Table Of Content

Duality Relation for Quantum Ratchets J. Peguiron1,2 and M. Grifoni2 1Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 2Institut fu¨r Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany (Dated: February 2, 2008) 5 0 A duality relation between the long-time dynamics of a quantum Brownian particle in a tilted 0 ratchet potential and a driven dissipative tight-binding model is reported. It relates a situation of 2 weak dissipation in one model to strong dissipation in the other one, and vice versa. We apply n thisdualityrelation toinvestigatetransportandrectification inratchet potentials: From thelinear a mobility we infer ground-state delocalization for weak dissipation. We report reversals induced by J adiabaticdrivingandtemperatureintheratchetcurrentanditsdependenceonthepotentialshape. 8 2 PACSnumbers: 05.30.-d,05.40.-a,73.23.-b,05.60.Gg ] ll Periodic structures with broken spatial symmetry, V)11 ∆ ~ 2 π h- ha knownasratchetsystems[1],presenttheattractiveprop- of 2V 1 L=ηL ertyofallowingtransportundertheinfluenceofunbiased s 2 es- forces. Theinterplayofdissipativetunneling[2]withun- (unit0 ϕ 2 .m bfeiaatsuedresdraibvsinengteinnriicthsecslatshseicaqlucaonutunmterrpaatrcthleitkee,ffee.cgt.,wcuitrh- V(q) -1 4π2V1 ∆2 t rent reversals as a function of temperature [3, 4]. Quan- -1 0 q/L 1 a m tum ratchetsystems have only recently been experimen- tallyrealizedinsemiconductor[4]andsuperconductor[5] FIG. 1: Dual relation between a dissipative ratchet system - d devices. Also from the theory side there are still few andatight-binding(TB)modelsketchedforatwo-harmonics n works[3,6,7,8,9,10]which,withtheexceptionof[7,8], ratchet potential (thick curve). Each harmonic (thin curves) o generates couplings to different neighbors in the TB system, arerestrictedtotheregimeofmoderate-to-strongdamp- c according to Eqs. (5) and (7). The periodicity L˜ of the TB ing. After the pioneering semiclassical work [3], further [ model is determined bytheviscosity η in theoriginal model. progresstowardsaquantumdescriptionwasmadein[9], 2 where the role of the band structure in ratchet poten- v tials sustaining few bands below the barrier was investi- 9 describes a ratchet system. The interaction of the sys- 5 gated. Recently,aquantumSmoluchowskitreatment[10] tem with a dissipative thermal environment is modeled 7 addedto the availablemethods. In this paper,we gener- 7 alize to an arbitrary ratchet potential a duality relation by the standard Hamiltonian HˆB of a bath of harmonic 0 oscillators whose coordinates are bilinearly coupled to put forward in [11] for a cosine potential. It provides a 4 the system coordinate qˆ [2]. The bath is fully charac- tight-binding description of quantum Brownian motion 0 terized by its spectral density J(ω). We consider strict / in a ratchet potential, and leads to an expressionfor the t Ohmic damping J(ω) = ηω, which reduces to instan- a ratchetcurrentvalidinawideparameterrangeincluding m weakdissipationandnonlinearadiabaticdriving. Weap- taneous viscous damping (viscosity η) in the classical limit. Insuchasystem,theratcheteffectischaracterized - ply this methodto discuss rectificationandground-state d delocalization occurring for weak dissipation in ratchet by a nonvanishing average stationary particle current n v∞ = lim t−1 tdt′v(t′) in the presence of unbiased co plimotietn.tials. Our results encompass correctly the classical dRriving, cth→a∞racterRiz0ed by limt→∞t−1 0tdt′F(t′) = 0, : We consider the Hamiltonian Hˆ of a quantum par- switched on at time t′ = 0. In thisRpaper, we shall v R ticle of mass M in a one-dimensional periodic potential consider the particular case of unbiased bistable driving i X V(q+L)=V(q) tilted by a time-dependent force F(t), switching adiabatically between the values F. We re- ± r portamethodtoevaluatethestationaryvelocityv∞ (F) a Hˆ (t)= pˆ2 +V(qˆ) F(t)qˆ. (1) in the biased situation of time-independent driviDnCg F, R 2M − which is also of experimental interest [4, 5]. The ratchet current in the presence of adiabatic bistable driving can The potential assumes in Fourier expansion the form be expressed as v∞ =v∞ (F)+v∞ ( F). R DC DC − ∞ V(qˆ)= V cos(2πlqˆ/L ϕ ), (2) Thewholeinformationonthesystemdynamicsiscon- l l Xl=1 − tainedinthereduceddensitymatrixρˆ(t)=Tr Wˆ(t),ob- B and can take any shape. Apart from special configu- tained from the density matrix Wˆ(t) of the system-plus- rations V sin(ϕ lϕ ) = 0 l of the amplitudes V bath Hˆ = Hˆ +Hˆ , with time-independent driving F, l l 1 l R B { − ∀ } andphasesϕ ,this potentialisspatiallyasymmetricand byperformingthetraceoverthebathdegreesoffreedom. l 2 Toevaluatetheevolutionoftheaverageposition q(t) = on the continuous coordinates q and q′. Here A[q] = Tr qˆρˆ(t) , the diagonal elements P(q,t)= q ρˆh(t)qiof exp (it/¯h)Hˆ is the propagator of the ratchet sys- R R { } h | | i {− } the reduced density matrix are needed, and can be ob- tem for a path q(t′), and F[q,q′] the Feynman-Vernon tainedbyreal-timepathintegralstechniques[2]. Theve- influence functional of the bath inducing nonlocal-in- locity followsby time differentiation. Atinitialtimet′ = time Gaussian correlations between the paths q(t′) and 0, we assume a preparation in a product form where the q′(t′) [2]. Due to the nonlinearity of the potential V(q), bath is in thermal equilibrium with the ratchet system these path integrals cannot be performed explicitly. For Wˆ(0) = ρˆ(0)e−βHˆB[Tr e−βHˆB]−1. The bath tempera- acosinepotential,FisherandZwerger[11]introducedan B ture is fixed by T =1/βk . This leads to a double path exactexpansioninthepropagatorA[q]whichtransforms B integral the path integrals into Gaussian ones that can be performed. Generalizing this idea for the arbitrary periodic P(qf,t)= (3) potential (2), we find the expansion qf qf dqi dqi′ qi ρˆ(0)qi′ q ∗q′A[q]A∗[q′]F[q,q′] Z Z h | | iZ D Z D qi qi′ i t ∞ m i∆ t tm t2 i t exp dt′V (q(t′))= − lj dt dt ... dt exp dt′ρ(t′)q(t′), (4) m m−1 1 −¯hZ0  mX=0X{lj}jY=1(cid:18) ¯h (cid:19)Z0 Z0 Z0 −¯hZ0      where ρ(t′)=(2π¯h/L) m l δ(t′ t ), and withtheseriesexpressionfortheexpectationvalueofthe ∆l = V2leiϕl Pfojr=1l>j 0, −∆j−l =∆∗l. (5) ptiogshitt-ibonindoipnegra(TtoBr)qˆsTyBste=m,L˜dPesc∞nr=ib−e∞dnb|yntihhne|Hoafmailtdorniviaenn The physical meaning of these new quantities will be ∞ discussed later. For each term of the sum on m in (4) Hˆ = (∆ n+l n +∆∗ n n+l ) Fqˆ , (7) we have introduced m intermediate times tj, and corre- TB n,lX=−∞ l| ih | l| ih | − TB spondingindicesl takinganyvalueamong 1, 2,... . j {± ± } and bilinearly coupled to a different bath of harmonic The sum runs on all configurations of these in- {lj} oscillators. The spectral density of this bath J (ω) = dices. A siPmilar expansionis performed for the propaga- TB torA∗[q′],involvinganewsetofm′timest′ andindicesl′ J(ω)/[1+(ω/γ)2] is still Ohmic but presents a cutoff at beingusedtodefineρ′(t′)similarlytoρ(t′)j. Thisenablejs the frequency γ set by dissipation. At initial time t′ =0 the TB system is prepared in the state n = 0 . The us to rewrite the average position q(t) = dqqP(q,t) | i h i calculationshowsthatthe ∆ introducedin(5)areiden- in terms of a series inthe amplitudes Vl ofthRe potential. l tifiedwiththecouplingsoftheTBsystem(7). Westress Thoughstillintricate,theresultingexpressionbecomes that the lth harmonic of the original potential results in easier to treat in the long-time limit we are interested acouplingto thelthneighborsinthe dualTBsystemas in. Quantitatively, the measurement time t should be very long on the time scale γ−1 = (η/M)−1 set by dis- sketched in Fig. 1. One can easily show that the spatial symmetryconditiononthephasesϕ isthesameinboth sipation. A second approximation is necessary to pro- l ceed: we neglect terms e−γtj, e−γt′j, e−γ(t−tj), e−γ(t−t′j), systems. The first three terms on the right-hand side e−ωBtj, e−ωBt′j, e−ωB(t−tj), and e−ωB(t−t′j), where ωB = of (6) reproduce exactly the classicalsolution for the average position q(t) of a free system, V(q) 0, at long 2πk T/¯h, in the integrands involved in the series ex- B h i ≡ times. Inthislinearcase,thequantumandclassicalsolu- pression for q(t) . We shall refer to this assumption h i tions should be identical, due to Ehrenfest theorem, and as the rare transitions (RT) limit and discuss its valid- they are,because the TB average q (t) vanishes in ity later. Generalizing[11], we consider a Gaussianwave TB TB h i theabsenceofthepotentialV(q). Weexpectthesamere- packet centered at position q =Tr [qˆρˆ(0)] and momen- 0 R sult when the potential is present but unimportant, e.g., tump =Tr [pˆρˆ(0)]asinitialpreparationfortheratchet 0 R for large driving F and/or high temperatures T. system. We obtain the important result The series expression for the diagonal elements of the p Ft q(t) q0+ 0 + qTB(t) TB. (6) reduced density matrix ρˆTB of the TB system, which h i t→R∼T∞ η η −h i leads to the series expression for qTB(t) TB, can be h i Partsoftheseriesexpressionfor q(t) hasbeensummed, written in terms of pairs of TB trajectories qTB(t′) = yielding the first three terms. Thhe reist can be identified η−1 t′dt′′ρ(t′′) [with ρ(t′) introduced above Eq. (5)], 0 R 3 n=0 1 2 interesting to notice that it was also derived in [13] for n´=0 the linear ac mobility in a cosine potential. However,we ∝|∆ |2 did not completely succeed in generalizingEq.(8) in the 1 1 a presence of time-dependent driving. ∝|∆ |2 c ∝∆2∆1*2 aryWveelsohcailtlynvo∞w(fFoc)u.sBoynstohlveinevgatluhaetgioenneorfaltizheedsmtaatisotner- 2 2 DC b equation mentioned above, one finds the stationary velocity v∞ = L˜ mΓ in the dissipative TB system. TB m m FIG.2: Representationofsomeofthesecond-order(a,b)and The duality relaPtion (6) can then be used to obtain third-order(c)pathscontributingtothediagonalelementsof the reduced density matrix of the tight-binding model, and v∞ (F)=F/η (L/α) ∞ m(Γ Γ ). (9) thecorresponding dependenceon the couplings ∆l. DC − m=1 m− −m X As discussed above, the rates Γ are power series in the m couplings∆ startingfromsecondorder. Foragivenm= athnadtqoT′nBe(etx′)trdaecfitsntehdessipmaitliaarllypeirniotdeircmitys Lõfoρf′t(ht′e).TBFrsoyms- 0,there arelonly twopossible second-ordercontribution6 s tem, yielding L˜ = L/α, where α = ηL2/2π¯h is the di- to Γm, which, after use of Eq. (5), sum up to [2] mThenesseiopnaleisrssdoifsstirpaajteicotnorpieasracmometbeirnoefitnhedoisrcirgeintealpsaytshtsemin. Γ(m2) = 4Vh¯m22γ Z ∞ dτ e−m2α˜Q(τ)+im(FL˜/h¯γ)τ. (10) the q q′ plane parametrizedby pairsof integers(n,n′). −∞ − Each path starting in the diagonal element (0,0) and The influence of the dissipative environment enters ending at time t in (m,m) contributes to mρˆ (t)m . through the dimensionless bath correlation function TB Each transition in the path brings a correhsp|onding|faci- Q(τ) = 2 ∞ dω [coth(ω/2θ)(1 cosωτ)+isinωτ] 0 ω(1+ω2) − tor ∆l and all paths involve at least two transitions (cf. with θ = kRBT/¯hγ. At zero bias F = 0 and in the scal- Fig. 2). Written in this form, the diagonal elements of ing limit h¯γ k T, the rates show a power-law de- B the reduced density matrix are a solution of a general- pendence on t≫emperature Γ(2) T2m2α˜−1. The linear m ized master equation[12] in terms of transitionratesΓm mobility µ is thus dominated∝by the rate Γ(2) at low from the TB site (n,n) to the site (n+m,n+m). Con- TB 1 temperatures, and vanishes at T = 0 for α < 1, which sequently, these rates are expressed in power series of corresponds to free dynamics µ = µ in the dual weak- 0 all the couplings ∆ , starting from second order. As the l binding system [14]. This suggests that the occurrence timest ,t′ introducedin(4)areidentifiedwiththetran- j j of a delocalization to localization transition at α=1 for sition times in the TB representation, the rates Γ give m the ground state of a cosine potential [11, 15] would not also a way to control our assumption of rare transitions. be affected in more general potentials (see also Fig. 3). Itcorrespondstoneglectthosepathswhichinvolvetran- In the remainder of the paper, we focus on the sitions on a time scale max(γ−1,ω−1) after the initial B ratchet current induced by adiabatic bistable driving time t′ =0 or before the final time t′ =t. As transitions v∞ = v∞ (F)+v∞ ( F). The second-order rates obey in the TB model happen on a time scale Γ−1, the du- R DC DC − m Γ(2)( F) = Γ(2) (F) and therefore cancel out in the ex- ality relation will be valid when the transitions are rare m − −m pressionfor the ratchetcurrent. Hence, we haveto focus onthe time scale max(γ−1,ω−1), i.e., whenall ratessat- B on contributions of at least third order to the rates Γ . isfy Γ min(γ,ω ). This condition is controlled by m m B ≪ Hereweneglecthigherorders. Thisisknowntoprovidea the dissipation through γ = η/M and the temperature goodapproximationinTBsystemswithlargedissipation through ω =2πk T/¯h. B B parameterα˜ and/orhightemperature[2]. Forsimplicity Due to the change of periodicity length between the we also consider a potential consisting of only two har- twosystems,the dissipationparameterαandthe energy monics. Thereisnoproblemofprincipletoincludemore drop per unit cell ǫ = FL become α˜ = 1/α and ǫ˜= ǫ/α harmonics [16]. We find, with m= 1, 2, in the TB system. Thus, weak dissipation in one system ± ± maps to strongdissipation inthe other one althoughthe V2V ∞ viscosity η in the spectral density does not change. The Γ(m3) = 4h¯13γ22 Im(cid:20)Z dτ G(|m3)|(τ) eim(FL˜/h¯γ)τ−isgn(m)ϕ(cid:21), −∞ asymptotic dynamics is usually described by the nonlin- (11) ear mobility µ = lim v(t)/F. With these notations, t→∞ where we have introduced ϕ=ϕ 2ϕ , and 2 1 the duality relation (6) can be rewritten in the form − ∞ G(3)(τ)= dρ e−2α˜Q(ρ) µ(α,ǫ) µ0 µTB(1/α,ǫ/α), (8) 1 −Z −R−→T − 0 e−2α˜Q(τ+ρ)+α˜Q(τ+2ρ)+e−2α˜Q(τ−ρ)+α˜Q(τ−2ρ) , w0.heInretµh0e=sp1e/ciηalisctahseemofobailciotysinofetphoetferneteiasyl,sttehmis,rVel(aqt)io≡n ×h ∞ i G(3)(τ)= dρ eα˜Q(ρ)−2α˜Q(τ+ρ/2)−2α˜Q(τ−ρ/2). (12) wasalreadyobtainedin[11]forthedcmobility. Ititalso 2 Z 0 4 3×10-5 2 4×10-4 ) ) η L L =F/ Ω0 Ω0 0 v0 0 01 of ∞/R f /v s 1v-4×10-4 o C t 0 5 10 s ∞D v∞ (F) ni (unit v0.95 -vD∞DCC(-F) ∞ (uDC v-v∞D∞C(F(-)F) ∞ R v DC v 0 10 20 -4×10-5 0 0 10 kBT/∆V 20 0 1 2FL/∆V3 4 5 FIG. 3: Ratchet current and stationary velocity (inset) as a FIG. 4: Stationary velocity and ratchet current (inset) as a function of temperature for the potential of amplitude ∆V functionofdrivingforthepotentialofamplitude∆V depicted depicted in Fig. 1. Weak dissipation is chosen with α = 0.2 in Fig. 1. The dashed line is the classical solution in the and h¯γ =0.76∆V. Drivingis set to FL=0.57∆V. absenceofpotential. Weakdissipationischosenwithα=0.2 and h¯γ =0.76∆V. Temperatureis set tokBT =0.076∆V. AtthirdordertheratesobeyΓ(3)(F,ϕ)=Γ(3) ( F, ϕ), m −m − − which is a consequence of parity. The dependence of the ratchet potential. We demonstrated its application to ratchetcurrentonthepotentialparametersisthenupto investigate rectificationof adiabatic driving and ground- third order in the potential amplitude state delocalization for weak dissipation. v∞ V2V sin(ϕ 2ϕ ). (13) R ∝ 1 2 2− 1 We thank U. Weiss for seminaldiscussions. This work The ratchet current vanishes for a symmetric potential was supported by the Dutch Foundation FOM. sin(ϕ 2ϕ )=0 as it should. 2 1 − The behavior of the particle and ratchet currents as function of temperature and driving is shown in Figs. 3 and 4 for a two-harmonics potential. In Fig. 3, the driv- [1] Appl. Phys. A 75, 167 (2002), special issue on Ratchets ing is set to FL = 0.57∆V, whereas in Fig. 4, the tem- and Brownian motors; P. Reimann, Phys. Rep. 361, 57 perature is fixed to kBT =0.23∆V. With V1 =4V2, the (2002); R. D. Astumian and P. Hänggi, Phys. Today 55 untiltedpotential,depictedinFig.1,hasabarrierheight (11), 33 (2002). ∆V = 2.2V . We choose α = 0.2 and h¯γ = 0.76∆V. It [2] U. Weiss, Quantum Dissipative Systems, 2nd ed.(World 1 means that the typical action is √2M∆VL2 2h¯, and Scientific, Singapore, 1999). the dissipation rate γ = η/M is about one-fou≈rth of the [3] P.Reimann,M.Grifoni,andP.Hänggi,Phys.Rev.Lett. 79, 10 (1997). classical oscillation frequency Ω = 2π V /ML2 in the 0 1 [4] H. Linke et al., Science286, 2314 (1999). untilted potential (weak dissipation). pIn this numerical [5] J. B. Majer et al.,Phys.Rev. Lett.90, 056802 (2003). application,noneoftheratesexceeds0.05γ and 0.08ωB, [6] R.RoncagliaandG.P.Tsironis,Phys.Rev.Lett.81,10 which means that the duality relation is valid for this (1998). system. Moreover,the third-orderratesstayatleastone [7] S. Scheidl and V. M. Vinokur, Phys. Rev. B 65, 195305 order of magnitude below the second-order ones. The (2002). ratchet current presents several reversals as a function [8] J. Lehmann et al.,Phys. Rev.Lett. 88, 228305 (2002). [9] M. Grifoni et al.,Phys. Rev.Lett. 89, 146801 (2002). both of the driving and the temperature. As expected [10] L. Machura et al.,Phys.Rev. E 70, 031107 (2004). for the small values of driving and dissipation used in [11] M. P.A.Fisher and W.Zwerger, Phys.Rev.B 32, 6190 Fig. 3, the stationary velocity is very close to the value (1985). of a free system v0 = F/η at T = 0, which corresponds [12] M. Grifoni, M. Sassetti, and U. Weiss, Phys.Rev.E 53, to localization v∞ 0 in the TB system [17]. Accord- R2033 (1996). TB ≈ ingly,v∞ 0inthisregime. Thestationaryvelocityalso [13] M. Sassetti, H. Schomerus, and U. Weiss, Phys. Rev. B tends toRv≈(dashed line in Fig. 4) for driving or temper- 53, R2914 (1996). 0 [14] Duetothebehavioroftherates,theraretransitionslimit atures much higher than the potential barrier, and the can hold down toT =0 for α<1. ratchet current vanishes correspondingly. If observed in [15] A. Schmid,Phys. Rev.Lett. 51, 1506 (1983). experiments, this linear behavior would provide a direct [16] A potential with few harmonics also comes naturally in estimation of dissipation. experiments with arrays of Josephson junctions [5]. Inconclusion,weobtainedadualityrelationyieldinga [17] We obtain the opposite behavior vTB → ∞ at low tem- tight-binding description of Brownian motion in a tilted peratures for α=1.26.