Theory of the Fabry-Perot Quantum Hall Interferometer
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Theory of the Fabry-Perot quantum Hall interferometer
Bertrand I. Halperin, Ady Stern, Izhar Neder, and Bernd Rosenow Physics Department, Harvard University, Cambridge MA 02138, USA Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel Institute for Theoretical Physics, Leipzig University, D-04009 Leipzig, Germany (Dated: September 4, 2018)We analyze interference phenomena in the quantum-Hall analog of the Fabry-Perot interferometer,exploring the roles of the Aharonov-Bohm effect, Coulomb interactions, and fractional statistics onthe oscillations of the resistance as one varies the magnetic field B and/or the voltage V G applied to aside gate. Coulomb interactions couple the interfering edge mode to localized quasiparticle states inthe bulk, whose occupation is quantized in integer values. For the integer quantum Hall effect, if thebulk-edge coupling is absent, the resistance exhibits an Aharonov-Bohm (AB) periodicity, where thephase is equal to the number of quanta of magnetic flux enclosed by a specified interferometer area.When bulk-edge coupling is present, the actual area of the interferometer oscillates as function of B and V G , with a combination of a smooth variation and abrupt jumps due to changes in the numberof quasi-particles in the bulk of the interferometer. This modulates the Aharonov-Bohm phase andgives rise to additional periodicities in the resistance. In the limit of strong interactions, the am-plitude of the AB oscillations becomes negligible, and one sees only the new “Coulomb-dominated”(CD) periodicity. In the limits where either the AB or the CD periodicities dominate, a color map ofresistance will show a series of parallel stripes in the B − V G plane, but the two cases show differentstripe spacings and slopes of opposite signs. At intermediate coupling, one sees a superposition ofthe two patterns. We discuss dependences of the interference intensities on parameters including thetemperature and the backscattering strengths of the individual constrictions. We also discuss howresults are modified in a fractional quantized Hall system, and the extent to which the interferometermay demonstrate the fractional statistics of the quasiparticles. PACS numbers: 73.43.Cd, 73.43.Jn, 85.35.Ds, 73.23.Hk
I. INTRODUCTIONA. Background
In the last few years there has been a surge of interestin electronic interference phenomena in the regime of thequantum Hall effect. This interest, both theoretical and experimental , results in large part from the hopeof utilizing interference to probe unconventional statisticsin various fractional quantum Hall states. Interestingly,interferometer experiments have led to puzzling resultseven in the integer regime, which have posed a challengeto our theoretical understanding.Arguably the simplest realization of a quantum Hallinterferometer is an analog to the optical Fabry-Perotdevice. It is constructed of a Hall bar perturbed by twoconstrictions, each of which introduces an amplitude forinter-edge scattering. (See Figure 1) The backscatter-ing probability of a wave packet that goes through theconstrictions is then determined by an interference of tra-jectories. In the limit of weak inter-edge scattering, twotrajectories interfere, corresponding to scattering acrosseach of the two constrictions. As the scattering ampli-tudes get larger, multiple reflections play a more signifi-cant role.In our analysis, we assume that the two constrictionsforming the interferometer are identical to each other,and that there is a single partially-transmitted edge chan-nel penetrating the two constrictions. This partially
FIG. 1: Fabry-Perot interferometer with f T = 1 totally trans-mitted edge modes. Filling factor at the center of the con-striction is in the range 1 < ν c < ν out = 1 and ν in = 2. A third edge mode is totallyreflected before entering the constrictions, as the bulk fillingfactor in the center of the interferometer lies in the range2 . < ν b < . transmitted channel separates two quantized Hall statescorresponding to rational filling factors ν in > ν out , with ν out being closer to the sample edge. In addition to theinterfering channel, there may be a number of outer edgechannels that are fully transmitted through the two con-strictions, whose number we denote by f T ≥
0. The situ-ations considered in this paper assume that the states ν in and ν out are either integer states or integers plus a frac-tion described in the composite fermion picture, wherethe partially filled Landau level is less than half full. Inparticular, this means that all edge states propagate inthe same direction. We refer to the cases where ν in isinteger or fractional as IQHE and FQHE interferometer,respectively.For non-interacting electrons, there will be an interfer-ence between electrons backscattered at the two constric-tions, with a relative phase determined by the Aharonov-Bohm effect. It is periodic in the magnetic flux Φ en-closed by the loop defined by the two interfering trajec-tories, with a period of one flux quantum Φ (we definethe flux quantum as Φ = h/ | e | >
0, where e < B theflux is Φ = BA I , with A I being the area of the inter-ference loop. Experimentally, it is customary to affectthis flux through two experimental knobs: B the mag-netic field, and V G , the voltage on a gate that affects thearea of the loop. The gate may be positioned above theinterference loop or to its side. For fractional quantumHall states, where electron-electron interaction is essen-tial, the relative phase is made of two contributions, anAharonov-Bohm phase that is scaled down by the chargeof the interfering quasi-particle, and an anyonic phase,accumulated when one quasi-particle encircles another.Experimentally, several remarkable observations weremade when interference was measured insmall Fabry-Perot interferometers, e.g. with an inter-ference loop whose area is around 5 µm . One obser-vation was that when the magnetic field is varied, thebackscattering current oscillates as a function of the mag-netic field, but the period ∆ B of the oscillations was notΦ /A I . Rather, it was given by Φ /f T A I , which means,in particular, that there was no dependence on B for f T = 0. The period ∆ B did not change when ν c , the fill-ing factor at the center of the constriction was varied inthe range between f T and f T +1, and the back scatteringprobability for the partially transmitted edge state variedfrom strong to weak. Second, when the lines of constantphase in the B − V G plane were examined , they werefound to have positive slope, which is opposite sign rela-tive to what one would naively expect for an Aharonov-Bohm interference effect (Similar lines were observed alsoin Ref. [20], where a scanning probe was used to probethe spectrum of excitations of a spontaneously formedquantum dot). By contrast, in interferometers that aresufficiently large (e.g., area around 17 µm ), where thecenter island is covered by a screening top gate, the con-ventional Aharonov-Bohm pattern was observed, withfield period Φ /A I and negative slope for the lines ofconstant phase. A similar Aharonov-Bohm behavior wasalso observed in some small interferometers .Previous works have explained that the periodicitiesand slopes in the Fabry-Perot interferometer are affectedby the Coulomb interactions and the discreteness of elec-tronic charges . The regime of parameters wherelines of constant phase have positive slope (or zero slope in the case f T = 0) will be referred to as the Coulomb-Dominated (CD) regime, in contrast with the Aharonov-Bohm (AB) regime.In this article, we present a general picture of the in-terplay of the AB and CD regimes in the Fabry-Perotinterferometer, and elucidate the way this interplay is de-termined by the combination of Coulomb interaction andcharge discreteness. We limit our analysis of the FQHEto abelian states. We hope to extend our present studyto the case of non-abelian states in a future publication. B. Summary of our results
Before we turn into a detailed discussion, we summa-rize our results and present a physical way of understand-ing them. Generally, when electron-electron interactionsare taken into account, we find that the area A I enclosedby the interfering edge state is not a smooth monotonicfunction of the magnetic field and gate voltage. Rather,we find that A I has the form A I = ¯ A ( B, V G ) + δA I , (1)where ¯ A is a slowly varying function of its arguments,while δA I has rapid oscillations, on the scale of one fluxquantum or on a scale of a change in V G that adds oneelectron. (We assume that the area A I is large enoughto enclose many electrons and flux quanta, so that theoscillations occur on a scale where there is only a smallfractional change in B or ¯ A . We shall also assume, unlessotherwise stated, that the secular area ¯ A is only weaklydependent on the magnetic field B , i.e., that B∂ ¯ A/∂B isnegligible compared to ¯ A .) The oscillatory dependenceof δA I on the magnetic field and V G can have strikingconsequences on the interference pattern, as we shall seebelow.Typically, experiments measure the “diagonal resis-tance” R D [23], which is essentially the two-terminal Hallresistance of the interferometer region. We find that R D has an oscillatory part δR , which is a periodic functionof B and δV G . In the limit of weak backscattering it maybe written as δR = Re ∞ X m = −∞ R m e πi ( mφ + α m δV G ) ! , (2)where φ ≡ B ¯ A/ Φ , (3)and the coefficients R m , α m are real and only slowly vary-ing functions of B, V G . The voltage V G affects the phases e πi ( mφ + α m δV G ) in (2) in two ways. First, it affects theflux φ through its effect on the area ¯ A . Second, it affectsthe density in the bulk of the interferometer, indirectlyaffecting the interference through interactions of the edgewith the bulk. The coefficients α m quantify the latter ef-fect, which we will analyze further below. FIG. 2: < δR > = Re (cid:0) R e πiφ + R − f T e − πif T φ (cid:1) as a color map in the plane of B and V G , for f T = 2, with the parameter γ chosen equal to 3 . β . Panels (a), (b), (c), (d), and (e) , have, respectively | R − /R | =0, 0.5, 1, 2, and ∞ , corresponding to tothe AB, mixed and CD regimes. All Fourier components other than m = 1 and m = − f T are neglected. Alternating red andblue regions represent positive and negative values, respectively, while white signifies a value close to zero. For non-interacting electrons (the extreme AB limit),the weak backscattering limit has only one non-zerocomponent in Eq. (2). That component is m = 1,with α = 0 . Since the area ¯ A should be a monotoni-cally increasing function of V G , we find that for smallchanges in B and V G the contours of of constant phaseare straight lines of negative slope in the B − V G plane.(When backscattering becomes stronger, multiple reflec-tions lead to more harmonics of m showing up, but still α m = 0, so the slope does not change.) When plottedas a color-scale map in a B − V G plane, the resistance R D forms a set of parallel lines, such as the dominantfeatures seen in Fig. [2a].Electron-electron interactions lead to two importantdifferences between the quantum Hall interferometer anda naive Aharonov-Bohm interference experiment. First,as mentioned above, the area A I of the interference loopis not rigidly constrained a priori, but can fluctuateslightly. Thus, the area of the interference loop varieswith magnetic field and the flux within the loop is gen-erally not a simple linear function of the magnetic field.The position of the edge is related to the charge it en-closes, and its variation in our model is a consequenceof considerations of energy. Second, we model the regionenclosed by the interference loop as one in which thereare localized states close to the chemical potential. Thenumber N L of electrons (in the IQHE regime) or quasi-particles (in the FQHE regime) that are localized in thebulk is an integer, and varies discretely. Due to consid-erations of energy, an abrupt change of occupation of alocalized state as the magnetic field is varied affects alsothe position of the interfering edge, and hence inducesan abrupt change in the flux enclosed by the interferenceloop.Thus, as B or V G vary, the phase accumulated by theinterfering particle, θ , evolves in two ways: continuousevolution for as long as N L does not vary, and abruptjumps for magnetic fields at which N L abruptly changes.The continuous change results from the variation of the magnetic flux in the interference loop, both directly as aconsequence of the varying B , and indirectly as a conse-quence of the variation of the loop’s area A I . The abruptchange results from the effect of a variation of N L on thearea of the interference loop, and, in the FQHE, fromthe anyonic phase accumulated when fractionally chargedquasi-particles encircle one another. Specifically, in theinteger case, θ is simply related to the field B and thearea A I by θ = 2 πBA I / Φ , (4)while, for the FQHE states that we consider, we have θ = 2 πe ∗ in BA I Φ + N L θ a , (5)where θ a is the phase accumulated when one elementaryquasi-particle of charge of the inner FQHM state ν in en-circles another, and e ∗ in is the charge of the quasiparticle.Here, and in the following, charge is to be measured inunits of the (negative) electron charge e .Within our model, both the rate of continuous evolu-tion of the phase, dθ/dφ , and the size 2 π ∆ of the phasejump associated with a change of N L by -1, vary onlyslowly with B and V G . The same holds for the magneticfield spacings between consecutive changes in N L .In the extreme Coulomb dominated regime, for integerand fractional states alike, we find that a change of N L isaccompanied by a change of the area of the interferenceloop in such a way that the phase jump ∆ θ is an un-observable integer multiple of 2 π . Coulomb interactionmakes the area vary in such a way that the continuousvariation of the phase follows dθdφ = − π ν out e ∗ out , where e ∗ out is the elementary charge of the outer ν out quantized Hallstate. Neglecting the unobservable phase jumps, then, θ = − π ν out e ∗ out φ for both the IQHE and the FQHE. Thislimit characterizes interferometers where the capacitivecoupling of the bulk and the edge is strong. By contrast,in the extreme Aharonov-Bohm case, where the bulk andthe interfering edge are not coupled, the area of the inter-ference loop does not vary with B at all. Moreover, A I does not vary when N L varies. Thus, for integer states θ = 2 πφ . The fractional case is more complicated due tothe anyonic phase θ a .In between these two extremes, θ is not proportionalto φ , and thus the Fourier transform of e iθ with respectto φ has more than one component. For fractional statesthis is the case even in the extreme AB limit, due tothe anyonic phase θ a . We find that for all the cases weconsider, the components that appear in Eq. (2) satisfy m = − ν out e ∗ out + g ν in e ∗ in , (6)where g is an integer. Note that the ratios ν out /e ∗ out and ν in /e ∗ in are always integers, so the allowed values of m are integers as well. Moreover, due to the interaction, α m is not proportional to m , leading to different slopesof the equal phase lines for the different m components.The Coulomb dominated limit and the Aharonov-Bohm limit are both defined in terms of the dominantvalues of g in (6). In the extreme CD limit the only termthat appears in the sum (2) is that of g = 0 in (6), bothfor integer and fractional states. In the extreme AB limitof integer states the only term that appears in (2) is thenaive Aharonov-Bohm term m = 1 (or g = 1 in (6)).For fractional states, however, there will be coupling dueto the phase jumps associated with the anyonic statis-tics of the quasi-particles, and one would not find pureAB behavior, (only g = 1), even when the Coulomb cou-pling between N L and A I can be neglected. Moreover,for FQHE states with ν in >
1, one finds that there is novalue of g that generates m = 1 in Eq. (6), so the naiveAB period is completely absent in the weak backscatter-ing limit.In between the extreme Aharonov-Bohm andCoulomb-dominated limits, all integers g appear in theFourier decomposition of δR , with the relative domi-nance of the AB and CD components being determinedby the value of ∆. We find, under plausible assumptions,that 0 < ∆ <
1, and that if 0 ≤ ∆ < / / < ∆ ≤ . When the sum (2) is dominated by one term, as is thecase in the CD limit and the AB limit of the IQHE, thecolor-scale plot of δR on the B − V G plane is characterizedby a set of parallel lines, as is the case in Figs. [2a] and[2e].The three figures [2b]-[2d] show the intermediate case,in which several values of m contribute, and α m is notproportional to m . Then the structure of R D in the B − V G plane assumes a form of a two-dimensional lat-tice, rather than a set of lines, as it would if α m stayedproportional to m . The periodic structure may be char-acterized by a unit cell in the B − V G plane, describedby two elementary lattice vectors b and v . In the mostgeneral case these vectors can have two arbitrary orien-tations in the plane. However, if the secular area ¯ A is only weakly dependent on the magnetic field B , that is if B∂ ¯ A/∂B ≪ ¯ A , we find that one of the elementary latticevectors will be parallel to the B axis. Specifically, if V G isheld constant, δR will be unchanged when B is changedby the amount that increases φ by one. (We emphasizethat this is true even if the interfering particles are frac-tionally charged.) In our later discussions, rather thanemploying the direct lattice vectors b and v , we shall usea description in terms of their reciprocal lattice vectors.The restriction of the Fourier harmonics to the val-ues (6) is valid only in the limit of weak backscattering.As the constrictions are further closed and the ampli-tude for backscattering becomes appreciable, all valuesof m appear in (2). In the limit where this amplitude isstrong, oscillations in the reflection probability turn intotransmission resonances. The spacing between these res-onances varies with the degree of coupling between thebulk and the edge. Generally, a transmission resonanceoccurs when the almost closed interfering edge has a de-generacy point, at which it may accommodate an extraelectron (for the IQHE) or quasi-particle (for the FQHE)at no extra energy cost. In the Aharonov-Bohm limit, itis the energy of the edge, decoupled from the localizedcharges it encloses, that should be invariant to adding anextra charge carrier. At the integer quantum Hall regime,that would give rise to one transmission resonance perevery flux quantum. In the Coulomb Dominated limit,when the introduction of localized charges affects the en-ergy of the edge through their mutual coupling, therewould be ν out /e ∗ out resonances per quantum of flux, inboth the IQHE and the FQHE. Thus, the distinction be-tween the AB and CD limits holds even in the limit of aclosed interferometer, where the interfering edge almostbecomes a quantum dot.As should be clear from the discussion above, the formof δR depends crucially on the continuous and abruptphase variations dθ/dφ and ∆ θ . Both of these quantitiesdepend on energy considerations, since the interferome-ter’s area is a property of thermal equilibrium. We modelthe energy of the interferometer in terms of a capacitornetwork. The parameters of the model, describing theself capacitance of the interfering edge, the self capaci-tance of the localized quasi-particles, the mutual capaci-tance of the two, and the capacitive coupling of the gateto the interferometer, depend on microscopic parameterswhich we cannot accurately calculate at this point. How-ever, we are able to give some insights into the way inwhich various parameters should vary with details of thesystems, including particularly the perimeter and area ofthe interference loop. C. The structure of the paper
The structure of the paper is as follows. In Sec. (II)we deal with the weak backscattering limit. We identifywhat we believe to be the important degrees of freedom inthe interferometer, express the phase θ in terms of these symbol short description section ν in ( ν out ) filling factor inside (outside) the interfer-ing edge state I A f T number of fully transmitted edge states I A B magnetic field I A∆ B magnetic field periodicity I A V G voltage applied to a gate I A A I area of the interference loop I A¯ A slowly varying part of A I I B δA I rapidly oscillating part of A I I B R D diagonal resistance I B δR oscillatory part of R D I B φ magnetic flux within the area ¯ A I B α m quantifies the effect of V G on the bulk ofthe interferometer loop I B N L number of electrons or quasi-particles lo-calized in the bulk of the interferometer I B θ the interference phase I B e ∗ in ( e ∗ out ) the charge of a quasi-particle in the ν in ( ν out ) state I B2 π ∆ jump in phase θ when N L varies by − θ a anyonic phase I B r , r reflection amplitudes at constrictions 1,2 II N ej , N hj integer number of localized electrons andholes in the j ’th Landau level II K I , K IL , K L coupling constants in the energy func-tional describing the interferometer II A¯ q effective bulk background charge II A β quantifies the effect of V G on the area ofthe interferometer II A γ quantifies the effect of V G on the bulkbackground charge II A∆ ν ν in − ν out II B C I , C L , C IL re-parametrization of K I , K L , K IL by ef-fective capacitances II C µ I , µ L electro-chemical potentials of the I and L regions II C w, L width and length of the region of non-uniform density near the loop’s edge II C Z partition function III B ~G gh reciprocal lattice vectors of the 2D de-scription of δR ( B, V G ) III E λ describes the variation of ¯ A with B III E η describes the variation of ¯ q with B III E P R reflection probability for the interferingedge state IV χ ± interferometer scattering phase shifts IV ρ ( ǫ ) density of states IV∆ φ flux spacing between resonances V N o total number of electrons in the highestLandau level enclosed by the interferingedge channels VTABLE I: List of symbols, their brief description, and thesection where they are defined degrees of freedom and introduce an energy functionalin terms of these degrees of freedom. In Sec. (III) wecalculate the thermal average of e iθ , which is the factorthat determines the interference contribution to R D inthe weak backscattering limit, and distinguish betweenthe Aharonov-Bohm and Coulomb-Dominated limits. InSec (IV) we extend the discussion to the regime of inter-mediate backscattering, and in Sec. (V) to the regime ofstrong backscattering. In Sec. (VI) we exemplify the wayin which the energy parameters for the interfering edgeand the localized states can be influenced by coupling toedge states that are fully transmitted, by solving in de-tail two simple models. In Sec. (VII) we compare ourfindings to earlier experimental and theoretical works.Finally, we summarize our results in Sec. (VIII).For the convenience of the reader, we include a tablewith a list of the main symbols used in the paper, theirbrief description, and a pointer to the Section in whichthey are defined. II. THE PHYSICAL MODEL - WEAKBACKSCATTERING CASE
In this section we introduce the physical model onwhich we base our analysis of the weak backscatteringlimit. We start with the IQHE interferometers, and thengeneralize to the FQHE ones.In the weak backscattering limit there should be anoscillatory part of the backscattered resistance given by δR ∝ Re[ r r ∗ < e iθ > ] (7)where r , r are the reflection amplitudes at the two con-strictions, and the angular brackets represent an averageover thermal fluctuations. We focus here on measure-ments in the limit of small source-drain bias, so we mayconsider all leads to be at the same electrochemical po-tential µ . We assume the change in B and V G to be smallenough that we may neglect any changes in r and r , andassociate oscillations in the resistance with oscillations inthe phase factor < e iθ > .Our analysis of the phase factor e iθ is based on thefollowing picture of the edge of a quantum Hall fluid inthe integer regime. We expect that any Landau level j which is more than half filled in the bulk of the systemwill have a single chiral edge state that circulates alongthe edge of the system. To the extent that the electrondensity varies smoothly near the edge of the sample, onthe scale of the magnetic length, we expect that the spa-tial location of the edge state will be close to the pointwhere the Landau level is half-full. In typical situations,we will not find that electronic states in the Landau levelare entirely empty at positions outside the edge state orentirely full inside the edge state. Rather, the Landaulevel will have a certain number of electrons N ej in local-ized states outside the edge state, and a certain number N hj of unoccupied localized states (holes) inside the edgestate. The quantities N hj and N ej are constrained to beintegers, as they represent the occupations of localizedstates.In our analysis we will neglect the electrons and holeslocalized between edge states, and consider only thosethat are localized in the bulk of the sample, where thefilling factor is ν in . This neglect is justified below, to-wards the end of this section. We shall also assume thatthe electrons that are localized in the ν in bulk regionare weakly conducting and compressible over long timescales, so that we can view them as forming a metal-lic region of a uniform electro-chemical potential, whosenumber of electrons is quantized to an integer N L . Ex-perimental support for this picture was found in [20].Both of these assumptions are further elaborated on to-wards the end of this section.Within this model, then, the interferometer has a sin-gle discrete degree of freedom, N L , and several continu-ous degrees of freedom A j , describing the area (relativeto a reference area) occupied by each of the edges thatare coupled to the leads (the subscript numbers the edgestate). As is always the case in the quantum Hall effect,charge density on the edge translates to an area enclosedby the Landau level. The phase θ is directly related tothe area A I enclosed by the interfering edge state, as de-limited by the points in the constrictions where there istunneling between the partially transmitted edge states.(The subscript I stands for “interfering”). Specifically,the relation is given in Eq. (4), above. Alternatively, wecan consider θ as a measurable quantity (mod 2 π ), anduse (4), to form a precise definition of A I . A. Macroscopic energy function
We will now formulate the way by which we will cal-culate (7) and its dependence on B and V G . Since thephase θ depends only on what happens in the ν in bulkregion, we find it useful to define an energy functional E ( N L , A I ) as the total energy of the system when N L and A I are specified, and the energy is minimized withrespect to all other variables, including the fluctuatingareas A j of any fully transmitted edge states. (The elec-trochemical potential µ of the leads is here taken to bezero.).Let us consider small variations of B about a giveninitial value B , at a fixed value of the gate voltage V G .For small variations in N L , A I , we may then expand theenergy E ( N L , A I ) to quadratic order, and write E = K I δn I ) + K L δn L ) + K IL δn I δn L , (8)where δn L is the deviation of the number of localizedelectrons from the value that would minimize the energyif there were no integer constraint on N L , and δn I is thedeviation of the charge on the interfering edge, in unitsof the electron charge, from the charge that would then minimize the energy. More precisely, δn L = N L + ν in φ − ¯ q (9)where ¯ q is the effective positive background charge, inunits of | e | , resulting from ionized impurities in thedonor layer and additional charges on the surfaces andon metallic gates, as well as any fixed charges in local-ized states outside the interference loop. We assume that¯ q depends monotonically on the gate voltage. Further-more, for weak backscattering, δn I = B ( A I − ¯ A ) / Φ = n I − φ, (10)where n I is the charge enclosed by the interfering edgestate, ignoring the charges of the localized electrons andholes.When the gate voltage V G is varied with B remainingfixed, the background charge ¯ q and the area ¯ A will vary.Their variation depends on the coupling of the gate to theinterferometer, and we characterize it by two parameters: β = ( B/ Φ ) d ¯ A/dV G , γ = d ¯ q/dV G . (11)The parameter β describes the extent to which a varia-tion of the gate voltage affects the area of the interfer-ometer A I (and indirectly φ ), while γ describes the waythe gate affects the background charge in the bulk of theinterferometer (and indirectly N L ).Note that the energy function (8) leads to an inter-ference phase that is unchanged when φ varies by one.This change in φ can be completely compensated in theenergy function by changing N L by the integer amount − ν in , while n I changes by one. The fixed value of δn I means that the area A I has not changed, but the phase θ has changed by 2 π . Such a phase change has no effecton the value of e iθ . B. Fractional quantized Hall states
Our considerations for the integer case can be easilyextended to fractional quantized Hall states of the form ν in = I + p ps + 1 , ν out = I + p − s ( p −
1) + 1 , (12)where p and s are positive integers, and I ≥ I ≤ ν < I + 1 /
2, and we assume that they are correctlydescribed by the standard composite fermion picture.Moreover, we assume that the backscattered excitation isthe elementary quasiparticle of the state ν in , with charge e ∗ in = 1 / (2 sp + 1) . We again use a quadratic energy func-tion of the form (8), but now we have to modify (9) and(10) and use (5) instead of (4) to describe the relationsbetween δn I , A I , N L and θ .Specifically, the phase θ accumulated by an interferingquasi-particle is θ π = e ∗ in BA I / Φ − N L se ∗ in . (13)The first term is the Aharonov-Bohm phase, scaled downby the charge of the interfering quasi-particle, and thesecond term is the anyonic phase accumulated when onecomposite fermion goes around another. The statis-tical phase θ a , which appeared in Eq. (5), is thus givenby θ a = − πse ∗ in .An increase of the magnetic flux by one flux quantumintroduces, on average, ν in /e ∗ in quasi-particles, hencemodifying (9) to be δn L = e ∗ in N L + φν in − ¯ q (14)Here N L is the net number of quasiparticles minus quasi-holes, of charge e ∗ in , inside the interfering edge state.The relation between the area enclosed by the interfer-ing edge and the charge contained in the correspondingcomposite fermion Landau level – the modified version of(10) – is, δn I = ∆ ν B ( A I − ¯ A ) / Φ , (15)where ∆ ν ≡ ν in − ν out . The normalizations of δn I and δn L have been chosen so that they are measured in unitsof the electron charge.As before, in the limit of weak back scattering, theresistance oscillation will be proportional to Re < e iθ > .Note that formulas for the fractional case reduce to thoseof the integer case if one sets s = 0. C. Comments on the energy function
The previous subsection has defined the model we willuse for analyzing the interference term (7) and its depen-dence on B and V G . Before carrying out this calculation,we pause to make some comments on the model.
1. An alternative parametrization of the energy function
The macroscopic energy function E may be alterna-tively described by an equivalent capacitor network. Ifwe introduce electrochemical potentials µ I = ∂E/∂ ( δn I ),and µ L = ∂E/∂ ( δn L ), then the quadratic part of E maybe rewritten as e E = C I µ I + C L µ L + C IL µ I − µ L ) , (16)where K I = e C L + C IL D , K L = e C I + C IL D , K IL = e C IL D ,D = ( C L + C IL )( C I + C IL ) − C IL . (17)The coefficients C L and C I may be interpreted as effec-tive capacitances to ground for the respective conductors,while C IL plays the role of a cross-capacitance. The ef-fective capacitances result from a combination of classicalelectrostatics and quantum mechanical energies. An advantage of rewriting the energy in this form isthat it may be easier to understand the dependenciesof the capacitance coefficients on the parameters of thesystem. For example, we would expect the coefficients C I and C IL to be proportional to the perimeter L of theinterferometer, if the structure of the edge is held fixed.The capacitance C L should be proportional to the area¯ A of the island, if the center region is covered by a topgate with a fixed set-back distance. On the other hand,we would expect C L to vary as L log L , if there is no topgate and the nearest conductors are gates along the edgesof the sample.In the situation where the edge state is connected toleads in equilibrium at zero voltage, the equilibrium valueof µ I will be zero. Then the ground state energy will begiven by E = ( C L + C IL ) µ L / e , and we have e δn L = µ L ( C L + C IL ) and e δn I = − µ L C IL .
2. Further justification for the model
The major simplification involved in our model is thereduction of the number of degrees of freedom in theproblem. In principle, the interferometer has edge statesthat form one-dimensional compressible stripes and a setof localized states between these stripes that may be ei-ther empty or full. Our model reduces the problem totwo degrees of freedom, A I and N L .We use one number, N L , to describe the number oflocalized states in the bulk of the interferometer basedon the assumption that electrons or holes in a local-ized state are localized in a one-body approximation, butare not completely immobile. At any finite temperature,they have a non-zero conductivity due to processes suchas multi-particle hopping, and we assume that they canreadjust their relative positions continuously. Thus, ona laboratory time scale, the interior of the island shouldbehave like a metal: the N L charges arrange themselvesto give a constant electrochemical potential in equilib-rium, within each class of localized states. As a resultof the integer constraints on the total occupation num-bers, however, there can be small differences between theelectrochemical potentials of the localized states and thatof the adjacent edge states or leads.We neglect the degrees of freedom associated with lo-calized states between edge states. We assume again thatthe width w of the region of non-uniform electron den-sity near the edge of the sample is small compared tothe overall radius to the island. The area available forlocalized electrons or holes in the Landau level that ispartially filled in the center of the interferometer shouldbe approximately ¯ A , while the areas available for local-ized electrons or holes in any other Landau levels shouldbe of order Lw , which is much smaller. Then the num-ber of localized electrons or holes in any of these regionswill be relatively small, and the energy cost of adding orsubtracting a particle from one of them should be rela-tively high. Thus we may generally neglect fluctuationsin these quantities at reasonably low temperatures. Thefluctuations in N L that do occur will arise normally fromchanges in the occupation of the innermost partially fullLandau level.If the magnetic field B or the gate voltage V G is var-ied by a sufficiently large amount, we do expect to en-counter discontinuous changes in the occupations of lo-calized states other than those N L of the inner-mostpartially full Landau level. These jumps should lead tojumps in the phase θ , which would appear as “glitches”in the interference pattern. The analysis of periodicitiesgiven above apply, strictly speaking, only in the inter-vals between glitches. The frequency of occurrence ofglitches should roughly correspond to the addition of oneelectron or one flux quantum in area Lw , which would berarer by a factor of Lw/ ¯ A than the oscillation frequencieswe are interested in. Also, in many cases, the couplingbetween the interfering edge state θ and a particular oc-cupation number N hj or N ej may be sufficiently small thatany glitches associated with changes in that occupationnumber would be unobservable.Finally, in replacing the full energy function by themacroscopic function E , we have minimized the energywith respect to all continuous variables n j other thanthat of the partially transmitted edge state, i.e., we haveignored the effects of thermal fluctuations in these vari-ables. This neglect is justified for the continuous vari-ables, because they enter the energy in a quadratic form,so their thermal fluctuations add only a constant to theenergy. III. AHARONOV-BOHM ANDCOULOMB-DOMINATED REGIMES IN THEWEAK BACKSCATTERING LIMIT
We now have Eq. (7) for the resistance in the weakbackscattering limit in terms of the interference phase θ .We also have Eqs. (4,5, 13) for θ in terms of the degrees offreedom A I , N L , and the energy function (8) for the en-ergy and its dependence on B and V G . In this section wemake use of these expressions to calculate several ther-mal averages. First, we calculate the abrupt phase jump2 π ∆ that occurs when the number of localized electrons(or quasi-particles) varies by −
1. Then, we calculate themagnetic field and gate voltage dependencies of < e iθ > at high temperatures, and show that in that limit the in-terferometer shows either AB or CD behavior, dependingon the value of ∆. Finally, we turn to the case where ABand CD behaviors mix together, and develop the toolsneeded to analyze this case, at low temperatures as wellas high. A. Continuous and abrupt phase evolution
As the energy function is quadratic with respect tothe continuous variable A I , the average A I is the one that minimizes the energy function. For a fixed number N L of localized charges, we obtain θ π = e ∗ in φ − se ∗ in N L − K IL K I e ∗ out [ e ∗ in N L + ν in φ − q ] . (18)The abrupt phase jumps 2 π ∆ associated with a changeof N L by − K IL K I = C IL C L + C IL . (19)When there is no bulk-edge coupling K IL = 0 the in-terference phase is unaffected by N L . When the bulk-edge coupling is strong, the jumps are unobservable, since∆ = 1.For an FQHE interferometer, we have∆ = K IL K I + 2 e ∗ in s (cid:18) − K IL K I (cid:19) (20)Now, if K IL = 0, then 2 π ∆ is the phase jump associ-ated with the fractional statistics of the quasi-particles.When the bulk and the edge are coupled, the phase jumpsreflect both the change of the area A I caused by the in-troduction of quasi-particles and the fractional statistics.In the limit of strong coupling, where K IL = K I , thephase jump becomes unobservable, just as in the integercase. Now, if there is a change of -1 in N L , correspondingto the introduction of a quasi-hole in the bulk, the area A I will increase by ( e ∗ in / ∆ ν )(Φ /B ). This is the areanecessary to accommodate the charge of the quasi-hole,and is also the area necessary for the accumulated phaseto grow by 2 π . B. Magnetic field dependence
Next, if the parameters entering (8) are known, we maycalculate the thermal expectation value < e iθ > = Z − X N L Z ∞−∞ dA I e − E/T e iθ , (21)with the partition function Z given by Z = X N L Z ∞−∞ dA I e − E/T . (22)Since E is a quadratic function of its variables, the in-tegration over A I is trivial. The sum over the discretevariable N L can be handled by using the Poisson sum-mation formula and taking the Fourier transform. Thuswe may write ∞ X N L = −∞ = Z ∞−∞ dN L ∞ X g = −∞ e − πiN L ( g − (23)Using this formula, one may perform the integrationsover N L in the numerator and denominator of (21). Theformulas simplify at high temperatures, where the parti-tion function Z becomes independent of φ , and we mayconcentrate on the numerator of (21). We then find thatthe expectation value can be written in the form < e iθ > = ∞ X g = −∞ D m e πi mφ , (24)where m ( g ) = − ν out e ∗ out + g ν in e ∗ in , as in Eq. (6).The coefficients D m may be written as D m = ( − g +1 | D m | exp (cid:20) πi ¯ q (cid:18) e ∗ in − mν in (cid:19)(cid:21) , (25)with | D m | = e − π T/E m (26)and 1 E m = 1( e ∗ out ) K I + ( g − K I ( e ∗ in ) ( K I K L − K IL ) (27)Remarkably, Eq. (27) identifies the most dominantFourier component of the resistance in the high temper-ature limit, and displays its relation to ∆: in the inte-ger case and for fractions where m = 1 is allowed (i.e.,for fractions in with ν in < / − / < ∆ < / g = 1 , m = 1. In contrast, if 1 / < ∆ < / . it is dominated by the CD component, with g = 0, m = 1 − ( ν in /e ∗ in )We note that the plausible assumption of a posi-tive cross capacitance C IL > < ∆ <
1. We will then find ∆ < / C IL < C L . We also remark that the energy E m for the CD term is related to the capacitances by E m = ( e ∗ in ) / ( C L + C I ). The denominator here may bethought of as an effective capacitance resulting from theelectrostatic and quantum capacitances of the combinedsystem of the localized charges and the interfering edgestate, if the edge state is disconnected from the leads. C. Gate voltage dependence
A variation of the gate voltage V G varies the phases ofthe Fourier components of < e iθ > through its effect on ¯ q and φ in the phases in Eqs. (24) and (25). There are twoorigins to this dependence - the effect of the gate voltageon the area of the interference loop ¯ A and its effect onthe charge density in the bulk, and through it, on N L .These two dependencies are described by the parameters β, γ of (11).For small variations δV G and δB, we see that D m e πimφ varies proportional toexp (cid:18) πi (cid:20) δV G ( α m + βm ) + mδB ¯ A Φ (cid:21)(cid:19) , (28) where the term proportional to β originates from the areachange induced by the gate, and the term proportionalto α m = γ ( e ∗ in − m ) /ν in . (29)originates from the effect of the gate on the bulk back-ground charge.For the integer case, we see that lines of constant slopein the AB regime will have dV G /dB = − ¯ A/ Φ β , whilein the CD regime, the lines of constant slope will have dV G /dB = f T ¯ A/ Φ ( γ − f T β ).We expect that applying positive voltage to a side gateshould tend to increase the area ¯ A , so that the coefficient β should be positive. To estimate γ , let us first considera model in which there is a constant electron density inthe interior of the interferometer, except for a thin regionaround the edge, and let us imagine that the effect of δV G is to alter the location of the edge, without changing itsdensity profile, and without changing the electron densityaway from the edge. In this case we would find γ =¯ νβ , where ¯ ν ≥ ( f T + 1 /
2) is the filling factor in theinterior. In reality, we would expect that positive δV G will increase the average density inside, so that γ shouldbe even larger. Thus we expect that the slope of theconstant phase lines will be negative for the AB stripesbut positive for the CD stripes. D. Low temperatures
Although at high temperatures we need only considerone Fourier component, at lower temperatures, particu-larly if ∆ is close to 1/2, the g = 0 and g = 1 componentsmay both be important. Then a color-scale map of theinterference signal versus B and V G will show lines ofboth slopes, with a resulting pattern of a checker-boardtype, as seen in Fig 2. Even if both slopes are present,however, the eye will tend to pick out only the strongercomponent, if there is a big difference in the amplitudes,as in panels 2b and 2d.At still lower temperatures, higher harmonics with g > g < Z in the denominator of (21) dependson φ . Let us expand Z as Z = ∞ X g = −∞ z g e πig ( ν in φ − ¯ q ) /e ∗ in . (30)(The coefficients z g fall off exponentially with increasingtemperature, except for z , which is simply proportionalto T .) The Fourier components of < e iθ > will thenbe a convolution of the Fourier components of Z − withthe Fourier coefficients obtained from the numerator of(21), which are given by (25) and (26). We see thatthis does not introduce any new Fourier components intothe function, but it can affect the relative weights of thedifferent harmonics.0In the limit of low temperatures, the phase θ becomes asaw-tooth function of φ , for fixed V G , and we can simplyevaluate the Fourier coefficients of e iθ . Up to a constantphase factor, we find that for the allowed values of m ,the coefficients D m may still be written in the form (25),but now | D m | = sin( π ∆) π (∆ + g −
1) (31)We see that the CD component ( g = 0) will be largest if1 / < ∆ ≤
1, and the component ( g = 1) will be largestif 0 ≤ ∆ < /
2, at T=0 as well as at high temperatures.In our discussions of the temperature-dependence ofthe interference signal, we have taken into accountonly classical fluctuations, ignoring quantum fluctua-tions, which can be important on energy scales largerthan k B T . In the FQHE case, quantum fluctuations leadto a renormalization of the tunneling amplitudes, whichwill typically cause the individual reflection amplitudes r , r to decrease with increasing temperature, as a powerof 1 /T , in the weak backscattering regime. At high tem-peratures, this decrease should be less important thanthe exponential decrease of the interference signal aris-ing from classical fluctuations, predicted by Eq. (26), butthe power-law dependence should be taken into accountat lower temperatures. If one defines a normalized inter-ference signal by dividing the interference term by the to-tal backscattered intensity, ∝ ( | r | + | r | ), then the lowtemperature power-law dependence should be cancelled. Quantum fluctuations do not lead to a power law depen-dence of the normalized interference signal on length ofthe interferometer, in the limit of vanishing temperatureand vanishing source-drain voltage. E. Two-dimensional description
For a proper analysis of the regime where the CD andAB lines co-exist, we need to introduce a two-dimensionalFourier transform of δR with respect to B and V G , ratherthan the Fourier transform with respect to φ at fixed V G , which we have employed so far. One finds that theperiodic pattern can be expanded in terms of a set of“reciprocal lattice vectors” ~G gh ≡ ( G ( b ) gh , G ( v ) gh ), where g and h are integers, with ~G gh = g ~G + h ~G (32) ~G = 2 π (cid:18) ν in e ∗ in ¯ A Φ , βν in − γe ∗ in (cid:19) (33) ~G = 2 π (cid:18) − ν out e ∗ out ¯ A Φ , γ − βν out e ∗ out (cid:19) (34)and δR ( B, V G ) = X gh R gh e i ( G ( b ) gh δB + G ( v ) gh δV G ) . (35) The reality of δR requires that R gh = R ∗− g, − h .The set of reciprocal lattice vectors may be derivedby first removing the restriction that N L is an inte-ger. Regardless of the values of K I , K L , K IL , the energycan then be minimized by choosing A I and N L so that δn I = δn L = 0. using (15) and (14). If we then calcu-late changes in θ using (13), we find that δθ = G ( b )11 δB + G ( v )11 δV G , while δN L = − ( G ( b )10 δB + G ( v )10 δV G ) / π , with ~G gh defined as in (32) - (34). Here, we have used therelations ∆ ν = e ∗ in e ∗ out and 2 s = ( e ∗ out − e ∗ in ) / ∆ ν .In the limit of weak back scattering, the only reciprocallattice vectors with non-zero amplitudes have h = ± . For h = 1, the coefficients R gh may be related to thecoefficients D m defined previously, with R g, ∝ r r D m ,where m is related to g by Eq. (6) and r , r are thebare reflection amplitudes at the two constrictions. For h = −
1, the coefficients are the complex conjugates of R − g, .When one goes beyond weak backscattering, as dis-cussed below, one finds harmonics at reciprocal latticevectors which are arbitrary sums of the ones present inthe weak back scattering limit. Thus, one may obtaincontributions at all integer values of h , including h = 0 . Using the two-dimensional description, we may read-ily extend our analysis to the situation where one can-not neglect the dependence of the secular area ¯ A on themagnetic field B . In this case, we should also take intoaccount the change in the “background charge” ¯ q result-ing from the change in ¯ A . We define two dimensionlessparameters, λ = − B ¯ A ∂ ¯ A∂B , η = − Φ ¯ A ∂ ¯ q∂B . (36)Then the formulas for the fundamental reciprocal latticevectors should be replaced by ~G π = (cid:20)(cid:18) ν in (1 − λ ) + ηe ∗ in (cid:19) ¯ A Φ , βν in − γe ∗ in (cid:21) (37) ~G π = (cid:20) − (cid:18) ν out (1 − λ ) + ηe ∗ out (cid:19) ¯ A Φ , γ − βν out e ∗ out (cid:21) (38)If the field B is varied while the gate voltage V G isheld fixed, the field periods associated with the AB term( g, h ) = (1 ,
1) and the CD term ( g, h ) = (0 ,
1) are given,respectively, by¯ A (∆ B ) AB = Φ (1 − λ ) (cid:16) ν in e ∗ in − ν out e ∗ out (cid:17) + η (cid:16) e ∗ in − e ∗ out (cid:17) , (39)¯ A (∆ B ) CD = − e ∗ out Φ η + ν out (1 − λ ) . (40)If η = 0, the two periods will generally be incommensu-rate. Then when the magnetic field is varied at constantgate voltage, the resistance will not be a periodic func-tion of B , but rather quasi-periodic. To obtain a periodic1variation, one must vary B and V G simultaneously, alonga line of appropriate slope.As a simple example, let us assume that ¯ A ( B, V G ) isdetermined by a contour in the zero-field electron density n ( ~r ) where n Φ /B = ( ν in + ν out ) /
2, and let us assumethat ¯ q is equal to the integral of this density inside thearea ¯ A . Then we find η = λ ( ν in + ν out )2 , (41) λ = 1¯ A I n ( ~r ) dr |∇ n | , (42)where the integral is around the perimeter of the area ¯ A .We see that η and λ will vanish in the limit where thelength scale for density variations at the edge is smallcompared to the radius of the island, (assuming that thedensity in the bulk is not too close to density at theinterfering edge state). IV. INTERMEDIATE BACKSCATTERING
If one goes beyond the lowest order in the backscat-tering amplitudes r and r , the above analysis must bemodified in several respects. In this Section we confineourselves to the IQHE case; we come back to the FQHE inthe next Section, for the regime of strong back-scattering.The most obvious change from the weak backscatter-ing limit is that the interference contribution to the resis-tance R D is no longer simply proportional to Re[r r ∗ e i θ ].To be specific, let us consider the case of symmetricconstrictions, so that r = r . We may write R − D =( f T + 1 − P R ) ( e /h ), where 0 < P R < θ as the phase accumu-lation around the interferometer loop for an electron atthe Fermi energy, then the full expression for P R is P R = 2 | r | θ | r | + 2 | r | cos θ (43)If we expand this in powers of r , we find terms of order | r | multiplying cos θ , etc., which we may understand ascontributions from electrons that undergo multiple reflec-tions and therefore traverse the circuit more than once.Such terms will add additional harmonics of e πiφ to thereflection coefficient, and in principle all harmonics willbe present. However, the underlying period will not beaffected. Moreover, at least at high temperatures, thehigher harmonics should fall off faster than the principalAB component ( ∝ e πiφ ) or the principal CD component( ∝ e − πif T φ ) and should not be very noticeable.In the presence of a significant reflection probability,one should also take into account the fact that in this casethe number of electrons enclosed by the partially trans-mitted edge state is no longer precisely equal to θ/ π . This follows from the the Friedel sum rule, which statesthat ρ ( ǫ ), the density of states for the Landau level insidethe interferometer at energy ǫ may be written as ρ ( ǫ ) = 1 π ∂ ( η + + η − ) ∂ǫ , (44)where η ± are the phase shifts, derived from the eigenval-ues e iη ± of the 2 × S -matrix for transmission throughthe interferometer. Explicitly, the eigenvalues are givenby e iη ± = (1 − | r | ) e iθ ± i | r | ( e iθ + 1)1 + | r | e iθ , (45)and the phase shifts are required to be continuous func-tions of the energy ǫ. For | r | = 0, this gives an oscilla-tory contribution to the phase shifts, and an oscillatorycontribution to the density of states. Since the electronnumber n I is the integral of ρ ( ǫ ) up to the Fermi energy,it will also acquire an oscillatory part. Specifically wemay write n I = π − ( η + + η − ) + const = (2 π ) − θ + f ( θ ) , (46)where the phase shifts are evaluated at the Fermi energy,and f ( θ ) is periodic, with period 2 π .The oscillatory contribution to n I will also be manifestwhen one varies the magnetic field, the gate voltage, orthe electrochemical potential µ . For an interacting sys-tem, where the number of electrons n I associated withthe interfering edge state is coupled to other variables,such as N L , or even to continuous variables such as thenumber of electrons in fully transmitted edge states, anoscillatory component of n I will lead to an additional os-cillatory component to the energy E , which should betaken into account when evaluating the thermal averageof P R . Again, we see that these effects can lead to addi-tional oscillatory contributions at harmonics of the basicperiods, giving rise to non-zero amplitudes at arbitraryreciprocal lattice vectors ~G gh , but they should not changethe fundamental frequencies ~G and ~G .We can treat the case of intermediate (or strong) backscattering within our general model if we make a fewmodifications of the definitions. We continue to use theenergy formula (8), with the definitions (4) and (9) for θ and δn L . We continue to define δn I ≡ n I − φ , as in(10) but we can no longer equate this to B ( A I − ¯ A ) / Φ .Instead, we must compute n I using (46). Finally, wemust calculate < P R > by averaging (43) with the weight e − E/T , integrating over A I and summing over N L . V. STRONG BACKSCATTERING
It is interesting to explore the behavior of the inter-ferometer at low temperatures in the limit of strongbackscattering, where the amplitude r is close to unity.For the case r = r , when θ is an odd integer multiple2of π a resonant tunneling occurs, and P R = 0. Then,for non-interacting electrons, at large r , we would findthat the the reflection probability P R is close to unitymost of the time, but there would be a series of valuesof the magnetic field, or of the gate voltage, where in anarrow interval, P R drops to zero. The actual vanishingof P R is special to the case where r = r , but even foran asymmetric case, one would find reductions in P R inthe vicinity of the points where θ is an odd multiple of π . We now analyze the effect of interactions between elec-trons on these transmission resonances, and in particularon the flux spacing ∆ φ between transmission resonances.Interestingly, we find that this spacing is different for in-terferometers in the AB and CD regimes.In the limit of strong backscattering the charge en-closed in the ν in area is almost quantized in units of e ∗ out .The condition for transmission resonance, that θ is anodd multiple of π , is also the condition for a degeneracyof the energy for two consecutive values of the charge onthe interfering edge. We now formulate this condition interms of our energy functional and explore the magneticfield spacings between such resonances.We start with the integer quantum Hall regime. Let N o = n I + N L be the total number of electrons in thehigher Landau levels enclosed by the (almost-closed) in-terfering edge channel, excluding electrons in the f T filledLandau levels that correspond to the totally transmittedchannels. The energy of the system is then E ( N o , N L ) = K I N o − N L − φ ) + K IL ( N o − N L − φ )( N L + ( f T + 1) φ − ¯ q )+ K L N L + ( f T + 1) φ − ¯ q ) (47)An increase of φ by one decreases N L by ( f T + 1) andincreases N o by f T . Resonant transmission occurs whenthere is a vanishing energy cost for adding one electronto the closed edge, that is, a vanishing energy cost forvarying N o by one while keeping N L fixed. Degeneracypoints where N L changes by ± N o is fixed willgenerally not lead to resonances, even though n I changesby ∓ θ will technicallypass through an odd multiple of π in this process, oneexpects that these transitions will generally happen dis-continuously, so there is no point at which the resonancecould be observed. Points where N L and N o increase si-multaneously do not involve a change in n I and do notlead to transmission resonances.In the extreme AB limit, where K IL = 0, there aredegeneracy points where E ( N o , N L ) = E ( N o + 1 , N L + 1)separated on the φ axis by spacings ∆ φ = 1 / ( f T + 1).These points do not, however, lead to resonances, sincethey involve a change in N L . Degeneracy points thatdo lead to resonances occur when E ( N o , N L ) = E ( N o +1 , N L ), and the spacings between those is ∆ φ = 1, theflux period that characterizes also the weak backscatter-ing regime of the AB limit. In the extreme CD regime, K I = K IL , and stabilityrequires K L > K I . Then jumps of N L are separated fromjumps of N o . In an interval where φ increases by 1, therewill be f T resonant events where N o decreases by one,while N L is fixed, and ( f T + 1) separate events where N L increases by one while N o is fixed. The resonancesare thus separated by ∆ φ = 1 /f T . Again, this is theflux period that characterized the CD regime in the weakbackscattering limit.The difference in ∆ φ between the AB and CD limitscharacterize also the fractional case. In this case the bulkaccommodates N L quasi-particles of charge e ∗ in , and thetotal charge in the ν in region is quantized in units of e ∗ out .The charge on the interfering edge is given by n I = N o e ∗ out − N L e ∗ in . (48)Then, the energy functional becomes, E ( N o , N L ) = K I e ∗ out N o − e ∗ in N L − ∆ ν φ ) + K IL ( e ∗ out N o − e ∗ in N L − ∆ ν φ )( e ∗ in N L + ν in φ − ¯ q )+ K L e ∗ in N L + ν in φ − ¯ q ) (49)In the Coulomb dominated limit, K IL /K I = 1, andthe number of transmission resonances that occur while φ changes by one is equal to ν out /e ∗ out . This leads to∆ φ = e ∗ out /ν out . The leading component in the Fouriertransform of P R ( φ ) would then correspond to the g =0 component of (6), just as in the weak backscatteringlimit.In the extreme Aharonov-Bohm limit, where K IL = 0,the structure of transmission resonances is more com-plicated, due to the difference between the elementarycharges e ∗ in and e ∗ out . Just as in the weak backscatteringcase for the FQHE at K IL = 0, there is no single domi-nant value of g . In the case of weak backscattering, thisoccurs because ∆ = 2 se ∗ in = 0, according to Eq. (20).Here we note that e ∗ out − e ∗ in = 2 se ∗ in e ∗ out .Over all, we see that in the limit of strong backscat-tering, in the CD regime, the number of peaks in thetransmission probability as we increase B by one fluxquantum is the same number ν out /e ∗ out as we obtained inthe weak backscattering regime, consistent with the pre-diction that the period of the CD oscillations would notchange as we vary r . The strong back scattering limitmay also be understood as a Coulomb-Blockade effect:maxima in the transmission probability occur at pointswhere the system consisting of the localized states andthe almost-totally-reflected edge state is about to changefrom one integer value to another.Typically, the reflection coefficient r should increasefrom near zero to near unity as one decreases the electrondensity in the constrictions through the range where thefilling factor ν c at the center of the constriction decreasesfrom slightly below ν in to slightly above ν out . For an idealconstriction, the variation in r should be smooth andmonotonic. In real constrictions, however, the variation3may be more complicated, as the Fermi-level may passthrough one or more resonances due to tunneling throughlocalized states in the constriction.Our discussion of the variation in r should also apply if ν c is varied by changing the magnetic field B rather thanby changing a gate voltage at the constriction. Again, thefield periods for the AB and CD oscillations should re-main fixed as long as ν out /e ∗ out does not change. However,the parameter ∆ which governs the relative strengths ofthe AB and CD contributions could conceivably changeas the other parameters are varied.Under some circumstances, if there is a large region ofintermediate electron density within a constriction, thenumber of localized states in the constriction may be-come so large that there is a large density of states atlow energies associated with rearrangements of electronsin these states. Then, backscattering through the con-striction could become incoherent, either because of in-elastic scattering from the low energy modes, or becausethe path length for tunneling is changed randomly due tothermally excited rearrangements of the localized states.We assume that this does not happen in the system ofinterest. VI. MODEL WITH MULTIPLE EDGE STATES
In order to better understand how the presence of mul-tiple edge states may affect the parameters entering theenergy function (8), we discuss here some simplified mod-els which may illustrate the physics.We consider the integer case, with f T fully transmittededge states. We define δn i to be the charge fluctuation as-sociated with a fluctuation in area of the i -th edge state,for 1 ≤ i ≤ N , where N = f T + 1, while δn i = δn L , for i = N + 1. The partially reflected edge state has i = N ,so δn I = δn N We may now write the quadratic part of the energy inthe form E = X ij κ ij δn i δn j , (50)where the sums go from 1 to N + 1. We assume that thecoupling constants κ ij are known, and we wish to findthe values of the coupling constants K L , K I , K IL whichentered our earlier computations. We wish to specify thevalues of δn L and δn I , and minimize the energy withrespect to the other variables. This means that for 1 ≤ j ≤ N −
1, we have X i κ ji δn i = 0 . (51)The resulting energy will be quadratic in δn I and δn L ,and the coefficients may be identified with K I , K L and K IL .We illustrate further with two examples. In our firstmodel, we consider a situation where κ ij = U + κ δ ij , (52) for 1 ≤ i, j ≤ N , and κ ij = κ L , for i = j = N + 1 ,κ ij = V , if either i or j = N + 1 but i = j. In this model, the interaction between the edge states isentirely determined by the total edge charge P j ≤ N n j ,and the interaction with N L involves only that charge.After some straightforward algebra, on obtains the re-sults K I = κ I + ˜ U , K IL = ˜ V , (53)where ˜ U ≡ U − f T U κ + f T U , (54)and ˜ V = V ˜ U /U . We see from these results that K IL /K I = ˜ V / ( κ + ˜ U ). If V ≤ U and f T >
0, thisratio is necessarily less than 1/2 , so the model will be inthe AB regime. For f T = 0, the model leads to the CDregime if and only if κ + U < V .The second model we consider is the opposite extreme,where edges are coupled only to their nearest neighbors.We choose the diagonal coupling constants κ jj as the pre-vious model, while for off diagonal couplings we choose κ ij = κ , if | i − j | = 1, and κ ij = 0 otherwise. Now, wefind that couplng to the fully transmitted edges renor-malizes the coefficient K I but has no effect on K L and K IL , which remain equal to κ L and κ , respectively. Inthe case f T = 1, we find K I = κ − κ /κ . (55)The value of K I will be reduced further with increasing f T , but the value remains finite in the limit of large f T ,where one finds K I → κ κ − κ ) / K I is reduced by up to afactor of 2 as a result of coupling to additional edges.Stability of the model, in the limit of large f T , requiresthat κ /κ < /
2, and we see that K IL /K I <
1. Atthe same time, if 2 / < κ /κ < /
2, the ratio K IL /K I will be greater than 1/2, for sufficiently large f T , so thesystem may be pushed from AB into the CD regime. Ofcourse, the CD regime could be reached more easily if themodel is modified so that the coupling κ N,N +1 betweenthe localized charge and the partially reflected edge stateis made larger than the other coupling energies, or if thediagonal element κ NN is made smaller than the coeffi-cients κ ii for the fully transmitted edge states.For a uniform edge of length L , we expect that the con-stants κ ij should be proportional to 1 /L , for 1 ≤ i, j Oscillations in the transport properties of quantumHall devices, associated with interference effects, werealready observed in the 1980’s, in both IQHE and FQHEregimes. The possible importance of Coulomb block-ade effects in these experiments, and of fractionalstatistics for the FQHE situation, was noted by the-orists around that time. Interpretation of the early ex-periments was difficult, however, as the interfering pathswere not the result of a deliberate construction but were,presumably, the result of random fluctuations in the dop-ing density, whose geometry was not known. In a typ-ical case, one might see oscillations in the resistanceof a micron-scale Hall bar, on the high-field side of aquantized Hall plateau, which might be attributed tobackscattering through a “dot” or an “anti-dot” inclu-sion, where the electron density was higher or lower thanin the surrounding electron gas. The strength of tun-neling into and out of the dot or anti-dot was generallyassumed to be weak, and the oscillations were associatedwith resonances as additional electrons or quasiparticleswere added to the inclusion. In later years, improvedexperiments were carried out using fabricated anti-dotswith controlled areas, in which one could investigate sys-tematically the dependence on magnetic field and on elec-tron density, controlled by a back gate. Quantum Hall interferometers with the Fabry-Perotgeometry studied in the present paper have been exploredexperimentally by several groups. In several early works ,Coulomb blockade effects in a dot weakly coupled to leadswere studied, in a region with several filled LLs. . Thecrossover between AB and CD regime for a weakly cou-pled dot was analyzed in [35]. Both integer and fractionalquantum Hall interferometers in the absence of chargingeffects were discussed in [1].In an earlier experiment , a strong dependence of themagnetic field period ∆ B on the constriction filling fac-tor was found but interpreted in terms of a magneticfield dependent interferometer area. In a reanalysis of that experiment, it was pointed out that under the as- sumption of a magnetic-field-independent interferometerarea, the data agree with ∆ B ∼ /ν in .More recently, several groups have conducted system-atic investigations of interferometers of different sizes,with and without top gates, in which they could setthe filling factor in the constriction independently of thedensity in the bulk, and data has been collected as acontinuous function of both magnetic field and side-gatevoltage. The AB regime, the CD regime and the in-termediate regime were all observed in these experiments.In the he CD regime, when lines of equal R D were plot-ted in the B − V G plane, they were found to have positiveslope for ν out = 0, or zero slope for ν out = 0. The CD fluxperiod, in the integer case, was found to be ∆ φ = 1 /ν out ,independent of the strength of the backscattering. TheAB regime, observed in the IQHE, was characterized bylines of equal R D that had negative slope in the B − V G plane and flux periodicity of ∆ φ = 1. Intermediateregimes, where AB and CD behaviors combined togetherwere also observed, giving a checkerboard pattern of thetype seen in Fig. [2c].In the fractional case flux periodicity of ∆ φ = 1 /ν out was observed in the cases ( ν in , ν out ) = ( , 0) and ( , ν out is an integer and ν in a fraction. In the case of( , ), a period of ∆ φ = e ∗ out /ν out = 1 was observed. In an earlier work , in which two of us analyzed theinterference patterns in a Fabry-Perot interferometer, aparameter ∆ x / ∆ characterizing the strength of bulk-edgecoupling was introduced. In the present notation, it cor-responds to the ratio K IL /K I . Here, we have gone be-yond the approach of [6] by studying the directions oflines of constant phase and the temperature dependenceof the interference terms, and by allowing for arbitrarystrength of backscattering.A first principles approach to the study of interferome-ters was described in [40]. Possibly due to the approxima-tions chosen in that approach, an influence of Coulombinteractions on the magnetic field period of resistance os-cillations was not found.A situation in which the area of the interfering loop issmall compared to the lithographic area, and where it ishighly dependent on the magnetic field was discussed in[41]. In this paraemter regime, the coefficient λ , definedin our Eq. (42), can be larger than 2, so (1 − λ ) − can benegative, with a magnitude smaller than 1. This wouldcause the Aharonov-Bohm constant-phase lines to havereversed slope, and a period smaller than one flux quan-tum. However, under this assumption, the magnetic fieldperiod would vary continuously, rather than being quan-tized at a flux quantum divided by an integer, so thismechanism does not seem to explain the experimentalfindings . Also, this would not explain the simulta-neous appearance of AB and CD lines, as observed inseveral cases.The influence of anyonic statistics on magnetic field pe-riodicities of Fabry-Perot interferometers was discussedin [39], although the results obtained there disagree insome cases with our findings.5Observations of a magnetic field superperiod, corre-sponding to an addition of five flux quanta to the in-terferometer area have been reported in [9,10,42] for asample in which the bulk is in a quantized Hall statewith ν = 2 / 5, while the constrictions have filling fraction1 / VIII. DISCUSSION AND CONCLUSIONS In this paper, we have presented a general frameworkfor discussing the electronic transport in a quantum HallFabry-Perot interferometer. Our aim was to understandthe oscillatory dependence of the interferometer resis-tance R D on the magnetic field B and voltage appliedto a side gate V G , when these parameters are varied byan amount large enough to change the number of fluxquanta or the number of electrons by a finite amount, butsmall enough so that there is not a large fractional changein either the flux or the electron number. A central as-sumption was that the resistance arises from the partialreflection of one quantum-Hall edge state in the two con-strictions. We also restricted our analysis to the inte-ger quantum Hall states or a subset of fractional states,where all modes at a given edge propagate in the samedirection. Our understanding of the physics of the prob-lem was described in general terms in the Introduction,Section I, and in detail in the body of the paper. In thissummary we focus on the results we obtained.We found that δR , the oscillatory part of R D , is, ingeneral, a two-dimensional periodic function in the planeof B and V G . It is useful to describe this function interms of its two-dimensional Fourier transform, whichmeans that we should specify a set of reciprocal latticevectors ~G gh and the associated amplitudes R gh , where g, h are arbitrary integers, and ~G gh = g ~G + h ~G . Ex-plicit formulas for the reciprocal lattice basis vectors ~G and ~G were given in Subsection III E, in terms of thesmoothly varying secular area ¯ A enclosed by the inter-fering edge mode, the filling factors ν in and ν out of thequantum Hall states separated by this edge mode, andparameters β, λ, γ, η describing the derivatives of ¯ A andthe enclosed “background charge” ¯ q with respect to V G and B . Our most general expression for δR is δR ( B, V G ) = X gh R gh e i ( G ( b ) gh δB + G ( v ) gh δV G ) , (58)with basis vectors given by (37) and (38). However, incases where the radius of the interferometer is large com-pared to the widths of the density transition regions at the edges, one may be able to neglect the magnetic-fielddependence of ¯ A and ¯ q , in which case λ and η may be setequal to zero. Then the basis vectors ~G and ~G aregiven by the simpler expressions (33) and (34).For the remainder of this summary, we shall limit our-selves to the case λ = η = 0. Then, if V G is held fixed, wefind that δR is a periodic function of the magnetic field,with a fundamental period corresponding to the additionof one flux quantum to the area ¯ A . However, the fun-damental period may not have the largest Fourier ampli-tude, so the most visible oscillations may correspond to aharmonic, with a period that is the fundamental perioddivided by an integer.For non-interacting electrons, in the integral quan-tized Hall effect, the observed interference pattern willreflect the fundamental Aharonov-Bohm period, wherethe phase increases by 2 π when the dimensionless mag-netic flux φ ≡ B ¯ A/φ changes by one, due to variationof B or of V G or both. In our current notation, thismeans that non-zero Fourier components R gh will corre-spond to reciprocal lattice vectors where g = h . In thecase of weak backscattering at the constrictions, or athigh temperatures, the oscillations are simply sinusoidal,and the dominant contributions are R and its conju-gate R − , − . On a a color-scale map of δR in the B − V G plane, AB oscillations would appear as a series of parallelstripes with negative slope. For stronger backscattering,at low temperatures, we may get higher Fourier compo-nents due to multiple scattering events across the twoconstrictions.In the case of weak backscattering, Fourier componentsat additional reciprocal lattice vectors can arise fromelectron-electron interactions. For the integer QHE, thisis due to the Coulomb interaction between electrons onthe interfering edge state and localized electrons or holeswhich exist in the bulk of the interferometer. Becausethe number of localized particles is required to be an in-teger, the net number of localized particles N L will jumpperiodically, as B or V G is varied. Interactions with theedge then cause small variations δA I in the area A I en-closed by the interfering edge state, which will cause theactual number of flux quanta enclosed by A I to fluctuateabout the nominal value φ , and thus lead to an additionalmodulation of the interference phase. For FQHE states,there is an additional jump θ a in the interference phase,arising from the fractional statistics, whenever there is achange in the number of localized quasiparticles.If the Coulomb coupling between N L and the edge-state charge is sufficiently strong, one finds that the dom-inant terms in the Fourier expansion have g = 0, both forthe IQHE and the FQHE. In this limit, the color-scalemap will show a series of stripes which have positive slopeand ∆ φ = e ∗ out /ν out , except when ν out = 0, in which casethe stripes will be horizontal on a B − V G plot.In between the AB and CD limits, the two sets ofstripes occur simultaneously, and a map of δR will showa checkerboard pattern, as seen in Fig. 2, above. Theabsolute strengths of the various Fourier coefficients will6depend on the backscattering amplitudes in the individ-ual constrictions and on the temperature, as well as onthree energy parameters, which we denote K I , K L , K IL .At high temperatures, the Fourier amplitudes will fall offexponentially with T with varying rates, so generally asingle pair of Fourier amplitudes will dominate at large T . This may be either the AB term ( g = h = ± 1) orthe CD term ( g = 0 , h = ± δR will be a simplesine function of magnetic field, with either the AB or CDperiod.At lower temperatures, where many Fourier compo-nents maybe present the situation is more complicated.We discuss here the limit of weak backscattering, where h is limited to h = ± 1. Then, at low temperatures, onefinds that the phase θ of the interference path is a saw-toothed function of the magnetic field, varying linearlywith B most of the time, but with periodic jumps byan amount 2 π ∆, which occur each time the number oflocalized quasiparticles N L changes by − 1. There willbe ν in /e ∗ in equally-spaced phase jumps per flux quantachange in the loop, which give rise to Fourier componentsof < e iθ > with arbitrary values of g . For h = 1, at T = 0,the Fourier amplitudes vary with g as ( g + ∆ − − , ac-cording to Eq. (25). At higher temperature the jumpswill be smeared, giving a more gradual change of < e iθ > as B is varied. This smearing causes the Fourier ampli-tudes at the higher g ’s to vanish exponentially.We see from the above that at low temperatures, theAB Fourier amplitude will be larger than the CD am-plitude if the jump parameter ∆ satisfies 0 < ∆ < / / < ∆ < 1. We find simi-larly at high temperatures that the AB component will belarger than the CD component if, and only if, ∆ < / K IL /K I . It vanishes in theextreme AB iimit ( K IL → 0) and approaches 1 in the ex-treme CD limit ( K IL → K I ). For the FQHE, the value of∆ depends on the statistical phase angle θ a and the ratio K IL /K I , according to Eq. (20), going from | θ a | / π , inthe absence of bulk-edge coupling, to 1, when this cou-pling is strong. This suggests the possibility that onecould obtain a direct measure of θ a by observing the dis-crete jump in the interference phase θ a as an additionalquasihole enters the interferometer at low temperatures.In order to extract the value of θ a , however, one wouldhave to independently find a measure of K IL /K I , or be able to vary K IL (say by varying the area of the inter-ferometer) and extrapolate to K IL = 0.We have said little about the actual values of the pa-rameters β and γ which determine the gate-voltage pe-riods of the AB and CD stripes, nor have we estimatedthe energy parameters K I , K L , K IL , which determine theratio between the AB and CD amplitudes and the tem-perature dependence of these amplitudes.One might try to estimate β and γ using a simplifiedmodel, where the electron density in the sample dependson V G but is independent of B . According to Eq. (11),this means that if one considers a sample with fixed gatevoltage, at various values of B, corresponding to differentbulk filling factors, the parameter β will be proportionalto B , while γ will be independent of B . Using Eq. (34) wefind that the V G period for a CD stripe should be equal to e ∗ out / ( γ − βν out ). The filling factor ν out will depend on themagnetic field, but also may be varied by changing thevoltage on the gates defining the quantum point contactconstrictions of the interferometer. It appears that thedependence of the gate period on B and V G predictedby these considerations is only partly in agreement withexperiment, and that significant effects are omitted fromthis simple model. Although the energy K L may be largely determined bythe geometric capacitance of the island, the parameters K IL and K I should be sensitive to the detailed struc-ture of the edge and are difficult to estimate without adetailed microscopic model and a numerical calculation.The values of these parameters should depend also onthe value of the magnetic field and on the setting of theconstrictions, which determines which edge mode is theinterfering one. For a dot of sufficiently large area, cov-ered by a top gate, the parameter K IL should decreaseinversely as the area, so for an integer quantized Hallstate, one would be necessarily in the AB regime. How-ever, the converse is not true; for a small area dot onecould be in the CD or AB regime depending on details.Further investigation of these points will be left for futurework.Acknowledgments: We acknowledge support from NSFgrant DMR-0906475, from the Microsoft Corporation,the BSF, the Minerva foundation, and the BMBF. Wehave benefited from helpful discussions with C.M. Mar-cus, D. McClure, Y. Zhang, A. Kou, M. Heiblum,N. Ofek, A. Bid, V. Goldman, and R. Willett. C. de C. Chamon, D.E. Freed, S.A. Kivelson, S.L. Sondhi,and X.G. Wen, Phys. Rev. B 55, 2331 (1997). E Fradkin, C. Nayak, A Tsvelik, and F. Wilczek, Nucl.Phys. B , 704 (1998). A. Stern and B.I. Halperin, Phys. Rev. Lett. , 016802(2006). P. Bonderson, A. Kitaev, and K. Shtengel, Phys. Rev. Lett. , 016803 (2006). P. Bonderson, K. Shtengel, and J.K. Slingerland, Phys.Rev. Lett. , 016401 (2006). B. Rosenow and B.I. Halperin, Phys. Rev. Lett. , 106801(2007). R. Ilan, E. Grosfeld, K. Schoutens, and A. Stern, Phys.Rev. B 79, 245305 (2009). Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu,and H. Shtrikman, Nature , 415 (2003). F.E. Camino, Wei Zhou, and V.J. Goldman, Phys. Rev.Lett. , 246802 (2005). F.E. Camino, Wei Zhou, and V.J. Goldman, Phys. Rev. B , 155305 (2007). F.E. Camino, Wei Zhou, and V.J. Goldman, Phys. Rev.Lett. , 076805 (2007). M.D. Godfrey, P. Jiang, W. Kang, S. H. Simon,K.W. Baldwin, L.N. Pfeiffer, and K.W. West, preprintarXiv:0708.2448 (2007). Y. Zhang, D.T. McClure, E.M. Levenson-Falk, C.M. Mar-cus, L.N. Pfeiffer, and K.W. West, Phys. Rev. B , 241304(2009). D.T. McClure, Y. Zhang, B. Rosenow, E.M. Levenson-Falk, C.M. Marcus, L.N. Pfeiffer, and K.W. West, Phys.Rev. Lett. , 206806 (2009). Ping V. Lin, F.E. Camino, and V.J. Goldman, Phys. Rev.B , 125310 (2009). N. Ofek, A. Bid, M. Heiblum, A. Stern, V. Umansky,D. Mahalu, preprint arXiv:0911.0794 (2009). R.L. Willett, L.N. Pfeiffer, and K.W. West, PNAS0812599106 (2009). R.L. Willett, L.N. Pfeiffer, and K.W. West, preprintarXiv:0911.0345 (2009). B.W. Alphenaar, A.A.M. Staring, H. van Houten,M.A.A. Mabesoone, O.J.A. Buyk, and C.T. Foxon, Phys.Rev. B , 7236 (1992). S. Ilani, J. Martin, E. Teitelbaum, J.H. Smet, D. Mahalu,V. Umansky, and A. Yacoby, Nature ,328 (2004). See, e.g. , J. A. Simmons, H. P. Wei, L. W. Engel, D. C.Tsui, and M. Shayegan, Phys. Rev. Lett. , 1731 (1989);and references therein. B. Hackens, F. Martins, S. Faniel, C.A. Dutu, H. Sell-ier, S. Huant, M. Pala, L. Desplanque, X. Wallart, andV. Bayot, Nature Commun. , 39 (2010). C.W.J. Beenakker and H. van Houten in: H. Ehrenreich and D. Turnbull, Editors, Solid State Physics , Aca-demic Press, New York (1992), p. 1. A. K. Evans, L. I. Glazman, and B. I. Shklovskii, Phys.Rev. B , 11120 (1993). B. I. Halperin, Phys. Rev. Lett. , 1583 (1984). B. Blok and X. G. Wen, Phys. Rev. B , 8145 (1990). Ady Stern, Annals of Physics , 204 (2008). X. G. Wen, Phys. Rev. B , 5708 (1991). P.A. Lee, Phys. Rev. Lett. , 2206 (1990). S. Kivelson, Phys. Rev. Lett. , 3369 (1990). J.K. Jain and S. Kivelson, Phys. Rev. Lett. , 1542(1988). V.J. Goldman and B. Su, Science 267, 1010 (1995). P.L. McEuen, E.B. Foxman, Jari Kinaret, U. Meirav,M.A. Kastner, Ned S. Wingreen, and S.J. Wind, Phys.Rev. B , 11419 (1992). C.M. Marcus, A.J. Rimberg, R.M. Westervelt, P.F. Hop-kins, and A.C. Gossard, Surf. Science , 480 (1994). C.W.J. Beenakker, H. van Houten, and A.A.M. Staring,Phys. Rev. B , 1657. B.J. van Wees, L.P. Kouwenhoven, C.J.P.M. Harmans,J.G. Williamson, C.E. Timmering, M.E.I. Broekaart,C.T. Foxon, and J.J. Harris, Phys. Rev. Lett. , 2523(1989). F.E. Camino, W. Zhou, and V.J. Goldman, Phys. Rev. B , 155313 (2005). M.W.C. Dharma-wardana, R.P. Taylor, and A.S. Sachra-jda, Solid State Commun. , 631 (1992). V.J. Goldman, Phys. Rev. B , 045334 (2007). S. Ihnatsenka and I.V. Zozoulenko, Phys. Rev. B ,235304 (2008). A. Siddiki, preprint arXiv:1006.5012 (2010). V. L. Ping, F. E. Camino, and V. J. Goldman, Phys. Rev.B80,