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Discontinuous conductance of bichromatically ac-gated quantum wires Tomasz Kwapin´ski,1,2,∗ Sigmund Kohler,1 and Peter Ha¨nggi1 1Institut fu¨r Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany 2Institute of Physics, M. Curie-Skl odowska University, 20-031 Lublin, Poland (Dated: January 16, 2009) We study the electron transport through a quantum wire under the influence of external time- dependentgate voltages. Thewire ismodelled bya tight-bindingHamiltonian for which weobtain 9 thecurrentfromthecorrespondingtransmission. Thenumericalevaluationofthedccurrentreveals 0 that for bichromatic driving, the conductance depends sensitively on the commensurability of the 0 driving frequencies. The current even possesses a discontinuous frequency dependence. Moreover, 2 wefind that theconductanceas a function of thewire length oscillates with a period that depends on the ratio between thedriving frequencies. n a PACSnumbers: 05.60.Gg, 73.23.-b,73.63.Nm J 6 1 I. INTRODUCTION mixing.25 Also dynamical localization effects in quantum dots may be affected by bichromatic driving fields, ] l in particular when the freuquencies of the latter are l Recent advancements in nanotechnology enabled the a commensurable.26,27 investigation of low-dimensional systems like quantum h A well established theoretical tool for treating period- - dots, quantum wires or few-atom systems. Thereby, s ically driven quantum system is Floquet theory, which techniques like mechanically controlled break junc- e is based on the discrete time-translation invariance of m tions, scanning tunneling microscopy, or angle-resolved the driving field. Higher harmonics of the fundamental photo-emission electron spectroscopy provide informa- . drivingfrequencydo notaffectthis invarianceand,thus, t tion about the electronic structure or transport proper- a can be included straightforwardly. As soon as the ratio m ties of these systems. Particularly interesting are the between the fundamental frequency and its higher har- transport properties of low-dimensional nanoscale con- - monicincreases,anumericaltreatmentmayrequirequite d ductorsin the presenceoftime-dependent externalfields some effort. The situation becomes even worse, when n forwhichawealthofeffectshasbeenpredictedlike,e.g., the frequencies are no longer commensurable. Then it o photon-assisted tunneling,1,2 and electron pumping.3,4 c External fields can be also used to control both the dc is rather desirable, even from a practical point of view, [ to find analytical expressions for the behavior of the current5 and the zero-frequency noise6,7 in mesoscopic driven quantum system, as for example in the recent 1 conductors. works 20,21,22. For the description of electron trans- v 2 In undriven one-dimensional quantum wires, con- port through monochromatically driven systems, com- 5 ductance oscillations — the oscillatory dependence of montransporttheorieshavebeencombinedwithFloquet 4 the conductance as a function of the wire length — theory.16,28,29,30 2 were predicted8,9 and experimentally confirmed.10 They In this paper we study the transport properties of a . emerge when the energy levels of a quantum wire match 1 bichromatically driven quantum wire, focussing on the the Fermienergyofthe systemandrelate tochargeneu- 0 influence of commensurability of the driving frequencies. 9 trality of the wire.11 The period of these oscillations Indoingso,wegeneralizeinSec.IIrecenttheoreticalap- 0 is usually two sites for undriven wires, but may be- proaches such that they allow one to treat two ac fields : come larger if the wire is exposed to a time-dependent v with arbitrary frequency ratio. Then we investigate in i field.12,13,14 Sec.IIItheinfluenceofbichromaticdrivingonthetrans- X A second harmonic added to an ac field that drives mission and the conductance oscillations. r a aquantumwiremaysignificantlyaltertransportproper- ties. Themainreasonofthedifferencesisthatthesecond harmonic may brake time-reversal symmetry and gen- II. THEORETICAL DESCRIPTION eralized parity dynamically, leading to effects like non- adiabatic electron pumping,15,16,17,18,19 i.e., to a dc current even in the absence of any net bias. However, if A. Wire-lead model the ac field acts as a gate voltage that shifts all wire levels uniformly, the pump current must vanish.16 Sim- Weconsiderthetime-dependentwire-leadHamiltonian ilar effects have been predicted for quantum Brown- H(t)=H (t)+H +H , where wire leads coupl ian motion in bichromatically driven infinitely extended periodic potentials.20,21,22 Moreover, two incommensu- N N−1 rable rectangular-shaped driving forces can cause a dc H (t)= [ε +f(t)]c+c + V c+c +h.c. (1) current23,24 and lead to interaction-induced frequency wire X n n n X n n n+1 n=1 n=1 2 models in a tight-binding description the wire with sites This means that the influence of the driving field is sub- |ni,whereneighboringsitesarecoupledviatunnelmatrix sumed in the phase elements V . The on-site energies ε are modulated by n n t an ac gate voltage f(t). The first and the lastsite of the F(t)= dsf(s). (7) wire are coupled via the tunnel Hamiltonian Z H = V c+ c + V c+ c (2) Defining the currentthroughthe wireasthe changeof coupl X~kL ~kL ~kL 1 X~kR ~kR ~kR N the electron number in the left lead, I = −ednL(t)/dt, we obtain for its expectation value the time-dependent to leads at the left (L) and at the right (R) end. Landauer-like expression Within a wide-band approximation, we assume that e 2thπe sp~ke|cVt~kr|a2lδ(dεen−siεtαy)o=f tΓhαe,wisireen-leeragdycinoudpeplienngd,eΓnαt.(εT)h≡e I(t)= hZ dε(cid:2)TRL(t,ε)fL(ε)−TRL(t,ε)fL(ε)(cid:3)−q˙L((t8)) leadPs are modelled as ideal Fermi gases by the Hamilto- with the time-dependent transmission nian H = ε c+ c . (3) TLR(t,ε)=ΓLΓR|G1,N(t,ε)|2 (9) leads ~kα ~kα ~kα X X α=L,R ~k and T defined accordingly. The function LR The operators c , c , and their Hermitian adjoints are ~kα n theusualannihilationandcreationoperatorsforanelec- G(t,ε) = dτeiετG(t,t−τ) (10) Z tron in lead α = L,R with wave vector ~k and for an electron at the nth wire site, respectively. We focus = e−iF(t) dτeiετ+iF(t−τ)g(τ) (11) on experiments like those described in Refs. 9,31,32, Z in which electron-electron interactions can be neglected. is the Fourier transformed of the Green function with Forphoton-assistedtransport,thisis,e.g.,thecasewhen respecttothetimedifferenceτ =t−t′. NotethatGstill the photon energyis much smaller thanboth the typical depends explicitly on the final time t. The displacement splitting between single-particle energies of the wire and current q˙ (t) describes charges oscillating between the the Coulomb interaction.33 L left lead and the wire and does not contribute to the dc currentconsideredbelow. Thus it will not be considered henceforth. B. Time-dependent scattering theory In order to evaluate the τ-integration in Eq. (11), we havetospecifythetime-dependentgatevoltagef(t). We FollowingthescatteringapproachofRef.16,wederive assume that it consists of K harmonics, such that fromthetime-dependentwire-leadHamiltonianH(t)the Heisenberg equations of motion for the creation opera- K tors. The wire part of their solution can be expressed in f(t) = ∆ cos(ω t), (12) k k terms of a retarded Green function G(t,t′) which, owing X k=1 to the time-dependence of the Hamiltonian, depends ex- K ∆ plicitly on both times and fulfills the equation of motion F(t) = k sin(ω t), (13) k ω X k k=1 d i i~ −H −f(t)+ Γ G(t,t′)=δ(t−t′). (4) h dt 0 2 i whereωk and∆k arethefrequencyandtheamplitudeof thekthharmonicoftheacfield. ThentheGreenfunction The Hamiltonian H = H −f(t)1 describes the 0 wire wire (11) reads wire in the absence of the time-dependent gating, while Γ=Γ |1ih1|+Γ |NihN|is aself-energystemming from L R G(t,ε)= J (∆ /ω )···J (∆ /ω ) the coupling of the wire to the leads. m1 1 1 mK K K m1,X···,mK The solutionofEq.(4)canbe tracedbacktothe solu- (14) tionofthetime-independentproblemforwhichtheGreen ×exp i mkωkt g ε− mkωk . function g obeys the equation of motion (cid:0) Xk (cid:1) (cid:0) Xk (cid:1) i~ d −H0+ iΓ g(t−t′)=δ(t−t′). (5) TsithieonBoefsstehlefuenxcptoinonenstJiaml isnteEmq.fr(6o)m, wahFiloeutrhieeredveacluomatpioon- h dt 2 i ofthe τ integralprovidedthe Fourier transformedof the Then it is straightforward to show that the Green func- Green function, g(ǫ). Note that we have ignored a time- tions G and g are related according to dependent but nevertheless irrelevant phase factor. Assumingmoreoversymmetricwire-leadcoupling,i.e., G(t,t′)=e−i[F(t)−F(t′)]g(t−t′). (6) Γ = Γ = Γ, the time-dependent transmission (9) be- L R 3 comes integers. Then we obtain TLR(t,ε)=Γ2 Jm1(∆1/ω1)···JmK(∆K/ωK) T(ε)= Γ2J (∆ /ω )J (∆ /ω ) m1,X···,mK <m1,mX2,m′1,m′2> m1 1 1 m2 2 2 ×m′1,X···,m′KJm′1(∆1/ω1)···Jm′K(∆K/ωK) ××gJm′1((ε∆−1/mω1)ωJm−′2(m∆2ω/ω)2) (19) 1N 1 1 2 2 ∗ ′ ×g1N ε− mkωk g1N ε− mkωk ×g∗ (ε−m′ω −m′ω ), (cid:0) Xk (cid:1) (cid:0) Xk (cid:1) 1N 1 1 2 2 ′ ×exp i (m −m )ω t (15) k k k wherethe angularbracketsrestrictthe sumto those val- (cid:0) Xk (cid:1) ues of m ,m ,m ,m that comply with the condition 1 2 3 4 =TRL(t,ε). (16) (m −m′)ω = −(m −m′)ω . Note that for commen- 1 1 1 2 2 2 surable frequencies, this condition can also be fulfilled The relation in the last line means that from the right for m 6= m′. Since for incommensurable frequencies k k lead to the left lead has the same probability as the re- the restriction to the sum is fulfilled only for m = m′ 1 1 versed process. This is particular for ac gating and does andm =m′,expression(19)incidentallyholdsforboth 2 2 not hold for general driving.16 Here it stems from the cases. factthatforthepresentmodelwithtime-dependentgat- The transmissions (18) and (19) represent the main ing,thetime-dependenttransmissioncanbeexpressedin analyticalresults of this work. In the caseof commensu- terms of the Green function g for the time-independent rable frequencies, the driving is periodic and, thus, can problem. be treated within Floquet scattering theory.6,12,16 Nev- The dc current,being the main quantity of interest, is ertheless, the explicit evaluation of the transmission can then obtained from Eq. (8) and reads be quite cumbersome. The case of incommensurable fre- quenciesgenerallyevenrequiresa“two-colorFloquetthe- e ory” which can be numerically demanding. Only the re- I0 = hZ dε fL(ε)−fL(ε) T(ε), (17) striction to models with uniform gating allowed us to (cid:2) (cid:3) obtainthe presentcompactexpressionsforthe transmission. where T(ε) denotes the time-average of the transmission T (t,ε), and its computation from Eq. (15) seems LR straightforward. This is however not the case, because the averaging procedure depends crucially on the commensurability of the frequencies ω . If the gate volt- k III. WIRE WITH EQUAL ON-SITE ENERGIES age f(t) consists of many spectral components, the ac- tual computation can be quite cumbersome. However, we below restrict ourselves to two frequencies, so that it In order to present explicit results, we have to spec- is sufficient to address two cases: all frequencies being ify the wire model. A N-site model that allows even commensurable or all being incommensurable. further analytical treatment is one with equal onsite energies ε = ε for all sites |ni, n = 1,...,N and equal For incommensurable frequencies, the phase factor in n 0 couplings V =V. For such a wire, the relevant element thelastlineofthetime-dependenttransmission(15)van- n ishes if and only if m = m′ for each mode k. Conse- of the Green function in the absence of driving reads k k quently, we obtain the average transmission (−1)N−1 g (ε)= , T(ε) = Γ2Jm21(∆1/ω1)···Jm2K(∆K/ωK) 1N VuN(x)+iΓuN−1(x)−Γ2uN−2(x)/4V m1,X···,mK (20) 2 where x = (ε − ε0)/2V, while un denotes the nth × g1N ε− mkωk . (18) Chebyschevpolynomialofthesecondkind. Foraderiva- (cid:12)(cid:12) (cid:0) Xk (cid:1)(cid:12)(cid:12) tion,wereferthereadertoRef.34. Thisrelationtogether (cid:12) (cid:12) with Eqs. (18) and (19) allows one to obtain explicit ex- The Green function of the undriven wire, g(ǫ), depends pressions for the transmission of the ac-gated wire (1). onthe specific wiremodelandwillbe determinedbelow. In our calculations we use the effective wire-lead cou- If the frequencies of the external fields are commensu- pling strength Γ as energy unit. Then for Γ = 0.1eV, rable, time-averaging unfortunately does not result in a the current unit becomes 2eΓ/~ ≃ 50 µA. Moreover, we less involved expression for the transmission. Therefore measure level energies with respect to the Fermi energy, weprovidetheexplicitresultonlyforthecaseoftwofre- whichimpliesE =0. Henceforth,wepresentresultsfor F quencies with ratio ω /ω =m/m′, where m and m′ are a bichromatic driving field. 1 2 4 000...666 000...222 ∆∆∆222===444 ωωω111===222 000...111555 000...444 ωωω ===444 TTT 000 000...111 222 III 000...222 000...000555 ωωω ===333...999999 222 000 000 000...666 ∆∆∆222=== 222 111 ///111 111 ///222 00..11 ωω ==33..9999 ωω11==44 22 ωω ==44 000...444 22 TTT 222///333 00 555///333 444///555 II 00..0055 444///111 000...222 222///111 444///333 000 00 000...555 111 111...555 222 222...555 333 333...555 444 --22 00 22 44 66 88 1100 ωωω εε 111 00 FIG. 1: Transmission for a wire with N =5 sites with equal FIG. 2: Average current for the chemical potentials µL = onsite energies ε0 = 0 as a function of the frequency ω1 for µR =0.1 as a function of the equal onsite energies ε0 for a − the driving amplitudes ∆2 = 4 (a) and ∆2 = 2 (b). The wire with N =3 sites and tunneling matrix elements V =4. tunnelcouplingbetween twoneighboring wire sites is V =4, Thefirstdrivingfieldiskeptconstant[∆1 =4withfrequency while the other driving parameters read ω2 = 2, ∆1 = 4. ω1=2(a) andω1 =4(b)]. Thesecond field hasdrivingam- Solid lines are obtained from expression 18 which is valid for plitude∆2 =4,whileitsfrequencyisω2 =4(thicksolidlines) incommensurablefrequenciesonly,whilethedashedlinesand andω2 =3.99(thinsolidlines),respectively. Thedashedline thecirclesareobtained from Eq.(19)and,thus,arevalidfor in panel (a) marks the undriven case which corresponds to alldrivingfrequenciesω1,ω2. Thefractionsinpanel(b)mark vanishingdriving amplitudes, i.e. ∆1 =∆2 =0. selected frequency ratios ω2/ω1. over,the2/1resonance,albeit“veryrational”,doeshere A. Discontinuous transmission and corresponding notleadtoasignificantdiscontinuity. Asimilarbehavior current has been observed for the transport in classical ratchet devices.24 It is worth noting that the transmission for In order to reveal the role of commensurability of the incommensurable frequencies [Eq. (18) and solid lines in driving frequencies, we investigate the transmission of a Fig.1]possesseslocalmaximaattheresonances1/2,1/1, wire with onsite energies at the Fermi energy level. Fig- and 2/1. ure1showstheresultasafunctionofthefrequencyofthe Inordertoinvestigatefurthertheroleofcommensura- first external field, ω , while the frequency of the second bility, we focus on the 2/1 resonance for ω =2 and the 1 1 field is kept at a constant value. As a most prominent 1/1 resonance for ω = 4. In doing so, we compare the 1 feature, the transmission turns out to be a discontinu- current for the commensurable case ω =4 with the one 2 ousfunction,whichassumesexceptionalvalueswhenever found for the “less commensurable” ω =3.99. The cur- 2 ω /ω is rational, i.e., when the driving frequencies are rentforawireoflengthN =3asafunctionoftheonsite 2 1 commensurable. A clearly visible effect is found for ra- energy ε is shown in Fig. 2. A common feature of both 0 tios that can be expressed by small natural numbers. In cases is the emergence of side peaks: The undriven wire particular,fortheratios1/1and2/1,the deviationfrom possesses the eigenenergies E0 = ε0 and E± = ε0±5.6. the transmissioninthe vicinity ofthe resonanceisofthe Whenever these eigenenergieslie within the voltagewin- order of 20%. For “less rational” numbers like e.g. 4/3 dow at the Fermi energy EF = 0, i.e. for ε0 = 0 and for or2/3,westillfindasmalldiscontinuitywhich,however, ε = ∓5.6, the current assumes a maximum in compli- 0 is hardly visible on the scale chosen. The values can de- ancewiththeBreit-Wignerformula;seeFig.1ofRef.34. pend sensitively on details like driving amplitudes and Inthe presenceofac gating,we findside peaks shifted wireparameters. Inthepresentcase,itisevensuchthat from the original peaks by multiples of the driving fre- the deviation at the prime resonance 1/1 turns with in- quencies ω and ω . Overlapping peaks may lead to de- 1 2 creasing amplitude ∆ from a remarkablenegative value structive interference. This is, e.g., the case for the data 2 toapositivevalue;cf.panels(a)and(b)ofFig.1. More- shown in Fig. 2(a) in the vicinity of ε = 0, where the 0 5 side peaks at ε ∓ω ∓ω for ε =±5.6 compensate the 11 0 1 2 0 MM==44 fundamental peak at ε = 0, such that the total current 0 00..88 M=6 is characterizedby the localminimum. The properloca- tion of the side peaks is most clearly visible for ω = 4 00..66 1 and ω ≈ 4 [Fig. 2(a)], i.e., when both frequencies are TT 2 00..44 practicallyindistinguishable. Thenweobservesidepeaks at ε0 = 4, 8 (main peak at ε0 = 0), at ε0 = 1.6, 9.6 00..22 (main peak at ε = 5.6), and at ε = 2.4 (main peak at 0 0 00 ε = −5.6). Let us emphasize that the location of the 0 22 44 66 88 1100 1122 1144 peaksdoesnotallowustodividethecommensurablecase from the incommensurable case, because the resonance 00..88 MM==1100 M=12 width Γ = 1 is much larger than the difference between the two values for ω which reads 4−3.99=0.01. 00..66 2 Despite the practically identical location of the peaks, TT 00..44 the peak heights are significantly different for the two cases. This correspondsto the discontinuity observedal- 00..22 readyinFig.1. Thedifferencecanevenbeuptoafactor 5,asisthecaseforω1 =4[Fig.2(b)]forthezeroth-order 00 resonances at ε = 0 and ε = 5.6. This means that in 44 88 1122 1166 2200 2244 0 0 a possible experiment, the currentmay be controlledvia NN a slight shift of one of the driving frequencies, provided thatonetunesthewireparameterssuchthatthecurrent in the absence of the driving assumes a large value. FIG. 3: Transmission as a function of the wire length N for ∆1 = ∆2 = 6, ε0 = 0, V = 4. (a) the driving frequencies are ω1 = √2V, ω2 = 2V (circles) and ω1 = V, ω2 = 2V (triangles) correspond to the oscillation periods M = 4 and B. Conductance oscillations M =6, respectively. (b)ω1=2V cos(π/5),ω2 =2V (circles) and ω1 =V, ω2 =√2V (triangles) corresponding to M =10 The conductance G of a tight-binding system with and M = 12, respectively. The lines serve as a guide to the onsite energies at the Fermi level obeys oscillations as eye. a function of the wire length. Typically these oscillations have period 2 (even-odd oscillations),8,9,10,11 but also larger periods have been predicted for both accordingtoG=(2e2/h)T]forbichromaticdriving. The static34,35,36,37 and monochromatically driven12 wires. results for various combinations of the driving frequen- We here investigate the corresponding behavior for cies are shown in Fig. 3. For all parameters used in the bichromatic gating. numericalevaluationofthetransmission(19),weobserve The oscillation period for the driven case can be esti- anoscillationperiodM >2incompliancewithEq.(22). mated along the lines followed in Ref. 12: For a static Conductance oscillations with period M = 6 have also wire, the emergence of conductance oscillation with pe- been found in Ref. 12 for wires under the influence of a riod M means that a wires with N sites possesses the monochromatic ac field. There the oscillation period is same conductance as a wire with N +M sites, i.e. also determined by Eq. (22) for k = 0. We like to em- 2 phasize that in the absence of the driving, by contrast, g1,N(ε)=g1,N+M(ε). (21) the oscillation period is M = 2, i.e., we find the usual even-odd oscillations. After some algebra, one obtains34 cos(mπ/M) = (ε − A remaining intriguing question is whether for bichro- ε )/2V, where m = 0,1,...,M −1. The energy differ- 0 matic driving, the commensurability of the frequencies ence on the right-hand side of this equation marks the influences the conductance oscillations. Therefore, we distance of the onsite energies ε to the Fermi energy of 0 compare for ω = V the commensurable ω = ω the leads attached. In the presence of ac fields, onsite 1 2 1 (which in fact represents monochromatic driving) with energies acquire shifts by multiples of the driving field the “practically incommensurable” ω = 0.99ω . For quanta. Thenthe oscillationperiodisdeterminedbythe 2 1 both cases, Eq. (22) predicts conductance oscillations condition with a period of approximately M = 6. Figure 4 de- mπ ε−ε +k ω +k ω picts the resulting transmissionas a function of the wire 0 1 1 2 2 cos = , (22) length. Note, that the oscillation period for the static (cid:16) M (cid:17) 2V conductor is M = 2, while it becomes M = 6 in the where again m = 0,1,...,M −1, while k and k are presence of the driving. Generally, the variation of the 1 2 integer numbers. conductance within one oscillation period is larger for In order to corroborate this condition, we have com- incommensurate driving frequencies. For the driving pa- putedthetransmission[whichrelatestotheconductance rameters used in Fig. 4, it is even such that for the com- 6 11 chromatically ac-gated quantum wire described by a ∆∆==44 ωω ==00..9999 ωω tight-binding model. If the gating acts uniformly on all 22 11 00..88 ωω ==ωω wire sites, we have provided an expression for the time- 22 11 dependenttransmissionthatdependsonthetransmission 00..66 TT in the absence of the driving and some time-dependent 00..44 prefactors. The time-average of the transmission for ac gatingdeterminesthedccurrentbeingthecentralexper- 00..22 imentally accessable quantity. In order to perform the 00 time-average,we had to distinguish between commensurable and incommensurate frequencies. We have found ∆∆==88 00..55 that the transmission and, thus, the dc current depends sensitively on the commensurability of the driving fre- 00..44 quencies. The difference is most noticeable close to the TT 00..33 resonanceswith the “very rational”frequency ratios 1/2 00..22 and1/1,wheretheconductanceasafunctionofonedriv- ing frequency is discontinuous and may suddenly change 00..11 by up to one order of magnitude. 00 Thecommensurabilityofthedrivingfrequenciesleaves 22 44 66 88 1100 1122 1144 also its fingerprints in the conductance oscillations: NN Whileforstaticconductors,oneusuallyobservesanoscil- lation period of two sites (even-odd oscillations), bichromatic driving may increase the period significantly. For FIG. 4: Transmission as a function of the wire length N for our model, we observed oscillation periods of up to 12 commensurable frequencies (ω1 = ω2 = 4, triangles) and in- sites. The physicalreasonforthe enlargedperiodis that commensurable frequencies (ω2 = 0.99ω1, ω1 = 4, circles) and the driving amplitudes ∆1 = ∆2 = ∆ = 4 (a) and the bichromatic driving influences the resonance condi- ∆1 = ∆2 = ∆ = 8 (b). The other parameters are ε0 = 0, tionthatdetermines the oscillations. We haveconfirmed V =4. The lines are a guideto the eye. this intuitive prediction by numerical studies. As a re- markable influence of the commensurability, we found that the growth of the oscillation period is more visible mensurate ω1 = ω2, the oscillations have almost period for incommensurate frequencies. M = 3 or M = 2 (solid lines) with a small variation of InapossibleexperimentwithSTMtipsthatcontacta every second peak, such that strictly speaking, the pe- quantum wire on a vicinal surface,31,32 the observed ef- riod is still M = 6. Moreover, we again observe one fect may be used for switching the current upon slightly significantdifference betweenthe commensurateandthe changing the frequency of the ac gate voltage. Alterna- incommensurate case: For given driving amplitudes, the tively, such an experiment may also be performed with conductance may change by almost one order of magni- a molecular wire that bridges a break junction10,38 or tude uponshiftingthe secondfrequencyfromω2 =ω1 to witha chainofcoherentlycoupledquantumdots.39,40 In ω = 0.99ω . This again allows one in an experiment to 2 1 particularthe latter type ofexperimentis wellsuited for switch the current by changing one driving frequency by applying time-dependent gate voltages. only 1%. Note also that when the driving amplitude becomes smaller, the situation tends towards the undriven limit forwhichthe oscillationperiodisM =2. Thetransition Acknowledgments is such that the conductance within the M = 6 period changes until eventually the oscillation period becomes practically M =2. This work has been supported by the Grant No. NN202146833 of the Polish Ministry of Science and Higher Education and the Alexander von Humboldt IV. CONCLUSIONS Foundation (TK). SK and PH acknowledge support by the DFG via SPP 1243,the excellence cluster “Nanosys- Using a Green’s function technique, we have deriveda tems Initiative Munich” (NIM), and the German-Israeli Landauer-like formula for the current through a poly- Foundation (GIF). ∗ Electronic address: [email protected] N.C.vanderVaart,andC.J.P.M.Harmans,Phys.Rev. 1 T. H. Oosterkamp, L. P. Kouwenhoven, A. E. A. Koolen, Lett. 78, 1536 (1997). 7 2 W.G.vanderWiel,S.DeFranceschiandJ.M.Elzerman, Physica A 338, 406 (2004). T. Fujisawa, S. 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