Braiding of anyonic quasiparticles in the charge transfer statistics of symmetric fractional edge-state Mach-Zehnder interferometer
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Braiding of anyonic quasiparticles in the charge transfer statistics of symmetricfractional edge-state Mach-Zehnder interferometer
Vadim V. Ponomarenko
Center of Physics, University of Minho, Campus Gualtar, 4710-057 Braga, Portugal
Dmitri V. Averin
Department of Physics and Astronomy, University of Stony Brook, SUNY, Stony Brook, NY 11794 (Dated: December 2, 2018)We have studied the zero-temperature statistics of the charge transfer between the two edges ofQuantum Hall liquids of, in general, different filling factors, ν , = 1 / (2 m , +1), with m ≥ m ≥ m states of the interferometer with different effectiveflux through it, where m ≡ m + m . For m >
1, this dynamics reflects both the fractional charge e/m and the fractional statistical angle π/m of the tunneling quasiparticles. Explicit expressions forthe second (shot noise) and third cumulants are obtained, and their voltage dependence is analyzed.
PACS numbers: 73.43.Jn, 71.10.Pm, 73.23.Ad
I. INTRODUCTION
Electronic Mach-Zehnder interferometer (MZI) [1–3]can be realized with the edge states of the Quantum Hallliquids (QHLs). Together with the quantum antidots[4, 5], MZIs in the regime of the Fractional Quantum Halleffect (FQHE) are expected [6–8] to be useful for obser-vation of the fractional statistics of FQHE quasiparticles.In contrast to fractional quasiparticle charge, which hasbeen confirmed in several experiments [4, 9, 10], thereis no commonly accepted observation of anyonic statis-tics of the quasiparticles, which remains a challengingexperimental problem. Currently, this problem attractsinterest in the context of solid-state quantum computa-tion, since individual manipulation of anyonic quasiparti-cles involving their braiding provides an interesting possi-ble basis for implementation of the quantum informationprocessing [11–13]. However, in typical interferometer-based experimental set-up the quasiparticles emerge asa continuum of gapless edge excitations that should bedescribed by a 1D field theory [14]. Individual quasipar-ticles can be realized in this theory only asymptoticallyin a special limit. In the fractional edge-states MZI, sucha limit occurs at large voltages, when the system is char-acterized by the Hamiltonian dual to the Hamiltonian ofthe initial electron tunneling model of the MZI. The lat-ter is perturbative in electron tunneling at low voltages,and is much better defined, since weak electron tunnelingis probably the most basic process in solid-state physics.In the dual description derived from the electronicmodel by the instanton technique [8, 15], MZI acquires m = m + m +1 different quantum states which differ bythe the effective flux Φ through it. This flux contains, inaddition to the external flux Φ ex , a statistical contribu-tion earlier also found [16] for the antidots. The tunneling of each quasiparticle changes this effective flux by ± Φ and therefore switches the MZI from one flux state intoanother. Since quasiparticle also carries the charge e/m ,the change of flux by ± Φ results in the change of the in-terference phase for the quasiparticles by 2 π/m and thecorresponding change of the rate of coherent tunnelingthrough the interferometer. Summation over the m fluxstates in calculation of the physical quantities restoresΦ periodicity of their Φ ex dependence. At low voltages,this periodicity is guaranteed by Φ periodicity of theelectron tunneling amplitude.In this work, we follow the approach to the MZI thatallows us to obtain the uniform description of its trans-port properties in both regimes of electron and quasipar-ticle tunneling. We consider the standard MZI geometry(Fig. 1) with two tunneling contacts between the twoeffectively parallel edges of QHLs, but allow for, in gen-eral, different filling factors, ν , = 1 / (2 m , + 1), with m ≥ m ≥
0, of these edges. In the symmetric caseof equal propagation times along the two edges betweenthe contacts, the corresponding 1D field theory permitsan exact Bethe ansatz solution [8]. Making use of thissolution, we calculate the zero-temperature full countingstatistics [17] of the charge transferred between the twoedges forming MZI. The transferred charge distributionis shown to reflect the anyonic braiding statistics of thetunneling quasiparticles, and decomposition of electronsinto quasiparticles with increase of voltage.The paper is organized as follows. In Section 2 weintroduce the model of symmetric MZI and its Betheansatz solution. In Section 3, we use this solution toderive the general expression for the generating function P ( ξ ) of the distribution of the transferred charge. Themethod employed in this calculation generalizes to theMZI transport through two point contacts the method (cid:81) U U x (cid:81) x x FIG. 1: Mach-Zehnder interferometer considered in this work:two point contacts with tunneling amplitudes U j formed atpoints x j , j = 1 , ν and ν . The edges are assumedto support one bosonic mode each, with arrows indicatingdirection of propagation of these modes. developed earlier by Saleur and Weiss [18] for a sin-gle point contact. The charge transfer statistics foundfor one contact [18] demonstrated fractionalization ofelectron charges with increasing bias voltage across thecontact. In the case of MZI, one can expect that inaddition to charge fractionalization, braiding propertiesof anyonic quasiparticles should emerge in the chargetransfer statistics. To see this, we analyze our resultfor the transfer statistics in different regimes. We findthat the logarithm of the generating function separatesinto two parts, which for very different absolute valuesof the electron tunneling amplitudes in the two con-tacts, U /U ≫
1, become identical with the cumulant-generating functions of the two separate contacts. In gen-eral, however, each “single-contact” term accounts alsofor the interference between the contacts. At low volt-ages, these single-contact terms are combined in sucha way that the charge transfer occurs only in electronunits, with the low-voltage asymptotic describing the reg-ular Poisson distribution of tunneling electrons. In Sec-tion 4, we develop analogous qualitative interpretationof the large-voltage asymptotics of the Bethe ansatz re-sult for the generating function. The interpretation isbased on the m -state model of the quasiparticle tunnel-ing described above. Following the method of Ref. 22,the cumulant-generating function is calculated from thekinetic equation governing the quasiparticle transitionsin the basis of the m flux states. Direct comparison ofthis generating function with the leading asymptotics ofthe Bethe ansatz result reveals their coincidence undera special choice of m quasiparticle interference phaseswhich are found as φ l = ( κ + ( m − π + 2 πl ) /m , where l = 0 , ..., m − l -dependent, statistical, part of these phasesagrees with the expected anyonic statistics of the quasi-particles, while the common phase is given by the elec-tron interference phase κ with an additional phase shift π/m for even m . The equivalence of the two distributionsin the large-voltage limit proves that the Bethe ansatzconstruction we implemented indeed describes the sta-tistical transmutation of the effective flux through the MZI. In Section 5, we use the obtained generating func-tion to find the first three cumulants of the transferredcharge distribution. We study, in particular, the volt-age dependence of the first and the second cumulantsproportional to the average current and the shot noise,respectively. The ratio of the two, the Fano factor, is anexperimentally observable [9, 10] characteristics of theelectron-quasiparticles decomposition. Finally, we con-sider the third cumulant, which determines the asymme-try of the transferred charge distribution around average.In Section 6, we look at the special case m = 2 which al-lows to obtain these cumulants in terms of elementaryfunctions for arbitrary voltages. The results of this workare summarized in the Conclusion. II. MODEL OF THE SYMMETRICMACH-ZEHNDER INTERFEROMETER ANDITS BETHE-ANSATZ SOLUTION
We start our discussion with the electronic model ofMZI (Fig. 1) formed by two single-mode edges with fillingfactors ν l = 1 / (2 m l + 1), l = 0 ,
1. Electron operator ψ l ofthe edge l is expressed using the standard bosonizationapproach [14] as ψ l = ( D/ πv l ) / ξ l e i [ φ l ( x,t ) / √ ν l + k l x ] . Here φ l are the two chiral bosonic modes propagat-ing in the same direction (to the right in Fig. 1),which satisfy the usual equal-time commutation relations[ φ l ( x ) , φ p (0)] = iπ sgn( x ) δ lp . The Majorana fermions ξ l account for the mutual statistics of electrons in differ-ent edges, and D is a common energy cut-off of the edgemodes. The Fermi momenta k l correspond to the averageelectron density in the edges, while the operators of thedensity fluctuations are: ρ l ( x, τ ) = ( √ ν l / π ) ∂ x φ l ( x, τ ).In the symmetric case of equal times of excitation prop-agation between the contacts along the two edges, thetwo combinations of the bosonic operators φ l that enterthe electron tunneling terms of the two contacts can beexpressed as the values at points x , of the same right-propagating chiral bosonic field: φ − ( x ) = √ ν φ − √ ν φ √ ν + ν . (1)The Langrangian describing electron tunneling in the twocontacts can then be written as: L t = X j =1 , ( DU j /π ) cos[ λφ − ( x j ) + κ j ] , (2)where U j and κ j are the absolute values and the phasesof the dimensionless tunneling amplitudes. The productsof the Majorana fermions ξ ξ were omitted from theLagrangian (2), since they cancel out in each perturbativeorder due to charge conservation. The phases κ j includecontributions from the external magnetic flux Φ ex andfrom the average electron numbers N , on the two sidesof the interferometer: κ − κ = 2 π [(Φ ex / Φ )+( N /ν ) − ( N /ν )]+const ≡ − κ. The factor λ = √ m in the Lagrangian (2) follows fromthe normalization of the bosonic field φ − , which in theabsence of tunneling is a free right-propagating chiralfield. This field undergoes successive scattering at thetwo contacts by the tunneling terms of the Lagrangian.The scattering breaks the charge conservation and there-fore creates tunneling current. The applied voltage canbe introduced at first as a shift of the incoming field of oneof the edges: φ − √ ν V t . As one can see from Eqs. (1),such a shift translates into the shift of the tunneling field φ − − V t/λ .The thermodynamic Bethe ansatz solution of the tun-neling model was developed [19] for a single-point tunnel-ing contact with λ = 2 m by application of one-particleboundary S -matrices [20] to a distribution of the bosonicfield excitations (kinks, antikinks, and breathers) intro-duced through the massless limit of the ”bulk” sine-Gordon model. This solution was generalized [8] to thetwo tunneling contacts relevant for the MZI problem bysuccessive application of two boundary S -matrices to thesame distribution of the excitations of the model. For thecharge transport, only the kink-antikink (and vice verse)transitions are important, and their boundary S -matricesare written as S ±± j,k = ( ak/T jB ) m − e iα k i ( ak/T jB ) m − , S − + j,k = e i ( α k − κ j ) i ( ak/T jB ) m − . (3)Here the standard energy scales T jB , j = 1 ,
2, are usedto characterize the tunneling strength at the individualcontacts, and a = v √ π Γ(1 / [2(1 − ν )]) ν Γ( ν/ [2(1 − ν )]) . The explicit relation between the energy scales T jB andelectron tunneling amplitudes is given below. III. CUMULANT-GENERATING FUNCTIONFOR THE CHARGE TRANSFER DISTRIBUTION
At zero temperature, dynamics of the liquid shouldbe described with only one type of quasiparticles, e.g.,kinks, which fill out all available states with the ”bulk”distribution ρ ( k ), with k being the quasiparticle momen-tum, up to some limiting momentum A defined by theapplied voltage. Each momentum- k quasiparticle under-goes successive scattering at the two tunneling contactsindependently of other quasiparticles. The overall scat-tering process is described by the product of the twoboundary S -matrices given by Eq. (3). Our goal is to findthe cumulant-generating function ln P ( ξ ) of the chargetransfer between the two branches of the interferometer, which is defined, as usual, as the logarithm of the Fouriertransform of the probability distribution function of thetransferred charge. We measure the charge in units of theelementary electron charge by setting e = 1. The inde-pendence of the scattering events of quasiparticles withdifferent momenta implies then that ln P ( ξ ) can be foundas a sum of logarithm of the generating functions of indi-vidual momentum states, and its long-time asymptoticsis ln P ( ξ ) = t Z A dkρ ( k ) ln p ( k, ξ ) . (4)The generating function of one state with momentum kp ( k, ξ ) = 1 + τ C ( k )( e iξ −
1) (5)is defined by the total transition probability τ C ( k ) ofthe momentum- k kink into antikink. Taking the prod-uct of the scattering matrices of the two contacts tofind the total scattering matrix ˆ S ˆ S , and using theparametrization of the contact tunneling strengths as( T jB /a ) ≡ exp { θ j / ( m − } , we can write the proba-bility τ C ( k ) from the corresponding matrix element ofˆ S ˆ S as τ C ( k ) = | ( ˆ S ˆ S ) − , + | = B ( τ ( θ , k ) − τ ( θ , k )) . (6)Here τ ( θ j , k ) are the transition probabilities in the indi-vidual point contacts: τ ( θ j , k ) = | ˆ S − , + j | = [1 + k m − e − θ j ] − , (7)and the factor B characterizes interference between thetwo contacts: B ( T jB , κ ) = | T m − B + T m − B e iκ | T m − B − T m − B . (8)Without a loss of generality, we assume below a specificrelation between the tunneling strength parameters of thetwo contacts, θ ≥ θ , and write them as θ , = ¯ θ ∓ ∆ θ ,with ∆ θ ≥ P ( ξ ) in terms of the twogenerating functions ln P S for charge transfer in an in-dividual point contact that was found from the Betheansatz solution by Saleur and Weiss [18]. This deriva-tion does not need the explicit expressions for ρ ( k ) and A which can be found in [19]. Following the approachfor one contact, we first relate ln P ( ξ ) in Eq. (4) to theeffective tunneling current. To do this, we introduce thegeneralized tunneling probability τ C ( u, k ): τ C ( u, k ) ≡ [1 + ( τ − C ( k ) − e − u ] − . (9)which is the solution of the following differential equationin the new parameter u : ∂ u τ C ( u, k ) = (1 − τ C ( u, k )) τ C ( u, k ) (10)that satisfies the initial condition τ C ( u, k ) | u =0 = τ C ( k ).One can extend Eq. (5) for p to include the parameter u through the substitution τ C ( k ) → τ C ( u, k ). Equation(9) shows then that the logarithm of p extended this waycan be expressed asln p = ln[1 + τ C ( k )( e u + z − (cid:12)(cid:12) z = iξz =0 . (11)Calculating the derivatives of Eq. (11) with respect to u and ξ , and using Eqs. (9), one can see that − i∂ ξ ln p = ∂ u ln p + τ C ( u, k ) = τ C ( u + iξ, k ) . (12)Combining Eq. (12) with (4) in which p is extended toinclude the parameter u , one sees that the cumulant-generating function satisfies the following relation ∂ iξ ln P ( u, ξ ) /t = Z A dkρ ( k ) τ C ( u + iξ, k ) ≡ I ( u + iξ, V ) , (13)which expresses it through the total tunneling current I ( u, V ) in the two contacts that is defined by the gener-alized tunneling probability τ C ( u, k ).As the next step, one substitutes Eq. (6) into (9) andcasts the total tunneling probability τ C ( u, k ) into the fol-lowing form τ C ( u, k ) = B e u sinh ∆ θ sinh ∆ θ ( u ) X ± τ (¯ θ ± ∆ θ ( u ) , k ) , (14)where ∆ θ ( u ) is defined by the conditions thatcosh ∆ θ ( u ) = cosh ∆ θ + B ( e u −
1) sinh ∆ θ , (15)and ∆ θ ( u ) >
0. Differentiation of Eq. (15) shows that thecoefficient in Eq. (14) in front of the sum can be writtenas the derivative of ∆ θ ( u ): ∂ u ∆ θ ( u ) = B e u sinh ∆ θ sinh ∆ θ ( u ) . (16)Using Eqs. (14) and (16) in the definition of the tunnelingcurrent I ( u, V ) (13) we obtain an important relation I ( u, V ) = ∂ u ∆ θ ( u ) X ± ± I /m (¯ θ ∓ ∆ θ ( u ) , V ) , (17)which expresses the derivative of the cumulant-generating function (13) for the charge transfer in a sym-metric interferometer in terms of the tunneling current I /m in one point contact between the two edges withthe effective filling factor ν = 1 /m . The current in onecontact has been calculated [19] from the Bethe ansatzsolution, and its tunneling conductance G /m [ V /T B ] = G /m [( V /a ) e − θ/ [2( m − ] = I /m ( θ, V ) /V is given at zero temperature by a universal scaling func-tion expressed in the form of the low- and high-voltageexpansion series. Integrating Eq. (17) over u , and using the result in Eq. (13), we express the generating functionln P ( ξ ) = ln P ( u, ξ ) | u =0 in the following form:ln P ( ξ ) = − V t n Z ¯ θ − ∆ θ ( iξ ) θ + Z ¯ θ +∆ θ ( iξ ) θ o dθ · G /m [( V /a ) e − θ/ [2( m − ] . (18)The explicit expansion series for G /m in Eq. (18) allowintegration in each order. The integration transformsthe generating function (18) into the sum of the two gen-erating functions ln P S for charge transfer in individualcontacts and gives our key result:ln P ( ξ ) = X j =1 , ln P S (cid:0) V /T jB , e ( − j (∆ θ − ∆ θ ( iξ )) (cid:1) . (19)The single-contact generating function ln P S is known interms of the low- and high-voltage expansion series [18]:ln P S ( s, e iξ ) σ V t = ∞ X n =1 c n ( m ) mn s n ( ν − ( e inξ − , s < e ∆ , = iνξ + ∞ X n =1 c n ( ν ) n s n ( ν − ( e − inνξ − , s > e ∆ ,c n ( ν ) = ( − n +1 Γ( νn + 1)Γ(3 / n + 1)Γ(3 / ν − n ) , (20)where e ∆ = ( √ ν ) ν/ (1 − ν ) √ − ν , and σ is the conductance quantum.Equation (19) representing the total generating func-tion ln P as the sum of the two single-contact generatingfunctions ln P S seems to suggest that the charge transferin the MZI is divided into two independent processes as-sociated with the two point contacts of the MZI. Indeed,one manifestation of this division is that dependence ofeach of these processes on the bias voltage V (throughthe function P S ) is determined, as in Eq. (20), only bythe characteristics energy scale T jB of the correspondingjunction. Such a division, however, is not complete. Thetotal generating function ln P depends also on the chargedynamics in the interferometer as a whole, since each in-dividual charge transfer process in one contact triggersmultiple charge transfers involving interference betweenboth contacts of the interferometer. Information aboutthis charge dynamics enters Eq. (19) through the function∆ θ ( u ) determined by Eq. (15) and sensitive to both in-terferometer contacts. Nevertheless, the interference canbecome irrelevant if the two contacts are strongly asym-metric. For T B ≫ T B , the generating function ln P iswell approximated at low voltages by the single-contactln P S defined by T B , i.e. by the strongest electron tun-neling amplitude U – see Eq. (24) below. At large volt-ages, the generating function ln P is approximated by thesingle-contact ln P S defined by T B , i.e., by the weakestelectron tunneling amplitude U . A. Low-voltage behavior of the generating function
At small voltages, when
V < T jB e ∆ for both j = 1 , P S (20) for both terms in Eq. (19) can becombined as follows:ln P ( ξ ) = σ V t ∞ X n =1 c n ( m ) mn X j =1 , (cid:0) V /T jB (cid:1) n ( m − · (cid:2) cosh( n ∆ θ ( iξ ))cosh( n ∆ θ ) − (cid:3) . (21)Since cosh( n ∆ θ ( iξ )) is a polynomial of cosh(∆ θ ( iξ )), andthe latter, according to Eq. (15), is a linear function of e iξ , this expansion of the MZI generating function showsthat at low voltages, the charge transfer between theedges of the MZI is quantized in units of electron charge.More explicitly, using the standard expansion ofcosh nx (see Eq. 1.331.4 in Ref. 21) and the relationsthat follow from Eqs. (8) and (15):cosh ∆ θ ( iξ ) = cosh ∆ θ [1 + R ( e iξ − ,R ≡ B tanh ∆ θ = | T m − B + T m − B e iκ | T m − B + T m − B , (22)cosh ∆ θ = T m − B + T m − B T B T B ) ( m − , we bring Eq. (21) into the following form:ln P ( ξ ) = σ V t ∞ X n =1 c n ( m ) mn (cid:2) X j (cid:0) V /T jB (cid:1) m − (cid:3) n · n [1 + R ( z − n + n [ n/ X l =1 ( − l C n − l − l − l (2 cosh ∆ θ ) l · (cid:2) R ( z − (cid:3) n − l o(cid:12)(cid:12)(cid:12) z = e iξ z =1 . (23)Equation (23) quantifies our previous conclusionsabout properties of the MZI charge transfer statistics (19)in the low-voltage regime. In the limit of strongly differ-ent contacts, T B ≫ T B , one finds that R → θ ≫
1, so that the charge transfer statistics (23)approaches that of one point contact [18] characterizedby T B (i.e., the strongest electron tunneling amplitude U ) with corrections in T B /T B also quantized in theelectron charge units. The n th term in the expansion ofthis statistics in powers of bias voltage V corresponds inthis case to tunneling of exactly n electrons. By contrast,the n th order term of the general MZI transfer statistics(23) involves transfer of all numbers of electrons up to n .In the lowest order in V , the MZI statistics reduces tothe Poisson distribution, with the coefficient in front of( z −
1) in Eq. (23) is equal to the average electron tun-neling current. One can check this starting from Eq. (2)by direct perturbative calculation [8], if the energy scales T jB are expressed through the electron tunneling ampli-tudes U j : T jB = 2 D [Γ( m ) /U j ] / ( m − . (24) B. Large-voltage behavior of the generatingfunction
At large voltages,
V > T jB e ∆ , the combinationlarge-voltage expansions of both single-contact generat-ing functions ln P S in Eq. (19) brings the total MZI gen-erating function into the following form:ln P ( ξ ) = σ V t ∞ X n =1 c n (1 /m ) n X j (cid:0) V /T jB (cid:1) n (1 − m ) /m · (cid:2) cosh( n ∆ θ ( iξ ) /m )cosh( n ∆ θ /m ) − (cid:3) . (25)In terms of the parameters introduced in Eq. (22), thisequation can be rewritten as:ln P ( ξ ) = σ V t ∞ X n =1 c n (1 /m ) n n/m h X j ( T jB /V ) m − i nm · X ± h R ( z − ± (cid:0) [1 + R ( z − − cosh − ∆ θ (cid:1) i nm (cid:12)(cid:12)(cid:12) z = e iξ z =1 . (26)We can see again that in the asymmetric limit, T B ≫ T B , when R → θ ≫
1, Eq. (26) for thecharge transfer statistics reduces to that of a single pointcontact [18]. The dominant contact is now character-ized by the larger quasiparticle tunneling amplitude W (i.e., smaller electron tunneling amplitude U ) and cor-responding energy scale T B , related to W as: T jB = 2 mD [ W j / Γ(1 /m )] m/ ( m − . (27)The n th order term in the expansion (26) of the gen-erating function corresponds to the transfer of the frac-tional charge n/m by n quasiparticles. One can see, how-ever, that the MZI transfer statistics (26) does not con-tain in general the terms e iξ/m that would corresponddirectly to transfer of individual quasiparticles of charge1 /m . In particular, the n = 1 term of the expansionthat gives the leading large-voltage contribution to thestatistics, can not be interpreted as a Poisson process oftunneling of independent quasiparticles, in contrast tothe leading low-voltage term that did represent Poissonprocess of individual electron tunneling events. The rea-son for this is the m -state dynamics of the effective fluxthrough the interferometer associated with the quasipar-ticle tunneling, which introduces correlations in the tun-neling process. These correlations can be most easily un-derstood in the description of the quasiparticle tunnelingbased on kinetic equation. Such an equation is discussedin the next Section. IV. KINETIC EQUATION FOR THELARGE-VOLTAGE CHARGE TRANSFER
To derive kinetic equation that reproduces the large-voltage asymptotics of the generating function (25), andtherefore provides a simple physical picture of the dy-namics of quasiparticle tunneling, we start by rewritingthis asymptotics in terms of the quasiparticle tunnelingamplitudes W j . From the last relation in Eq. (22), wehavecosh(∆ θ /m ) = 12 h(cid:0) T B T B (cid:1) m − m + (cid:0) T B T B (cid:1) m − m i . (28)Using this equation to transform the leading, n = 1, termin Eq. (25), and replacing the energy scales T jB by thequasiparticle amplitudes W j with the help of Eq. (27),we have:ln P ( z ) = tK ( V ) (cid:2) W W cosh (cid:0) ∆ θ ( z ) m (cid:1) − X j W j (cid:3) , (29)where K ( V ) = σ V (2 mD/V ) m − /m c (1 /m ) / Γ (1 /m ).Kinetic equation describing the quasiparticle tunnelingcan be written down based on the following considera-tions. The general picture of the quasiparticle dynamicsin the MZI discussed in the Introduction implies that thequasiparticles create statistical contribution to the effec-tive flux through the interferometer. Because of this sta-tistical contribution, the MZI can be found in m separatestates which differ by the effective flux, with each tun-neling quasiparticle changing successively the state l into l − m . Since the total rates of the quasiparticletunneling in the MZI depend on the effective flux, suchdynamics of flux introduces correlations into quasiparti-cle transitions, separating naturally the processes of suc-cessive quasiparticle tunneling events into the groups of m transitions. As usual, to cast the kinetic equation gov-erning this flux dynamics into the form appropriate forthe calculation of the cumulant-generating function, wemultiply transition probabilities by a factor z /m = e iξ/m that keeps track of the transferred charge. Then, we in-troduce the probabilities d l,n ( t ) that at time t the MZIis in the state l and n quasiparticles have been trans-ferred through it. Combining these probabilities into an m -dimensional vector Q l ( z, t ) = P n d l,n ( t ) z n/m , one canwrite the kinetic equation in the following matrix form: ∂ t Q l ( z, t ) = X l ′ M ( z ) l,l ′ Q l ′ ( z, t ) . (30)According to the qualitative picture of quasiparticletunneling discussed above, the transition matrix has asimple form, with the only non-vanishing elements arethose on the main diagonal, l = l ′ , and those with l = l ′ − M ( z ) l,l ′ = − γ l δ l,l ′ + γ l ′ z /m δ l,l ′ − . (31)Here the Kronecker symbol δ l,l ′ is defined modulo m . Theleading large-time asymtotics of the generating function for the probability distribution evolving according to thekinetic equation (30) is (see, e.g., [22]): ln P ( z ) = t Λ,where Λ is the maximum eigenvalue of the transition ma-trix (31). The structure of this matrix shows directly thatthe characteristic equation det( M − Λ) = 0 has the form: m − Y l =0 ( γ l + Λ) − z m − Y l =0 γ l = 0 . (32)The maximum eigenvalue Λ is the solution of this equa-tion which goes to zero when z →
1, since all other eigen-values of the matrix M ( z = 1) are negative.Before trying to establish the general relation betweenthe generating function obtained from this equation andthe generating function (29), we consider the simple case m = 2. In this case, Eq. (29) for the large-voltage asymp-totics of the generating function (25), can be simplifiedfurther. First, we have from Eq. (22):cosh( ∆ θ ( z )2 ) = h cosh ( ∆ θ R z −
1) cosh ∆ θ i / . Using this relation, Eq. (22), and Eq. (28) with m = 2,we transform Eq. (29) into:ln P ( z ) = tK ( V ) nh ( X j W j ) + | W + W e iκ | · ( z − i / − X j W j o . (33)On the other hand, in the kinetic-equation approach,Eq. (32) be solved readily for m = 2 giving the followingexpression for Λ:Λ = (cid:16)(cid:2)(cid:0) X j γ j (cid:1) + 4 γ γ ( z − (cid:3) / − X j γ j (cid:17) / . (34)This equation describes the generating function for thestatistics of any transfer process consisting of the twosteps with the rates γ , , e.g., incoherent charge transferthrough a resonant level [22]. Comparison of Eqs. (34)and (33) shows that the generating function (34) ob-tained from the kinetic equation reproduces the large-voltage asymptotics (33) of the generating function (25),if the tunneling rates are taken as γ , = K ( V ) | W + W e iφ , | , φ l = ( κ + π ) / πl . (35)Equation (35) for the tunneling rates agrees with thephysical picture of quasiparticle tunneling discussedabove. Statistical contribution to the effective fluxthough the MZI means that tunneling of each quasipar-ticle changes the phase between the interferometer con-tacts by 2 π/m = π in agreement with the quasiparticleanyonic exchange statistics. Equation (35) also showsthat the quasiparticles see a phase shift π/ κ/ m taking the tunneling rates γ l as: γ l = K ( V ) | W + W e iφ l | , φ l = φ + 2 πl/m , (36)with some unknown φ . With these tunneling rates,Eq. (32) reads m − Y l =0 (cos φ l + λ + W + W W W ) = z m − Y l =0 γ l K ( V ) W W , (37)where λ ≡ Λ /K ( V ). To further transform Eq. (37), weuse the basic identity: x m − m − Y l =0 ( x − e i πl/m ) , (38)and the two identities that follow directly from it: m − Y l =0 φ l φ m + π m −
12 ) , m − m − Y l =0 [cos φ l + cosh( ∆ θm )] = cosh ∆ θ (39) − ( − ) m cos( mφ ) . The first one is obtained essentially by taking x = − e iφ in Eq. (38), while the second one follows from the first ifthe sum of the cosines is transformed into their product.By direct comparison of the second identity in (39) withEq. (37) we see that Λ as defined by ln P ( z ) /t in Eq. (29)indeed solves Eq. (32) ifcosh ∆ θ = ( − ) m cos( mφ ) + z m − Y l =0 γ l K ( V ) W W . (40)Making use of Eq. (38) one more time, we calculate theproduct on the right-hand-side of Eq. (40): m − Y l =0 γ l K ( V ) W W = | W m e im ( φ − π ) − W m | W m W m . (41)One can see that with this expression for the prod-uct, Eq. (40) precisely coincides with the definition ofcosh ∆ θ ( z ) by Eq. (22). Indeed, combining all three re-lations in Eq. (22) one can express cosh ∆ θ ( z ) ascosh ∆ θ ( z ) = − cos κ + | T m − B + T m − B e iκ | T B T B ) ( m − z . (42)Replacing T jB in Eq. (42) with the amplitudes W j through Eq. (27), we see that Eq. (42) precisely coincideswith Eqs. (40) and (41), if cos κ = ( − ) ( m +1) cos( mφ ), i.e.,if the phase φ is chosen to satisfy the condition mφ = κ + ( m − π . (43) The deviation of the phase φ from κ/m in this equationis important only for even m . In this case, it producesthe shift ( m − π/m of the interference phase in thequasiparticle tunneling rates from the value induced bythe external magnetic field. This shift coincides with thephase acquired by one of the two Klein factors of the MZIquasiparticle tunneling action of the MZI [15] in the flux-diagonal representation due to the m -power periodicitycondition for the Klein factors, which corresponds physi-cally to the requirement of the proper exchange statisticsbetween electrons and quasiparticles. This phase shift en-sures, actually, that there is no shift in the interferencepattern of the tunnel current in the interferometer (seethe final results below) between the regimes of electronand quasiparticle tunneling. V. CUMULANTS OF THE CHARGETRANSFER DISTRIBUTION
So far, we have established the interpretation of thelow- and high-voltage asymptotic behavior of the chargetransfer statistics in terms of, respectively, tunneling ofindividual electrons and quasiparticles. In this Section,we calculate the charge transfer cumulants in these twolimits, with the emphasis on the quasiparticle limit whichexhibits the non-trivial behavior of the cumulants. Wealso will use the generating function found in Sec. IIIfor arbitrary voltages to calculate the full voltage de-pendence of the cumulants, and to study the crossoverbetween the two asymptotic regimes of electron andquasiparticle tunneling. The cumulants of the charge N ( t ) transferred through the interferometer during alarge time interval t can be found from the cumulant-generating function (13) by the standard relation: h N j ( t ) i c /t = ∂ ju ln P | u =0 /t = ∂ j − u I ( u, V ) | u =0 , (44)where the current I ( u, V ) is given by Eq. (17).The first cumulant gives the average tunneling current: I ( V ) ≡ I (0 , V ) = h N ( t ) i /t . At arbitrary voltages, theaverage current was calculated before in [15]. Its large- V asymptotics can also be obtained directly from Eqs. (26)and (27): I ( V ) = K ( V ) m B ( W − W ) = 1 P l γ − l , (45)where the interference factor B (8) can be expressed interm of the quasiparticle amplitudes W , as B ( W j , κ ) = | W m + W m e iκ | W m − W m . (46)The second equality in Eq. (45) provides direct interpre-tation of the asymptotics of the average current in termsof the quasiparticles transitions [7]. It can be proven for-mally by making use of the identity X l γ − l = W ∂ W − W ∂ W K ( V )( W − W ) ln[ Y l γ l ] , (47)which can be obtained by directly differentiating indi-vidual rates γ l in this equation. On the other hand,differentiating the product of all γ l s together, as givenby Eq. (41), and using Eq. (43), we obtained the sec-ond equality in (45). This equality agrees naturally withthe simple solution of the quasiparticle kinetic equation,which gives the average tunneling rate as inverse of theaverage tunneling times in the different states of the in-terferometer.The second cumulant h N ( t ) i defines the spectral den-sity of the current fluctuations at zero frequency S I (0) = h N ( t ) i c /t , which at zero temperature reflects the shotnoise associated with the charge transfer processes. FromEq. (17), one can write the first derivative of the currentas ∂ u I ( u, V ) = I ( u, V ) × ∂ u ln[ ∂ u ∆ θ ( u )] − ( ∂ u ∆ θ ( u )) X ± ∂ ¯ θ I /m (¯ θ ∓ ∆ θ ( u ) , V ) . (48)Substituting this formula into Eq. (44) and calculatingthe derivatives of ∆ θ ( u ) (16) for u → S I (0) = (1 − B coth ∆ θ ) I − B X j =1 , ∂ θ I /m ( θ j , V ) . (49)It is convenient to characterize the short noise repre-sented by this spectral density through the Fano factor F defined as F = S I (0) /I . In the case of MZI, the Fanofactor reflects both the charge and statistics of the tun-neling excitations and illustrates the transition betweenthe electron and quasiparticle regimes. In the low-voltagelimit, F = 1 as a result of the regular Poisson process ofelectron tunneling. To find the Fano factor in the quasi-particle, large-voltage, limit, we start with Eq. (49) whichgives the following general expression for F : F = 1 − B n coth ∆ θ − X j ∂ θ I /m ( θ j , V ) · (cid:2) X j ( − j I /m ( θ j , V ) (cid:3) − o . (50)Using Eq. (22), the fact that in the large-voltage limitonly one quasiparticle tunneling term ∝ W can be keptin the current I /m , cf. Eq. (45), and that with theparametrization of the energy scales T jB with θ intro-duced above, W ∝ e θ/m , we get from Eq. (50): F = 1 − B n W m + W m W m − W m − m W + W W − W o . (51)The Fano factor (51) corresponds to the dynamics ofquasiparticle tunneling as described by the kinetic equa-tion (30). This can be seen by following the steps similarto that taken above for the average current. Applying thedifferential operator from Eq. (47) to individual terms in the sum of the inverse tunneling rates γ l , one obtainsdirectly the following identity: − W ∂ W − W ∂ W K ( V )( W − W ) X l γ − l = X l γ − l . On the other hand, replacing the sum of inverse γ l s inthis equation with the corresponding expression fromEq. (45): X l γ − l = m/ [ BK ( V )( W − W )] , and performing differentiation, we see that the large-voltage asymptotics (51) of the Fano factor can be writ-ten in terms of the tunneling rates γ l as F = X l γ − l / ( X l γ − l ) , This result agrees with the calculation [23] based directlyon the kinetic equation. Because of the complex natureof the quasiparticle tunneling dynamics characterized by m different tunneling rates γ l , F is not equal simply tothe quasiparticle charge 1 /m but varies as a function ofparameters, e.g. the interference phase κ , between 1 /m and 1. T ) F m=3T /T =13 5 10 FIG. 2: The zero-temperature Fano factor F of the tunnelcurrent in the Mach-Zehnder interferometer formed by two ν = 1 / m = 3, as a function of the bias voltage V for different degrees of asymmetry of the tunneling strengthof the two contacts characterized by the T B /T B ratio. Thesolid curves corresponds to the case of complete constructiveinterference, κ = 0; for the dashed curves, κ = π . In the lattercase, F = 1 identically for identical contacts, T B /T B =1. The curves illustrate the transition between the electronregime F = 1 at small voltages to the quasiparticle m -statetunneling dynamics at large voltages. The transition regionis characterized by the Fano factor F reaching the minimumbelow the quasiparticle minimum 1 /m = 1 / At arbitrary bias voltage V , the Fano factor F shouldbe plotted numerically. Figure 2 shows F in the case m = 3 which corresponds, e.g., to tunneling between thetwo ν = 1 / κ = 0, and completedestructive interference, κ = π . The range of variation of F with the interference phase κ decreases with increas-ing junction asymmetry. In general, the curves showthe transition between electron tunneling with F = 1at small voltages V to quasiparticle tunneling at largevoltages. In the quasiparticle regime, F is still can besignificantly different from 1 / γ l ∝ cos ( φ l / κ ≃ π , the tun-neling rate in one of the flux states of the interferometer, l = 0, is much smaller than the rates in the two otherstates. This mean that on the relevant large time scaleset by the smallest rate, the three quasiparticles transi-tion that transfer interferometer from state l = 0 backto itself happen almost simultaneously, so that the threequasiparticle charges 1 / F back to 1.Finally, we study the third charge transfer cumulantthat characterizes the asymmetry around average of thetransferred charge distribution, and has been measuredexperimentally for electron tunneling in metallic tunneljunctions – see, e.g., [24, 25], and in quantum point con-tacts [26]. As for the other cumulants, the large-timeasympotic of the third cumulant is linear in time, and itcan be characterized by the coefficient C ≡ h N i c /t . Tocalculate this coefficient, we first find the second deriva-tive of the tunnel current from Eq. (48): ∂ u I ( u, V ) = (cid:0) ∂ u ln ∂ u ∆ θ ( u ) + [ ∂ u ln ∂ u ∆ θ ( u )] (cid:1) I − X ± (cid:2) ∂ u ∆ θ∂ u ∆ θ∂ ¯ θ ∓ ( ∂ u ∆ θ ) ∂ θ (cid:3) I /m (¯ θ ∓ ∆ θ, V ) . (52)Derivatives of ∆ θ ( u ) here can be found from Eq. (16). Inparticular, the coefficient in front of I in Eq. (52) can beexpressed as: ∂ u ln ∂ u ∆ θ ( u ) + [ ∂ u ln ∂ u ∆ θ ( u )] = 1 − θ ( u ) × ∂ u ∆ θ ( u ) + [3 coth ∆ θ ( u ) − ∂ u ∆ θ ( u )) . (53)Substitution of Eqs. (52), (53), and (16) into Eq. (44)gives us the coefficient C of the third cumulant: C = (cid:2) − B coth ∆ θ + B (3 coth ∆ θ − (cid:3) I − X j =1 , (cid:2) B (1 − B coth ∆ θ ) ∂ θ I /m ( θ j , V ) (54)+( − ) j B ∂ θ I /m ( θ j , V ) (cid:3) . The ratio F = C /I has also been suggested [27] asa possible alternative to the Fano factor to characterize the charge of the tunneling particles. Indeed, in a Poissonprocess, F is equal to the Fano factor multiplied by thetunneling charge. Therefore, in the case of MZI, F (54)reduces to 1 in the low-voltage limit, as a result of theregular Poisson electron tunneling. In the quasiparticle,large-voltage, limit, repeating the calculation similar tothat leading to Eq. (50), we can relate both factors asfollows: F = 3 F − B n m + 12 W m W m ( W m − W m ) − m ( W + W )( W − W ) ( W m + W m )( W m − W m ) o . (55) VI. CHARGE TRANSFER STATISTICS FOR m = 2 . For general m , the results for the cumulants of thecharge transfer statistics in the MZI discussed above cannot be presented in a finite analytical form for arbitrarybias voltages. The situation is simpler for m = 2, whenthe kink-quasiparticles of the ”bulk” sine-Gordon modelthat provide the basis for the Bethe-ansatz solution ofthe MZI transport are the regular fermions (though car-rying charge 1 / ρ ( k ) in Eq. (4)is the Fermi-Dirac step-function. In practice, the m = 2regime should take place in the MZI formed by the twoedges of different Quantum Hall liquids, with filling fac-tors ν = 1 / ν = 1. Individual tunneling contactsof this type have been realized experimentally [28]. Us-ing the Fermi-Dirac property of the distribution function ρ ( k ) for m = 2, one can obtain the cumulant generatingfunction ln P ( ξ ) either by integration in Eq. (4), or equiv-alently, by direct substitution into Eq. (18) of the knownsingle-contact tunnel conductance G / , which can be ex-pressed simply as G / ( s ) = σ [1 − arctan(2 s ) / (2 s )] / P ( ξ ) = σ V t X j =1 , (cid:16) y j ( u ) arctan[1 /y j ( u )]+(1 /
2) ln[1 + y j ( u )] (cid:17)(cid:12)(cid:12)(cid:12) u = iξu =0 , (56)where y j ( u ) ≡ y j (0) exp { ( − j (∆ θ ( u ) − ∆ θ ) / } , and y j (0) ≡ T jB / (2 V ).This generating function can be combined withEq. (44) to calculate the cumulants of the transferredchange distribution for m = 2 using the same steps asin the previous section. Alternatively, one can substituteEq. (4) into (44) and perform the integration directly.Below we briefly discuss the first three cumulants ob-tained in this way. The first cumulant gives the averagetunneling current in the MZI as [15]: I = σ B (Γ I − Γ I ) , (57)where I j ≡ arctan( V / j ), and Γ j ≡ T jB / DW j /π are the characteristic quasiparticle tunneling rates in sep-arate contacts. Using the fact that K ( V ) = σ D for0 m = 2, one can see explicitly that the current agrees atlarge voltages with Eq. (45) that follows from the quasi-particle kinetic equation with the tunneling rates (35): I = πσ | Γ + Γ e iκ | Γ + Γ = γ γ γ + γ . (58)It is interesting to note that this agreement relies stronglyon the shift of the interference phase (43) from theexternally-induced phase κ . ) F m=2 / =1 3 10010 FIG. 3: The zero-temperature Fano factor F of the tunnelcurrent for m = 2, i.e., in the Mach-Zehnder interferometerformed by edges with ν = 1 / ν = 1, as given by Eq. (59).The solid curves corresponds to the case of complete construc-tive interference, κ = 0; for the dashed curves, κ = π . In thelatter case, F = 1 identically for identical contacts. In gen-eral, the transition from the electron regime F = 1 at smallvoltages to the 2-state tunneling dynamics of quasiparticles atlarge voltages is characterized by the Fano factor F reachingthe minimum in the crossover region. The second cumulant gives the following expression forthe Fano factor at arbitrary voltages, including the tran-sition region between electron and quasiparticle tunnel-ing: F = 1 − | Γ + Γ e iκ | Γ I − Γ I n Γ (Γ I − Γ I )(Γ − Γ ) +12(Γ − Γ ) X j =1 , Γ j (cid:2) I j + V / j V / j ) (cid:3)o . (59)This equation is plotted in Fig. (3) and describes thetransition from F = 1 for electron tunneling at smallvoltages to F = 1 − | Γ + Γ e iκ | (Γ + Γ ) for quasiparticle tunneling at large voltages. One can seethat the quasiparticle charge e/ κ = 0, when the total quasiparticle tunneling rates (35) coincide, γ = γ , regardless of the relationbetween the individual rates Γ j . Similarly to the case m = 3 illustrated in Fig. 2, the Fano factor reduces toelectron value 1 even in the quasiparticle regime, if Γ ≃ Γ and κ = π . In this case, one of the total quasiparticaltunneling rates γ is much smaller than the other, so onthe relevant large time scale set by the smaller rate thequasiparticles effectively tunnel together, restoring F to1. ) =1 F m=2 / =3 0.31000.5 FIG. 4: Alternative “Fano factor” F = C /I (60) related tothe third cumulant of the tunnel current noise in the Mach-Zehnder interferometer with m = 2 and zero temperature,as a function of the interferometer phase κ . The curves areplotted for several bias voltages V between the interferometeredges, and illustrate the transition between the electron andquasiparticles tunneling with increasing voltage. The tran-sition is characterized by the non-monotonous change of F ,which reaches minimum at the intermediate voltages. Calculating the third cumulant, we find the followingexpression for the “alternative” Fano factor: F = 3 F − | Γ + Γ e iκ | I − Γ I )(Γ − Γ ) X j =1 , n ( − j +1 · j I j (Γ j + 10Γ j Γ j ′ + 5Γ j ′ )(Γ − Γ ) + 3 V − Γ ) (60) · Γ j + 3Γ j ′ V / j ) + ( − j +1 V [1 + ( V / j ) ] o , where j ′ is defined by j, j ′ = 1 , j ′ = j . Equation(60) is plotted in Fig. 4, which shows F as a functionof the interference phase κ at several voltages. Voltagedependence of F is qualitatively very similar to that ofthe Fano factor shown in Fig. (3): it approaches 1 atsmall voltages, in agreement with the underlying Poissontunneling process of electrons. At large voltages, Eq. (60)reduces to the following form F = 3 F − | Γ + Γ e iκ | (Γ + Γ ) . F is non-monotonic, with aminimum between the regimes of electron and quasipar-ticle tunneling. The main qualitative difference betweenthe noise-related Fano factor and its third-cumulant al-ternative is that the minimum of F can become nega-tive for some values of parameters (regime not shown inFig. 4). VII. CONCLUSION
Starting from the exact solution of the tunneling modelof symmetric Mach-Zender interferometer in the FQHEregime, we have calculated the statistics of the chargetransfer between interferometer edges. The obtainedstatistics shows the transition from electron tunneling atlow voltages to tunneling of anyonic quasiparticles of the fractional charge e/m and statistical angle π/m at largevoltages. Deep in the electron tunneling regime, the dy-namics of charge transfer is represented by the standardPoisson process. Dynamics of quasiparticle tunneling ismore complicated and reflects the existence of m effec-tive flux states of the interferometer. The interferencephase between the quasiparticle tunneling amplitudes intwo contacts of the interferometer contains a contributionfrom the quasiparticle exchange statistics, making thequasiparticle tunneling rates in different interferometerstates different. In general, the transition from electronto quasiparticle tunneling is reflected in the Fano factor F or its third-cumulant alternative F , which both reachminima in the transition region. However, in the regimeclose to complete destructing interference (interferome-ter phase κ = π and equal tunneling strength in the twocontacts), both F and F have electron value 1 for allvoltages.V.P. acknowledges support of the ESF Science Pro-gram INSTANS, and the grant PTDC/FIS/64926/2006. [1] Y. Ji, Nature , 415 (2003); I. Neder, M. Heiblum, Y.Levinson, D. Mahalu, and V. Umansky, Phys. Rev. Lett. , 016804 (2006).[2] L.V. Litvin, H.-P. Tranitz, W. Wegscheider, and C.Strunk, Phys. Rev. B , 033315 (2007).[3] P. Roulleau et al. , Phys. Rev. B , 161309(R) (2007).[4] V.J. Goldman and B. Su, Science , 1010 (1995).[5] D.V. Averin and J.A. Nesteroff: Phys. Rev. Lett. ,096801 (2007).[6] C.L. Kane, Phys. Rev. Lett. , 226802 (2003).[7] K.T. Law, D. E. Feldman, and Y. Gefen Phys. Rev. B , 045319 (2006).[8] V.V. Ponomarenko and D.V. Averin, Phys. Rev. Lett. , 066803 (2007).[9] L. Saminadayar et al. , Phys. Rev. Lett. , 2526 (1997).[10] R. de-Picciotto et al. , Nature , 162 (1997).[11] D.V. Averin and V. J. Goldman, Solid State Commun. , 25 (2001).[12] S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev.Lett. , 166802 (2005).[13] A. Stern, Ann. Phys. , 204 (2008).[14] X.G. Wen, Adv. Phys. , 405 (1995).[15] V.V. Ponomarenko and D.V. Averin, Phys. Rev. B , 045303 (2009).[16] V.V. Ponomarenko and D.V. Averin, Phys. Rev. B ,241308(R) (2005).[17] L.S. Levitov and G.B. Lesovik, JETP Lett. , 230(1993).[18] H. Saleur and U. Weiss, Phys. Rev. B , 201302(R)(2001).[19] P. Fendley, A.W.W. Ludwig, and H. Saleur, Phys. Rev.Lett. , 3005 (1995); Phys. Rev. B , 8934 (1995).[20] S. Ghoshal and A.B. Zamolodchikov, Int. J. Mod. Phys. A9 , 3841 (1994).[21] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Se-ries, and Products , (Academic Press, 2007).[22] D.A. Bagrets and Yu.V. Nazarov, Phys. Rev. B ,085316 (2003).[23] D.E. Feldman et al. , Phys. Rev. B , 085333 (2007).[24] A.V. Timofeev et al. , Phys. Rev. Lett. , 207001 (2007).[25] Q. Le Masne et al. , Phys. Rev. Lett. , 067002 (2009).[26] G. Gershon et al. , Phys. Rev. Lett. , 016803 (2008).[27] L.S. Levitov and M. Reznikov, Phys. Rev. B , 115305(2004).[28] S. Roddaro et al. , Phys. Rev. Lett.103