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Running-mode resonance in A.C.-biased periodic potential Lance Labun∗ and Marcelo Gleiser† Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA We investigate the stochastic dynamics of a particle in the presence of a modulated sinusoidal potential. Using the time derivative of the winding number, we quantify the particle’s motion according to its running time, the time it runs monotonically to the left or right. For a range of modelparameters,weshowthat,intheoverdampedregime,theparticle’smotioninthismodulated 8 washboard potential exhibits stochastic resonance. We briefly suggest possible applications of our 0 results, including the amplification of signals for measurement devices and in stimulated tunneling 0 of Bose-Einstein condensates. 2 PACSnumbers: n a J INTRODUCTION ommendstherelevanceofthismodel. Inthenextsection 1 we introduce the model and explore some of its proper- 2 Stochastic resonance (SR), originally proposed in ref- ties. In section 3 we discuss our results, while in section ] 4 we present a summary. h erence to two equally-favored climate states, has only c more recently been discussed as a more general concept e independent of an explicitly bistable potential function m MODEL (see [1] for a review). It has now been identified in - a variety of different contexts, from lasers and optical t a traps to neurologicalsystems and biochemical reactions. Theequationofmotionistheusualoneassociatedwith st Quantum versions of stochastic resonance have also re- studies of Brownian motion and stochastic resonance, . t ceived attention, especially in superconducting quantum a x¨=ξ(t) γx˙ V′(x,t) (1) interference devices (SQUIDs) and nanomechanical res- m − − onators due to their location near the boundary of clas- - in rescaled units. The first two terms on the right d sicalandquantumdynamics. IthasbeenshownthatSR represent the thermal bath at temperature T via the n can be an efficient amplification mechanism, and similar fluctuation-dissipation theorem: Gaussian white noise o ideas are of interest to controlled diffusion, for instance c of zero mean ξ(t) = 0 and δ correlation function by applying various noise models in symmetric or asym- h i [ ξ(t)ξ(t′) =2γkTδ(t t′) combined with viscous damp- metric periodic potentials to obtain directed transport h i − ing γ. k is Boltzmann’s constant. The last term is the 1 (such as described by “Brownian motors” [2]). v potential, which has both a static anda periodic driving The interest in nonlinear resonance phenomena in pe- 1 component riodic potentials is not due solely to models of diffusion 2 2 and transport, as such potentials underlie other inter- V(x,t)= (A/2π)cos(2πx)+xF cosΩt. (2) 3 esting physical systems, for example, sphalerons in high − 1. energyphysics,whichhavebeeninvokedinthecontextof Previous work [9, 10, 11] has maintained a constant 0 the electroweakphase transition[3]. Sticking to the con- biasthroughoutanytime-dependentmodulationsofbar- 8 crete languageof diffusion, many authors have discussed rier heights, so that the driving component appeared as 0 SR in the basic sinusoidal potential, starting with the : F(t) = F0 +F1cosΩt for F0 > 0. Our study departs v simplest (static, symmetric) case [4, 5], implementing a by setting F0 = 0, thereby tilting the whole washboard i static bias [6, 7, 8], and finally tackling actively modu- X rather than merely modulating its corrugation. lated systems [9, 10, 11, 12]. Borromeo and Marchesoni Weconsideralowfrequencyinputandtheoverdamped r a recentlygeneralizedstochasticresonancetodynamically- limit, in which the equation of motion (1) can be adia- defined states in a tilted washboardpotential [13]. batically reduced to One case not yet treated in the literature is a rocking washboard, in which a periodic potential is subjected to x˙ = Asin(2πx)+F cosΩt+ξ(t) (3) − a global modulation, producing alternately positive and negativebiases. Thissystemoffersasimplemodelfordif- withtnormalizedtoabsorbγ. Thepotentialwasdefined fusion in the presence of agitation, such as experienced so that integer values of the position correspond to the byadatomsorparticulatesonavibratedsurface,oreven winding number ω. electrons on the surface of a dielectric and subject to In the limit of slow modulation, Ω 1, transitions ≪ a radiation field. Periodic potentials are also known in between wells are thermally induced for any value of the particle physics, so the possibility of an oscillatory bias, damping. Atvariouspointsinitsevolution,thepotential e.g., through a background scalar field [14], further rec- has a nearly constant bias in the positive or negative 2 direction,andthe differenceinducedinthe rightandleft We retain the larger flux I(E−) in order to strengthen barrier heights is given by former inequality (6a), since it expressesthe suppression of the flux during the unbiased parts of the period. For V(1/2,t) V( 1/2,t)=F cosΩt. (4) − − the effect to become pronounced, the inequality should For phases near Ω = π/2,3π/2, regular diffusion across clearly be as strong as possible. However, in the context an unbiased periodic potential applies, while around of stochastic resonance,small input biases are of greater Ω = 0,π, the dynamics are effectively those of activated interest, and in the following F <A. diffusion. In the latter case, the extensive characteri- In a biased periodic potential, overdamped systems zation of the jump length and time statistics given by represented by (3) can exhibit a consistently running [5,15]forthetiltedwashboardpotentialmaybestraight- state only for F = 2A, as the barriers are completely forwardly applied. Jump lengths and times, respectively suppressedin this case. However,many results for diffu- defined as the number of wells and the total period of sion in a tilted periodic potential are quite general [17], monodirectionalmotion,arecontrolledbytheratioF/γ, anddirecteddiffusionstillariseswithanaveragevelocity as discussed in [13]. given by The activated behavior near maximum bias will only (kT/γ)(1 exp( F/kT) dominatetheoverallstatisticsofthesystemiftheparticle x˙ = − − . (8) 1 is effectively locked into a specific well for the remainder h i R R0 dxdyexp[(V(x)−V(x−y)−yF)/kT] of the phase. Letting E− A/π F/2, the reduced Considering shorter time dynamics, the modulation of ≡ − barrier height in the direction of the bias, left at Ω = 0 the barrier heights introduces a phase bias for diffusive and right at Ω = π, we may derive an intuitive con- transitions. For small F, a simple correlation should be straint in the spatially-diffusion-controlled regime (large apparentintheresidencetime distribution,asjumpsbe- damping in our normalization) via Kramers rates [16]. tweenwells willtend to occur inphase with the modula- On the one hand, the particle should experience the tion. tilted washboard potential for longer than the proba- We therefore expect a strong correlation between the ble escape period, Tmin < TΩ/4 with Tmin = 1/rk = directionoftransportandthephaseoftheforcing. Given (2πγ/ω0ωb)exp(βE−) the inverse of the Kramers rate that transitions are suppressed when the biasing force near maximum bias and the quarter period being a rea- F(t)=F cosΩtisnearzero,transitionsagainsttheprof- sonable choice. (ω0 and ωb correspond to the oscillating feredbiasareevenmorestronglysuppressed(thoughve- frequencies at the metastable well and top of the bar- locity inversions occur less often than initially expected rier, respectively, and β = 1/kT.) On the other hand, in a statically biased potential according to [8]). the particle should also have a suppressed probability We observed a dynamic resonance between the time of escape near zero bias, and hence Tmax > TΩ/4 for derivative of the winding number w˙ and the modula- Tmax = (2πγ/ω0ωb)exp(βA/π). Rearranging and com- tion F(t), now understood as an input signal. Defining bining the inequalities, we find “left-running” and “right-running” states corresponding tow˙ <0andw˙ >0,respectively,the formalismandlan- 0<(kT/A)ln(4α )+1/π <F/2A (5) o guage of stochastic resonance can be applied. All of the where αo Ωγ/ω02 is a dimensionless time parameter in generally accepted ingredients are present: (i) an activa- ≡ which we used the fact that ωb = ω0 for the sinusoidal tion barrier, E− 2A F/2; (ii) a (weak) input signal, potential. Because this inequality locates the region of the global modul≡ation−of the potential, F(t) in (4); and parameterspaceinwhichweexpecttheactivatedbehav- (iii)asourceofnoise,namelythestochastictermin(1). ior to dominate the overall dynamics, we clearly desire F/A as large as possible. Averysimilar constraint,more accuratein the caseof RESULTS lowdamping,canbederivedfromtheKramersrateinthe energy-diffusion-limited regime, γI(E) kT (in which The most direct connection to previous observations ≪ I(E) is the energy-dependent flux in action space). An of SR lies in residence times, defined as the time spent extra step is required in combining the inequalities due in a given well: we found that the distribution of res- to the different values I(E−) and I(A/π), idence times for the left- and right-running states—or, I(A/π) ω0 eβA/π < 1, (6a) moreappropriatelytooursituation,therunningtimes— 4γΩ arepeakedaroundthehalfintegermultiplesoftheforcing 1 I(E−) ω0 eβA/π > eβF/2. (6b) period, (n+ 2)TΩ. Running times for various values of 4γΩ the temperature are plotted in figure 1, and the qualita- tive similarity to plots for resident times such as found We then use the monotonicity of I(E), from which in [1] is easily verified. I(E−)>I(A/π), to write Wenotethattheusualresonanceintheresidence time 0<(kT/A)ln(4α )+1/π <F/2A. (7) distribution(timespentlocalizedinagivenwell)canstill u 3 FIG. 2: The Fourier transform of position x in an over- damped system, driven at Ω = 10−3. The primary peak at thedrivingfrequencyisclear, accompanied byan extrapeak at the first harmonic, 3Ω, exactly as predicted by [18]. Pa- rameters: A=1,F =.3,2πkT =.325 FIG.1: Theresidence (top)and running(bottom) timedis- tributions in the overdamped model for three temperatures 2πkT/A=0.25,0.35,0.45. Theverticalaxisdenotesthenum- berofinstancesofameasuredperiod,normalizedonthehori- zontalaxisbytheforcingperiod,TΩ. Theresidencetimealso displays a (barely-discernable) resonance, showing a prefer- encefortransitionsonbothintegerandhalf-integermultiples oftheperiodatsufficientlylowtemperature. Thedistribution ofthejumplength∆w intheinsetshowstheexpectedexpo- nential die off [5]. Parameters: A=104,F =2000,Ω=10 beidentifiedatlowtemperatures,althoughnotasclearly due to the scale of the figure. Below, we will present results in the underdamped regime, where the resident time resonance is more apparent (see Fig. 4). With γ normalized to 1, F/γ is always less than 8√A aFgIaGin.s3t: teTmhpeernaotrumrea,lizoebdtadiniffeudsifornomcoetffihecireengtreDss/iDon0 opflohtxte2di (thethresholdforthelockedregime[13]),sojumplengths from 1000 particles. Parameters: A=1,F =.3,TΩ=1000 display the expected exponential distribution, as can be seen in the inset of figure 1 [5]. Note that ‘running’ refers only to the monotonicity of dicted by [18]. the winding, as the particle almost alwaysresidesa time In [12], it is noted that diffusion is somewhat sup- comparable to the relaxation time ω0−1 even when only pressed in a tilted periodic potential (as compared to transiting a well. For this reason, in order to obtain the Einstein diffusion D0 = kT/γ), and the global modula- cleanresultsdisplayedinfigure1,weconsideredachange tion may be expected to suppress diffusion even further inwindingnumbertooccuronlyaftertheparticlebegins by introducing an extra coherence in the transport of itsdescentintotheadjacentwell,notmerelywhenithas particles in the potential. As can be seen in figure 3 we surmounted the barrier. This definition is the cause of haveinvestigatedthisfact,findingaverysignificantsup- the low frequency cutoff apparentupon close inspection. pression, which decreases with increasing temperature. Further support for characterizing the running time In underdamped systems γ ω0, the left- and right- ≪ as a resonance comes from the Fourier transform of x. running states have an obvious counterpart in the jump The signal-to-noiseratio should show a maximum in the times and lengths discussed in [13]. However, the corre- temperature range suggested by (5). Figure 2 displays lation of w˙ to the phase of the drive cannot be properly the Fouriertransformofa systeminthe suggestedrange called a resonance phenomenon, as inertial effects rein- andfurthermoreexhibitsthefirstharmonicat3Ωaspre- force rather than suppress the peaked distribution, as 4 FIG. 5: The power in the delta peaks of the first through fourth harmonics, (2n + 1)Ω for n = 1..4. The power in each peak increases nonlinearly with temperature, indicating an increase in signal to noiseratio—behavior contrary tothe resonantcharacterofSR.Eachharmonicappearstosaturate FIG. 4: Residence (bottom) and running (top) time distri- around2πkT =0.4. TheinsetdisplaystheFouriertransform butionsforanunderdampedsystemattemperatures2πkT = ofanexperimentalrunat2πkT =0.35inwhichthepeaksat 0.2,0.25,0.3. The response of w˙ does not show a resonance- the higher harmonics can be clearly identified. Parameters: like behavior, as the distribution remains sharply peaked at A=1,F =0.3,γ =0.015,TΩ =100 the half integer multiples even while temperature increases. The residence time is also dramatically peaked near zero, thoughthehigherharmonicsherealsogainpowerastemper- damped regime. ature increases. The steadiness of the distribution, however, Looking ahead, our results are of interest to measure- highlights that it is entirely an inertial effect. Parameters: ment considerations in the context of bifurcation ampli- A=1,F =0.3,TΩ=100 fication[19], wherebyasmallinputsignalisenhancedso astoeffectthedistinctionbetweenthe(bistable)dynam- ical states arising just beyond the bifurcation threshold demonstrated in figure 4. Physically, if γ is of the same of a nonlinear system. This immediately suggests a pos- order as 2Ω, sufficient kinetic energy may be conserved sibledirectioninwhichtheseideasmaybetaken: instead throughthehalfperiodofopposingbiassuchthatnove- ofF(t)being anunvaryinginput mode,the particlemay locity inversionoccurs [8]. Evidence for this explanation becoupledto asecondsystemwhoseoscillationsprovide is derived from the Fourier transform of x: the power the input signal. From this perspective, (1) represents in peaks at higher harmonics should generally increase the high mass-ratio limit of this second system, so that with temperature, as lower frequency velocity inversions theobservedvariable(modeledhere)effectsnosignificant become more common. Simulations support this, to a back-reactionon the input signal. For example, starting point, as we find that the peaks actually become satu- fromtheequationsofmotionforabilinearlycoupledpair ratedbeyondacertaintemperature,asshowninfigure5. Although the saturation indicates a decrease in signal- m1x¨21 = V1′(x1) γ1x˙1+λx2 to-noise ratio, it does not produce a profile sufficiently − − (9) peaked as to justify its characterizationas a resonance. m2x¨22 =−V2′(x2)−γ2x˙2+λx1 we may consider the perturbative case in which the so- lution x2(t) is harmonic, being limited to a region near SUMMARY one of its potential minima. Secular perturbation the- ory on the full system produces the expected condition We described a dynamic resonance exhibited by an for the validity of this Ansatz, namely λ/m2 1. Then ≪ overdamped particle in a rocking washboard potential. with(1)amodelofsucha bipartitesystem,the dynami- Defining “running” states according to the sign of w˙, calresonancedescribedabovebecomesalargeamplitude wherebyaparticlerunsmonotonicallytotheleftorright signalfortheoscillationoftheunderlyingsystem(x2)be- beforereversingitsdirection,wewereabletocharacterize tween two of its states. These states may be dynamical the phenomenon as a new kind of stochastic resonance. or static, and the coupling engineered accordingly. Numerical results are consistent with the range of pa- Asecondconnectioncanbemadetorecentadvancesin rameters derived from naive expectations. We have also quantumtunnelingdynamics. Experimentswererecently shownthatnosuchresonancesarepossibleinthe under- undertaken in which resonantly enhanced tunneling was 5 clearlyobservedinaBose-Einsteincondensateinatilted [1] L. Gammaitoni, P. H¨anggi, P. Jung, and F. Marchesoni, washboardpotential [20]. Although the proper quantum Rev. Mod. Phys. 70, 223 (1998). mechanical calculations should be carried out in detail, [2] P. H¨anggi, F. Marchesoni, and F. Nori, Ann. Phys. 14, 51 (2005). theclassicalresultsheresuggestthatbytuningtheoscil- [3] P. Arnold and L. McLerran, Phys. Rev. D 37, 1020 lation of the washboard to the tunneling rate, a second (1988). orderresonanceshould arise. The loss of particles to the [4] V. Melnikov, Phys.Rep. 209, 1 (1991). continuum following the first tunneling event could be [5] R. Ferrando, R. Spadacini, and G. Tommei, Phys. Rev. drasticallyreducedbyreversingthe tiltonthe potential. E 48, 2437 (1993). The discretizationofthe energylevelsprovidesaninher- [6] P. Reimann, C. V. den Broeck, H. Linke, P. H¨anggi, ent suppression of tunneling events during the inversion J. Rubi, and A. P´erez-Madrid, Phys. Rev. E 65, 031104 (2002). of the bias. Testing this hypothesis might more easily [7] F. Marchesoni, Phys.Lett. A 231, 61 (1997). be achieved in a circularly symmetric optical potential [8] M. Borromeo and F. Marchesoni, Phys. Rev. Lett. 84, such as was used in [21], so that the modulated bias is 203 (2000). alternately positively and negatively radial. [9] L. Machura, M. Kostur, P. Talkner, J. Luczka, and This paper demonstratesthe utility ofstochastic reso- P. H¨anggi, Phys.Rev. E 73, 031105 (2006). nanceasageneralmodelofphenomenaforwhichobvious [10] S. Miret-Artes and E. Pollak, J.Phys. Cond. Mat. 17, localizedstatesarenotnecessarilypresentedbytheform S4133 (2005). [11] K. Lindenberg, A. Lacasta, and J. S. et al, New J.Phys. of the potential. Clearly, with ideas progressing in such 7, 29 (2005). directions as bifurcation amplification [19] and the abil- [12] M. Borromeo and F. Marchesoni, Chaos 15, 026110 itytoeasilyconstructandmanipulateperiodicpotentials (2005). thanks to laser trapping, the languageand the theory to [13] M. Borromeo and F. Marchesoni, Surf. Sci. Lett. 465, describe the possible dynamics should develop as well. L771 (2000). Stochastic resonance in the context of more generalized [14] S. Digal, R. Ray, S. Sengupta, and A. M. Srivastava, states will almost certainly appear and be fruitfully dis- Phys. Rev.Lett. 84, 826 (2000). cussed in ways similar to those presented above. [15] R. Ferrando, R. Spadacini, and G. Tommei, Phys. Rev. E 51, 126 (1995). [16] P.H¨anggi,P.Talkner,andM.Borkovec,Rev.Mod.Phys. We thank Miles Blencowe for his thoughts and sup- 62, 251 (1990). port. This work was supported by a summer research [17] P. Reimann, C. V. den Broeck, H. Linke, P. H¨anggi, grant from the Department of Physics and Astronomy J. Rubi, and A. P´erez-Madrid, Phys. Rev. Lett. 87, at Dartmouth College. MG is supported in part by a 010602 (2001). National Science Foundation grant PHY-0653341. [18] P. Jung and P. H¨anggi, Europhys.Lett. 8, 505 (1989). [19] I. Siddiqi, R. Vijay, F. Pierre, C. Wilson, M. Metcalfe, C.Rigetti,L.Frunzio,andM.Devoret,Phys.Rev.Lett. 93, 207002 (2004). [20] C. Sias, A. Zenesini, H. Lignier, S. Wimberger, ∗ Electronic address: [email protected]; Now at: D. Ciampini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. 98, 120403 (2007). Department of Physics, University of Arizona, Tucson, [21] S. Tatarkova, W. Sibbett, and K. Dholakia, Phys. Rev. AZ 85721, USA † Electronic address: [email protected] Lett. 91, 038101 (2003).

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