Current correlations of Cooper-pair tunneling into a quantum Hall system
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Current correlations of Cooper-pair tunneling into a quantum Hall system
Andreas B. Michelsen,
1, 2
Thomas L. Schmidt, and Edvin G. Idrisov Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK (Dated: April 23, 2020)We study Cooper pair transport through a quantum point contact between a superconductorand a quantum Hall edge state at integer and fractional filling factors. We calculate the tunnellingcurrent and its finite-frequency noise to the leading order in the tunneling amplitude for dc and acbias voltage in the limit of low temperatures. At zero temperature and in case of tunnelling intoa single edge channel both the conductance and differential shot noise vanish as a result of Pauliexclusion principle. In contrast, in the presence of two edge channels, this Pauli blockade is softenedand a non-zero conductance and shot noise are revealed.
PACS numbers: 73.43.-f, 73.43.Lp, 73.43.Jn, 73.63.-b, 85.75.Nn, 74.78.-w
I. INTRODUCTION
The quantum Hall (QH) effect [1, 2] is one of themost important effects of modern mesoscopic physics. Itsmain observable feature is the precise quantization of theHall conductance to the value G H = νe /h , where ν isthe so-called filling factor. In a two-dimensional elec-tron gas (2DEG) at integer ( ν ∈ N ) or certain fractional[ ν = 1 / (2 n + 1) where n ∈ N ] filling factors, electrontransport occurs through one-dimensional (1D) channelslocated close at the edges of the sample [3]. The electronmotion in these 1D channels is chiral, i.e., the electronspropagate in one direction with a speed of the order of10 to 10 m/s [4–6]. Electrons in such edge channelspropagate ballistically without backscattering, in a waysimilar to photons in wave guides. This analogy has ledto the emergence of the field of electron quantum opticswhich aims to realize quantum-optics-type experimentswith electrons [7].Recent progress in experimental techniques at thenanoscale has allowed experimentalists to create hybridmesoscopic systems where QH edge states are coupled toother edge states [7–9], to quantum dots [6, 10–13], toOhmic contacts [14–17], or to superconductors [18–22].This development has provided a successful platform tostudy some of the fundamental questions of mesoscopicphysics, such as phase-coherence [23–29], charge [14, 30–32] and heat quantization [16, 33], equilibration [34–38]and entanglement [9, 39]. A particularly important setupfor studying the transport properties of hybrid meso-scopic systems is based on QH edge states coupled toa metal via a quantum point contact (QPC), a narrowregion between two electrically conducting systems. SuchQPCs allow for tunneling experiments in the presence ofan applied dc or ac bias voltage. In particular, the cur-rent and shot noise through a QPC connecting a QH edgestate have been investigated in many experiments [7].These experiments have made it possible to study thecrossover from Fermi liquid to non-Fermi liquid phasesin the I − V (current-voltage) characteristics and in thecorresponding noise measurements.To study the transport in mesoscopic devices based on QH edge states, the low-energy effective theory de-veloped by Wen is commonly used [40]. This bosoniza-tion approach shows that fractional edge states of theLaughlin series [ ν = 1 / (2 n + 1)] can be modelled as Lut-tinger liquids with Luttinger parameter K = ν . Thistheory has allowed the interpretation of the experimen-tal data [7, 9] obtained for transport properties of 1Dchiral edge states. Moreover, the tunnelling current andconductance, as well as the zero-frequency and finite-frequency non-equilibrium noise between edge states werestudied theoretically [41–57].In these works, it was already shown that the typicalbehavior of the tunneling conductance of Laughlin frac-tional QH chiral edge states at low temperatures followsa power law, i.e., G ( T ) ∝ T g − , where T is the temper-ature and the parameter g is equal to ν or 1 /ν dependingon the geometry of QPC. Additionally it was shown thatthe behavior of the dc I − V characteristic at zero temper-ature, low bias and g = 1 is non-Ohmic, I dc ( V ) ∝ V g − ,which is associated with the non-Fermi (Luttinger) liq-uid phase. In the case of a time-dependent bias voltage˜ V ( t ) = V + V ac ( t ) with frequency Ω and amplitude V inthe periodic ac part, the dc component of the current wasfound to have the form I dc ∝ P n a n ( κ ) | e ∗ V + n ~ Ω | g − ,where κ ∝ e ∗ V / ~ Ω, e ∗ is the effective charge of the tun-neling particle and the coefficients of the sum depend onthe form (cosine, Lorentzian [7]) of the ac part of the biasvoltage. Apart from the I − V characteristic, the studyof the zero- and finite-frequency noise in these referencesrevealed a power-law dependence of the noise on the fre-quency at low temperatures. For instance, to the lowestorder in the tunnel coupling, the finite-frequency sym-metric noise at frequency ω is proportional to the sumof two terms | ω ± ω | g − , which exhibit singularities atfrequencies ω = e ∗ V / ~ . In the case of a time-dependentbias voltage, the result gets modified similarly to the cur-rent to | ω ± ( ω + n Ω) | g − , and again exhibits singular-ities at certain frequencies. The noise thus provides oneof the most straightforward methods to measure the ef-fective charge e ∗ of tunneling Laughlin quasiparticles [7].In the recent past, it has become possible to investigatesuch transport problems not only between identical bal-listic chiral QH states but also between distinct systems,such as QH edge states and superconductors, both theo-retically [58–67] and experimentally [19, 20, 22, 68, 69].Motivated by this progress, we investigate the noise prop-erties of the tunneling current between a superconductorand QH edge states at integer and Laughlin filling factors.We show that the previously demonstrated Pauli block-ade [58] in the tunneling current at filling factor ν = 1also manifests itself in shot noise experiments. We expectthat one can investigate shot noise and finite-frequencynoise experimentally, as was done in Refs. [70] and [7]where the authors measured the dependence of noise ontemperature and applied bias.The rest of this article is structured as follows. InSec. II, we introduce the model of a QPC in the spirit ofRef. [58]. In Sec. III, we calculate the tunneling currentand the conductance perturbatively for a finite dc bias,which we will need in the following section. In Sec. IV,we calculate the finite-frequency noise in the dc regime.Sec. V is devoted to the derivation of the tunneling cur-rent and the finite-frequency noise for a periodic ac biasvoltage. Finally, we present our conclusions and somefuture perspectives in Sec. VI. Details of the calculationsand additional information are presented in the Appen-dices. Throughout the paper, we set | e | = ~ = k B = 1. II. THEORETICAL MODEL OF A QUANTUMPOINT CONTACT
We start by introducing the Hamiltonian of a QPC be-tween a QH edge state at filling factor ν and an s -wavesuperconductor (see Fig. 1). To describe this system the-oretically, we use the phenomenological model presentedin Ref. [58]. The effective Hamiltonian of the quantumHall system is the sum of a term describing the unper-turbed QH edge and a term describing the tunneling be-tween the edge state and the superconductor,ˆ H = ˆ H QH + ˆ H T . (1)The exact form of the term ˆ H QH depends on the fillingfactor and the cases of integer and fractional states as wellas of a pair of co-propagating states will be presented innext sections.Tunneling between the superconductor and the edgestate is a two-step process. A Cooper pair from the su-perconducting condensate first splits into two electronswith opposite spins in a singlet state, both of which tun-nel into the QH system. However, since the edge stateis spin-polarized, a further spin-flip process, which canbe brought about by spin-orbit coupling, is necessary toreach the final state which contains two electrons withthe same spin propagating in the edge state.At temperatures much smaller than the superconduct-ing gap, the Cooper pairs can be described as the meanvalue of the bosonic field ˆ c describing the superconduct-ing condensate, ∆ = h ˆ c i . It was shown in Ref. [58] thatin this regime, the combination of electron pair tunneling SC QH B SC QH V ν =1 ν =2 τ τ BV FIG. 1. Schematic representation of the system: a QPC withtunneling amplitude τ connects a superconductor (SC) to thechiral edge states of an integer QH phase at filling factor ν . At ν = 1 both electrons of the Cooper pair would have to occupythe same state, leading to a Pauli blockade, while at ν = 2the electrons can enter different states. The bias is appliedbetween the chiral edge channel and the superconductor. and spin flip can be phenomenologically described by aneffective pairing Hamiltonian in the edge state,ˆ H T = τ ∆ ˆ ψ †↑ ( x = ξ ) ˆ ψ †↑ ( x = 0) + H.c. , (2)where τ is the effective tunneling amplitude which is offirst order in both tunneling amplitude and spin-flip am-plitude and ψ ↑ ( x ) is the annihilation operator for a spin-up electron in the edge state at position x . The effectivepairing is suppressed at short distances by the Pauli prin-ciple and vanishes exponentially at distances larger thanthe superconducting coherence length ξ ∝ v F / ∆ [71],where v F is the Fermi velocity. This is why the effectivepairing can be approximated using a fixed distance ξ be-tween the electrons in the final state. From here on wewill suppress the spin index.The Hamiltonian (1) gives a complete description ofthe system under consideration. In the following, therelevant energy scales are assumed to be small comparedto the Fermi energy, allowing us to use the effective low-energy theory to take into account the strong electron-electron interaction in edge states for the cases of fillingfactor ν = 2 and ν = 1 / (2 n + 1) ( n ∈ N ) [40, 41]. Thetunneling term (2) is considered perturbatively. III. TUNNELING CURRENT IN THE DCREGIME
The operator for the tunneling current is given by ˆ J = d ˆ N QH /dt = i [ ˆ H, ˆ N QH ], where ˆ N QH = R dx ˆ ψ † ( x ) ˆ ψ ( x ) isthe electron number operator in the QH channel. It canbe expressed as ˆ J = 2 iτ ∆( ˆ A † − ˆ A ) , (3)where the operator ˆ A = ˆ ψ (0) ˆ ψ ( ξ ) consists of twofermionic fields. In the interaction picture the averagetunneling current is given by the expression I ( t ) = h ˆ U † ( t, −∞ ) ˆ J ( t ) ˆ U ( t, −∞ ) i , (4)where the average is taken with respect to the dc bi-ased ground state of QH edges and superconductor. Thecurrent becomes time-independent once the system hasreached a steady state. At the lowest order of tunnelingcoupling, the time evolution operator is given byˆ U ( t , t ) ≈ − i Z t t dt ˆ H T ( t ) . (5)One then finds that the average tunneling current can bewritten in term of a commutator of A operators [72–74] I dc ( V ) = 2( τ ∆) Z ∞−∞ dte iV t Dh ˆ A † ( t ) , ˆ A (0) iE , (6)where V is the applied dc bias voltage. The average istaken with respect to the ground state of the uncoupledsystem, i.e., with respect to the equilibrium density ma-trix ˆ ρ ∝ exp[ − ( ˆ H QH + ˆ H SC ) /T ] where T is the tem-perature. The integrand of Eq. (6) only depends on onetime variable due to time translation invariance in pres-ence of dc bias. The pre-factor 2 reflects the charge 2 e of the Cooper pairs. The perturbative result is valid aslong as the tunneling current is small compared to theHall current. Restoring the natural units, the Hall cur-rent is given by the relation I H = e V / π ~ . It is worthmentioning that here we have used the real-time Keldyshapproach, in contrast to Ref. [58] where the equilibriumMatsubara technique was used to calculate the conduc-tance at zero frequency. A. Filling factor ν = 1 As an illustration of our approach based on Eq. (6), wefirst start by considering a system at filling factor ν = 1and described byˆ H QH = − iv F Z dx ˆ ψ † ( x ) ∂ x ˆ ψ ( x ) . (7)Without loss of generality we consider right-movingfermions and focus on a positive applied dc bias voltage V >
0. In the case of finite temperature T , an analyticalcontinuation in the complex plane is applied to Eq. (6).One finds the following result for the tunneling current, I dc ( V ) I ( V ) = T V sinh (cid:18) VT (cid:19) (8) × (cid:20) F (cid:18) , VπT (cid:19) − F (cid:18) πξTv F , VπT (cid:19)(cid:21) , where I ( V ) = 2( τ ∆) V /πv F is a normalization factorand the terms in square brackets are given by the integral F ( a, b ) = Z + ∞−∞ cos( bz ) dz cosh( a ) + cosh( z ) . (9)Here one can check that the tunneling current vanishesat V → ξ →
0. In the general case, the result ofEq. (8) can be expressed in terms of hypergeometric func-tions. However, we are mainly interested in the regime oflow temperature compared to the superconducting gap, ξT /v F ≪
1. Moreover, as we are mainly interested inthe linear conductance, we also assume low voltages com-pared to the temperature scale,
V /T ≪
1. In this casethe result simplifies to I dc ( V ) /I ( V ) ≃ (2 / πξT /v F ) .A direct calculation of the conductance G = ∂I dc ( V ) /∂V at V → G ( T ) G = 1 − πξT /v F sinh(2 πξT /v F ) , (10)where the normalization is equal to G = 2( τ ∆) /πv F .In the low-temperature limit ξT /v F ≪ G ( T ) /G ≃ (2 / πξT /v F ) .Next, we will discuss the results at zero temperature.Using Eq. (8), we obtain the expression for the tunnelingcurrent at T = 0, I dc ( V ) I ( V ) = 1 − sin(2 V ξ/v F )2 V ξ/v F . (11)In the limit of small bias voltage V ξ/v F ≪
1, we findnon-Ohmic behaviour I dc ( V ) ∝ ( τ ∆) V ξ /v F . The lin-ear QPC conductance associated with tunneling currentis given by G = ∂I dc ( V ) /∂V at V →
0. The direct cal-culation gives G = 0 at zero temperature. According toRef. [58], the vanishing conductance and the non-Ohmicbehavior of the tunneling current is related to the Pauliexclusion principle. At low energy scales, Pauli exclusiondiminishes the effective density of states for electron-pairtunneling, namely ρ DOS ∝ ( V ξ/v F ) at zero temperatureand ρ DOS ∝ ( T ξ/v F ) at finite temperature. Physicallythis means that after the first electron has tunneled, thetunneling of a second electron is strongly suppressed upto times t ∼ ξ/v F . B. Filling factor ν = 2 In this subsection, we consider the QH edge at fill-ing factor ν = 2. First, we describe the non-interactingcase. A pair of electrons from the superconductor cannow tunnel simultaneously into two different edge chan-nels [58], denoted by 1 and 2. To model this process,the electron operator in Eq. (3) can be represented as asuperposition of independent fermionic fields ˆ ψ , ( x ) asˆ ψ = √ p ˆ ψ + √ − p ˆ ψ , where p is the probability of anelectron tunneling into edge state 1, and 1 − p is the prob-ability of tunneling into edge state 2. To calculate thetunneling current (6) we need the two-point correlationfunctions G j ( x − x ; t − t ) = h ˆ ψ † j ( x , t ) ˆ ψ j ( x , t ) i ,where j = 1 , v F . A difference in Fermi velocities could beabsorbed into a redefinition of p .At finite temperatures, similar steps as for filling factor ν = 1 lead to the following expression for the tunnelingcurrent I dc ( V ) I ( V ) = T V sinh (cid:18) VT (cid:19) (12) × (cid:20) F (cid:18) , VπT (cid:19) − N ( p, kξ ) F (cid:18) πξTv F , VπT (cid:19)(cid:21) , with F ( a, b ) defined as in Eq. (9). We have introducedthe interference factor N ( p, kξ ) = 1 − p (1 − p )[1 − cos( kξ )] , (13)where k = Bl is the momentum difference between thetwo edge channels when separated by a length l in a mag-netic field of strength B . This reflects the inherent re-lationship between momentum and position of QH edgestates[75], where taking the difference avoids all depen-dence on the choice of gauge. The result for the zero-biasconductance at finite temperature reads G ( T ) G = 1 − πξT N ( p, kξ ) v F sinh(2 πξT /v F ) . (14)For ξT /v F ≪ G ( T ) /G ≃ − N ( p, kξ ) +(2 N ( p, kξ ) / πξT /v F ) . The leading order generallydoes not vanish and does not depend on temperature.Physically this is due to a circumvention of the Pauliblockade by allowing the electrons to tunnel simultane-ously into different channels.Employing Eq. (12), we get the result for the tunnelingcurrent at zero temperature I dc ( V ) I ( V ) = 1 − N ( p, kξ ) sin(2 V ξ/v F )2 V ξ/v F . (15)As before, the current vanishes if either V → ξ →
0. For tunneling into a single edge state ( p = 1 or p = 0) one has N = 1 and thus recovers the result fromEq. (11). In the limit V ξ/v F ≪
1, we find that hav-ing two edge channels available and thus the possibilityto avoid the Pauli blockade restores Ohmic behaviour: I dc ( V ) ∝ ( τ ∆ /v F ) (1 − N ) V , whereas the sub-leadingterm is proportional to ( τ ∆) N V ξ /v F . A straightfor-ward calculation leads to the zero-temperature conduc-tance G ( T = 0) /G = 1 − N ( p, kξ ).We now introduce electron-electron interactions bothwithin a given edge state as well as between the two edgestates. To study the effects of these interactions on thePauli blockade, we start in the blockaded regime and thusassume that electrons only tunnel into one edge chan-nel, corresponding to p = 1 or p = 0. To describe the edge states in the presence of interactions we use an ef-fective field theory [40, 41]. The edge state excitationsare then described as collective fluctuations of the chargedensity ˆ ρ j ( x ) = (1 / π ) ∂ x ˆ φ j ( x ), where the index j = 1 , φ j ( x ) is a bosonic field oper-ator which satisfies the standard commutation relations[ ˆ φ i ( x ) , ˆ φ j ( y )] = iπδ ij sgn( x − y ). The Hamiltonian of theQH edge states is then given byˆ H QH = 12 X ij =1 , Z dx Z dy ˆ ρ i ( x ) V ij ( x, y )ˆ ρ j ( y ) , (16)where the interaction kernel is given by V ij ( x, y ) =( U +2 πv F δ ij ) δ ( x − y ) with U > H QH can be di-agonalized by the unitary transformation [72]ˆ φ = 1 √ χ + ˆ χ ) , ˆ φ = 1 √ χ − ˆ χ ) , (17)which conserves the bosonic commutation relations[ ˆ χ i ( x ) , ˆ χ j ( y )] = iπδ ij sgn( x − y ). Substituting these fieldsinto the Hamiltonian (16) we obtainˆ H QH = 14 π X j =1 , v j Z dx ( ∂ x ˆ χ j ) , (18)which now contains a fast charge mode ( j = 1) and aslow dipole mode ( j = 2), with velocities v = U/π + v F and v = v F , respectively. This bosonization procedureallows us to take into account electron-electron interac-tions with arbitrary strength explicitly and shows thatthe spectrum is split into two modes. Now, it is straight-forward to calculate the four-point correlation functionsusing this diagonal Hamiltonian and the unitary trans-formation (17) (see App. B). Substituting the correlationfunctions from Eq. (B3) into Eq. (6), we get the follow-ing expression for the tunneling current at finite temper-atures I dc ( V ) I ( V ) = TV sinh (cid:18) VT (cid:19) Y j =1 , sinh (cid:18) πT ξv j (cid:19) × J (cid:18) πT ξv , πT ξv , VπT (cid:19) , (19)where I ( V ) = 2( τ ∆) V /πv v is the normalization co-efficient and the last factor has the integral form J ( a , a , b ) = Z ∞−∞ dy cosh − ( y ) cos( by ) Q i =1 , p cosh(2 y ) + cosh( a i ) . (20)At low temperatures T ξ/v ≪
1, the asymptotic formof the conductance is G ( T ) / G ≃ (2 / πξT / √ v v ) ,where G = 2( τ ∆) /πv v is a normalization coefficient.At zero temperature and V ξ/v ≪ v > v we get I dc ( V ) ∝ (2 / π )( V ξ ) / ( v v ) , resulting in vanishingzero-bias conductance. Thus we get the same result asin case of filling factor ν = 1 in Eq. (11) at V ξ/v F ≪
1. The Pauli blockade persists even with cross-channelinteraction. As one can see, the interaction renormalizesthe Fermi velocity, so to obtain the current at T → v F to √ v v in asymptotics of Eq. (11). C. Filling factor ν = 1 / (2 n + 1) The fractional QH edge state with Laughlin filling fac-tor ν = 1 / (2 n + 1) , n ∈ N , consists of a single channelwith a free bosonic field ˆ φ ( x ) propagating with velocity v . The electron operator is given by the vertex operatorˆ ψ ( x ) ∝ e i ˆ φ ( x ) / √ ν [40, 41]. We can then repeat the stepsof the previous sections to get the tunneling current atfinite temperature I dc ( V )˜ I ( V ) = 2 /ν T V (cid:18) rT v (cid:19) /ν − × sinh /ν (cid:18) πξTv (cid:19) sinh (cid:18) VT (cid:19) Q (cid:18) πξTv , VπT (cid:19) , (21)where r is an ultraviolet cut-off, ˜ I ( V ) = 2( τ ∆) /πv isthe normalization coefficient, and we use the dimension-less integral Q ( a, b ) = Z ∞−∞ dy cosh − /ν ( y ) cos( by )[cosh(2 y ) + cosh( a )] /ν . (22)At T ξ/v ≪ G ( T )˜ G ≃ √ π /ν )Γ(1 / /ν ) (cid:18) rT v (cid:19) /ν − (cid:18) πξTv (cid:19) /ν , (23)where Γ( x ) denotes the gamma function and ˜ G =2( τ ∆) /πv . Further, at zero temperature and low volt-ages V ξ/v ≪
1, using Eq. (21), we find that the currenthas the form I dc ( V )˜ I ( V ) ≃ π v r (cid:18) rξπv (cid:19) /ν V /ν − Γ(4 /ν ) . (24)Consequently, the conductance vanishes as in the case offilling factor ν = 1, i.e. G = 0. This result can also beobtained from Eq. (23) at T →
0. Even though we havetunneling between two effectively bosonic systems, thePauli blockade persists and makes the QPC an insulatorat zero bias. At filling factor ν = 1 the results of thissubsection coincide with the results of subsection (III A). IV. FINITE-FREQUENCY NOISE IN THE DCREGIME
In this section, we consider the finite-frequency noise inthe case of an applied dc voltage. The exact experimen-tally measurable current noise depends on the details of the setup, so we calculate the non-symmetrized currentcorrelation function from which other forms of noise, e.g.,the symmetrized noise, can be obtained [47, 74, 76, 77].It is defined as S dc ( ω, V ) = Z + ∞−∞ dte iωt h δ ˆ J ( t ) δ ˆ J (0) i , (25)where δ ˆ J ( t ) = ˆ J ( t ) −h ˆ J ( t ) i and the average is taken withrespect to the dc biased ground state of QH system andsuperconductor. Using the time translation invarianceof the vertex operators (see App. C), the noise can bewritten to the lowest order of the tunneling coupling as S dc ( ω, V ) = g ( ω + 2 V ) + g ( ω − V ) , (26)where the correlation function on the right is given by g ( ω ) = 4( τ ∆) Z + ∞−∞ dte iωt h ˆ A † ( t ) ˆ A (0) i . (27)It is worth pointing out that the shot noise at ω = 0 isdetermined by the anti-commutator of ˆ A operators, incontrast to the tunneling current in Eq. (6), namely [74] S dc (0 , V ) = 4( τ ∆) R + ∞−∞ dte iV t h{ A † ( t ) , A (0) }i . Thenoise can be symmetrized as the even combination ofthe two non-symmetrized terms, [ S ( ω ) + S ( − ω )] /
2, andwhether measuring the non-symmetrized or the sym-metrized noise is possible depends on the experimentaldetector [76].
A. Filling factor ν = 2 We start again by considering a system with positivebias voltage
V >
0, no interactions and Cooper pairstunneling simultaneously into both edge channels (seeSec. III B). Using Eq. (25) and the two-point correlationfunctions from App. A, the noise at finite temperaturebecomes S dc ( ω, V ) I ( V ) = X σ = ± T V exp (cid:18) ω + 2 σV T (cid:19) (28) × (cid:20) F (cid:18) , ω + 2 σV πT (cid:19) − N ( p, kξ ) F (cid:18) πξTv F , ω + 2 σV πT (cid:19)(cid:21) , where F ( a, b ) is defined in Eq. (9) and the normalizationcoefficient I ( V ) is given after Eq. (8).At zero temperature we can use Eq. (28) to obtain theexpression S dc ( ω, V ) I ( V ) = X σ = ± θ ( σ + ω/ V ) (29) × (cid:26)(cid:12)(cid:12)(cid:12) σ + ω V (cid:12)(cid:12)(cid:12) − N ( p, kξ ) v F V ξ sin (cid:18) V ξv F (cid:12)(cid:12)(cid:12) σ + ω V (cid:12)(cid:12)(cid:12)(cid:19)(cid:27) , where N ( p, kξ ) is given in Eq. (15) and θ ( x ) is the Heavi-side step function. Furthermore, we calculate the deriva-tive of the shot noise with respect to the applied bias at V → G − ∂S dc (0 , V ) ∂V (cid:12)(cid:12)(cid:12)(cid:12) V → = 2[1 − N ( p, kξ )] , (30)where the prefactor G on the left is given after Eq. (10).Here, as with the corresponding conductance, we see thatat filling factor ν = 2 the Pauli blockade is lifted. At lowtemperatures ξT /v F ≪
1, the sub-leading correction toEq. (30) is given by (4 / N ( p, kξ )( πξT /v F ) , i.e. thetemperature independence is only to leading order.Furthermore, using Eqs. (12) and (28), and F ( a, − b ) = F ( a, b ), one can see that the Fano factor has the well-known form [7] S dc (0 , V ) I dc ( V ) = 2 coth (cid:18) VT (cid:19) . (31)Therefore, the current fluctuations satisfy a classicalPoissonian shot noise form [7]. At zero temperature,we have coth( V /T ) →
1, so the Fano factor becomes S dc (0 , V ) /I dc ( V ) = 2, which can be taken as an in-dication that the elementary charge carriers tunnelingthrough the QPC are indeed charge-2 e Cooper pairs.The result for filling factor ν = 1 can be obtainedsetting N ( p, kξ ) = 1 at p = 1 or p = 0. In particular,at low temperatures and bias voltage, ξT /v F ≪ V /T ≪
1, we get S dc (0 , V ) I ( V ) ≃ (cid:18) πξTv F (cid:19) . (32)Thus, the right hand side of Eq. (32) becomes propor-tional to the differential conductance (10) at V →
0, i.e. ∂S dc (0 , V ) /∂V ∝ ( ξT /v F ) .Taking into account electron-electron interactions, as-suming tunneling into only one channel, and repeatingthe steps leading to Eq. (19) and (28) for finite temper-atures yields the following result S dc ( ω, V ) I ( V ) = X σ = ± TV exp (cid:18) ω + 2 σV T (cid:19) Y i =1 , sinh (cid:18) πT ξv i (cid:19) × J (cid:18) πT ξv , πT ξv , ω + 2 σVπT (cid:19) , (33)where I ( V ) is given in Eq. (19), and J is given inEq. (20). Calculating the zero-frequency noise, one findsthat the Fano factor is given by Eq. (31), as it is ex-pected. It is worth mentioning here that at V →
0, thedifferential shot noise vanishes.At zero temperature, using Eq. (33), for the shot noiseat ω = 0 and V ξ/v , V ξ/v ≪ S dc (0 , V ) I ( V ) ≃ V ξ v v . (34)Direct calculations using the asymptotic result inEq. (34) or the exact expression at ω = 0 in Eq. (33),give that ∂S dc (0 , V ) /∂V = 0 at V →
0, which is causedby the Pauli blockade.
B. Filling factor ν = 1 / (2 n + 1) Repeating the steps of the previous subsections, in caseof finite temperature we get the following result for noiseat fractional filling factors S dc ( ω, V )˜ I ( V ) = X σ = ± /ν T V (cid:18) rT v (cid:19) /ν − sinh /ν (cid:18) πξTv (cid:19) × exp (cid:18) ω + 2 σV T (cid:19) × Q (cid:18) πξTv , ω + 2 σVπT (cid:19) , (35)where Q ( a, b ) is given in Eq. (22) and the normaliza-tion factor is given in Eq. (21). Taking this equation at T ξ/v ≪ ∂S dc (0 , V ) /∂V ∝ ( rT / v ) /ν − ( πξT /v ) /ν . This expres-sion vanishes at T →
0. In particular we find that at zerotemperature, zero frequency, ω = 0, and V ξ/v ≪ S dc (0 , V )˜ I ( V ) ≃ π Γ[4 /ν ] v r V (cid:18) rξV πv (cid:19) /ν . (36)This expression results in vanishing differential shotnoise, namely ∂S dc (0 , V ) /∂V = 0 at V →
0. It is worthmentioning that at ν = 1 the result of this subsectionagrees with the expressions of the previous subsectionand Eq. (31) for Fano factor at finite temperatures issatisfied. V. TUNNELING CURRENT AND FINITEFREQUENCY NOISE IN THE AC REGIME
To study the case of time-dependent voltage, we as-sume a periodic bias of the form ˜ V ( t ) = V + V cos(Ω t ),where Ω is the driving frequency. The time-dependentpart of such bias averages to zero over one period T =2 π/ Ω, and the dc part of the time averaged tunnelingcurrent in the case of ac bias is given by I = 2 T Z T dt Z t −∞ dt ′ Re n e i R tt ′ ˜ V ( t ′ ) h [ ˆ A † ( t ) , ˆ A ( t ′ )] i o . (37)With the exact form of the vertex operators from App. Band App. C, and using an expansion in terms of Besselfunctions, exp[ iλ sin ϕ ] = P ∞ n = −∞ J n ( λ ) exp[ inϕ ], wefind I = + ∞ X n = −∞ J n (2 V / Ω) I dc ( V + n Ω / . (38)At Ω → V →
0, the sum of Floquet factors goesto one, i.e P ∞ n = −∞ J n (2 V / Ω) → T . Due tothe drive, this can be regarded as noise due to photonassisted electron transport across the QPC. The time-averaged photon assisted finite-frequency noise is givenby the following Wigner transformation S ( ω ) = 1 T Z T dτ Z + ∞−∞ dτ ′ S ( τ + τ ′ / , τ − τ ′ / e iωτ ′ , (39)where we have introduced the “center of mass” and “rel-ative” time variables, τ = ( t + t ′ ) / τ ′ = t − t ′ ,respectively. The integrand includes the current-currentcorrelation function S ( t, t ′ ) = h δ ˆ J ( t ) δ ˆ J ( t ′ ) i with δ ˆ J ( t ) =ˆ J ( t ) − h ˆ J ( t ) i and the average is performed with respectto the biased ground state of the system. Using again anexpansion of the exponent in terms of Bessel functions,the time invariance of the vertex correlation functions(see App. C) and J − n ( x ) = J n ( x ) we get the final resultfor finite-frequency noise S ( ω ) = ∞ X n = −∞ J n (2 V / Ω) S dc ( ω, V + n Ω / . (40)Here again, as Ω → V →
0, the sum of Floquetfactors goes to one, i.e P ∞ n = −∞ J n (2 V / Ω) → P n = ∞ n = −∞ J n ( V / Ω)Λ dc ( V + n Ω), holds for electroncurrent, heat current and shot noise under ac bias overa QPC contact between two edge states. According toEq. (40), this statement holds in one more general case,namely for finite-frequency noise. This result can be usedto interpret the experiments on dynamical response ofLaughlin anyons in presence of time-dependent bias [78].
VI. CONCLUSION
In this paper we have studied tunneling between a su-perconductor and a QH edge state at different filling fac-tors, namely ν = 1, ν = 2 and ν = 1 / (2 n +1). To accountfor electron-electron interaction in the QH edge state, weused a low-energy effective theory based on bosonization.In the bosonic picture of collective excitations, the spec-trum splits into two modes, namely the fast charge modeand slow dipole mode. Exact diagonalization allows us tocalculate the two- and four-point equilibrium correlationfunctions, which are necessary to evaluate the transportproperties of system, such as current and noise. We in-vestigated the tunneling between the QH edge states andthe superconductor to the lowest order in the tunnelingcoupling under the dc and ac biases.For filling factor ν = 1, at zero temperature and V ξ/v F ≪
1, we found that the tunneling current isproportional to I dc ( V ) ∝ ( τ ∆) ξ V /v F , which is amanifestation of non-Ohmic behavior. This scaling of the tunneling current with the applied dc bias resultsin a vanishing conductance. At finite temperatures,at ξT /v F ≪ V /T ≪
1, the current is propor-tional to the applied bias, and the density of states isrenormalized by the dimensionless factor ξT /v F , namely I dc ( V ) ∝ ( τ ∆ /v F ) ( ξT /v F ) V . In addition to thetunneling current, we presented results for the finite-frequency current noise. The ratio between shot noiseand tunneling current, known as the Fano factor, wasfound to be S dc (0 , V ) /I dc ( V ) = 2. Thus, the differen-tial shot noise, ∂S dc (0 , V ) /∂V at V → S dc (0 , V ) /I dc ( V ) = 2 coth( V /T ). As a result, at ξT /v F ≪
1, the differential shot noise at V → T → ν = 2, in case of simultaneous tun-neling of a Cooper pair into different QH channels, thesituation changes drastically. At zero temperature and V ξ/v F ≪
1, the current manifests Ohmic behaviorto leading order, I dc ( V ) ∝ ( τ ∆ /v F ) (1 − N ) V , where0 < N < S dc (0 , V ) /I dc ( V ) = 2, so the differential shot noiseat V → ξT /v F ≪ N ( ξT /v F ) . In the presence of electron-electron inter-action the results are qualitatively similar, but one hasto replace the Fermi velocity by the geometric average ofthe velocities of the charged and dipole modes, √ v v .For filling factor ν = 1 / (2 n + 1), the power-law behav-ior of transport quantities depends on ν . At V ξ/v ≪ I dc ( V ) ∝ V /ν − and the con-ductance vanishes. At low temperatures, ξT /v ≪
1, wehave I dc ( V ) ∝ T /ν − V and the conductance depends ontemperature. The behavior of the differential shot noiseat V → ∝ T /ν − .We also provided a general expression for the tunnel-ing current and the finite-frequency noise in the presenceof a periodic ac bias voltage. This result, valid for allfilling factors considered, demonstrates that the currentand finite-frequency noise can be expressed as the sumof dc currents and noise terms with Floquet coefficients.Recently, it was found that an expression similar to ourresult (40) holds for shot noise [78]. We have found thatthis statement holds in the more general case of finite-frequency noise.At Laughlin filling factors, in addition to the Coulombblockade [79], it has been found that the vanishing con-ductance [58] and differential shot noise at low tempera-tures is a consequence of an additional suppression mech-anism called the Pauli blockade: after the tunneling ofthe first electron of a Cooper pair the tunneling of thesecond electron into the QH edge state is suppressed upto times ξ/v F , where v F is the velocity of the edge ex-citations, due to the Pauli exclusion principle. At fillingfactor ν = 2, in the case of simultaneous tunneling ofa Cooper pair into both channels, the Pauli blockade ispartially removed. Electron-electron interactions do notchange the physics qualitatively but result in a renor-malization of the Fermi velocity. Finally, as a futureperspective, it would be interesting to consider a similarproblem in the context of levitonic physics [9], where theinjection of single particles due to tailored voltage pulsesis investigated. ACKNOWLEDGMENTS
The authors acknowledge financial support from theNational Research Fund Luxembourg under GrantsCORE C19/MS/13579612/HYBMES and ATTRACT7556175.
Appendix A: Two-point correlation function
In this Appendix we calculate the two-point correlation function of right-moving fermions G j ( x, t ; x ′ , t ′ ) at fill-ing factor ν = 1 and at finite temperature T . We use bosonization technique, which is further necessary to takeinto account the electron-electron interaction. Here the subscript j = 1 , G j ( x, x ′ ; t, t ′ ) = h ˆ ψ † j ( x, t ) ˆ ψ j ( x ′ , t ′ ) i = 1 r h e − i ˆ φ j ( x,t ) e i ˆ φ j ( x ′ ,t ′ ) i = 1 r e M ( x,t ; x ′ ,t ′ ) (A1)where r is an ultra-violet cut-off, and in Gaussian approximation under consideration the exponent is given by M ( x, t ; x ′ , t ′ ) = − h ˆ φ j ( x, t ) i − h ˆ φ j ( x ′ , t ′ ) i + h ˆ φ j ( x, t ) ˆ φ j ( x ′ , t ′ ) i . (A2)Using the expansion of bosonic field in terms of creation and annihilation operators of bosons, we get the followingexpression (zero modes are ignored) M ( x, t ; x ′ , t ′ ) = Z ∞ dkk e − rk π h (1 + f B ( k )) (cid:16) e ik [ X ( t ) − X ′ ( t ′ )] − (cid:17) + f B ( k ) (cid:16) e − ik [ X ( t ) − X ′ ( t ′ )] − (cid:17)i , (A3)where X = x − v j t , X ′ = x ′ − v j t ′ and f B ( k ) = ( e v j k/T − − = P ∞ n =1 e − v j βkn is an equilibrium bosonic distributionfunction with inverse temperature β = 1 /T . Further integration with respect to momentum variable k gives M ( x, t ; x ′ , t ′ ) = log (cid:20) ir/ πX ( t ) − X ′ ( t ′ ) + ir/ π (cid:21) − ∞ X n =1 log (cid:20) π ( X ( t ) − X ′ ( t ′ ) + ir/ π ) /v j β ] π n (cid:21) . (A4)Next, exponentiating the above relation and using the definition of hyperbolic sinesinh( z ) = z ∞ Y n =1 (cid:18) z π n (cid:19) , (A5)we finally get the result for two-point correlation function at finite temperature G j ( x, x ′ ; t, t ′ ) = − iT v j πT ( t − t ′ − ( x − x ′ ) /v j − iγ )] , γ → +0 . (A6)The correlation function at zero temperature is obtained, using that sinh( x ) ∼ x , namely G j ( x, x ′ ; t, t ′ ) = − i πv j t − t ′ − ( x − x ′ ) /v j − iγ . (A7) Appendix B: Four-point correlation function
In this Appendix we derive the expression for four-point correlation function [80]. We again use the Gaussiancharacter of theory to calculate it, namely the average of four vertex operators is written as the exponent of combinationof averages of bosonic field. To demonstrate this, we use bosonization technique to rewrite the four-point correlationfunction, namely L = h ˆ ψ † ( x , t ) ˆ ψ † ( x , t ) ˆ ψ ( x , t ) ˆ ψ ( x , t ) i = 1 r h e − i ˆ φ ( x ,t ) e − i ˆ φ ( x ,t ) e i ˆ φ ( x ,t ) e i ˆ φ ( x ,t ) i , (B1)where we have omitted the arguments of L and the average is taken with respect to equilibrium zero density matrix,ˆ ρ . Next, using the Eq. (17) from main text, the above expression can be rewritten as a product of two four-point vertexcorrelation functions corresponding to charged and dipole modes in presence of interaction, namely L = L × L ,where ˆ χ j ( x, t ) = ˆ χ ( x − v j t ) and consequently L j = 1 r h e − i √ ˆ χ j ( x − v j t ) e − i √ ˆ χ j ( x − v j t ) e i √ ˆ χ j ( x − v j t ) e i √ ˆ χ j ( x − v j t ) i . (B2)Further in Gaussian approximation [80], in term of new bosonic fields, the above correlation function takes the from L j = 1 r exp " − X i =1 λ i h ˆ χ j ( x i − v j t i ) i − X i In this Appendix we show the time invariance of vertex correlation functions, namely that h ˆ A † ( t ) ˆ A (0) i = h ˆ A ( t ) ˆ A † (0) i = h ˆ A † (0) ˆ A ( − t ) i . (C1)To do this, we represent these vertex correlation function though two-point correlation functions G j ( x − x ′ , t − t ′ )defined in previous Appendix. For filling factor ν = 1 it is obvious because of Wick’s theorem. 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