Spectroscopy of electron flows with single- and two-particle emitters
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Spectroscopy of electron flows with single- and two-particle emitters
Michael Moskalets , , and Markus B¨uttiker D´epartement de Physique Th´eorique, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland Institut f¨ur Theorie der Statistischen Physik, RWTH Aachen University, D-52056 Aachen, Germany Department of Metal and Semic. Physics, NTU ”Kharkiv Polytechnic Institute”, 61002 Kharkiv, Ukraine (Dated: October 17, 2018)To analyze the state of injected carrier streams of different electron sources, we propose to usecorrelation measurements at a quantum point contact with the different sources connected via chiraledge states to the two inputs. In particular we consider the case of an on-demand single-electronemitter correlated with the carriers incident from a biased normal reservoir, a contact subject toan alternating voltage and a stochastic single electron emitter. The correlation can be viewed as aspectroscopic tool to compare the states of injected particles of different sources. If at the quantumpoint contact the amplitude profiles of electrons overlap, the noise correlation is suppressed. In theabsence of an overlap the noise is roughly the sum of the noise powers due to the electron streamsin each input. We show that the electron state emitted from a (dc or ac) biased metallic contactis different from a Lorentzian amplitude electron state emitted by the single electron emitter (aquantum capacitor driven with slow harmonic potential), since with these inputs the noise correlationis not suppressed. In contrast, if quantized voltage pulses are applied to a metallic contact insteadof a dc (ac) bias then the noise can be suppressed. We find a noise suppression for multi-electronpulses and for the case of stochastic electron emitters for which the appearance of an electron atthe quantum point contact is probabilistic.
PACS numbers: 73.23.-b, 72.10.-d, 73.50.Td
I. INTRODUCTION
The experimental realization of an on-demand, high-frequency, single-electron source (SES) makes it possi-ble to inject into a solid state circuit single particles,electrons and holes, in a controllable way. Using sev-eral uncorrelated single-electron sources mesoscopic cir-cuits were proposed which permit to vary the amountof fermionic-correlations and to produce controllablyorbitally entangled pair of particles . Similar high-frequency sources of single electrons were realized us-ing dynamical quantum dots without or with a per-pendicular magnetic field . The principal advantage ofon-demand, single-electron sources over the usually usedmetallic contacts (MCs) as electron sources is the pos-sibility, in the former case, to switch on and off quan-tum correlations between particles initially emitted fromuncorrelated sources. An example of correlations gen-erated by normal metallic contacts is the two-particleAharonov-Bohm effect in the solid state Hanbury Brown-Twiss interferometer discussed theoretically and foundexperimentally . In contrast, with single-electron sourcesthe two-particle interferometer, as it is discussed inRef. 3, can show or not show the Aharonov-Bohm effectdepending on whether sources are driven in synchronismor not.The appearance of quantum correlations (fermionic, inthe case of electrons) between initially uncorrelated par-ticles is due to the overlap of wave-packets on the wavesplitter. For electrons in solid state circuits the split-ter is a quantum point contact (QPC), (see Fig. 1, theQPC labeled C ). Such correlations are well known inoptics, see, e.g., Ref. 10. The overlap of fermions was dis-cussed in Ref. 11 and in Ref. 12. The overlap depends on the spatial extend of wave-packets and also on the timeswhen they arrive at the wave splitter. Thus the resultingcorrelations can be used to access information about thespace-time extend of quantum states. For MCs work-ing as electron sources such information is rather hiddensince the mentioned correlations are always present. Incontrast with on-demand single-electron sources controlof the emission time can be achieved, i.e. the appear-ance or disappearance of correlations can be controlled.Thus with such sources the space time extend of quantumstates becomes accessible.In mesoscopics physics shot noise is the natural quan-tity that can be used to find information on two-particlecorrelations. The shot noise of carriers emitted by S I (t) VC I (t)
FIG. 1: (color online) A mesoscopic electron collider circuitwith a single-electron source S , a circular edge state, and ametallic contact source biased with a voltage V . At the quan-tum point contact C the particles emitted by the two sourcescan collide if the times of emission are adjusted properly. Solidblue lines are edge states with direction of movement indi-cated by arrows. Short dashed red lines are quantum pointcontacts connecting different parts of a circuit. Black rectan-gles are metallic contacts. two SES’s is suppressed if the wave-packets overlap atthe QPC connecting edge states in which emitted par-ticles propagate. If two sources are identical and theyemit particles at the same time, the emitted particles arein identical quantum states and the shot noise is sup-pressed down to zero. This effect is similar to the Hong-Ou-Mandel effect in optics with the evident differencethat electrons are rather forced go into different outputchannels while photons are bunched into the same outputchannel.The aim of this paper is to use the shot noise suppres-sion as a spectroscopic tool allowing comparison of thequantum states emitted by the different electron sources.As the test state we will use the one emitted by the SES.The SES is made of a quantum capacitor in the quan-tum Hall effect regime. The SES is connected to one ofthe arms of the mesoscopic electron collider, Fig. 1. Un-der the action of a potential U ( t ) = U cos(Ω t ) periodicin time the SES emits a sequence of alternating electronsand holes. In a certain range of amplitudes, in the quan-tized emission regime, the SES emits one electron and onehole. At low driving frequency, in the adiabatic regime,the emitted state by the SES is close to the state gen-erated by voltage pulses of Lorentzian form with a timeintegral equal to a flux quantum: such a quantized volt-age pulse produces a single-particle state on top of theFermi sea. . In the second arm we put the source ofinterest and investigate the resulting shot noise.The paper is organized as follows: In Sec. II we calcu-late the zero-frequency cross-correlator of currents flow-ing into two outputs in the collider circuit with a SES inone input and a biased metallic contact in other one. InSec. III we address the effect of stochastic single-particleemitters which emit or do not emit a particle in a givenperiod that arrives at the QPC. We demonstrate the irrel-evance of such stochastic emission to the shot noise sup-pression effect. In Sec. IV the shot noise suppression ef-fect is found for colliding single- and two-electron pulses.A discussion of our results is given in Sec. V. Much ofthe analysis is grouped into three Appendices. In Ap-pendix A we present a detailed model of a single electronsource. In Appendix B we calculate a current correlationfunction for a periodically driven mesoscopic scattererconnected to reservoirs biased with periodic voltages. InAppendix C the zero-frequency cross-correlation functionfor a circuit with two SESs is expressed it terms of single-and two-particle probabilities.
II. SINGLE ELECTRON EMITTER ANDBIASED METALLIC CONTACT AS ANELECTRON SOURCE
We consider an electron collider with a SES in onebranch and a MC with a potential V ( ∼ )2 ( t ) = V ( ∼ )2 ( t + T )periodic in time in another branch, see, Fig. 1. The po-tential U ( t ) driving the SES and V ( ∼ )2 ( t ) have the sameperiod, T = 2 π/ Ω, which is assumed to be large enough to consider adiabatic transport and neglect relaxationand decoherence processes relevant for high-energyexcitations. In addition the MC is biased with a constantpotential V with respect to the other contacts which allhave the same chemical potential µ . The temperature istaken to be zero.We utilize the scattering matrix approach to trans-port in mesoscopic systems and describe this circuit withthe help of the frozen scattering matrixˆ S ( t ) = e ikL S S SES ( t ) r C e ikL V t C e ikL S S SES ( t ) t C e ikL V r C , (1)with S SES ( t ) the scattering amplitude of the SES, seeAppendix A, Eq. (A5), r C /t C the reflection/transmissionamplitude for the central quantum point contact C , and L jX the length from the SES ( X = S ) or the MC( X = V ) to the contact j = 1 , I j ( t ) is measured. At zero temperature we needall quantities at the Fermi energy µ only.We are interested in the zero-frequency correlationfunction P of the currents I ( t ) and I ( t ) flowing intothe contacts 1 and 2, see Fig. 1. The corresponding cal-culations are presented in Appendix B. In the adiabaticregime and at zero temperature we have, P ≡ P ( sh,ad )12 ,Eq. (B26): P = −P ∞ X q = −∞ (cid:12)(cid:12)(cid:12)(cid:8) S SES Υ ∗ (cid:9) q (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) eV ~ Ω − q (cid:12)(cid:12)(cid:12)(cid:12) , (2)where P = e R C T C / T is the shot noise produced byone particle (either an electron or a hole) emitted by theSES during the period T . The oscillating potential atcontact 2 appears in the form of a phase factorΥ ( t ) = e − i e ~ t R −∞ dt ′ V ( ∼ )2 ( t ′ ) , (3)which multiplies the scattering amplitude of the SES.The symbol { .. } q indicates the q-th Fourier component(in time) of these two amplitudes. Eq. (2) illustratesthat the correlation tests the coherence properties of thetwo sources.Note the decoherence processes, which we neglect inthe present work, can lead to suppression of coherence A. Quantized voltage pulse
With a voltage pulse of Lorentzian shape and a timeintegral quantized to a single flux quantum one canexcite an electron from a Fermi sea without any other dis-turbance to the Fermi sea. The state for an excited elec-tron has a Lorentzian density profile (the time-dependentcurrent is a Lorentzian pulse) which is similar to the one emitted by the SES in the adiabatic regime. Thus we canexpect a shot noise suppression effect if an electron ex-cited out of a metallic contact with a quantized voltagepulse and an electron emitted by the SES collide at thecentral QPC. Below we show that this is really the case.Thus let us assume that a periodic pulsed potential isapplied to the MC, eV ( t ) = 2 ~ Γ( t − t ) + Γ , < t ≤ T , (4)where Γ ≪ T is the half-width of the pulse, and t is thetime when the electron is excited. Such a pulse excitesone electron during the period T .The potential V ( t ), Eq. (4), has a dc component, eV = h/ T , and a component which is periodic in time, eV ( ∼ )2 ( t ) = 2 ~ Γ( t − t ) + Γ − h T , < t ≤ T . (5)The corresponding phase factor Υ ( t ), Eq. (3), (for 0 5. Inset: The noise at theminimum as a function of Γ. an electron and a hole, are emitted by the SES. However,if the MC and the SES emit electrons at the same time, t = t ( − )0 , then after colliding at the central QPC theseelectrons become correlated and do not contribute effec-tively to the shot noise. The remaining value, −P , isdue to the hole emitted by the SES.The shot noise suppression (ShNS) effect depends onthe overlap of wave-packets in time (hence the times ofemission should be the same) and in space (hence thewidth of wave-packets should be the same). If the wavepackets have different width, see inset to Fig. 2, there issome extra noise. Therefore, the ShNS effect provides adirect tool to compare the states of particles emitted fromthe sources of different types, not only from the similarsources. The ShNS of two SES’s was already discussedin Ref. 2.The most used source of electrons in mesoscopics is abiased (with dc or ac voltage) metallic contact. Now weshow that the electron collider circuit with SES and abiased MC as an electron source does not show what wecall the shot noise suppression effect. Therefore, the stateof electrons emanating from a biased MC is different fromthe state of an electron emitted by the SES. B. DC bias If no ac bias is applied, V ( ∼ )2 ( t ) = 0, the phase factor is | Υ ( t ) | = 1 and only the Fourier coefficients (cid:12)(cid:12) S SESq (cid:12)(cid:12) =4Ω Γ exp( − | q | ) enter Eq. (2). We recall that weassume an adiabatic limit ΩΓ ≪ 1, where 2Γ is thetime during which an electron (a hole) is emitted by theSES. Since | S q | = | S − q | , we conclude from Eq. (2) thatin this case the shot noise is independent of the sign ofthe voltage. Therefore, the result will be the same nomatter whether the MC emits electrons, eV > 0, or holes, eV < 0. For definiteness we will use eV > P = −P (cid:26) eV ~ Ω + 2 e − [ eV ~ Ω ] (cid:18) (cid:20) eV ~ Ω (cid:21)(cid:19)(cid:27) . (7)where we have introduced the integer part [ eV / ( ~ Ω)] . This correlator has the following asymptotics, P = − P , eV ≪ ~ Ω , − ( e V /h ) R C T C , eV ≫ ~ Γ − . (8)Here the first line is the shot-noise due to the SES emit-ting one electron and one hole during the period. Thesecond line is the shot noise due to a dc biased metalliccontact alone. The latter noise is due to scattering atthe quantum point contact C of extra electrons flowingout of a biased contact above the Fermi sea with chem-ical potential µ . These electrons are emitted with rate eV /h . Therefore, one could naively expect that if therate of emission of electrons from the SES and from theMC is the same, ~ Ω = eV , then each emitted electronwill collide at the central QPC with an electron propa-gating within another edge state and the shot noise getssuppressed.This is not the case! As follows from Eq. (7), the shotnoise has no strong feature at eV ∼ ~ Ω. The shot noiseis a monotonous function of the dc bias V . A possiblereason for this is that the states of electrons emitted fromthe SES and from the dc biased MC are quite different:The electrons emitted from the SES can be thought aswave-packets with spatial extend proportional to the du-ration of emission Γ . In contrast the electrons emittedby the metallic contact are rather plane-wave like ex-tended along the whole edge state. Thus their overlap atthe central QPC is minute hence they do not acquire anysignificant correlations. The shot noise remains roughlythe sum of the noises produced independently be the SESand by the dc biased MC.Next we show that the noise suppression effect is alsoabsent if the metallic contact is driven by an ac bias. C. AC bias Consider next the case of a metallic contact with anac bias, V ( ∼ )2 ( t ) = V ( ∼ )2 cos(Ω t ), V = 0. In this casethe metallic contact emits both electrons and holes. Thecross-correlator, Eq. (2), as a function of amplitude V ( ∼ )2 is given in Fig. 3. There is a small feature at eV ( ∼ )2 = ~ Ωvisible in Fig. 3 but this feature is minute compared tothe huge dip of interest in this work. Therefore, there isno indication of a shot noise suppression at eV ( ∼ )2 ∼ ~ Ωwhen the rate of emission of particles from the SES andfrom the MC is the same.This is in contrast to the case when sinusoidal voltages eV ( ∼ )2 ( ~ Ω) P ( P ) FIG. 3: The noise P , Eq. (2), as a function of the amplitude eV ( ∼ )2 of the ac potential applied to the metallic contact. Theparameters of the single electron source in Eq. (A1), are: T = 0 . U = 0 . U = 0 . are applied to both inputs . Then the discussion canbest be cast into excitations of electron-hole pairs whichcreate a shot noise which has been measured . Fortwo oscillating voltages theory predicts significant two-particle correlations due to the Hanbury Brown-Twisseffect which depend on the phase delay of the two oscil-lating voltages .One can wonder whether the absence of the shot noisesuppression effect is possibly due to fluctuations in emis-sion of electrons from the biased metallic contact. Ourexpectation is that neither fluctuations nor a possiblepresence of multi-electron (multi-hole) states play a cru-cial role. To show it we consider next two circuits. III. SHOT-NOISE SUPPRESSION EFFECTWITH STOCHASTIC SINGLE-ELECTRONSOURCES In Fig. 4 we show a circuit with two single-electronsources, S L and S R each emitting one electron and onehole per period T . Initially the particle stream is reg-ular. However, say for particles emitted by the source S L , at the quantum point contact L an electron (a hole)can be either reflected to the metallic collector 3 or betransmitted to the central part of a circuit. Thus, thesingle-electron source S j together with a correspondingquantum point contact j = L, R comprise a stochasticsingle-electron source which can either inject into the cen-tral part of a collider one electron (hole) during a givenperiod or not. S L1 I (t) CL RS R I (t)21 43 FIG. 4: (color online) A mesoscopic electron collider circuitwith two stochastic single-particle streams originated from thequantum point contacts L and R . In the case two particlesenter the central part of a circuit they can collide at the quan-tum point contact C if the times of emission by the sources S L and S R were adjusted properly. Solid blue lines are edgestates with direction of movement indicated by arrows. Shortdashed red lines are quantum point contacts connecting dif-ferent parts of the circuit. Black rectangles numbered by 1 to4 are metallic contacts. We assume that all the metallic contacts are groundedand calculate the zero temperature cross-correlator P ≡ P ( sh,ad )12 , Eq. (B26), P = e Ω4 π ∞ X q = −∞ | q | X γ,δ =1 { S γ S ∗ δ } q { S γ S ∗ δ } ∗ q . (9)Since there is no bias all the phase factors are Υ δ = 1in Eq. (B25) and V γδ = 0 in Eq. (B26). The elementsof the frozen scattering matrix are expressed in termsof the transmission/reflection amplitudes t i /r i for thequantum point contacts i = L, R, C , time-dependentamplitudes S SESj ( t ) for sources S j , j = L, R , and cor-responding phase factors e ikL αβ with L αβ a length be-tween metallic contacts α and β . For instance, S ( t ) = e ikL S SESL ( t ) t L r C . For S SESj ( t ) we use Eq. (A5) withemission times t ( ± )0 and a pulse half-width Γ replacedby t ( ± ) j and Γ j . respectively. Then we find: P = − P (cid:26) ( T L − T R ) (10)+ T L T R h γ (cid:16) ∆ t ( − ) (cid:17) + γ (cid:16) ∆ t (+) (cid:17)i (cid:27) , where ∆ t ( ± ) = t ( ± ) L − t ( ± ) R with t ( ± ) j ( j = L, R ) the timeof an electron ( − )/hole (+) emission by the SES j , andthe suppression function γ (∆ t ) = (∆ t ) + (Γ L − Γ R ) (∆ t ) + (Γ L + Γ R ) , (11)If the single electron sources emit particles at differenttimes, ∆ t ( ± ) ≫ Γ L , Γ R , then the correlation is, P = − P (cid:8) T L + T R (cid:9) . (12)This expression is due to the shot noise produced bythe four uncorrelated particles (two electrons and twoholes) emitted by the two sources during the period T .Apparently the single-particle contribution (to the cross-correlator) is negative. We call this regime classical , sincethe shot noise can be explained in terms of single-particleprobabilities only, see Appendix C..On the other hand, if the pulses of the same width,Γ L = Γ R , are emitted at the same time, t ( − ) L = t ( − ) R (forelectrons) and t (+) L = t (+) R (for holes), then the cross-correlator is suppressed: P = − P ( T L − T R ) . (13) If in addition the circuit is symmetric, T L = T R , then thecross-correlator is suppressed down to zero.This suppression is due to a positive two-particle con-tribution arising (in addition to negative single-particlecontributions which are also present) when particles (ei-ther two electrons or two holes) collide at the quantumpoint contact C . Due to such collisions each of the par-ticles loses information about its origin (i.e., about thesource that emitted it) and the pair of particles prop-agating to contacts 1 and 2 in Fig. 4 becomes orbitallyentangled. We call this regime a quantum regime , sinceto describe a shot noise we additionally need to takeinto account the existence of both direct and exchangetwo-particle quantum mechanical amplitudes for collid-ing particles, see Appendix C, Eq. (C6).Thus with this circuit we showed that the shot noisesuppression effect is sensitive to a space-time confinementof electron states rather than to a regularity in appear-ance of electrons at the place (the QPC C ) where theycan overlap. IV. SHOT-NOISE SUPPRESSION EFFECTWITH SINGLE- AND TWO-PARTICLE SOURCES Now we consider a circuit (see Fig. 5) which containsboth a single particle emitter S and a two-particle emit-ter S . As a two-particle source we use two single-electron sources placed close to each other and emittingin synchronism. In the adiabatic case of interest herethe scattering amplitude S T ES ( t ) is the product of scat-tering amplitudes of single-electron sources comprisinga two-particle source. For simplicity we assume bothsources to be identical. Then S T ES is the square of theamplitude given by Eq. (A5) with t ( ± )0 and Γ replacedwith t ( ± )2 and Γ , respectively. At the time t ( − )2 ( t (+)2 ) S I (t) I (t) C S FIG. 5: (color online) A mesoscopic electron collider circuitwith a single-electron source S and a two-particle source S .At the quantum point contact C the particles emitted bydifferent sources can collide if the times of emission were ad-justed properly. Solid blue lines are edge states with directionof movement indicated by arrows. Short dashed red lines arequantum point contacts connecting different parts of a circuit.Black rectangles are metallic contacts. the pair of electrons (holes) is emitted by the source S .The cross-correlator P , Eq. (B26), reads P = −P ∞ X q = −∞ | q | (cid:12)(cid:12)(cid:12)(cid:12)n S SES (cid:0) S T ES (cid:1) ∗ o q (cid:12)(cid:12)(cid:12)(cid:12) , (14)where S SES ( t ) is the scattering amplitude, Eq. (A5), fora single-electron source S and S T ES ( t ) is the scatteringamplitude for the two-electron (two-particle) source S shown in Fig. 5.Simple calculations yield: P = −P (cid:26) γ (cid:16) ∆ t ( − ) (cid:17) + 2 γ (cid:16) ∆ t ( − ) (cid:17) + χ (cid:16) ∆ t ( − ) (cid:17) (15)+ γ (cid:16) ∆ t (+) (cid:17) + 2 γ (cid:16) ∆ t (+) (cid:17) + χ (cid:16) ∆ t (+) (cid:17) (cid:27) . where ∆ t ( ± ) = t ( ± )0 − t ( ± )2 . The function χ (∆ t ) is, χ (∆ t ) = 16Γ Γ (cid:16) (∆ t ) + (Γ + Γ ) (cid:17) , (16)and the suppression function γ (∆ t ) is given in Eq. (11)with Γ L and Γ R replaced by Γ (for a SES) and Γ (fora two-particle source), respectively.If all the particles are emitted at different times,∆ t ( ± ) ≫ Γ , Γ , the cross-correlator, P = − P , isdue to contributions of six uncorrelated particles (threeelectrons and three holes) emitted during the period T .While for simultaneous emission, ∆ t ( ∓ ) = 0, the cross-correlator is partially suppressed. If Γ = Γ , the cross-correlator is suppressed down to the level due to two-particles, P = − P . So when the two-electron wave-packet collides with a single-electron wave-packet, twocolliding electrons, one from each side, produce no noisewhile the remaining electron produces noise as if it prop-agated alone through the QPC. The same holds for holewave-packets. V. CONCLUSION A method to compare quantum states of initially un-correlated electrons in mesoscopic circuits was proposed.The electron streams should be directed onto a quantumpoint contact from different sides and the cross-correlatorof currents flowing out of the QPC should be measured.In general two uncorrelated streams produce additivenoises. However, if the particles overlap at the QPCthey become correlated and the noise gets suppressed.The closer the quantum states of particles resemble eachother the better the overlap that can be achieved, hencethe noise is suppressed more strongly. We considered several sources of electrons, in particu-lar, (i) a metallic contact, emitting a rather continuousstream of electrons with a rate proportional to the bias,and (ii) a periodically driven quantum capacitor, a single-electron source, emitting traveling wave-packets of elec-trons which are rather localized in space and alternatewith wave packet of holes. We found that the streamsproduced by the MC biased with a dc (ac) voltage andby the SES remain almost uncorrelated after passing theQPC even if the electrons are emitted with the samerate. Therefore, we conclude that the electrons of thesestreams are in quite different quantum states. On theother hand, if the periodic sequence of quantized voltagepulses is applied to an MC, then the resulting electronstream can be easily correlated with a stream emitted bythe SES resulting in a complete suppression of the shotnoise. From this we can conclude that the electrons ofthese streams are in the same quantum statesIf the streams are fluctuating then the shot noise canbe suppressed by the amount proportional to the aver-age number of particles overlapping at the QPC. We alsofound a partial suppression of the shot noise in the caseof pulses carrying different number of particles. Basi-cally the remaining noise is due to the difference of thenumbers of particles carried by the colliding pulses. Acknowledgments We thank M. Albert, P. Degiovanni and G. F`eve fordiscussion and communication. M. M. thanks the councilof the doctoral school program of Western Switzerlandfor an invitation to present a series of lectures. M. B. issupported by the Swiss NSF, MaNEP, and the EuropeanNetworks NanoCTM and Nanopower. Appendix A: Scattering amplitude and current of aSES As a single-electron source we use a quantumcapacitor , , described by a model in which a sin-gle circular edge state of circumference L in a cavity iscoupled via a quantum point contact (QPC) with trans-mission probability T to a linear edge state (see theleft upper corner of the Fig. 1). A potential U ( t ) = U + U cos(Ω t + ϕ ) periodic in time is induced uniformlyover the cavity with the help of a top gate. In the case ofa slow potential, Ω τ ≪ T , where τ is the time of one turnaround the cavity, the (frozen ) scattering amplitude ofa capacitor for an electron with incident energy E andpropagating in the linear edge state at time t is: S SES ( t, E ) = e iθ r √ − T − e iφ ( t,E ) − √ − T e iφ ( t,E ) . (A1)Here θ r is the phase of the reflection amplitude r = √ − T e iθ r of the QPC connecting the circular edgestate in the cavity to the linear edge state. φ ( t, E ) = θ r + φ ( E ) − πeU ( t ) / ∆ is the phase accumulated byan electron with energy E during one trip along thecavity and ∆ is the level spacing in the cavity. Thephase φ ( E ) = k F L + ( E − µ ) L/ ( ~ v D ) with k F a con-stant and v D a drift velocity can be taken to dependlinearly on the energy. In the following, we considerthe scattering amplitude for electrons with Fermi energy, S SES ( t ) ≡ S SES ( t, µ ). We are interested in the limitof a small transparency, T → 0, when the width of thelevels in the cavity is much smaller than the level spac-ing ∆. The amplitude U of the oscillating potential ischosen in such a way that during a period only one levelof the cavity crosses the Fermi level µ in the linear edgestate. The time of crossing t is defined by the condition φ ( t , µ ) = 0 mod 2 π . Introducing the deviation of thephase from its resonance value, δφ ( t ) = φ ( t, µ ) − φ ( t , µ ),we obtain the scattering amplitude, S SES ( t ) = − e iθ r T + 2 iδφ ( t ) T − iδφ ( t ) + O ( T ) . (A2)We keep only terms to leading order in T ≪ t ( − )0 and t (+)0 , respectively. At the time t ( − )0 one electron is emit-ted by the cavity into the linear edge state, while at thetime t (+)0 one electron enters the cavity, a hole is emitted.We suppose that the constant part of the potential U accounts for a detuning of the nearest electron level E n inthe SES from the Fermi level. Then the resonance timescan be found from the following equation: E n + eU (cid:16) t ( ∓ )0 (cid:17) = µ ⇒ U + U cos (cid:16) Ω t ( ∓ )0 + ϕ (cid:17) = 0 . (A3)For | eU | < ∆ / | eU | < | eU | < ∆ − | eU | we find, t ( ∓ )0 = ∓ t (0)0 − ϕ Ω , t (0)0 = 1Ω arccos (cid:18) − U U (cid:19) . (A4)The deviation from the resonance time, δt ( ∓ ) = t − t ( ∓ )0 ,can be related to the deviation from the resonance phase, δφ ( ∓ ) = ∓ M Ω δt ( ∓ ) , where ∓ M = dφ/dt | t = t ( ∓ )0 / Ω = ∓ π | e | ∆ − p U − U . With these definitions we canrewrite Eq. (A2) as follows: S SES ( t ) = e iθ r t − t (+)0 − i Γ t − t (+)0 + i Γ , (cid:12)(cid:12)(cid:12) t − t (+)0 (cid:12)(cid:12)(cid:12) . Γ ,t − t ( − )0 + i Γ t − t ( − )0 − i Γ , (cid:12)(cid:12)(cid:12) t − t ( − )0 (cid:12)(cid:12)(cid:12) . Γ , , (cid:12)(cid:12)(cid:12) t − t ( ∓ )0 (cid:12)(cid:12)(cid:12) ≫ Γ . (A5)Here Γ is (half of) the time during which the level risesabove or sink below the Fermi level:Ω Γ = T ∆4 π | e | p U − U . (A6)Eq. (A5) assumes that the overlap between the reso-nances is small, (cid:12)(cid:12)(cid:12) t (+)0 − t ( − )0 (cid:12)(cid:12)(cid:12) ≫ Γ . (A7)The basic equation for the time-dependent current is(see, e.g., Ref. 32), I ( t ) = − ie π Z dE (cid:18) − ∂f ∂E (cid:19) S SES ∂ (cid:0) S SES (cid:1) ∗ ∂t . (A8)Using Eq. (A5), we find the adiabatic current at zero tem-perature (for 0 < t < T ): I ( t ) = eπ Γ (cid:16) t − t ( − )0 (cid:17) + Γ − Γ (cid:16) t − t (+)0 (cid:17) + Γ . (A9)In each time interval 2 π/ Ω the current, shown in Fig. 6,consists of two pulses of Lorentzian shape with half-widthΓ . The pulses correspond to the emission of an electronand a hole. Integrating over time it is easy to check thatthe first pulse carries a charge e while the second pulsecarries a charge − e . Appendix B: Current correlation function1. General formalism Let the scatterer be connected via one-channel leadsto reservoirs having different potentials, V α ( t ) = V α + V ( ∼ ) α ( t ) . (B1) π/Ω) I ( t ) ( e Ω / π ) FIG. 6: The time-dependent current, Eq. (A8), generated bythe single-electron source at zero temperature. The posi-tive (negative) peak corresponds to emission of an electron(a hole). The parameters of the single electron source de-scribed by , Eq. (A1), are: T = 0 . U = 0 . U = 0 . ϕ = 0. Following the approach developed in Refs. 33,34, we in-clude the potential V ( ∼ ) α ( t ) = V ( ∼ ) α ( t + T ), T = 2 π/ Ω,oscillating with frequency Ω into the phase of the wavefunction for electrons injected into the circuit from reser-voir α . The constant part of the potential changes theFermi distribution function in contact α , f α ( E ) = 11 + exp E − µ α k B T α , µ α = µ + eV α . (B2)We introduce the second quantization operator ˆ a ′ α ( E )annihilating an electron in the state with energy E carry-ing a unit flux in reservoir α . Then the correspondingdistribution function is, (cid:10) a ′† α ( E ) a ′ α ( E ′ ) (cid:11) = f α ( E ) δ ( E − E ′ ) . (B3)If the reservoir α is subject to a periodic in time potential V ( ∼ ) α ( t ), then the wave function for particles described bythe operators ˆ a ′ α is a Floquet type function having side-bands with energies E n = E + n ~ Ω, n = 0 , ± , ± , . . . The amplitudes of side-bands are,Υ α,n = T Z dt T e in Ω t Υ α ( t ) , (B4)Υ α ( t ) = e − i e ~ t R −∞ dt ′ V ( ∼ ) α ( t ′ ) . We suppose that there is no oscillating potential in theleads connecting the reservoirs to the scatterer. Then theoperator for particles in lead α is, ˆ a α ( E ) = ∞ X n = −∞ Υ α,n ˆ a ′ α ( E − n ) . (B5)If the scatterer is driven periodically then it is char-acterized by the Floquet scattering matrix ˆ S F . We as-sume that the scatterer is driven with the same period T as the reservoirs. The element S F,αβ ( E n , E ) is a cur-rent scattering amplitude for an electron incoming fromthe lead β with energy E to be scattered with energy E n = E + n ~ Ω into the lead α . With these amplitudeswe find the operators for scattered particles, ˆ b α ( E ) = X β ∞ X m = −∞ S F,αβ ( E, E m )ˆ a β ( E m ) (B6)= X β ∞ X m = −∞ ∞ X n = −∞ S F,αβ ( E, E m )Υ β,n ˆ a ′ β ( E m − n ) . Now we calculate the symmetrized current correlationfunction in frequency representation, P αβ ( ω , ω ) = 12 D ∆ ˆ I α ( ω )∆ ˆ I β ( ω ) + ∆ ˆ I β ( ω )∆ ˆ I α ( ω ) E , (B7)where h· · · i stands for quantum-statistical averaging overthe (equilibrium) state of reservoirs, ∆ ˆ I α ( ω ) = ˆ I α ( ω ) − D ˆ I α ( ω ) E , and ˆ I α ( ω ) is the operator for the current in lead α ,ˆ I α ( ω ) = e ∞ Z dE n ˆ b † α ( E )ˆ b α ( E + ~ ω ) − ˆ a † α ( E ) a α ( E + ~ ω ) o . (B8)Using Eqs. (B3), (B5) – (B8) we find P αβ ( ω , ω ) = ∞ X l = −∞ π δ ( ω + ω − l Ω) P αβ,l ( ω ) , (B9) P αβ,l ( ω ) = e h Z dE ( δ αβ f αα ( E, E + ~ ω ) − f αα ( E, E + ~ ω ) X n X p,q S F,βα ( E l + n , E p ) Υ α,p S ∗ F,βα ( E n + ~ ω , E q + ~ ω ) Υ ∗ α,q − f ββ ( E, E + ~ ω ) X n X p,q S F,αβ ( E l + n , E p ) Υ β,p S ∗ F,αβ ( E n + ~ ω , E q + ~ ω ) Υ ∗ α,q + X γ,δ X n.m.s X p,q,p ,q f γδ ( E n , E m + ~ ω ) S F,βγ ( E l + s , E n + q ) Υ γ,q S ∗ F,αγ ( E, E n + p ) Υ ∗ γ,p × S F,αδ ( E + ~ ω , E m + q + ~ ω ) Υ δ,q S ∗ F,βδ ( E s + ~ ω , E m + p + ~ ω ) Υ ∗ δ,p ) . Here f αβ ( E , E ) = 12 n f α ( E ) [1 − f β ( E )] + f β ( E ) [1 − f α ( E )] o . (B10)We are interested in the zero-frequency limit of the equation given above, when the noise can be convenientlyrepresented as the sum of the thermal noise P ( th ) αβ (vanishing at k B T α = 0, ∀ α ) and the shot noise P ( sh ) αβ (vanishing atΩ = 0 and eV α = eV , ∀ α ). 2. Zero frequency noise power At l = 0 and ω = ω = 0 equation (B9) can be represented as follows: P αβ = e h Z dE n P ( th ) αβ ( E ) + P ( sh ) αβ ( E ) o , (B11a) P ( th ) αβ ( E ) = δ αβ ( f αα ( E, E ) + X γ F αγ ( E ) ) − F αβ ( E ) − F βα ( E ) , (B11b)with F αγ ( E ) = f γγ ( E, E ) X n,p,q S F,αγ ( E n , E q ) Υ γ,q S ∗ F,αγ ( E n , E p ) Υ ∗ γ,p , P ( sh ) αβ ( E ) = 12 X γ,δ X n.m.s X p,q,p ,q n f γ ( E n ) − f δ ( E m ) o (B11c) × S F,βγ ( E s , E n + q ) Υ γ,q S ∗ F,αγ ( E, E n + p ) Υ ∗ γ,p S F,αδ ( E, E m + q ) Υ δ,q S ∗ F,βδ ( E s , E m + p ) Υ ∗ δ,p . Now we show how Eq. (B9) was obtained. 3. Derivation of the current correlation function To make the calculations more transparent it is con-venient to represent the current as a sum, ˆ I α ( ω ) =0ˆ I ( out ) α ( ω ) + ˆ I ( in ) α ( ω ), of a current I ( out ) α carried by thescattered particles and a current I ( in ) α carried by the in-cident particles:ˆ I ( out ) α ( ω ) = e ∞ Z dE ˆ b † α ( E )ˆ b α ( E + ~ ω ) , (B12)ˆ I ( in ) α ( ω ) = − e ∞ Z dE ˆ a † α ( E )ˆ a α ( E + ~ ω ) . Then P αβ ( ω , ω ), Eq. (B7), can be represented as thesum of four terms, P αβ ( ω , ω ) = X i,j = in,out P ( i,j ) αβ ( ω , ω ) , (B13) P ( i,j ) αβ = 12 D ∆ ˆ I ( i ) α ( ω )∆ ˆ I ( j ) β ( ω ) + ∆ ˆ I ( j ) β ( ω )∆ ˆ I ( i ) α ( ω ) E . We evaluate each of these four contributions separately. a. Correlator for incoming currents The first term in Eq. (B13) reads, P ( in,in ) αβ ( ω , ω ) = e ∞ Z Z dE dE J ( in,in ) αβ + J ( in,in ) βα , (B14)where J ( in,in ) αβ = Dn ˆ a † α ( E ) ˆ a α ( E + ~ ω ) − D ˆ a † α ( E ) ˆ a α ( E + ~ ω ) Eo × n ˆ a † β ( E ) ˆ a β ( E + ~ ω ) − D ˆ a † β ( E ) ˆ a β ( E + ~ ω ) E oE . In the correlation J ( in,in ) βα with the indices interchanged,the order of operators in each of the products contribut-ing to J ( in,in ) βα is interchanged. Using Wick’s theorem, werepresent the average of the product of four operators viathe average of pair products and find, J ( in,in ) αβ = Π ( in,in ) αβ Ξ ( in,in ) αβ , (B15)Π ( in,in ) αβ = D ˆ a † α ( E ) ˆ a β ( E + ~ ω ) E , Ξ ( in,in ) αβ = D ˆ a α ( E + ~ ω ) ˆ a † β ( E ) E . Then using Eq. (B5) we obtain after straightforward buta little bit lengthy calculations, P ( in,in ) αβ ( ω , ω ) = 2 π δ ( ω + ω ) P ( in,in ) αβ ( ω ) , (B16) P ( in,in ) αβ ( ω ) = δ αβ e h Z dE f αα ( E , E + ~ ω ) . This is exactly what could be expected for equilibriumelectrons. Therefore, uniform oscillating potentials at thereservoirs in themselves do not produce additional noise. b. Correlator between incoming and outgoing currents The next term in Eq. (B13) is, P ( in,out ) αβ = − e ∞ Z Z dE dE J ( in,out ‘) αβ + J ( out,in ) βα , (B17)where J ( in,out ) αβ = Π ( in,out ) αβ Ξ ( in,out ) αβ , (B18)Π ( in,out ) αβ = D ˆ a † α ( E ) ˆ b β ( E + ~ ω ) E , Ξ ( in,out ) αβ = D ˆ a α ( E + ~ ω ) ˆ b † β ( E ) E . In the correlation J ( out,in ) βα the order of operators in theaverages of pairs is interchanged. Using Eqs. (B5) and(B6) we find,Π ( in,out ) αβ = X n,m,p Υ ∗ α,n S F,βα ( E + ~ ω , E ,m + ~ ω ) × Υ α,p f α ( E , − n ) δ ( E , − n − E ,m − p − ~ ω ) , Ξ ( in,out ) αβ = X n ,m ,p Υ α,n Υ ∗ α,p [1 − f α ( E , − n + ~ ω )] × S ∗ F,βα ( E , E ,m ) δ ( E , − n − E ,m − p + ~ ω ) . Next we integrate over energy E using the Dirac delta-function in Π ( in,out ) αβ . In the reminder we use E ,m − p = E , − n − ~ ω and find,1 P ( in,out ) αβ = − e ~ Z dE X n,m,p X n ,m ,p × f αα ( E , − n , E , − n + ~ ω ) Υ ∗ α,n Υ α,p Υ α,n Υ ∗ α,p × δ ( ω + ω − Ω [ p − n − m − p + n + m ]) × S F,βα ( E ,p − n − m , E ,p − n ) × S ∗ F,βα ( E ,p − n − m + ~ ω , E ,p − n + ~ ω ) . We shift (under the integral over E ): E → E + n ~ Ω.Then we introduce w = n − n instead of n . The sumover w gives us δ w . Then we introduce l = p − m − p + m instead of m and r = p − m instead of m . Finally weget, P ( in,out ) αβ ( ω , ω ) = ∞ X l = −∞ π δ ( ω + ω − l Ω) P ( in,out ) αβ,l , (B19) P ( in,out ) αβ,l ( ω ) = − e h Z dE f αα ( E , E + ~ ω ) × X r,p,p S F,βα ( E ,l + r , E ,p ) Υ α,p × S ∗ F,βα ( E ,r + ~ ω , E ,p + ~ ω ) Υ ∗ α,p . With similar steps we find that P ( out,in ) αβ can be ob-tained from P ( in,out ) αβ if one replaces: α ↔ β , E ↔ E ,and ω ↔ ω . Therefore, from Eq. (B19) we immediatelyobtain, P ( out,in ) αβ ( ω , ω ) = ∞ X l = −∞ π δ ( ω + ω − l Ω) P ( out,in ) αβ,l , (B20) P ( out,in ) αβ,l ( ω ) = − e h Z dE f ββ ( E , E + ~ ω ) × X r,p,p S F,αβ ( E ,l + r , E ,p ) Υ β,p × S ∗ F,αβ ( E ,r + ~ ω , E ,p + ~ ω ) Υ ∗ β,p . To compare subsequently Eqs. (B19) and (B20) withEq. (B9) we need additionally to redefine: r → n and p → q . c. Correlator between outgoing currents The last term in Eq. (B13) reads, P ( out,out ) αβ = − e ∞ Z Z dE dE J ( out,out ) αβ + J ( out,out ) βα , (B21)where J ( out,out ) αβ = Π ( out,out ) αβ Ξ ( out,out ) αβ , Π ( out,out ) αβ = D ˆ b † α ( E ) ˆ b β ( E + ~ ω ) E , Ξ ( out,out ) αβ = D ˆ b α ( E + ~ ω ) ˆ b † β ( E ) E . In the correlation J ( out,out ) βα the order of operators in thepair averages is interchanged.Using Eqs. (B5) and (B6) we calculate,Π ( out,out ) αβ = X γ X n,m,p,q δ ( E ,n − p − E ,m − q − ~ ω ) × f γ ( E ,n − p ) S ∗ F,αγ ( E , E ,n ) × Υ ∗ γ,p S F,βγ ( E + ~ ω , E ,m + ~ ω ) Υ γ,q , Ξ ( out,out ) αβ = X γ X n ,m ,p ,q δ ( E ,n − p + ~ ω − E ,m − q ) × [1 − f γ ( E ,n − p + ~ ω )] S ∗ F,βγ ( E , E ,m ) × Υ ∗ γ ,q S F,αγ ( E + ~ ω , E ,n + ~ ω ) Υ γ ,p . Then we integrate over energy E using the Diracdelta-function in Π ( out,out ) αβ . In the rest we use E = E ,n − p − m + q − ~ ω = E ,n + q − p − m + ~ ω and find, P ( out,out ) αβ = e ~ Z dE X γ,γ X n,m,p,q X n ,m ,p ,q × f γγ ( E ,n − p , E ,n − p + ~ ω ) × δ ( ω + ω − Ω [ n + q − p − m − n − q + p + m ]) × S ∗ F,αγ ( E , E ,n ) Υ ∗ γ,p S F,βγ ( E ,n − p − m + q , E ,n − p + q ) × S ∗ F,βγ ( E ,n + q − p − m + ~ ω , E ,n + q − p + ~ ω ) × Υ γ,q Υ ∗ γ ,q S F,αγ ( E + ~ ω , E ,n + ~ ω ) Υ γ ,p . To simplify we introduce t = n − p instead of n , w = n − p instead of n , l = n + q − p − m − n − q + p + m instead of m , and s = n + q − p − m instead of m .Then we get,2 P ( out,out ) αβ ( ω , ω ) = ∞ X l = −∞ π δ ( ω + ω − l Ω) P ( out,out ) αβ,l , (B22) P ( out,out ) αβ,l ( ω ) = e h Z dE X γ,γ X s,t,w X p,q,p ,q × f γγ ( E ,t , E ,w + ~ ω ) × S ∗ F,αγ ( E , E ,t + p ) Υ ∗ γ,p S F,βγ ( E ,l + s , E ,t + q ) Υ γ,q × S ∗ F,βγ ( E ,s + ~ ω , E ,w + q + ~ ω ) Υ ∗ γ ,q × S F,αγ ( E + ~ ω , E ,w + p + ~ ω ) Υ γ ,p . To compare subsequently with Eq. (B9) we need addi-tionally to redefine: t → n , w → m , p ↔ q , and γ → δ .Collecting together equations (B16), (B19), (B20), and(B22) we arrive at Eq. (B9). 4. Adiabatic regime In the adiabatic regime the Floquet scattering matrixelements to leading order in Ω → S ( t, E ), S F,αβ ( E n , E m ) = S αβ,n − m ( E ) . (B23)Within this approximation we find from Eqs. (B11b),(B11c) the following, P ( th,ad ) αβ ( E ) = − f αα ( E, E ) | S βα ( E ) | − f ββ ( E, E ) | S αβ ( E ) | + δ αβ ( f αα ( E, E ) + P γ f γγ ( E, E ) | S αγ ( E ) | ) , (B24a) P ( sh,ad ) αβ ( E ) = 12 X γ,δ ∞ X q = −∞ { f γ ( E q ) − f δ ( E ) } × Φ ( γδ ) α,q Φ ( γδ ) ∗ β,q , (B24b)where Φ α,q is a Fourier transform ofΦ ( γδ ) α ( t ) = S ∗ αγ ( t, E )Υ ∗ γ ( t ) S αδ ( t, E )Υ δ ( t ) . (B25)Here the over bar stands for a time average, X = R T dtX ( t ) / T . Calculating the shot noise we made a shiftof E → E − m ~ Ω and introduced q = n − m instead of m .One can see that the potentials oscillating at reservoirshave no effect on the thermal noise. Their effect on theshot noise in the adiabatic regime can be taken into ac-count formally by changing the phase of the scatteringelements S ϕρ ( t, E ) by the factor Υ ρ ( t ), Eq. (B4). 5. Zero-temperature adiabatic regime At zero temperatures there is no thermal noise. Cal-culating the shot noise we take into account that in theadiabatic regime the frequency Ω is so small that wecan neglect the energy dependence of the scattering ma-trix elements over the interval of order several ~ Ω. , In addition we assume also that all the potential differ-ences V αβ = V α − V β are small compared to the sig-nificant energy scales of the scattering matrix. Thenwith Eq. (B24b) the integral over energy in Eq. (B11a)becomes trivial and we find, P ( sh,ad ) αβ = e Ω4 π X γ,δ ∞ X q = −∞ (cid:12)(cid:12)(cid:12)(cid:12) eV γδ ~ Ω − q (cid:12)(cid:12)(cid:12)(cid:12) Φ ( γδ ) α,q Φ ( γδ ) ∗ β,q . (B26)Note the dc bias and ac bias enter this equation in astrongly non-equivalent way. Appendix C: Probability description of the currentcross-correlator for a circuit with SESs The single-electron source emits electrons and holeswhich are uncorrelated. Hence electrons ( e ) and holes( h ) contribute to noise independently, P = P ( e )12 + P ( h )12 .In the adiabatic regime we can neglect the energy de-pendence of the scattering matrix. Therefore, electronsand holes contribute to the noise equally, P ( e )12 = P ( h )12 =0 . P . Below we restrict ourself to the electron contribu-tion. We assume that the circuit has two inputs and twooutputs 1 and 2. In each input there is a SES emittingone electron per period. 1. Classical versus quantum regimes It was noticed in Ref. 3 that the cross-correlator P ( e )12 isrelated to the electron number correlator δ N as follows, P ( e )12 = e Ω2 π δ N , (C1)where δ N = N − N N . (C2)Here N is the probability to find one electron in output1 and one electrons in output 2 during the period T ,whereas N j is the probability to find an electron in output j = 1 , A ij for an electron emitted by the source j = L, R to arrive at the output i = 1 , A L = e ik F L L t L r C , A R = e ik F L R t R t C , (C3) A L = e ik F L L t L t C , A R = e ik F L R t R r C . With these amplitudes we find single-particle probabili-ties, N = |A L | + |A R | = T L + T C ( T R − T L ) , (C4) N = |A L | + |A R | = T L − T C ( T R − T L ) . The calculation of the two-particle probability N de-pends crucially on whether electrons collide at the centralQPC or not.If electrons pass the QPC C at different times,∆ t ( − ) ≫ Γ L , Γ R , then there are two independent pro-cesses contributing to N with amplitudes A (2) I = A L A R and A (2) II = A R A L . Since the two-particleamplitudes factorize into the product of single-particleamplitudes, we term this the classical regime. With theseamplitudes we find, N = (cid:12)(cid:12)(cid:12) A (2) I (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) A (2) II (cid:12)(cid:12)(cid:12) = T L T R (cid:0) R C + T C (cid:1) . (C5)Using Eqs. (C4) and (C5) in Eq. (C2) we find the cross-correlator P ( e )12 , Eq. (C1), to be the same as the one givenin Eq. (12) (times 0 . C ,∆ t ( − ) = 0, then the two particle amplitude is given bythe Slater determinant, A (2) = det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A L A R A L A R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (C6)This is why we call this regime quantum . Then the two-particle probability reads, N = (cid:12)(cid:12)(cid:12) A (2) (cid:12)(cid:12)(cid:12) = T L T R . (C7) Note this equation is independent of the parameters ofthe central QPC, that can be used as an indication ofa quantum regime. We emphasize that in the quantumregime the two-particle probability becomes the Glauberjoint detection probability , since electrons after colli-sion of the QPC C arrive at the outputs 1 an 2 simultane-ously (disregarding a possible difference in arrival timesdue to the different distances). With Eqs. (C7), (C4),(C2), and (C1) we recover the result given in Eq. (13). 2. Positive two-particle correlations in quantumregime Let us show that in the quantum regime colliding elec-trons are positively correlated. To this end we representthe single-particle probabilities as the sum of contribu-tions due to each of sources, N i = N ( L ) i + N ( R ) i with N ( j ) i = |A ij | , i = 1 , j = L, R . Then we split theparticle number correlator δ N , Eq. (C2), into the sumof three contributions, δ N = δ N ( LL )12 + δ N ( RR )12 + δ N ( d LR )12 . (C8)Here the first two terms are contributions due to eithersource alone, δ N ( jj )12 = −N ( j )1 N ( j )2 , j = L, R . Sincethe source emits single particles this contribution to thecross-correlator δ N is definitely negative. The thirdcontribution is due to a joint action of both sources, δ N ( d LR )12 = N − N ( L )1 N ( R )2 − N ( R )1 N ( L )2 . (C9)In the classical regime we use Eq. (C5) and find, δ N ( d LR )12 = 0, i.e., the particles emitted by differentsources remain uncorrelated. In contrast in the quantumregime, using Eq. (C7), we get, δ N ( d LR )12 = 2 T L T R R C T C . (C10)Therefore, in this regime the particles emitted by twosources and colliding at the central QPC C , see Fig. 4,become positively correlated. We stress the total overallcorrelation N remains negative. G. F`eve A. Mah´e, J.-M. Berroir, T. Kontos, B. Pla¸cais, D.C. Glattli, A. Cavanna, B. Etienne, Y. Jin, Science , 1169 (2007). S. Ol’khovskaya, J. Splettstoesser, M. Moskalets, and M.B¨uttiker, Phys. Rev. Lett. , 166802 (2008). J. Splettstoesser, M. Moskalets, and M. B¨uttiker, Phys.Rev. Lett. , 076804 (2009). M. D. Blumenthal, B. Kaestner, L. Li, S. Giblin, T. J.B. M. Janssen, M. Pepper, D. Anderson, G. Jones, D. A.Ritchie, Nature Physics , 343 (2007). S. J. Wright, M. D. Blumenthal, Godfrey Gumbs, A. L.Thorn, M. Pepper, T. J. B. M. Janssen, S. N. Holmes, D.Anderson, G. A. C. Jones, C. A. Nicoll, and D. A. Ritchie,Phys. Rev. B. , 233311 (2008). M. B¨uttiker, Phys. Rev. Lett. 68, 843, (1992); B. Yurkeand D. Stoler, Phys. Rev. A , 2229 (1992). P. Samuelsson, E. V. Sukhorukov, and M. B¨uttiker, Phys.Rev. Lett. , 026805 (2004). P. Samuelsson, I. Neder, and M. B¨uttiker, Phys. Rev. Lett. , 106804 (2009); Proceedings of the Nobel Symposium2009, Qubits for future quantum computers, May 2009 inGoteborg, Sweden, Phys. Scr. T , 014023 (2009). I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu, V.Umansky, Nature , 333 (2007). L. Mandel, Rev. Mod. Phys. , S274 (1999). R. Loudon, in: J.A. Blackman, J. Taguena (Eds.), Disorderin Condensed Matter Physics, Clarendon Press, Oxford,1991, p. 441. Ya. M. Blanter and M. B¨uttiker, Physics Reports , 1(2000). M. B¨uttiker, Phys. Rev. Lett. , 2901 (1990). P. Samuelsson and M. B¨uttiker, Phys. Rev. B , 041305R (2006) . C. W. J. Beenakker, in ”Quantum Computers, Algorithmsand Chaos”, International School of Physics Enrico Fermi,Vol. 126 IOS Press, Amsterdam, 2006; See also cond-mat/0508488. C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. , 2044 (1987). There is an interesting proposal how to produce separateelectron and hole streams into edge states : F. Batista, P.Samuelsson, arXiv:1006.0136 (unpublished). D. A. Ivanov, H.-W. Lee, and L. S. Levitov, Phys. Rev. B , 6839 (1997) J. Keeling, I. Klich, and L. S. Levitov, Phys. Rev. Lett. , 116403 (2006). J. Zhang, Y. Sherkunov, N. dAmbrumenil, andB. Muzykantskii, Phys. Rev. B , 245308 (2009). G. F´eve, P. Degiovanni, and Th. Jolicoeur, Phys. Rev. B P. Degiovanni, Ch. Grenier, and G. F´eve, Phys. Rev. B ,241307(R) (2009). M. B¨uttiker and M. Moskalets, Int. J. of Mod. Phys. B ,1555 (2010). V. S. Rychkov, M. L. Polianski, and M. B¨uttiker, Phys.Rev. B , 155326 (2005). M. Moskalets and M. B¨uttiker, Phys. Rev. B , 245305(2004). L.-H. Reydellet, P. Roche, D. C. Glattli, B. Etienne, andY. Jin, Phys. Rev. Lett. , 176803 (2003). J. Splettstoesser, S. Ol’khovskaya, M. Moskalets, and M.B¨uttiker, Phys. Rev. B , 205110 (2008). A. Prˆetre, H. Thomas, and M. B¨uttiker, Phys. Rev B. ,8130 (1996). J. Gabelli, G. F`eve, J.-M. Berroir, B. Pla¸cais, A. Cavanna,B. Etienne, Y. Jin, D.C. Glattli, Science , 499 (2006). M. Moskalets, P. Samuelsson, and M. B¨uttiker, Phys. Rev.Lett. , 086601 (2008). M. Moskalets and M. B¨uttiker, Phys. Rev. B , 205316(2004). M. B¨uttiker and M. Moskalets, Lecture Notes in Physics, , 33 (2006). A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B. , 5528 (1994). M. H. Pedersen and M. B¨uttiker, Phys. Rev. B. , 12993(1998). M. B¨uttiker, Phys. Rev. B. , 12485 (1992). M. Moskalets and M. B¨uttiker, Phys. Rev. B , 205320(2002). G. J. Glauber, Phys. Rev.130