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Mode locking in Quantum Hall effect point contacts Hsiu-Hau Lin Department of Physics, University of California Santa Barbara, CA 93106-9530 8 Matthew P. A. Fisher 9 Institute for Theoretical Physics, University of California 9 1 Santa Barbara, CA 93106-4030 (February 1, 2008) n a Westudytheeffectofanacdriveonthecurrent-voltage(I-V)characteristicsofatunneljunction J betweentwofractionalQuantumHallfluidsatfillingν−1anoddinteger. WithinthechiralLuttinger 1 liquid model of edge states, the point contact dynamics is described by a driven damped quantum 3 mechanicalpendulum. Inasemi-classicallimitwhichignoreselectrontunnelling,thismodelexhibits mode-locking, which corresponds to current plateaus in the I-V curve at integer multiples of I = ] l eω/2π, with ω the ac drive angular frequency. By analyzing the full quantum model at non-zero l a ν using perturbative and exact methods, we study the effect of quantum fluctuation on the mode- h locked plateaus. For ν = 1 quantum fluctuations smear completely the plateaus, leaving no trace - of theac drive. For ν ≥1/2 smeared plateaus remain in theI-V curve,but are not centered at the s e currents I =neω/2π. For ν <1/2 rounded plateaus centered around the quantized current values m are found. Thepossibility ofusing modelocking in FQHEpoint contactsas a current-to-frequency standard is discussed. . t a PACS numbers: 72.10.-d 73.20.Dx m - d I. INTRODUCTION vanishing rapidly for currents below the critical current, n I . Moreover,thezerobiasresistanceisexpectedtovan- o J ishexponentiallyasT ,dV/dI exp( E /k T),with c The conductance through a tunnel junction is pro- → ∼ − J B [ portional to the electron density of states in the two energybarrierEJ =φIJ. Underexchangeofcurrentwith voltage, the behavior is similar to the vanishing conduc- 2 electrodes. For metallic electrodes, which have a non- tance in the FQHE point contact. In a Josephson junc- v zero density of states at the Fermi energy, the tunnel tion, the phase difference between the superconducting 6 junction current-voltage (I-V) characteristics are Ohmic electrodes is behaving classically, whereas in the FQHE 5 at low bias. In marked contrast, recent theories1,2 1 junction the classical variable is the transferred electron havepredictedstronglynon-ohmicbehaviorfortunneling 0 charge. through a point contact separating two fractional quan- 1 One of the most striking manifestations of the ac 5 tum Hall effect (FQHE) fluids. Specifically, for filling Josephson effect, is the presence of quantized voltage 9 factor ν = 1/m, with odd integer m, the tunnel current steps(Shapirosteps)inanappliedmicrowavefield4. The / at zero temperature is predicted to vary with voltage as at I V2/ν 1. At finite temperatures, Ohmic behavior is applied radiation at angular frequency ω mode locks to − m rec∼overed at small voltages, with a zero bias differential the discretephase slipeventsleadingto plateausatvolt- - conductance varying as, dI/dV T2/ν−2. A tempera- ages V = n(h¯/2e)ω, for integer n. In the plateaus, d ∼ the voltageis so accuratelyquantizedthat Shapirosteps ture dependence consistent with this has been seen in a n recent experiment by Milliken et. al.3 for the tunnelling serve as a voltage-to-frequency standard. o conductancebetweentwoFQHEfluidsatfillingν =1/3. The duality between Josephson junctions and FQHE c : The non-Ohmic tunneling conductance is due to the junctions, suggests that the latter might also exhibit in- v strangepropertiesoftheedgestatesintheFQHE.FQHE teresting behavior in the presence of an applied ac field. i X edgestatesareabeautifulrealizationofone-dimensional In this paper, we study in detail the effect of an ac drive r Luttingerliquids1. Incontrasttometallicelectrodes,the on a FQHE tunnel junction, focussing on the structure a tunneling density of states in a Luttinger liquid vanishes inducedintheI V characteristics. One anticipatesthe − at the Fermi energy, which leads to the vanishing tunnel possibility of mode locking between the ac drive and the conductance between two FQHE fluids. Thus, in con- electron tunnelling events. This could lead to steps in trast to conventional metallic tunnel junctions, a FQHE the junction current, quantized at integer multiples of tunnel junction is an insulator. I =eω/2π - the analog of Shapiro steps. An insulating point contact junction is, in many re- Quantized current plateaus for metallic tunnel junc- spects, the dual of a superconducting point contact - tionswereproposedseveralyearsback5. DuetoCoulomb namely a Josephson junction. In a Josephson junction blockade effects, it was arguedthat normalmetal tunnel the I-V curve is also strongly non-Ohmic, with voltage junctions with sufficiently high resistanceswould exhibit 1 thephenomenaofBlochoscillations-anoscillatoryvolt- the I-V curves is completely wiped out! For ν = 1/2, age in the presence of a dc current - the dual of the ac thesolutionrevealsremainingstructure,butthesmeared Josephson effect. Moreover, it was suggested that an current plateaus are not centered at integer multiples of appliedacdrivewouldmodelocktotheseoscillationsre- I =eω/2π. sulting in current plateaus. A more favorable geometry InSectionVwecompute the I V curvesinapertur- − forcurrentplateaus,consistsofmultipletunneljunctions bativeapproach,whichleadsusto conjecturethe follow- inseries,whichcanbeseparatelytweakedbyanacdrive, ing general form for the I-V curves at arbitrary ν: therebytransferringtheelectronsone-by-onethroughthe circuit. Such an electron “turnstile” was realized ex- I(Vsd,Vac)= cn 2 Idc(νVsd+nω). (1.1) | | perimentally, by a number of groups, both in metallic Xn systems6–8 and in semiconductor heterostructures9–11. Here Idc(V) I(V,0) is the tunnel current in the ab- Due to the multiple junction geometrythe tunstiles only sence of the a≡.c. drive, and c 2 = J (νV /ω)2, with work well at rather low frequencies, below tens of Mega- | n| | n ac | J (X)n’thorderBesselfunctions. Thesecoefficientssat- hertz. At higher frequencies the electrons take too long n isfy the sum rule c 2 = 1. This form has a simple to pass across the junctions, and do not “keep up” with n| n| physical interpretaPtion: Charge ν quasiparticles absorb the ac drive. n-quanta from the a.c. field with probability c 2, and Ina Josephsonjunction Shapirostepsareveryrobust, | n| are transmitted throughthe point contactwith totalen- and do not need complicated multiple junction geome- ergy νV +nω. tries. Moreover, Shapiro steps are observed up to quite sd Equation (1.1), which is also consistent with our ex- frequencies,comparabletothesuperconductinggap. The act solutions, gives a simple explanation as to why all reason for this is that the junction phase difference is a plateaus are wiped out at ν = 1. For ν = 1 the edge classical field, so that phase slip processes are classical states are describable in terms of non-interacting elec- events which readily lock to an ac drive. In “insulating” trons(Fermi liquid). Under the assumptionof anenergy FQHEpointcontactstheelectronchargeisagoodquan- independent transmission probability through the junc- tum number, which suggests that mode locking might tion, the d.c. I-V curves are linear (Ohmic). Since the also be possible in a single junction configuration. How- transmission is independent of energy, the a.c. drive has ever,quantumfluctuationsintheelectronchargetransfer noeffectonthe I-Vcurves,whichremaincompletelylin- are expected to be more important than quantum phase ear. slip processes in the Josephson junction, as reflected in For ν < 1 the d.c. I-V curves are non-linear, and the power law voltage and temperature dependences in plateau-like features show up with an a.c. drive. Re- theI-VcurvesoftheFQHEjunction. (Becausethephase cently, Fendley et. al.15 have obtained exact solutions of the superconducting wavefunction exhibits true long- for the d.c. I-V curve at arbitrary integer ν 1. These rangedorder,low frequency quantum phase slips are ex- − curves, together with the conjecture (1.1), enable us to pected to be completely absent.) This paper is devoted construct the I-V curves with a.c. drive present for the to studying the effect of such quantum fluctuations in experimentally relevant cases of ν = 1/3 and ν = 1/5. washing out mode-locked steps. Forthesecases,inthelimitofweakpinchoffatthepoint Theorganizationandcentralresultsofthepaperareas contact, the I-V curves exhibit smearedcurrentplateaus follows. In Section II we introduce the edge state model centered at integer multiples of I =eω/2π. for a FQHE tunnel junction at filling ν = 1/m, in the SectionVIisdevotedtoadiscussionoftheexperimen- presence of both a dc source-to-drain voltage, V , and sd tal consequences. an a.c drive voltage, V sinωt. While the model is only ac appropriateforFQHEedgeswhenν 1 isanoddinteger, − it is well defined for general ν. II. MODEL FOR POINT CONTACT WITH AC In Section III we consider a semi-classicallimit, which DRIVE ignoresquantumtunnellingofthe electron. Inthis limit, the model reduces to the classical dynamics of a period- ConsiderthenaFQHEstateatfillingν 1 anoddinte- ically driven overdamped pendulum, with the phase of − ger. ForthisclassofHallfluidsonlyasingleedgemodeis the pendulum representing the charge transferredacross expected1. FortheIQHEatν =1afree-fermiondescrip- the junction. This classical model is equivalent to the tion of the edge mode is possible16, but more generally resistively-shunted junction (RSJ) model of Josephson junction dynamics12–14. Not surprisingly, robust mode the edge mode is expected to be a (chiral) Luttinger liquid, describable in terms of a bosonic field. locked current plateaus are found in this semi-classical Letρ andρ denotetheelectrondensitiesintheright limit. R L and left moving edge modes, on the top and bottom of In Section IV we study the full quantum model, and the sample, as shown schematically in Figure 1. These derive exact solutions for the I-V curves at two special densities are written as gradients of bosonic fields, values, ν = 1 and ν = 1/2. At ν = 1, appropriate for the integer quantum Hall effect, quantum fluctuations 1 are so strong that all of the mode-locked structure in ρR/L =±2π∂xφR/L (2.1) 2 which satisfy the Kac-Moody commutation relations1: For later convenience it will be useful to introduce a gauge field A(t), defined via, V(t) = ∂ A(t). A useful t [φR/L(x),∂xφR/L(x′)]= i2πνδ(x x′). (2.2) identity is ∓ − Herexis aone-dimensionalpositioncoordinate,running ∞ eiνA(t) =eiνVsdt c e inωt, (2.7) along the edge. The appropriate Hamiltonian density n − describing propagationof edge modes is16,2 n=X−∞ = vF [(∂ φ )2+(∂ φ )2]. (2.3) where cn =(−i)nJn(νVωac), with Jn(X) Besselfunctions. 0 x R x L The full Hamiltonian density is = + + . H 4πν 0 1 V H H H H In the absence of backscattering at the point con- Here vF is the velocity of edge propagation. tact, the total source-to-draincurrent, I =vF(ρR ρL), − upon averaging over time is appropriately quantized: = νV /2π. Backscattering will reduce this cur- sd Vsd rent, 1 = νV , (2.8) 2π sd− B where I is the backscattering current operator. An ex- B pression for I follows upon functional differentiation: B Source Drain δH 1 I =∂ dx(ρ ρ ). (2.9) B t R L ≡−δA 2Z − For later convenience it will be useful to define new boson fields which propagate in the same direction: φ (x) φ (x), φ (x) φ ( x). (2.10) 1 R 2 L ≡ ≡ − FIG. 1:Schematic representation of a point contact in a The commutators become, FQHE fluid. The lines with arrowsrepresentedge states whichcanscatteratthepointcontact. Thevoltagedrop between source and drain is denoted Vsd. [φi(x),∂xφj(x′)]=−iδij2πνδ(x−x′). (2.11) The Hamiltonian density has the same form as before, At the point contact, the right and left moving edge v modes are brought into close proximity, and tunneling = F (∂xφi)2+vδ(x) ei(φ1−φ2)+h.c. between them becomes possible. In the limit of weak H 4πν h i tunneling, the dominant backscattering process at low 1 +(ρ ρ ) V(t) (2.12) 1 2 temperatures is of fractionally charged (Q=eν) Laugh- − 2 lin quasiparticles17. The appropriate tunneling term is provided the densities are defined as, ρ = ∂ φ /2π. i x i Upon using the continuity equations, ∂ ρ +v ∂ ρ =0, =vδ(x) ei(φR φL)+h.c. , (2.4) t i F x i 1 − validawayfromthepointcontactatx=0,thebackscat- H h i tering current operator can be re-expressedas: where v is the local tunneling amplitude, at the point contact, x=0. I = (v /2) dx∂ (ρ ρ ) B F x 1 2 In the presence of an applied source-to-drain voltage, − Z − theincomingedgemodeswillbeatdifferentchemicalpo- =(v /2)(ρ ρ ) x=0+ (2.13) tentials. Inter-edge tunneling processes will thus change F 1− 2 (cid:12)x=0− the energy. Denoting the applied voltage as V(t), the (cid:12) Here we have used the fact that(cid:12)the only backscattering energy change can be written: is at the origin, x=0. 1 It is worth emphasizing that the above model is only V =(ρR ρL) V(t). (2.5) appropriate for a FQHE point contact at filling factor H − 2 ν 1 an odd integer. For FQHE states at other filling − In addition to a dc source-to-drain voltage, V , we will factors,multiple edge modes areexpected. Nevertheless, sd consideranappliedacfield,arisingfromelectro-magnetic it will prove useful below to study the above model for radiation illuminating the point contact. The total volt- arbitrary ν. age drop between edges is written, The current voltage characteristics of the point contact follow upon computing the backscattering current V(t)=V +V sinωt. (2.6) (2.13). Before attempting this, we consider briefly a sd ac 3 semi-classicallimitofthemodelwhichdescribesanover- particular, the phase of the Cooper pair field has long- damped driven classical pendulum. Under exchange of ranged order in the bulk superconducting electrodes, in currentwithvoltage,thisisidenticaltothestandardRSJ contrast to the power law correlations described by the model of Josephson junction dynamics12–14. This classi- 1d edge modes in (2.3). cal model has been studied intensively, both because of Sinceweareinterestedinthenon-equilibriumcurrent- it’srelevancetoJosephsonjunctions,butalsoasasimple voltage characteristics, we need a real time formulation, example of a classical dynamical system which exhibits such as Keldysh21. In the Keldysh approach a generat- mode locking and a devil’s staircase19. ing functional is introduced as a path integral sum over two paths propagating forwards and backwards in time, θ (t): III. SEMI-CLASSICAL LIMIT ± Z = Dθ Dθ e S(θ±). (3.7) + − To take the semi-classical limit we first review the Z − equivalencebetweenthequantumHallpointcontactand In terms of new fields, theCaldeira-Leggettmodel20 forthequantummechanics of a damped pendulum. To this end, it is first useful to θ(t)= 1[θ (t)+θ (t)], θ˜(t)=θ (t) θ (t) (3.8) performagaugetransformationtoeliminate V in(2.5). 2 + − + − − H Since the equations of motion for x=0 take the form, 6 the appropriate real time action is S =S0+S1 with, ν (∂t±vF∂x)φR/L =±2V(t), (3.1) S = 1 dωα (ω)θ˜(ω)2 i dtθ˙θ˜, (3.9) 0 R 2Z | | − 2πν Z this can be achieved via the transformation: ν 1 φR/L →φR/L± 2A(t). (3.2) S1 = Z dt(±iv)cos(θ∓ 2θ˜+νA). (3.10) X ± After the gauge transformation, the full Hamiltonian reads, Here we have defined αR(ω) = 2πων coth(12βω). The above gives a general quantum-mechanical formulation v = F (∂ φ )2+(∂ φ )2 ofthe model. Tocomplete the descriptionwemustiden- x R x L H 4πν tify the source-to-draincurrent operator. From (3.4) we (cid:2) (cid:3) +vδ(x)cos(φ φ +νA). (3.3) 0 R L seethatθ(x=0)=2π ρ dx,whereρ =ρ +ρ . − tot tot R L Thusθ(x=0)/2π canbRe−∞identifiedasthetotalchargeto Sincetheinteractiontermonlydependsonthedifference, theleftofthepointcontact. Thesource-to-draincurrent φ φ , it is useful to define new fields: R− L through the point contact is thus simply: ϕ=φ +φ , θ=φ φ . (3.4) R L R− L I =∂tθ(x=0,t)/2π. (3.11) Since the transformed Hamiltonian is quadratic in ϕ, An instanton in θ(t) of magnitude 2π corresponds to it can be integrated out, giving for the Euclidean La- thetransferofoneelectronthroughthepointcontact. In grangian, the classical limit these charge transfer processes occur overthebarrier,ratherthanbyquantummechanicaltun- 1 E = (∂µθ)2+vδ(x)cos(θ+νA). (3.5) neling. In the Keldysh formulation, quantum tunneling L 8πν ˜ processescorrespondtoinstantonsinθ(t)-inwhichonly Here we have set v =1 in the first term. Finally, upon the forward path tunnels, say. Thus the semi-classical F integrating out θ(x) for x = 0, we arrive at an effective limit canbe obtainedby forbidding suchprocesses. This 6 Euclidean action in terms of θ(x=0,τ): can be implemented by expanding the cosines in (3.10) for small θ˜, and retaining only the leading term 1 dω S = ω θ(ω)2+ dτ vcos(θ+νA) (3.6) E 4πν Z 2π | || | Z S =iv dtθ˜ sin(θ+νA)+O(θ˜3). (3.12) 1 Z This action can be recognized as a Caldeira-Leggett model of a damped driven quantum pendulum20. It This expansion destroys the periodicity in θ˜. The full shouldbeemphasizedthattheOhmicdampingthatchar- action can now be written acterizes the Caldeira-Leggett model can be traced to 1 the 1d Luttinger liquid behavior of the edge modes. Al- S = dω α (ω)θ˜(ω)2 R though this model has been used to describe quantum 2Z | | dynamics in Josephson junctions, it is unclear that it 1 i dtθ˜ θ˙ vsin(θ+νA) , (3.13) describes the appropriate low frequency dynamics. In − Z (cid:20)2πν − (cid:21) 4 whichcanberecognizedastheMartin-Siggia-Roseaction Underexchangeofcurrentandvoltage,(3.17)becomes foraclassicalstochasticdifferentialequation22. Uponin- equivalent to the equation which describes Josephson troducing a stochastic noise term ξ(t) the action can be junctions12–14, except for the colored stochastic noise re-expressed as term which is non-vanishing even at zero temperature. However, if we take the semi-classical limit ν 0, with S = 1 dω 1 ξ(ω)2 νv and νV(t) held fixed, the noise term drop→s out. In 2Z α (ω)| | this classical limit, the FQHE point contact is exactly R 1 dual to a Josephson junction, and should exhibit similar i dtθ˜ θ˙ vsin(θ+νA)+ξ(t) . (3.14) mode-lockingunderexchangeofcurrentandvoltage. So- − Z (cid:20)2πν − (cid:21) lutionsof(3.17)inthenoiselesslimitarewellknown23–25. For a Josephson junction they give mode-locked voltage The integration over θ˜ then gives a delta function, en- plateausatintegermultiplesofV =(h¯/2e)ω. Physically, forcing the classical equation of motion: there is a mode locking between the discrete phase slip events and the a.c. drive. For the FQHE point contact, 1 θ˙=vsin(θ+νA)+ξ(t), (3.15) the mode-locked plateaus are in the current, at integer 2πν multiplesofI =e2πω. Thediscreteprocessisanelectron tunnelling through the point contact. with stochastic noise ω 1 ξ(ω)2 = coth( βω). (3.16) After rescaling the time in (3.17) via t ωt, it is | | 2πν 2 → (cid:10) (cid:11) clear that the I-V curves are characterized by two inde- A final gauge transformation, θ θ νA brings the pendent dimensionless parameter: 2πνv/ω and νVac/ω. equation into the familiar form, → − Representative current-voltage characteristics computed numericallyfrom(3.17)inthenoiselesslimitareshownin 1 ν Figure2. As expected, the I V curvesexhibit plateaus θ˙ =νvsin(θ)+ V(t)+νξ(t). (3.17) − 2π 2π in the current which are “flat” and quantized at integer multiples of I = eω/2π. Sub-harmonic plateaus are ab- sentforthemodel(3.17),butwouldbepresentiftheperi- odicfunctionsin(θ)includedhigherharmoniccontent26. Withinclusionofstochasticnoise,oneanticipatesthat these plateaus will be rounded slightly, as shown in the 2πΙ/ω 4 I-VcurvesinFigure3,obtainedbynumericallyintegrat- ing (3.17) with colored noise. When the noise is weak, the roundingis most visible atthe edges ofthe plateaus. 3 For large enough noise the plateaus become completely smeared out. The effects of colored noise are qualita- tively similar to stochastic white noise, which has been 2 studied extensively in the past. 1 It is worth commenting here on the validity of the νV/ω semi-classicalapproximationtothefullquantumdynam- ics. Asevidentfrom(3.12),thesemi-classicalapproxima- 0 tion involves discarding all electron tunnelling events, in 0 1 2 3 4 whichθ˜changesby2π. Onecanarguefromthequadratic action(3.9)that the typicalvarianceofθ˜is proportional to ν, even when v = 0: θ˜2 νln(ω /T), with cutoff c ∼ frequency ω . This suggests that the semi-classical ex- c pansionin(3.12)mightbecomeexactintheν 0limit. → FIG. 2: Current voltage characteristic within the semi- In the absence of an ac drive this is in fact the case. Re- classical approximation, obtained from (3.17) with no cently, Fendley et. al. have obtainedexactI V curves, − stochastic noise. Both the current and voltage are plot- with no ac drive, for arbitraryodd integer ν 1. One can − tedinunitsoftheacdrivefrequency. Thebackscattering analyze these I V curves in the limit ν 0, with νv − → amplitude has been chosen to be 2πνv = ω/4 and the and νV held fixed. In this limit, the I V curves be- − ac drive amplitude is νV = 1.6ω. Notice the current come equivalent to those which follow from the classical ac plateaus at integer multiples of I = eω/2π, indicating a equationofmotion(3.17),withwhitenoisereplacingthe mode-locking to the ac drive. stochastic colored noise. 5 free fermion16. When the a.c. drive is present, an exact solution for the I-V curve is possible. When ν = 1/2 a free fermion representation is also possible18. Although the theory is not directly applicable to the FQHE edge 2πΙ/ω states for ν = 1/2, the exact solution is nevertheless il- 4 luminating, revealing plateau-like structure in the I-V curve, in contrast to ν = 1 (see below). Moreover, the general structure of the solutions in these two soluble 3 cases, leads to a natural conjecture for more general ν, discussed in Section VI. 2 A. The ν =1 solution 1 νV/ω For ν =1 the edge modes have a free fermion descrip- tion, simpler than the general bosonized representation 0 of Section II. Upon defining fermion fields for the two 0 1 2 3 4 modes, Ψ= ψ1 = 1 eiφ1 (4.1) (cid:18)ψ2 (cid:19) √a0 (cid:18)eiφ2 (cid:19) with a a short length-scale cutoff, the full bosonized 0 FIG. 3: An I V curve in the semi-classicalapproxima- Hamiltoniancanbeexpressedasaquadraticfermionthe- − tion, obtained from (3.17) with stochastic colored noise. ory: As in Figure 2, we choose 2πνv =ω/4 and νV =1.6ω. ac 1 v The colored noise has strength ν = 0.1 and the cut-off = Ψ i∂ + V(t) σ Ψ+ δ(x)Ψ σ Ψ. (4.2) † x z † x frequency for generating the noise is ωc = 60ω. Notice H − (cid:20) 2 (cid:21) ωc that the current plateaus are rounded, due to the pres- Here,wehaveputtheFermivelocityv =1,andthecut- F ence of the stochastic noise. off frequency ω 1 . The backscattered current (2.13) c ∼ a0 takes the simple form: However, with an ac drive present, it is unlikely that t(h3.e17ν).→W0 iltimhitacisderqivueivaplreenstentot tthheerseemarie-cltawssoicpalarliamlleitl IB =(1/2)Ψ† σzΨ(cid:12)xx==00+− (4.3) (cid:12) processes which allow charge to be transported across Theequationofmotionwhichf(cid:12)ollowsfromthefermion the junction. In addition to electron tunnelling “under Hamiltonian is the barrier”, the electron can absorb quanta of energy i v ∂ +∂ V(t) σ Ψ= i δ(x) σ Ψ. (4.4) from the ac drive field. Once the electron energy is high (cid:20) t x− 2 z(cid:21) − ω x c enough, it can pass over the washboard barrier. In the The I-V curve can be obtained by solving this equation, classicallimit,bothoftheseprocessesaremodified: Elec- with appropriate boundary conditions, and extracting tron tunnelling is suppressed completely, and energy is the backscattered current, I . not absorbedfrom the ac drive in discrete quanta. How- B Our solution proceeds in two steps. Away from the ever, in the ν 0 limit, while the electron tunnelling is also complete→ly suppressed (since I V2/ν 1 0 as point contact at x=0, the equation describes free prop- − ∼ → agation with a uniform time-dependent potential, V(t). ν 0), energy is still absorbed in discrete quanta from → This can be eliminated by defining a new gauge trans- the ac drive. Thus, once the ac drive is present, one an- formed fermion field, which is assumed to be incident ticipates that mode locking features obtained from the semi-classicallimit (3.17), will not serve as a good guide upon the point contact with a Fermi-Dirac distribution. UpontransformingbacktotheoriginalFermionfield,the for the full quantum model, even for very small ν. This Fermi distribution function is modified, involving a sum will be confirmed by more detailed analysis in Section V over processes involving absorption and emission of the below. a.c. field. We refer to this distribution as an “excited Fermifunction”. Atthe pointcontact(x=0),backscattering takes place, which is characterized by reflection IV. EXACT SOLUTIONS FOR ν =1,1 2 andtransmissioncoefficients(anS-matrix)whicharein- dependent of the incident distribution function. The to- Inthissectionwestudythefullquantumdynamicsfor tal backscatteredcurrent, I , is an appropriateconvolu- B two special values of ν, for which simple exact solutions tion of the S matrix with the “excited Fermi” distri- − are possible. For ν =1 the edge mode is equivalent to a bution function. 6 Consider first scattering at the point contact. The S an “excited Fermi function”. Notice that the d.c. volt- matrixrelatestheincomingfieldΨ totheoutgoingfield ageV simplycausesashiftintheenergyoftheincident sd Ψ via − electron. The a.c. drive shifts the energy by nω, corre- + sponding to absorption or emission of n quanta, with Ψ+ =Sψ , (4.5) probability cn 2. − | | Finally we can obtain the backscattered current from whereΨ (t)=Ψ(x=0±,t). Integratingthe equationof (4.3), which can be re-expressed using (4.5) solely in motion (±4.4) through the origin, x=0, gives, terms of the incident fields as IB = S12 2Ψ† σzΨ . v After Fourier transforming to energy t−hi|s be|com−es, − Ψ (t) Ψ (t)= i σ Ψ(0,t), (4.6) + x − − − ωc IB = e−i(E−E′)t S12 2 Ψ† (E′) σzΨ (E) . where Ψ(0,t)= 1[Ψ (t)+Ψ (t)]. h i −ZE,E′ | | D − − E From this one re2adil+y obtains−the S matrix: (4.13) 1 (v/2ω )2 In addition to a time-independent piece, the backscat- c S =S = − , (4.7) 11 22 1+(v/2ω )2 teredcurrentwillhaveoscillatorycontributionsatmulti- c plefrequenciesofω,asisapparentfrom(4.13). Wefocus onlyonthetimeindependentpiece,whichisfinallygiven iv/ωc by S =S = − . (4.8) 12 21 1+(v/2ω )2 c 1 I = dE S 2[fex(E) fex(E)]. (4.14) The probability for the incoming field to be scattered h Bitime 2π Z | 12| 2 − 1 fromoneedgetotheotheris S 2,whereas S 2 isthe | 12| | 11| This resulttakes a familiarform, involvinganenergyin- probabilitytobetransmittedwithoutscattering. Proba- tegral of the reflection probability, weighted by energy bility conservationdictates a unitary S-matrix,S S=1, † distribution functions. Due to the a.c. drive, however, which is satisfied here. Notice that the S-matrix is inde- thesearenotsimplyFermifunctions,butratherthe“ex- pendent of the energy of the incident carriers, a conse- cited Fermi functions” given in (4.12). quence of the assumed delta-function point scatterer. Since the reflection probability is energy independent, Outside the scattering region, the right side of (4.4) the backscattered current can be seen to be completely vanishes. Transforming to a new fermion field independent of the a.c. drive. This follows by inserting Ψ(x,t)=e2iA(t) σzΨ˜(x,t), (4.9) tohfeindtiesgtrriabtuiotniontofuelnimctiinonat,e(4t.h1e2)d,raivnedfsrheqifutienngcythωe.enSeinrgcye c 2 = 1, the backscattered current is then exactly with V(t) = ∂tA(t) as before, then eliminates the time Pequna|lnto| theresultwithoutanya.c. drivepresent. Atzero dependence. Thenewfieldsatisfiesthesimplewaveequa- temperature this gives = (1/2π)S 2V , tion: (∂t+∂x)Ψ˜ =0,whichdescribesfreefermionsatzero or for the total transmittedBcurrteimnte(using (2.|8)1)2|upsodn chemical potential. This field is assumed to be incident restoring units: with an ordinary Fermi distribution function, e2 ψ˜i(E)†ψ˜i(E′) =2πδ(E E′)f(E), (4.10) I = h|S11|2Vsd. (4.15) D E − The I-V curve is linear, with conductance given by the where f(E) = (exp(βE)+1)−1 and Ψ˜(E) denotes the transmission probability, just as without any a.c. drive. Fourier transform of Ψ˜(x=0−,t). The quantum fluctuations have completely washed out The distribution function for the original incident the current plateaus seen in the semi-classical limit of Fermion, Ψ (t), can now be obtained by relating the SectionIII.TheabsenceofstructureintheI-Vcurvecan transformΨ−(E)to Ψ˜(E) using(4.9)andthe expansion be traced to the energy independent transmission prob- (2.7). This g−ives ability. The a.c. drivechangesthe energyofthe incident electron, via absorption or emission of quanta, but since 1 ψ1−,2(E)= cnψ˜1,2(E−nω± 2Vsd), (4.11) thhaestnroanesffmecitssoionntphreonbeatbtilriatynsims ietnteedrgcyuirnrdenept.endent, this Xn It is worth mentioning that the total transmitted cur- where c is defined in (2.7). The distribution func- rent can be cast into the suggestive form: n tion for the original Fermion, < ψj−†(E)ψj−(E′) >= I(V ,V )= c 2 Idc(νV nω), (4.16) 2πδ(E−E′)fjex(E), then takes the simple form: sd ac Xn | n| sd− fex = c 2f(E nω 1V ), (4.12) where Idc(V) I(V,0) is the current in absence of 1,2 | n| − ± 2 sd a.c. drive. As ≡we shall now show, this form also holds Xn 7 when ν = 1/2, even though in that case Idc(V) shows withψ(0)=(ψ(x=0+)+ψ(x=0 ))/2,whereasψ(x,t) − non-Ohmic structure. Moreover, as discussed in Sec- satisfies, tion V, this form is also valid perturbatively in the weak backscattering limit, for general ν. i v ∂ +∂ V(t) ψ =i δ(x)(a+a ). (4.24) t x † (cid:20) − 2 (cid:21) √ω c B. the ν = 1 solution We now proceed by direct analogy with the ν = 1 2 case. Away from the point contact, the right side of (4.24) vanishes, and the time dependent potential V(t) Consider now the model (2.12) with ν = 1/2. In can be eliminated by gauge transforming to a new field. this case one can show using the commutation relations At the point contact, we compute the S-matrix, which (2.11), that the operator exp(φ φ ), that enters in 1 2 − relates the amplitude of the incoming fermion (x = 0 ) the Hamiltonian, satisfies Fermi statistics. In order to − to the outgoing fermion (x=0+). fermionize this operator, it is convenient to define new boson fields15,18: TocomputetheS-matrix,firstintegrate(4.24)through the origin (x = 0), and then eliminate the local fermion φ(x,t)=[φ1(x,t) φ2(x,t)], (4.17) term a+a† using (4.23). This gives the local equation, − Φ(x,t)=[φ (x,t)+φ (x,t)]. (4.18) 1 2 v2 When the Hamiltonian (2.12) is re-expressedin terms of ∂t(ψ+−ψ−)= ωc [ψ+† +ψ−† −ψ+−ψ−], (4.25) thesenewfields,thefieldΦdecouplesandcanbeignored. where we have defined incoming and outgoing fields The remaining Hamiltonian becomes, ψ (t) = ψ(x = 0 ,t). This can be converted to an ± = vF(∂ φ)2+vδ(x) eiφ+h.c. al±gebraic equation by Fourier transformation: x H 4π 1 (cid:2) (cid:3) v2 +2V(t)(∂xφ), (4.19) ψ+(E)−ψ−(E)= iEωc[ψ+(E)+ψ−(E) where we have set ν =1/2. −ψ+†(−E)−ψ−†(−E)]. (4.26) Sinceeiφ hasFermistatistics,wecanfermionizethere- Uponcombiningthisequationwithit’sHermitianconju- maining boson field, via Ψ = 1 eiφ, with lattice cutoff a . The firsttermdescribesaf√reae0 chiralfermionandthe gate,wecaneliminateψ+†(−E),andexpresstheoutgoing 0 field ψ (E) in terms of the incoming fields ψ (E) and + thirdtermis alsoquadraticin Ψ,howeverthe tunnelling − termislinearinΨ. Toconvertthistermintoaquadratic ψ−†(−E), form, we introduce a local fermion field a as, ψ+(E)=S++(E)ψ (E)+S+ (E)ψ†( E). (4.27) − − − − Ψ(x)=(a+a )ψ(x), (4.20) † Here the energy dependent S-matrix elements are given wherebothaandψ(x)satisfyfermionanti-commutation by relations. The full Hamiltonian then becomes, α i E S (E)= , S (E)= , (4.28) =ψ (i∂ + 1V(t))ψ+ v δ(x)[ψ (a+a )+h.c.]. ++ αE +i +− αE +i † x † † H 2 √ω c with α Eωc. As required by current conservation, (4.21) the S-mEatr≡ix s2av2tisfies S (k)2+ S (k)2 =1 . ++ + To obtain the distri|bution f|unct|ion−for |the incident Here we have set v = 1, and the cutoff frequency F fermion, we follow the procedure used for ν = 1, and ω 1/a . Tocompletethefermionization,were-express c ∼ 0 define a new fermion field which eliminates the time de- the backscattering current from (2.13) in terms of the pendent potential in (4.24): fermion fields: IB = 21ψ†ψ(cid:12)xx==00+− (4.22) with V = ∂ Aψ(.x,At)ft=ereF2iAou(tr)ieψ˜r(xtr,at)n.sformation th(is4.2b9e)- (cid:12) t SincetheHamiltonian(4.21)isquadratic,itcanberead- comes, ily solved, and the current computed, as we now show. To this end, consider first the equations of motion for ψ (E)= c ψ˜ (E nω+ Vsd). (4.30) the fermion fields which follow from the Hamiltonian. − n − − 2 Xn The local fermion satisfies v Since the new field, ψ˜, satisfies the free wave equation, ∂ (a+a )=2i [ψ(0) ψ (0)], (4.23) t † √ωc − † (∂t +∂x)ψ˜ = 0, for x < 0, we assume again that it is 8 incident upon the point contact with a Fermi distribu- Noticethatincontrasttothe caseν =1,the S-matrix tionfunction,f(E)=(exp(βE)+1) 1. Thedistribution heredependsontheenergyoftheincidentfermion. Asa − function for the original fermion, < ψ†(E)ψ (E′) >= result, the I-V curve is non-Ohmic. The differential con- 2πδ(E E )fex(E), is thus given again−by th−e “excited ductance at zero temperature as obtained from (4.35) ′ fermi fu−nction”: in the absence of the a.c. drive is plotted in Figure 4. At small bias, the current vanishes with the cube of the V fex(E)= cn 2f(k nω+ sd). (4.31) source-drainvoltage. This is consistent with the general | | − 2 Xn result, I V2/ν−1, obtained from perturbation theory ∼ sd Finally, the backscattered current averaged over time in the strong backscattering limit. For large bias the I- follows from (4.22) as, V curve is linear with an offset voltage. Again, this is consistent with the general perturbative result for small 1 dE hIBitime = 2Z 2π Dψ+†(E)ψ+(E)−ψ−†(E)ψ−(E)E. bfoarckaslclaνtte<rin1g/;2(,(tνh/e2πI)-VVsdcu−rvIe)∼atvl2aVrgs2deν−v1o.ltNagoeticise tthhuast (4.32) expected to asymptote to I =(ν/2π)V , with no offset. sd After re-expressing the outgoing waves in terms of in- With the a.c. drive present, the I-V curve can be ob- coming, using the S-matrix (4.28), the averagesover the tained by summing in (4.34), with weighting c 2 = n incident distribution can be performed, giving J2(Vac). Since the I-V curve with no a.c. drive| is|non- n 2ω Ohmicatsmallbias,thiswillgivefeaturesinthefullI-V dE 1 <IB >time= S+ (E)2( fex(E)). (4.33) curve which resemble smeared current plateaus. These Z 2π | − | 2 − plateauscanbe morereadilyrevealedby plottingdI/dV The total transmitted current (2.8) can once again be versus V . sd cast into the form: Anticipating the analysis of the I-V curves for general I(Vsd,Vac)= cn 2Idc(νVsd+nω), (4.34) ν in the next section, it is convenient at this stage to | | Xn define an effective backscattering energy or temperature with ν = 1/2. Here the current in the absence of a.c. scale. Following Fendley et. al., we define a backscat- drive, Idc(Vsd) I(Vsd,0), is given by teringtemperatureTB =g(ν)ωc(v/ωc)1/(1−ν),wherethe ≡ function g(x) is, 1 dE Idc(V )= V S (E)2 sd sd + 4π −Z 2π(cid:26)| − | 1 1 4√π 1 Γ( 1 ) (2 −f(E+ 2Vsd)). (4.35) g(x)= x x2−12x(x −1)12Γ(2−x2x). (4.36) 2 2x − In the ν 0 limit, one has T = 2πv, which is the B → appropriate backscattering energy scale entering in the semi-classical equations of motion (3.17). For ν = 1/2, 1.5 (2π/ν)dI/dV TB = 4v2/ωc which is the energy scale that enters in (4.28). TheI V curvesatT =0arethencharacterized − by two dimensionless parameters, νV˜ = νV /ω and ac ac νT˜ =νT /ω. B B 1.0 In Figure 5 we plot the differential conductance versus voltage, obtained from (4.34) with νV˜ = 1.6 and ac νT˜ = 1/4. Notice the minima, which correspond to B 0.5 smeared plateaus in the I V curves. The differential conductance at the nth min−ima is 1 J2(V /2ω). The − n ac νV widths of the minima depend primarily on the backscattering energy scale, T , becoming narrower as T de- B B 0.0 creases. In contrast to the semi-classical approximation −4 −2 0 2 4 (3.17),the“plateaus”herehaveanon-vanishingdifferen- tialconductance everywhere- they arenot“flat”. More- over, the I-V curve is everywhere analytic, even at the “plateau” centers, since the I-V curve without a.c. drive is analytic even at zero bias. Evidently, quantum fluctu- FIG. 4: Differential conductance with no ac drive at ations are quite effective at smearing the semi-classical ν =1/2. We choose νT =1 in this plot. current plateaus, even for ν =1/2. B 9 It is also instructive to plot the differential conductance, dI/dV, versus the current, as shown in Figure 6. This shows clearly that the smeared “plateaus” are not centered at the quantized current values I = neω/2π. (2π/ν)dI/dV 1.5 This is due in part to the finite offset voltage at large bias, mentioned above. However, if we choose a smaller backscattering strength, T 0+, these “plateaus” B → become centered at the quantized current values, I = 1.0 neω/2π, as shown in Figure 6. We next consider the I-V curve for general ν. 0.5 V. GENERAL ν νV/ω In the previous section we showedthat for both ν =1 0.0 0 1 2 3 4 and ν = 1/2, the I-V curve with a.c. drive, could be re- latedtothe I-Vcurveinthe absenceofanya.c. drive,as a weighted sum over absorption and emission of quanta, see(4.34). Herewemaketheconjecturethatthisrelation holds in general, for arbitrary ν. If correct, this conjec- tureallowsustousetherecentresultsofFendleyet. al.15 FIG.5: Differentialconductancewithacdriveatν =1/2 to extractI-Vcurveswith a.c. drive forarbitrary integer obtained from (4.34), plotted versus voltage. We have ν 1. Before doing so, we show that the conjecture does − put νTB = ω/4, νVac = 1.6ω. The minima correspond hold for general ν in the limit of weak backscattering. to smeared plateaus in the I-V curve, centered around To this end, consider calculating the I-V curve with νVsd =nω. a.c. drive present, as a perturbation expansion in pow- ers of the backscattering amplitude v. The leading non- vanishing correction appears at order v2. This correction can be readily obtained from the Keldysh action discussed in Sec III. The back scattered current opera- torfollowsfrom(3.6)uponfunctionaldifferentiationwith respect to the gauge field A(t), (2π/ν)dI/dV 1.5 Iˆ =νvsin(θ+νA). (5.1) B 1.0 Within the Keldysh approach, the average over this operator can be performed by putting it into the forward path, so that 1.0 <IB >=<sin(θ+(t)+νA)>, (5.2) where the average is taken with respect to the generat- 2πΙ/ω ing functional (3.7). Upon expanding the exponential exp( S ) to second order in v, one obtains 0.5 00 11 22 33 44 − 1 i I = νv2 dt B ′ −2 Z hsin[θ+(t)−θ−(t′)+ν(A(t)−A(t′))]i0, (5.3) wherethe subscript0denotes anaveragewithrespectto FIG. 6: Differiential conductance with ac drive at ν = the quadratic action S in (3.9). Keeping only the con- 0 1/2 at two different back scattering strength. In the stant time-independent piece, it is then straightforward lower part of the figure, we plot the same differential to show that, conductance as in Figure 5 but versus current. As one can see that the smeared plateaus are not centered at = c 2 Idc(νV +nω), (5.4) I = neω/2π due to the finite offset. In the upper part, B time | n| B sd Xn we put νT = ω/100 and νV = 1.6ω. In this weak B ac backscattering limit, the “plateaus” are centered at the whereIdcisthebackscatteredcurrenttoorderv2without B quantized values, I =neω/2π. the a.c. drive. Since the total current is I = νV I , sd B − 10