1 ⬁ vn关f兴, vn关f兴=具cn共t1, ...,tn兲f共t1兲¯f共tn兲典,共7兲 where 具X典⬅T−n兰0 Tdt1¯兰0 TdtnXand v0=v关0兴. This functional Taylor expansion is a rigorous result of functional analysis valid for a very wide class of functionals on Banach spaces 共see 关31–33兴for details兲.Asv关−f兴=−v关f兴for any force f共t兲, c2n共t1,...,t2n兲⬅0, so only odd terms appear in expansion 共7兲. On the other hand, the functions cn共t1,...,tn兲can be taken Tperiodic in each variable and can always be chosen totally symmetric under any exchange of their arguments. Notice in passing that only if cn共t1,...,tn兲⬀ ␦ 共t1 −t2兲¯ ␦ 共tn−1−tn兲can vbe expanded in moments of f共t兲—thus the moment expansion is only a particular case of Eq. 共7兲. Let us now specialize Eq. 共7兲for biharmonic force 共1兲. First of all, vis not affected by the choice of time origin; thus v关T ␶ f兴=v关f兴, where 共T ␶ f兲共t兲=f共t+ ␶ 兲for any ␶ . But f共t + ␶ 兲= ⑀ 1cos共qt+ ␾ ˜ 1兲+ ⑀ 2cos共pt+ ␾ ˜ 2兲, with ␾ ˜ 1= ␾ 1+q ␶ and ␾ ˜ 2= ␾ 2+p ␶ ,sov关f兴must depend on the phases only through the combination ␪ =p ␾ 1−q ␾ 2=p ␾ ˜ 1−q ␾ ˜ 2. Now we must compute vn关f兴for any odd n⬎0. By expanding Eq. 共1兲in complex exponentials, vn关f兴=兺 兩n兩=n,nⱖ0 A共n兲 ⑀ 1 n1+n2 ⑀ 2 n3+n4ei关共n1−n2兲 ␾ 1+共n3−n4兲 ␾ 2兴,共8兲 where n=共n1,n2,n3,n4兲,兩n兩⬅n1+n2+n3+n4, and nⱖ0de- notes a componentwise inequality. Besides, because of the symmetry of the functions cn共t1,...,tn兲, A共n兲=n!2−n 兿 i=1 4 ni! 具cn共t1, ...,tn兲ei ␻ v·共t1,...,tn兲典,共9兲 where v ⬅共 q , ..., q ,− q , ...,− q ,p, ...,p,− p, ...,− p n 1 n 2 n 3 n 4 兲 . The complex number A共n兲=B共n兲ei ␺ 共n兲, where ␺ 共n1,n2,n3,n4兲=− ␺ 共n2,n1,n4,n3兲, B共n1,n2,n3,n4兲=B共n2,n1,n4,n3兲.共10兲 Let us focus now on the complex exponential in Eq. 共8兲. We know that vmust be a function of ␪ , so the only nonzero terms in this sum are those satisfying 共n1−n2兲 ␾ 1+共n3 −n4兲 ␾ 2=k ␪ for some k苸Z, i.e., n1−n2=kp and n4−n3=kq. But that means 兩k兩共p+q兲=兩n1−n2−n3+n4兩ⱕ兩n1+n2+n3+n4兩 =n. Suppose n⬍p+q; then k=0, which implies n1=n2and n3=n4, and therefore nmust be even. Thus vn关f兴must be zero for any odd n⬍p+q, but since there are no even terms in Eq. 共7兲, this means that no term with n⬍p+qcontributes to v. Suppose now nⱖp+q; then 兩k兩⬎0共k=0 is excluded because it would lead to an even n兲and so there will be nonzero terms in Eq. 共8兲corresponding to powers ⑀ 1 2n2+兩k兩p ⑀ 2 2n3+兩k兩q or ⑀ 1 2n1+兩k兩p ⑀ 2 2n4+兩k兩q. The lowest order is n=p+q, and is obtained either when n2=n3=0, n1=p, and n4=q共i.e., k=1兲,or when n1=n4=0, n2=p, and n3=q共i.e., k=−1兲. Because of Eq. 共10兲, the contribution of these two terms to Eq. 共8兲is vp+q=B ⑀ 1 p ⑀ 2 qcos共 ␪ + ␪ 0兲,共11兲 where B=2B共p,0,0,q兲and ␪ 0= ␺ 共p,0,0,q兲. Let us now assume that the equation is invariant when the force satisfies 共4兲. Then, since 具cn共t1,...,tn兲f共−t1兲¯f共−tn兲典 =具cn共−t1,...,−tn兲f共t1兲¯f共tn兲典, and v关Rf兴=−v关f兴, hence cn共−t1, ...,−tn兲=−cn共t1, ...,tn兲.共12兲 Applied to Eq. 共9兲this means that B共n兲e−i ␺ 共n兲=−B共n兲ei ␺ 共n兲; i.e., ␺ 共n兲= ␲ /2 for all n. On the other hand, if the equation is invariant when the force satisfies Eq. 共6兲, then v关−Rf兴=−v关f兴,so共recall that n is odd兲 QUINTERO, CUESTA, AND ALVAREZ-NODARSE PHYSICAL REVIEW E 81, 030102共R兲共2010兲 RAPID COMMUNICATIONS 030102-2 cn共−t1, ...,−tn兲=cn共t1, ...,tn兲.共13兲 Applied to Eq. 共9兲this means that B共n兲e−i ␺ 共n兲=B共n兲ei ␺ 共n兲; i.e., ␺ 共n兲=0 for all n. What we have just shown is that the mean velocity, v,of the ratchet current induced by biharmonic force 共1兲in an equation which is invariant under Eq. 共5兲always has form 共2兲if the amplitudes ⑀ 1and ⑀ 2are small. The coefficients B and ␪ 0depend on the frequency and on the remaining parameters of the system but not in a universal way that can be predicted under symmetry arguments such as these ones. This shape for the current has been observed, mostly for p =2 and q=1, in experimental, numerical, and theoretical results in several seemingly unrelated systems 关4,7,15,17,21–23,25兴. For p=4 and q=1, the collective coordinate on soliton ratchets developed in 关26兴also confirms Eq. 共2兲. If the equation is also invariant under Eq. 共4兲关7,15兴, then ␪ 0= ␲ /2 and we recover the form v⬃ ⑀ 1 p ⑀ 2 qsin ␪ , whereas if the equation is invariant under Eq. 共6兲, then ␪ 0=0 and v ⬃ ⑀ 1 p ⑀ 2 qcos ␪ , in agreement with the vortex motion observed in JJs 关5兴, with the overdamped stochastic dynamic of particles studied in 关11,21兴, and with the ratchet mobility of breathers in the discrete nonlinear Schrödinger equation computed for p=2 and q=1 in 关27兴. Notice that formula 共2兲does not imply that Bmust be nonzero 共Curie’s principle兲. It only proves, under symmetry arguments, that the leading term of vcan be of that precise form. It might well happen, for some specific equation, that B=0. In this case this analysis shows that the leading term must have a dependence on the amplitudes through powers higher than pand q. It is likely that if this occurs it will be the fingerprint of a hidden symmetry which, properly broken, will restore result 共2兲. This analysis provides a direct way to quantitatively relate the causes and the consequences of phenomena through Curie’s principle. For instance, our study can be extended to the so-called gating effect, i.e., when the amplitude of spatial or field potentials for particles or solitons, respectively, is modified by a multiplicative force g共t兲= ⑀ 2cos共p ␻ t+ ␾ 2兲as well as an additive force, f共t兲= ⑀ 1cos共q ␻ t+ ␾ 1兲, with pand q coprimes, acts on the system 关34–36兴. In such systems, if both f共t兲and g共t兲satisfy Eq. 共4兲共in the nondamped limit兲or Eq. 共6兲共in the overdamped limit兲,orf共t兲fulfills Eq. 共5兲,a ratchet transport cannot be induced. A similar procedure shows that, when these symmetries are broken, the average ratchet velocity is also given by Eq. 共2兲, where ␪ 0=0 or ␪ 0 = ␲ /2 in the nondamped or overdamped limits, respectively 关37兴. We acknowledge financial support through Grant Nos. 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Lett. 100, 040603 共2008兲. 关37兴N. R. Quintero, J. A. Cuesta and R. Álvarez-Nodarse 共unpublished兲. 关38兴See supplementary material at http://link.aps.org/supplemental/ 10.1103/PhysRevE.81.030102 for an exactly solvable example. QUINTERO, CUESTA, AND ALVAREZ-NODARSE PHYSICAL REVIEW E 81, 030102共R兲共2010兲 RAPID COMMUNICATIONS 030102-4