Hidden symmetries, instabilities, and current suppression in brownian ratchets

Hidden Symme ies, Ins abili ies, and Cu en Supp ession in B ownian Ra che s Da id Cube o1,* and Fe uccio Renzoni2,† 1Depa amen o de Física Aplicada I, EUP, Uni e sidad de Se illa, Calle Vi gen de Á ica 7, 41011 Se illa, Spain 2Depa men o Physics and As onomy, Uni e si y College London, Gowe S ee , London WC1E 6BT, Uni ed Kingdom (Recei ed 12 Sep embe 2015; published 7 Janua y 2016) The ope a ion o B ownian mo o s is usually desc ibed in e ms o ou -o -equilib ium and symme y- b eaking se ings, wi h he ele an spa io empo al symme ies iden i ied om he analysis o he equa ions o mo ion o he sys em a hand. When he app op ia e condi ions a e sa is ied, symme y- ela ed ajec o ies wi h opposi e cu en a e hough o balance each o he , yielding supp ession o anspo . The di ec ion o he cu en can be p ecisely con olled a ound hese symme y poin s by inely uning he d i ing pa ame e s. He e we demons a e, by s udying a p o o ypical B ownian a che sys em, he exis ence o hidden symme ies, which escape iden i ica ion by he s anda d symme y analysis, and which equi e di e en heo e ical ools o hei e ela ion. Fu he mo e, we show ha sys em ins abili ies may lead o spon aneous symme y b eaking wi h unexpec ed gene a ion o di ec ed anspo . DOI: 10.1103/PhysRe Le .116.010602 Mo ion a he nanoscale p esen s ea u es e y di e en om hose encoun e ed in he mac oscopic wo ld. Noise is a dominan p ocess a such a scale, and may con ibu e cons uc i ely o he dynamics a he han playing he usual ole o a dis u bance. New mechanisms o anspo eme ge a he nanoscale; in pa icula , di ec ed mo ion may occu in he absence o an applied bias o ce. B ownian a che s [1–3], he a che ypal model sys em cap u ing he mecha- nisms behind such a anspo p ocess, ep esen a key o unde s anding se e al biological p ocesses [4,5]; hey also ha e inspi ed a ple ho a o new nanode ices displaying di ec ed mo ion [6–21]. All hese sys ems a e usually desc ibed in e ms o ope a ion away om he mal equi- lib ium, wi h di ec ed mo ion ollowing om he b eaking o ce ain spa io empo al symme ies, which a e iden i ied om he analysis o he equa ions o mo ion o he sys em a hand. He e we p o e he exis ence o hidden symme ies, which escape iden i ica ion by he s anda d symme y analysis [2,3,22,23], and equi e di e en heo e ical ools o hei e ela ion. The main assump ion o he s anda d symme y analysis—i.e., ha wo ajec o ies connec ed by a symme y ans o ma ion ca y he same s a is ical weigh , a easoning ha can be aced back o Loschmid ’s pa adox [24]—yields inco ec p edic ions in hese dissipa i e sys ems, ailing o accoun o sys em ins abili ies ha lead o spon aneous symme y b eaking. Resul s.—A la ge class o B ownian mo o sys ems, which includes pa icles in solu ion [8], o ices in supe - conduc o s [14], and a oms in dissipa i e op ical la ices [21], co esponds o a B ownian pa icle di using in a pe iodic po en ial unde he ac ion o a d i ing o ce wi h ze o a e age. The pa icle’s mo ion is desc ibed by he ollowing Lange in equa ion: m x¼−γ_ xþFðx; Þþξð Þ;ð1Þ whe e γis he ic ion coe icien , Fðx; Þis a gene ic de e minis ic o ce, and ξð Þis a luc ua ing o ce, modeled as a Gaussian whi e noise wi h au oco ela ion hξð Þξð 0Þi ¼ 2Γδð − 0Þ, wi h he noise s eng h Γ ela ed o he empe a u e To he en i onmen ia he luc ua ion- dissipa ion ela ion Γ¼γkBT. The di ec ed cu en is de ined as h i¼lim →∞hxð Þi= , whe e he angle b acke s deno e he a e age o e noise ealiza ions. Fo ini e noise s eng hs Γ>0, e godici y implies h i¼lim →∞xð Þ= .In e y small sys ems, om he nanoscale o he mic oscale, he B ownian dynamics o small pa icles is equen ly in he o e damped egime, whe e ine ia e ec s— he e m m x in (1)—can be neglec ed. This is he egime o in e es he e. The s anda d symme y analysis [2,3,22,23] elies on he iden i ica ion o ans o ma ions ha lea e he equa ion o mo ion (1) unchanged and e e se he sign o he pa icle momen um. T ajec o ies wi h opposi e momen um a e equi alen , wi h a ne null con ibu ion o he di ec ed cu en , which hus u ns ou o be ze o. We will show ha his pic u e does no ully cap u e he basic p inciples behind he ope a ion o B ownian a che s. To do his, we conside a mo e gene al app oach [25], and ega d he di ec ed cu en h ias a gene ic unc ional o he d i ing o ce F, hus using he no a ion ½Fðx; Þ. Se e al p ope - ies ollow om symme y conside a ions. Fi s , due o he ec o ial na u e o bo h he o ce Fand he cu en , he ans o ma ion x→−xyields he ollowing p ope y: ½−Fð−x; Þ ¼ − ½Fðx; Þ:ð2Þ Second, an a bi a y ansla ion along he xo axis does no al e he cu en , i.e., ½Fðx; Þ ¼ ½Fðxþx0; Þ ¼ ½Fðx; þ 0Þ:ð3Þ PRL 116, 010602 (2016) PHYSICAL REVIEW LETTERS week ending 8 JANUARY 2016 0031-9007=16=116(1)=010602(6) 010602-1 © 2016 Ame ican Physical Socie y Le us conside now a o ced a che , i.e., Fðx; Þ¼ ðxÞþFð Þ, whe e ðxÞ¼−∂UðxÞ=∂xis a conse a i e o ce and Fð Þis a d i ing o ce. I he sys em is spa ially symme ic wi h espec o a ce ain poin x0, hen he po en ial sa is ies Uðxþx0Þ¼ Uð−xþx0Þ. Wi hou loss o gene ali y, we choose he coo dina e’s o igin such ha x0¼0. Then − ð−xÞ¼ ðxÞ, which oge he wi h (2) yields a cha ac e is ic p ope y o spa ially symme ic sys ems, ½ ðxÞ−Fð Þ ¼ − ½ ðxÞþFð Þ:ð4Þ A shi -symme ic o ce is de ined as Fð þ 0Þ¼−Fð Þ— o pe iodic d i es 0¼τ=2, whe e τis he pe iod, Fð þτÞ¼Fð Þ. The di ec applica ion o p ope ies (4) and (3) yields no cu en o shi -symme ic o ces in spa ially symme ic sys ems, ½ ðxÞ−Fð Þ ¼ ½ ðxÞþFð þ 0Þ ¼ ½ ðxÞþFð Þ ¼− ½ ðxÞþFð Þ:ð5Þ This is a well-known esul o spa ially symme ic sys ems, al eady cap u ed by he s anda d symme y analysis [2,3,22,23]. Howe e , ou cu en app oach e eals wo addi ional symme ies o o e damped one-dimensional sys ems, which a e no cap u ed by he s anda d app oach. They a e ½ ð−xÞþFð Þ ¼ ½ ðxÞþFð Þ;ð6Þ ½ ðxÞþFð− Þ ¼ ½ ðxÞþFð Þ:ð7Þ A p oo o (6) and (7) based on he Smoluchowski equa ion is gi en in he Supplemen al Ma e ial [26]. The symme ies (6) and (2), oge he wi h (3), yield he ollowing p ope y o shi -symme ic po en ials: ½ ðxÞþFð Þ¼ ½ ð−xÞþFð Þ¼− ½− ðxÞ−Fð Þ ¼− ½ ðxþL=2Þ−Fð Þ¼− ½ ðxÞ−Fð Þ: ð8Þ This is he same p ope y as Eq. (4) and, p oceeding as be o e, i implies cu en supp ession when combined wi h a shi -symme ic d i ing o ce. The e o e, qui e coun e - in ui i ely, in one-dimensional o e damped sys ems, he condi ion o cu en supp ession o sys ems wi h shi - symme ic po en ials—like he one shown in Fig. 1(b)—is, despi e being spa ially asymme ic, he same as o spa ially symme ic po en ials. Figu e 1(c) con i ms his unexpec ed beha io o la ge enough ic ions. In he unde damped egime his p ope y is no sa is ied exac ly. Ne e heless, e en in his egime he o e damped symme y (6) iden i ied he e has a las ing e ec : The ze o-cu en poin de e mined by he o e damped symme y is displaced o a lowe alue o he symme y pa ame e ain he unde damped egime. Thus, he o e damped symme y (6) de e mines a cu en e e sal in he unde damped egime. The disco e y o hidden symme ies epo ed abo e does no ep esen he only depa u e om he conclusions ha can be d awn om he s anda d symme y analysis. The p esence o ins abili ies may also al e he pic u e, as ajec o ies ha a e solu ions o he equa ions o mo ion wi h opposi e momen a may ha e e y di e en s abili y p ope ies, and, hus, esul in o a o al nonze o con ibu ion o he sys em cu en . Such a scena io o spon aneous symme y b eaking is bes illus a ed ia a speci ic case s udy. In he o e damped egime, om e e y solu ion xð Þ, he ajec o y ~ xð Þ¼xð− ÞþL=2is also a solu ion o (1) p o ided he po en ial is shi symme ic. I co esponds o a ans o med andom o ce ~ ξð Þ¼−ξð− Þ, which is s a is ically equi alen o ξð Þ, and a d i ing o ce ~ Fð Þ¼−Fð− Þ. Following he s anda d symme y analy- sis, no cu en is expec ed when an isymme ic d i ing o ces Fð þ 0Þ¼−Fð− þ 0Þa e applied [3,22,23,41]— an app op ia e choice o he ime o igin yields 0¼0. This p edic ion is co ec in one-dimensional sys ems, as eadily e i ied by nume ical simula ions. Howe e , he same easoning also p edic s no cu en in he case o highe dimensions, a esul ha is con adic ed by ou nume ical 0.6 0 0.2 0.4 0.8 1 a -0.005 0 0.005 0.01 γ=30 γ=20 γ=10 γ=5 U(x) o U(x) o a=0 a=1 (a) (b) (c) FIG. 1. Shi -symme ic po en ials ac like spa ially symme ic ones in one-dimensional o e damped sys ems. (a) Ra che po- en ial U a ðxÞ¼−U0½sinðkxÞþð1=4Þsinð2kxÞ wi h pe iod L¼2π=k. (b) Shi -symme ic po en ial de ined om U a ðxÞ as UssðxÞ¼U a ðxÞin he i s hal -pe iod, and UssðxÞ¼ −Ussðx−L=2Þin he second hal -pe iod. (c) Di ec ed cu en o a B ownian pa icle subjec o he mixed po en ial UðxÞ¼ U a ðxÞð1−aÞþUssðxÞaand o a shi -symme ic o ce de ined by Fð Þ¼gð Þ≡A½sinðω Þþð1=4Þsinð2ω Þ in he i s hal - pe iod, and Fð Þ¼−gð −τ=2Þin he second hal , whe e τ¼2π=ω. Reduced uni s a e de ined such as m¼L¼10ω¼1. O he pa ame e s a e A¼4,U0¼10=2π, and Γ¼10. The di ec ed cu en anishes in he o e damped limi (la ge ic ions γ) o he shi -symme ic po en ial (a¼1) because o hidden symme ies. PRL 116, 010602 (2016) PHYSICAL REVIEW LETTERS week ending 8 JANUARY 2016 010602-2 simula ions, as shown in Fig. 2 o a wo-dimensional po en ial and an applied spli biha monic d i e, as well as by independen esul s by Reimann’s g oup (see Re . [43], p. 16). The p esence o ins abili ies is he key o unde - s anding such an unexpec ed, spon aneous symme y- b eaking beha io . The s anda d analysis ails o accoun o he ac ual ins abili y o he ans o med solu ions ~ xð Þ, which makes hem e y unlikely. E en in he noiseless limi , he abo e ans o ma ion maps s able oscilla ions abou he po en ial minima in o highly uns able oscilla ions abou po en ial maxima [44]. We ha e e i ied ia nume i- cal simula ions ha , gi en a s able solu ion xð Þ, he ans o med solu ion ~ xð Þis uns able and hus quickly collapses on o xð Þ. This occu s bo h in one dimension as well as in highe -dimensional sys ems [26]. Gi en ha ins abili ies des oy he mechanisms o cu en supp ession due o he con ibu ions o a ajec o y and he ans o med one, he obse ed supp ession o di ec ed anspo in one- dimensional sys ems mus be associa ed o a di e en mechanism. This supp ession unde an isymme ic o ces is ac ually a consequence o he symme y (7), which yields no cu en o sys ems—which include spa ially symme ic as well as spa ially shi -symme ic sys ems o in e es he e—sa is ying he p ope y (4) ½ ðxÞþFð Þ ¼ − ½ ðxÞ−Fð Þ ¼ − ½ ðxÞþFð− Þ ¼− ½ ðxÞþFð Þ:ð9Þ A consequence o his analysis is ha uly spa ially symme ic sys ems should also exhibi no cu en in one-dimensional o e damped sys ems when an isymme ic o ces a e d i ing he sys em. This phenomenon is illus- a ed in Fig. 3. I was al eady expe imen ally obse ed in Re . [45], bu i emained unexplained un il he p esen Le e . These esul s a e a con i ma ion o he alidi y o he app oach based on a mo e gene al symme y analysis ha does no ely on he di ec analysis o he solu ions o he equa ion o mo ion. I is wo h s essing ha in he p esen discussion he dimensionali y o he sys em co esponds o he numbe o spa ial deg ees o eedom aking pa in o he ec i ica ion mechanism, and no necessa ily o he dimensionali y o he po en ial landscape. The iola ion o he symme ies (6), (7) in he abo e 2D o e damped se up is due o a ec i ica ion mechanism aking place in he wo pe pendicula di ec ions. Howe e , he symme ies (6), (7) a e no es ic ed o s ic ly one-dimensional sys ems; hey a e s ill p esen in highe -dimensional o e damped sys ems p o ided ha he ec i ica ion mechanism in ol es one spa ial dimension only. Fo example, he dashed line in Fig. 3(c) shows he supp ession o cu en o an isym- me ic d i ing o he same 2D sys em shown in Fig. 2 when he biha monic d i ing o ce is applied in he y di ec ion only. Addi ional examples a e shown in [26]. 0246810 12 14 16 18 ky/ kx -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 〈 y〉 x / L y / L kykx =4 FIG. 2. B eaking o symme ies (6),(7) in a 2D o e damped sys em. The d i ing o ce is Fð Þ¼A½cosðω Þexþ cosð2ω þπ=2Þey, i.e., a biha monic d i e spli [42] in wo pe pendicula di ec ions. The po en ial is Uðx; yÞ¼ U0cosðkxxÞ½1þcosðkyyÞ, which is spa ially symme ic in bo h di ec ions, and shi symme ic along he xdi ec ion. The cu en is p oduced in he ydi ec ion only—due o he symme y (5) in he xdi ec ion— h ough he coupling wi h he dynamics in he x di ec ion. Though he d i ing o ce is an isymme ic, a nonze o cu en is obse ed when kxand kya e compa able. Reduced uni s a e de ined such ha m¼kx¼ω¼1. O he pa ame e s a e U0¼γ¼50,A¼2γ, and Γ¼0.1γ2. The inse illus a es he po en ial landscape o ky¼4kx, wi h L¼2π=kx. 0π/2 π φ -1.5 -1 -0.5 0 0.5 1 1.5 〈 〉 γ=20 γ=15 γ=10 γ=5 U(x) F( ) (φ=π/2) (a) (b) (c) FIG. 3. Cu en supp ession in one-dimensional o e damped sys ems wi h a spa ially symme ic po en ial and applied an i- symme ic o ces. The d i ing o ce has a biha monic shape, Fð Þ¼A½cosðω Þþcosð2ω þϕÞ. (a) Spa ially symme ic po- en ial UðxÞ¼U0½cosðkxÞþcosð2kxÞ. (b) The d i ing o ce Fð Þis an isymme ic when ϕ¼π=2. (c) Di ec ed cu en as a unc ion o he d i ing phase ϕ, o di e en le els o damping. Reduced uni s a e de ined such ha m¼k¼ω¼1. O he pa ame e s a e U0¼20 and A¼Γ¼40. The dashed line shows, o compa ison, he cu en (h yi) o he same d i ing o ce applied in he ydi ec ion and a wo-dimensional po en ial Uðx; yÞ¼U0cosðkxÞ½1þcosð4kyÞ in he o e damped egime (γ¼U0¼50,A¼2γ,Γ¼0.1γ2). PRL 116, 010602 (2016) PHYSICAL REVIEW LETTERS week ending 8 JANUARY 2016 010602-3 Discussion.—The hidden symme ies iden i ied in he p esen Le e a e o ele ance o cu en expe imen s, and hey also allow us o ecas known esul s wi hin a mo e gene al heo e ical amewo k. This is well exempli ied by he wo speci ic case s udies ha a e p esen ed below. The i s case s udy co esponds o he sys em o ac-d i en o ices apped in a supe conduc o ha was expe imen ally s udied in Re . [14]. He e, in e pa icle in e ac ions p o ide an addi ional pa h o escape om he symme ies (6),(7). Ou esul s o Fig. 4 e e p ecisely o he one-dimensional sys em o in e ac ing B ownian pa icles ha was success ully used in Re . [14] o explain he mul iple cu en e e sals obse ed on ac-d i en o - ices apped in a supe conduc o . Despi e no being s ic ly sa is ied, he in luence o he symme ies (6),(7) is qui e no iceable, canceling he a che e ec and mos o he cu en e e sals in egions o he pa ame e space whe e he appea ance o a cu en is no di ec ly ela ed o pa icle in e ac ions. Fo a second case s udy, we e e o he celeb a ed lashing a che model [8–11,46,47], whe e he a che po en ial is pe iodically swi ched on and o in he absence o any addi ional addi i e d i ing Fð Þ—i.e., he e Fðx; Þ¼−∂Uðx; Þ=∂x—and mo e speci ically o he known esul [48] ha a lashing shi -symme ic po en ial canno p oduce di ec ed mo ion. The heo e ical amewo k and he ela ed new symme ies ha we in oduce he e allow o a simple explana ion o such a esul . In one-dimensional o e damped sys ems, he ollowing symme y is gene ally sa is ied [26]: ½Fð−x; − Þ ¼ ½Fðx; Þ:ð10Þ In wo-s a e sys ems ha a e pe iodically swi ched, e e sing he di ec ion o ime has no e ec , ½Fðx; − Þ ¼ ½Fðx; Þ; his ac , oge he wi h (2),(10), and (3), yields no cu en o shi -symme ic po en ials, ½Fðx; Þ ¼ − ½−Fð−x; Þ ¼ − ½−Fðx; − Þ ¼− ½−Fðx; Þ ¼ − ½−FðxþL=2; Þ ¼− ½Fðx; Þ:ð11Þ The e o e,a lashing a che wi hashi -symme icpo en ial, ega dless o whe he i is spa ially asymme ic, canno p oduce di ec ed mo ion; his hus shows ha in o e damped sys ems shi -symme ic po en ials beha e like spa ially symme ic ones. Conclusions.—The p esen Le e add esses he ou - s anding issue o p o iding a gene al heo e ical amewo k o he iden i ica ion o symme ies no cap u ed by he s anda d symme y analysis, examples o which we e al eady gi en in p e ious wo ks [44,48] wi h ad hoc ea men s. We ha e p o en he exis ence in a p o o ypical 1D o e damped sys em o hidden symme ies, which escape iden i ica ion by he s anda d symme y analysis and equi e di e en heo e ical ools o hei e ela ion. Though no igo ously sa is ied in highe -dimensional sys ems, he e ec s o hidden symme ies ha e been shown o be s ill no iceable in hem. Ou esul s pa e he way o new mechanisms o manipula ing anspo . In ac , he hidden symme ies de e mine cu en e e sals, which can be used o p ecisely con ol anspo and implemen mechanisms o pa icle sepa a ion. Speci ic ealiza ions o op ical weeze s and cold a om se ups a e discussed in he Supplemen al Ma e ial [26]. Financial suppo om he Royal Socie y (G an No. IE130734) (D. C. and F. R.), and he Le e hulme T us (G an No. RPG 2012 809) (F. R.) is acknowledged. *[email p o ec ed] †[email p o ec ed]k [1] R. D. As umian, The modynamics and kine ics o a B ow- nian mo o , Science 276, 917 (2000). [2] P. Reimann, B ownian mo o s: Noisy anspo a om equilib ium, Phys. Rep. 361, 57 (2002). [3] P. Hänggi and F. Ma chesoni, A i icial B ownian mo o s: Con olling anspo on he nanoscale, Re . Mod. Phys. 81, 387 (2009). [4] L. Mahade an and P. Ma sudai a, Mo ili y powe ed by sup amolecula sp ings and a che s, Science 288,95 (2000). 4 3 2 1 0 0 1 2 3 4 U /E n -10 -5 0 5 10 p1 0 4 3 2 1 0 0 1 2 3 4 U /E p1 0 0 E /L 10 -3 ( ) U(x) U(x)-U(x+L/2) ~U p1 〈 〉 FIG. 4. Cancella ion o anspo , ia he use o shi -symme ic po en ials, o he one-dimensional o e damped sys em o in e ac ing pa icles om Re . [14]. The bo om panels show he ne chain cu en as a unc ion o he numbe o pa icles pe pe iod, n, and he po en ial dep h Up1=E0, wi h he le panels e e ing o he o iginal one-pa icle po en ial UðxÞ(depic ed in he uppe panel), and he igh panels o a shi -symme ic po en ial buil om he o me as UssðxÞ¼UðxÞ−UðxþL=2Þ. The in e ac ion be ween he pa icles is accoun ed o by he pai po en ial Vin ð Þ¼−E0lnð Þ, wi h he pa icle sepa a ion. The sys em is d i en by a single-ha monic o ce ac ing on each pa icle, which is bo h shi symme ic and an isymme ic. The pa ame e s a e he same as in Fig. 2 o [14]. PRL 116, 010602 (2016) PHYSICAL REVIEW LETTERS week ending 8 JANUARY 2016 010602-4 [5] M. Schliwa and G. Woehlke, Molecula mo o s, Na u e (London) 422, 759 (2003). [6] J. V. He nández, E. R. Kay, and D. A. Leigh, A e e sible syn he ic o a y molecula mo o , Science 306, 1532 (2004). [7] M. V. Cos ache and S. O. Valenzuela, Expe imen al spin a che , Science 330, 1645 (2010). [8] J. Roussele , L. Salome, A. Ajda i, and J. P os , Di ec ional mo ion o B ownian pa icles induced by a pe iodic asym- me ic po en ial, Na u e (London) 370, 446 (1994). [9] J. S. Bade , R. W. Hammond, S. A. Henck, M. W. Deem, G. A. McDe mo , J. M. Bus illo, J. W. Simpson, G. T. Mulhe n, and J. M. Ro hbe g, DNA anspo by a mic o- machined B ownian a che de ice, P oc. Na l. Acad. Sci. U.S.A. 96, 13165 (1999). [10] C. F. Chou, O. Bakajin, S. W. P. Tu ne , T. A. J. Duke, S. S. Chan, E. C. Cox, H. G. C aighead, and R. Aus in, So ing by di usion: An asymme ic obs acle cou se o con inuous molecula sepa a ion, P oc. Na l. Acad. Sci. U.S.A. 96, 13762 (1999). [11] A. an Oudenaa den and S. G. Boxe , B ownian a che s: Molecula sepa a ions in lipid bilaye s suppo ed on pa - e ned a ays, Science 285, 1046 (1999). [12] S. Ma hias and F. Mülle , Asymme ic po es in a silicon memb ane ac ing as massi ely pa allel b ownian a che s, Na u e (London) 424, 53 (2003). [13] J. E. Villegas, S. Sa el’e , F. No i, E. M. Gonzalez, J. V. Angui a, R. Ga ca, and J. L. Vicen , A supe conduc ing e e sible ec i ie ha con ols he mo ion o magne ic lux quan a, Science 302, 1188 (2003). [14] C. C. de Souza Sil a, J. Van de Vondel, M. Mo elle, and V. V. Moshchalko , Con olled mul iple e e sals o a a che e ec , Na u e (London) 440, 651 (2006). [15] H. Linke, T. E. Humph ey, A. Lö g en, A. O. Sushko , R. Newbu y, R. P. Taylo , and P. Omling, Expe imen al un- neling a che s, Science 286, 2314 (1999). [16] T. Salge , S. Kling, T. Hecking, C. Geckele , L. Mo ales- Molina, and M. Wei z, Di ec ed anspo o a oms in a Hamil onian quan um a che , Science 326, 1241 (2009). [17] C. D exle , S. A. Ta asenko, P. Olb ich, J. Ka ch, M. Hi me , F. Mülle , M. Gmi a, J. Fabian, R. Yakimo a, S. La a-A ila, S. Kuba kin, M. Wang, R. Vaj ai, P. M. Ajayan, J. Kono, and S. D. Ganiche , Magne ic quan um a che e ec in g a- phene, Na . Nano echnol. 8, 104 (2013). [18] V. Se eli, C. F. Lee, E. R. Kay, and D. A. Leigh, A molecula in o ma ion a che , Na u e (London) 445, 523 (2007). [19] T. R. Kelly, H. de Sil a, and R. A. Sil a, Undi ec ional o a y mo ion in a molecula sys em, Na u e (London) 401, 150 (1999). [20] J. Siegel, In en ing he nanomolecula wheel, Science 310, 63 (2005). [21] R. Gomme s, S. Be gamini, and F. Renzoni, Dissipa ion- Induced Symme y B eaking in a D i en Op ical La ice, Phys. Re . Le . 95, 073003 (2005). [22] S. Flach, O. Ye ushenko, and Y. Zolo a yuk, Di ec ed Cu en Due o B oken Time-Space Symme y, Phys. Re . Le . 84, 2358 (2000). [23] S. Deniso , S. Flach, and P. Hänggi, Tunable anspo wi h b oken space ime symme ies, Phys. Rep. 538,77 (2014). [24] W. Thomson (Lo d Kel in), The kine ic heo y o he dissipa ion o ene gy, P oc. Roy. Soc. Edinbu gh 8, 325 (1874). [25] J. Casado-Pascual, J. A. Cues a, N. R. Quin e o, and R. Al a ez-Noda se, Gene al app oach o dealing wi h dynamical sys ems wi h spa io empo al pe iodici ies, Phys. Re . E 91, 022905 (2015). [26] See Supplemen al Ma e ial a h p://link.aps.o g/ supplemen al/10.1103/PhysRe Le .116.010602, which in- cludes Re s. [27–40], o a p oo o Eqs. (6) and (7), a discussion on he ole o ins abili ies in a 2D o e damped sys em, and on he con ol o anspo a he nanoscale abou he symme y poin s iden i ied in he p esen wo k. [27] H. Risken, The Fokke -Planck Equa ion (Sp inge , Be lin, 1984). [28] L. P. Faucheux, L. S. Bou dieu, P. D. Kaplan, and A. J. Libchabe , Op ical The mal Ra che , Phys. Re . Le . 74, 1504 (1995). [29] S.-H. Lee, K. Lada ac, M. Polin, and D. G. G ie , Obse - a ion o Flux Re e sal in a Symme ic Op ical The mal Ra che , Phys. Re . Le . 94, 110601 (2005). [30] A. V. A zola, K. Volke-Sepul eda, and J. L. Ma eos, Expe imen al Con ol o T anspo and Cu en Re e sals in a De e minis ic Op ical Rocking Ra che , Phys. Re . Le . 106, 168104 (2011). [31] O. M. Ma ago, P. H. Jones, P. G. Guccia di, G. Volpe, and A. C. Fe a i, Op ical apping and manipula ion o nano- s uc u es, Na . Nano echnol. 8, 807 (2013). [32] C.-S. Lee, B. Jankó, I. De ényi, and A.-L. Ba abási, Reducing o ex densi y in supe conduc o s using he ‘ a che e ec ’,Na u e (London) 400, 337 (1999). [33] D. E. Shalóm and H. Pas o iza, Vo ex Mo ion Rec i ica ion in Josephson Junc ion A ays wi h a Ra che Po en ial, Phys. Re . Le . 94, 177001 (2005). [34] J. F. Wambaugh, C. Reichha d , C. J. Olson, F. Ma chesoni, and F. No i, Supe conduc ing Fluxon Pumps and Lenses, Phys. Re . Le . 83, 5106 (1999). [35] D. Cole, S. Bending, S. Sa el’e , A. G igo enko, T. Tamegai, and F. No i, Ra che wi hou spa ial asymme y o con olling he mo ion o magne ic lux quan a using ime-asymme ic d i es, Na . Ma e . 5, 305 (2006). [36] C. Menne a -Robillia d, D. Lucas, S. Guibal, J. Tabosa, C. Ju czak, J.-Y. Cou ois, and G. G ynbe g, Ra che o Cold Rubidium A oms: The Asymme ic Op ical La ice, Phys. Re . Le . 82, 851 (1999). [37] M. Schia oni, L. Sanchez-Palencia, F. Renzoni, and G. G ynbe g, Phase Con ol o Di ec ed Di usion in a Symme ic Op ical La ice, Phys. Re . Le . 90, 094101 (2003). [38] R. Gomme s, S. Deniso , and F. Renzoni, Quasipe iodically D i en Ra che s o Cold A oms, Phys. Re . Le . 96, 240604 (2006). [39] D. Cube o, V. Lebede , and F. Renzoni, Cu en e e sals in a ocking a che : Dynamical s symme y-b eaking mech- anisms, Phys. Re . E 82, 041116 (2010). [40] A. Wickenb ock, P. C. Holz, N. A. Abdul Wahab, P. Phoon hong, D. Cube o, and F. Renzoni, Vib a ional Mechanics in an Op ical La ice: Con olling T anspo ia Po en ial Reno maliza ion, Phys. Re . Le . 108, 020603 (2012). PRL 116, 010602 (2016) PHYSICAL REVIEW LETTERS week ending 8 JANUARY 2016 010602-5 [41] P. Reimann, Supe symme ic Ra che s, Phys. Re . Le . 86, 4992 (2001). [42] S. Deniso , Y. Zolo a yuk, S. Flach, and O. Ye ushenko, Vo ex and T ansla ional Cu en s due o B oken Time- Space Symme ies. Phys. Re . Le . 100, 224102 (2008). [43] D. Spee , Ph.D. hesis, Uni e si ä Biele eld, 2011. [44] S. Deniso , S. Flach, A. A. O chinniko , O. Ye ushenko, and Y. Zolo a yuk, B oken space- ime symme ies and mechanisms o ec i ica ion o ac ields by nonlinea (non)adiaba ic esponse, Phys. Re . E 66, 041104 (2002). [45] S. Ooi, S. Sa el’e , M. B. Gai ullin, T. Mochiku, K. Hi a a, and F. No i, Nonlinea Nanode ices Using Magne ic Flux Quan a, Phys. Re . Le . 99, 207003 (2007). [46] A. Ajda i and J. P os , Mou emen indui pa un po en iel pé iodique de basse symmé ie: Diélec opho èse pulsée, C.R. Acad. Sci. Pa is II 315, 1635 (1992). [47] G. P. Ha me and D. Abbo , Losing s a egies can win by Pa ondo’s pa adox, Na u e (London) 402, 864 (1999). [48] R. Kanada and K. Sasaki, The mal a che s wi h symme ic po en ials, J. Phys. Soc. Jpn. 68, 3759 (1999). PRL 116, 010602 (2016) PHYSICAL REVIEW LETTERS week ending 8 JANUARY 2016 010602-6