Collisional interferometry of Levitons in quantum Hall edge channels at ν = 2

Giacomo Rebora,1, 2 Matteo Acciai,3, 1 Dario Ferraro,1, 2 and Maura Sassetti1, 2

1Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy2CNR-SPIN, Via Dodecaneso 33, 16146, Genova, Italy

3Department of Microtechnology and Nanoscience (MC2),Chalmers University of Technology, S-412 96 Göteborg, Sweden

(Dated: July 6, 2020)

We consider a Hong-Ou-Mandel interferometer for Lorentzian voltage pulses applied to QuantumHall edge channels at filling factor ν = 2. Due to inter-edge interactions, the injected electronicwave-packets fractionalize before partitioning at a quantum point contact. Remarkably enough,differently from what theoretically predicted and experimentally observed by using other injectiontechniques, we demonstrate that, when the injection occurs through time-dependent voltage pulses(arbitrarily shaped), the Hong-Ou-Mandel noise signal always vanishes for a symmetric device,and that a mismatch in the distances between the injectors and the point of collision is needed inorder to reduce the visibility of the dip. We also show that, by properly tuning these distancesor by applying different voltages on the two edge channels in each arm of the interferometer, it ispossible to estimate the intensity of the inter-edge interaction. The voltage pulses are chosen of theLorentzian type because of their experimental relevance.

I. INTRODUCTION

The progresses in the experimental control of individ-ual electronic degrees of freedom ballistically propagatingin mesoscopic devices led to the birth of a new branch ofcondensed matter physics known as Electron QuantumOptics (EQO)1–3. In this framework, intensity interfer-ometers such as the Hanbury-Brown-Twiss (HBT)4 andthe Hong-Ou-Mandel (HOM)5 have been realized by par-titioning electronic wave-packets6 and making them col-lide with a tunable delay7 at a Quantum Point Contact(QPC). These seminal experiments realized with voltage-driven mesocopic capacitors8–10, first proposed theoreti-cally in Refs.11–15, have been realized by means of pe-riodic trains of electrons and holes with wave-packetspeaked in energy.

One of the main differences between the photonic andthe electronic case is represented by the fact that elec-trons are charged interacting particles. This leads tomany-body effects which strongly affect the dynamics ofexcitations and play a major role in various experimen-tal situations. In particular this is true when experimentsare carried out in Quantum Hall (QH) edge channels atfilling factor ν = 2, where inter-channel interaction can-not be neglected16. This emerges dramatically in HOMexperiments realized with a driven mesoscopic capacitorin the non-adiabatic regime12,14,17,18, where the visibil-ity of the predicted dip in the auto-correlated noise as afunction of the injection delay12,14, signature of the anti-bunching of electrons, is strongly reduced due to electron-electron interactions19–21.

An alternative protocol for the injection of electronsconsists in the application of a train of well designedtime-dependent voltage pulses22. According to what hasbeen discussed by Levitov and coworkers23–25, a properlyquantized Lorentzian drive leads, in a non-interactingsystem, to the injection of purely electronic wave-packetswithout any additional electron-hole pair contribution.

This prediction has been validated experimentally26,27

through the realization of HBT and HOM collisional ex-periments in non-interacting narrow constrictions real-ized in two-dimensional electron gases. These low-energyexcitations, usually called Levitons, are predicted to berobust with respect to interaction-induced decoherence28

and anomalous correlations among electrons29,30. More-over, this robustness survives also in very strong inter-acting environments such as Fractional QH states25,31–33.Here, remarkable features related to a crystallization ofLevitons in the time domain have been reported by someof the authors34–36 and can be observed in HOM inter-ferometers.

As stated above, EQO in QH edge states at filling fac-tor ν = 2 has been investigated so far in the case of theemitted excitations generated via driven mesoscopic ca-pacitors (as experiments in this regime typically involvethis kind of source)16,19–21,37. Some theoretical workshave also addressed the case of injection at ν = 2 via volt-age pulses, focusing on the evolution of excitations due tointeractions on the HBT noise signal28,37–39. Thus, a de-tailed theoretical analysis of collisional HOM setups forvoltage pulses and in particular for Levitons in QH edgechannels at ν = 2, even if relevant for the interpretationof forthcoming experiments, is still missing. This paperintends to fill this gap by studying the signatures of inter-edge interactions in the profile of the HOM noise signalemphasizing the difference with the mesoscopic capacitorset-up. We demonstrate that the visibility of the centraldip is always maximal (the dip goes to zero) when theexcitations are injected through time-dependent voltagesof arbitrary shape and the setup is symmetric, namelythe distance of the two injectors from the QPC is thesame for both sides of the interferometer. This can beseen as a signature of the robustness of voltage signalsagainst decoherence, differently from what observed forthe driven mesoscopic capacitor16,20,21. Our results willbe discussed in detail for the particular case of Lorentzian

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We will also observe that the visibility can be stronglyreduced only in an asymmetric device, where a mismatchin the lengths of the two arms of the interferometer ispresent. Moreover, from the evolution of the visibilityas a function this mismatch it is also possible to extractinformation about the strength of the electron-electroninteraction. We will also propose a more direct measure-ment of the interaction based on the fact that, when prop-erly tuned voltages are applied on both the edge channelsof the two arms of device, it is possible to cancel all theinteraction-dependent features of the HOM noise recov-ering the non interacting case. Such kind of fine tuningallows to deduce the value of the interaction between thechannels.

The paper is organized as follows. Section II describesthe model for two interacting QH edge channels in termsof the edge-magnetoplasmon scattering matrix formal-ism. In Section III we describe the general aspects of theHOM interferometry for electrons injected by means ofvoltage drives. In Section IV we focus on the injectionof Levitons discussing the essential features of the HOMnoise as a function of the delay in the injection for botha symmetric and an asymmetric set-up. In Section V wedemonstrate that a collisional HOM experiment allows tomeasure the inter-edge interactions by studying the evo-lution of side dips in the HOM signal when the two edgechannels are driven independently. Finally, Section VIis devoted to the conclusions. Technical details of thecalculations are reported in three Appendices.

FIG. 1. Schematic view of QH channels at integer fillingfactor ν = 2. The shaded red area represents the interactionregion, which has a finite length L and is described by thescattering matrix S(L, ω). a) After passing this region, theincoming bosonic fields φ̃1,2(0, ω) are transformed into the

outgoing ones φ̃1,2(L, ω). b) The input voltages Vin1/2, applied

to the edge channels. Due to interactions, the excitationsemerging after the propagation from x = 0 to x = L areequivalent to those that would be generated by the outputvoltages V out1/2 applied to the channels directly at the end ofthe interaction region. These output voltages are related tothe incoming ones by Eq. (15).

II. MODEL

We consider a QH bar at filling factor ν = 2. Thetwo copropagating edge channels are assumed to inter-act along a region of finite length L via a screened (δ-like) Coulomb repulsion19,28,40–43 which correctly repro-duces the experimental observations at low enough ener-gies16,44,45. This mechanism is well described within thechiral Luttinger liquid theory based on bosonic collectiveexcitations called edge-magnetoplasmons28,38,42,43. Here,the Hamiltonian density H describing the two copropa-gating channels along each edge is the sum of a kineticterm H0 and an interaction contribution Hint. Follow-ing Wen’s hydrodynamical model46 they are written as(~ = 1)

H0 =∑i=1,2

vi4π

(∂xφi(x))2

(1)

Hint =u

2π(∂xφ1(x)) (∂xφ2(x)) , (2)

where the index i = 1, 2 labels inner and outer chan-nels respectively (see Fig. 1a), while φi are chiral bosonicfields satisfying the commutation relations

[φi(x), φj(y)] = iπ sign(x− y)δij (3)

and related to the particle density operator by47

ρi(x) =1

2π∂xφi(x) (4)

where the fermionic field ψi(x) is related to the bosonicone φi(x) via

ψi(x) =Fi√2πa

e−iφi(x) (5)

where a is a short distance cut-off and Fi are the Kleinfactors46,47.

In Eqs (1)-(2) vi are the bare propagation velocities ofthe two edge channels (here, without loss of generality, weassume v1 ≥ v2) and u is the intensity of the inter-edgecoupling. The full interacting problem can be diagonal-ized through a rotation in the bosonic fields space by anangle θ satisfying

tan(2θ) =2u

(v1 − v2). (6)

This parameter encodes the interaction strength, θ = 0being the non-interacting limit and θ = π/4 representingwhat in the literature is usually indicated as the stronginteracting regime16,19,41. However, the stability of themodel, namely the request that both eigenvelocities arepositive48, imposes a constraint on the maximum admis-sible value of u (see below), therefore strictly speakingthis limit can be properly obtained only for v1 = v2 bykeeping u fixed49. Experimentally, values of θ ranging

3

from θ ≈ π/650,51 to θ ≈ π/416 have been reported, indi-cating that this parameter strongly depends on the spe-cific details of the considered set-ups.

The rotation leads to two new bosonic fields, definedby

φρ(x) = cos θ φ1(x) + sin θ φ2(x) (7)

φσ(x) = − sin θ φ1(x) + cos θ φ2(x) , (8)

in terms of which the full diagonalized Hamiltonian den-sity becomes

H =∑β=ρ,σ

vβ4π

(∂xφβ(x))2 . (9)

These fields are associated with two new collective modes:a slow dipolar and a fast charge mode propagating re-spectively with velocities vσ and vρ, where

vρ/σ =

(v1 + v2

2

)± 1

cos(2θ)

(v1 − v2

2

). (10)

The dynamics of the edge channels can be solvedwithin a scattering formalism42,52. As depicted inFig. 1a, the fields outgoing from a scattering region offinite length L are related to the incoming ones throughthe edge-magnetoplasmon scattering matrix S(L, ω) as(

φ̃1(L, ω)

φ̃2(L, ω)

)= S(L, ω)

(φ̃1(0, ω)

φ̃2(0, ω)

). (11)

Here, φ̃1/2(x, ω) is the Fourier transform, with respect to

time, of φ1/2(x, t) and42,43

S =(

cos2 θ eiωτρ + sin2 θ eiωτσ sin θ cos θ(eiωτρ − eiωτσ )sin θ cos θ(eiωτρ − eiωτσ ) sin2 θ eiωτρ + cos2 θ eiωτσ

)(12)

where τρ/σ = L/vρ/σ are the times of flight associatedwith fast and slow modes respectively.

Following Refs.26,27,53 we can consider an electronsource modeled as an ohmic contact coupling each chan-nel to a time-dependent voltage source and allowing us tocontrol the injection of electrons through voltages V1,in(t)and V2,in(t) applied to the inner and the outer channel re-spectively, according to the conventional coupling Hamil-tonian

HU = −e∫ρi(x)Ui,in(x, t)dx , (13)

where i = 1 (2) labels the inner (outer) channel and −e(e > 0) is the electron charge and Ui,in(x, t) describesthe effect of the voltage source connected to the chan-nels. We write it as Ui,in(x, t) = Θ(−x)Vi,in(t) whereVi,in(t) is the time-dependent voltage of the source andthe Heaviside step function Θ(−x) specifies the regionwhere this potential is applied32,33.

This classical potential, coupled to the charge densityalong the edge according to the above Equation can then

FIG. 2. Schematic view of the HOM interferometer. Twopairs of copropagating and interacting edge states, on oppo-site sides of a QH bar, meet at a QPC. The electrons sourcesA and B are modeled as ohmic contacts which are used todrive each edge with time-dependent voltages (here VA/B(t)

is a compact notation to indicate VA/B

1/2,in(t), which are the

voltages shown in Fig. 1b and that are mixed by the interac-tion region indicated with the shaded red area). A detectorD is placed just after the QPC in order to measure currentcorrelations. Notice that the region of the QPC is brighter toindicate the fact that here the electron-electron interaction isscreened.

be seen as an external classical forcing for a quantum har-monic oscillator leading to the generation of a coherentstate of the edge-magnetoplasmons along the edge chan-nels. The displacement parameter associated to this co-herent state is derived by solving the equations of motionfor the bosonic fields (considering the complete Hamilto-nianH+HU ) and is proportional to the Fourier transformof the voltages Ṽi,in(ω)

39,54. In the frequency domain theinteracting region acts as a beam-splitter for this coher-ent state through the edge-magnetoplasmon S in exactlythe same way as for the bosonic modes in absence ofvoltage, namely(

Ṽ1,out(ω)

Ṽ2,out(ω)

)= S(L, ω)

(Ṽ1,in(ω)

Ṽ2,in(ω)

). (14)

In the time domain (see Fig. 1b) this leads to

V1,out(t) = cos2 θ V1,in(t− τρ) + sin2 θ V1,in(t− τσ)

+ sin θ cos θ[V2,in(t− τρ)− V2,in(t− τσ)]V2,out(t) = sin θ cos θ[V1,in(t− τρ)− V1,in(t− τσ)]

+ sin2 θ V2,in(t− τρ) + cos2 θ V2,in(t− τσ)

(15)

clearly showing that, at the end of the interaction region,the two incoming voltages are mixed.

III. GENERAL ASPECTS OF HOMINTERFEROMETRY

We now consider the effect of interaction in a HOMexperiment where electronic wave-packets, generated by

4

means of applied voltage pulses, collide at a QPC witha controlled delay in time. Unlike the injection with adriven mesoscopic capacitor7,8,20, this case still lacks of adetailed investigation from both the theoretical and theexperimental point of view.

Fig. 2 shows the HOM interferometer. Here, excita-tions emitted by the voltage sources A and B fraction-alize when going through the interacting regions and arethen partitioned at a QPC. For the moment, we assumethat the injection only occurs into the inner channelsof each edge, postponing the analysis of a more general

case to Sec. V. Therefore we set VA/B2,in (t) = 0, where the

notation now takes into account the fact that one canapply a voltage both to the A and the B source. It isworth noting that, as far as QH edge states in the in-teger regime are concerned, the different edge channelscan be addressed independently by means of additionalupstream QPCs55,56 or quantum dots with high trans-parency7. Moreover, we assume that the partitioning atthe QPC involves the inner channels only, which we la-bel as 1A (right-moving) and 1B (left-moving), relatedto the incoming fermionic fields ψI1A and ψ

I1B that are

evaluated immediately before the QPC. Such a situationcan be implemented by properly tuning the QPC trans-parency in such a way that the outer channels are com-pletely transmitted, while the inner ones are also par-tially reflected7,19,20. Thus, the brighter region of theQPC (see Fig. 2) is not included in the interacting regionand fermions are locally free at this location. Accordingto this and assuming a local tunneling, the free fermionicfields, outgoing from the QPC, are related to the incom-ing ones through a scattering matrix(

ψ1A(t)ψ1B(t)

)O

=

(√R i√T

i√T√R

)(ψ1A(t)ψ1B(t)

)I

(16)

where T and R = 1− T are positive real parameters de-scribing the probability for a particle to be transmitted orreflected, respectively. These probabilities are assumedas energy independent, a condition which is typically wellfulfilled in experiments6,7,20. This scattering approachfor fermionic fields is justified, in our specific case of in-teracting channels at ν = 2, as long as both the inter-edgeinteraction and the tunneling are local (see for exampleSupplementary Material of Ref.19). According to the chi-rality and locality of the coupling we can consider theinteraction region extending from just after the injectionpoint to just before the QPC28,42,43. This mathemati-cal description is physically motivated by the fact thatboth the contacts used to apply the voltage and the gatesthat realize the QPC locally enhance the screening of theinteraction, that can be therefore assumed as negligiblein these two regions. This theoretical approach alreadyshowed a very good agreement with the experimental ob-servation for HOM interferometers realized using drivenmesoscopic capacitors as single electron sources16,20.

Following what is usually investigated in EQO experi-ments, we focus our attention on the zero-frequency auto-correlated noise SHOM, which we evaluate just after the

QPC. This quantity is defined as57–59

SHOM =

∫[〈ID(t)ID(t′)〉 − 〈ID(t)〉 〈ID(t′)〉] dtdt′, (17)

where ID(t) is the total current arriving at the detectorD (see Fig. 2) and it is composed of the currents flowingin the channels 2A and 1B:

ID(t) = I2A(t) + I1B(t) . (18)

The current operator on a given channel j = 2A, 1B reads

Ij(t) = −evF : ψ†j (t)ψj(t) :, where : · · · : denotes the nor-mal ordering with respect to the Fermi sea and fermionicfields are evaluated at the level of the detector D.

In full generality, the HOM noise can be expressed as

SHOM = S2A,2A + S2A,1B + S1B,2A + S1B,1B , (19)

where (i, j = 2A, 1B)

Sij =

∫[〈Ii(t)Ij(t′)〉 − 〈Ii(t)〉 〈Ij(t′)〉] dtdt′ . (20)

The notation SHOM is chosen to emphasize that we aredealing with the zero-frequency noise in the HOM config-uration, i.e. when both sources are on. We note that inEq. (19) the first contribution S2A,2A consists only of thecurrent auto-correlations of the totally transmitted ex-ternal channel but this does not affect the measurementsbecause its contribution is zero. Also the terms S2A,1Band S1B,2A do not contribute, due to the fact that av-erages involving current operators in different channelsfactorize because no interaction occurs at the level ofthe QPC. Therefore, the only relevant contribution inEq. (19) is S1B,1B which involves terms referring to bothinner channels. This is because, according to Eq. (16),the fermionic field ψO1B at the output of the QPC is ex-pressed in terms of both incoming fields ψI1A and ψ

I1B . In

order to simplify the notation, in the following discussionwe will refer to the inner channels 1A and 1B just as Aand B.

By using Eqs. (16) and (17) we can express the totalnoise SHOM as

60

SHOM = −(evF )2RT∫

∆Q(t, t′)dtdt′ (21)

where

∆Q(t, t′) =∆G(e)A (t′, t)∆G(h)B (t

′, t) + ∆G(h)A (t′, t)∆G(e)B (t

′, t)

+∆G(e)A (t′, t)G(h)F,B(t

′, t) + ∆G(h)A (t′, t)G(e)F,B(t

′, t)

+G(e)F,A(t′, t)∆G(h)B (t

′, t) + G(h)F,A(t′, t)∆G(e)B (t

′, t).

(22)

In Eq. (22) ∆G(e/h)A/B are the non-equilibrium excess firstorder coherence functions15,61,62

∆G(e/h)A/B (t′, t) = G(e/h)A/B (t

′, t)− G(e/h)F,A/B(t′ − t) , (23)

5

where

G(e)A/B(t′, t) =

〈ψ†A/B(t)ψA/B(t

′)〉

(24a)

G(h)A/B(t′, t) =

〈ψA/B(t)ψ

†A/B(t

′)〉

(24b)

are correlators evaluated over the non-equilibrium state

induced by the voltage injection, whereas G(e/h)F,A/B are thecorrelation functions for the equilibrium states (i.e. whenno drive is applied) and are evaluated over the Fermi sea.The channel label (A/B) will be dropped in the followingwhen referring to the equilibrium correlation functions,as they are assumed identical for both channels. Theeffect of the external voltage drive can be properly takeninto account with a phase factor39,60, in such a way thatEq. (23) is rewritten as

∆G(e/h)A/B (t′, t) = G(e/h)F (t

′ − t)(e∓iϕA/B(t,t

′) − 1)

(25)

where

ϕA/B(t, t′) = e

∫ tt′VA/B1,out (τ)dτ (26)

is the phase contribution due to the time dependent volt-age, carrying information about interaction effects ac-cording to Eq. (15). It is worth noting that, limited tothe injection through voltage and under the assumptionof local interaction acting over a finite length, the co-herence functions can be written as the free fermionicones times phase factors encoding the effect of the ap-plied voltage and the interaction. Therefore, our systemcan be mapped onto a free fermion problem subject to amodified voltage which takes into account the fractional-ization effects39. Incidentally this fact can be seen as afurther validation of Eq. (16).

By replacing Eq. (25) into Eq. (22), the correlationfunction ∆Q(t, t′) can be expressed as

∆Q(t, t′) = 2G(e)F (t′ − t)G(h)F (t

′ − t)×[1− cos(ϕA(t, t′)− ϕB(t, t′))

].

(27)

If one of the two sources is switched off, the above for-mula simplifies and the HBT noise associated with thepartitioning of excitations incoming only in one arm ofthe interferometer is recovered (i = A,B)60:

SHBT,i = −2(evF )2RT∫dt dt′G(e)F (t

′ − t)G(h)F (t′ − t)

× [1− cos(ϕi(t, t′))] .(28)

In the following, according to what is usually done inconventional HOM experiments with voltage pulses26,27,we consider the two sources A and B to be driven byidentical signals apart from a controlled time delay δ,namely

V B1,in(t) = VA1,in(t+ δ). (29)

An important consequence arises when we consider theinteraction strengths and the distances between thesources and the QPC to be equal in both arms of the in-terferometer (symmetric configuration with θA = θB = θ

and LA = LB = L). In this case, the voltages VA/B1,out

after the interacting regions are the same for both arms.This can be easily seen from Eq. (15) where it is clearhow these voltages depend on the interaction strength θand on the interaction length L (via the times of flightτρ/σ). As a result, at zero injection delay δ = 0 one hasϕA(t, t

′) = ϕB(t, t′), leading to ∆Q(t, t′) = 0. Therefore

we arrive at the consequence that, even in the presenceof interactions, the HOM noise in a symmetric configu-ration always vanishes for a synchronized emission in thetwo incoming channels (δ = 0), regardless of the particu-lar form of the signal used for the time-dependent voltageinjection. Notice that these considerations still hold alsoin the case of a long-range interaction21,37,39 as long asit preserves the symmetry of the set-up.

The injection via the mesoscopic capacitor occurs ata well defined energy above the Fermi level and it hasbeen shown28 that in this case the emitted wave-packetsundergo a relaxation towards low-energy degrees of free-dom before the process of fractionalization takes place.On the contrary, voltage-generated excitations are robustin this respect, as the energy relaxation does not occurfor them28,37 and they are only affected by the fraction-alization process during their propagation through theinteracting region.

This qualitative difference is consistent with our re-sults, showing that the excitations injected via voltagepulses are robust and do not display any suppression ofthe HOM dip at zero delay.

We recall that a standard experimental procedure con-sists in normalizing the measured HOM signal with re-spect to the HBT ones16, thus defining the ratio

R(δ) = SHOM(δ)SHBT,A + SHBT,B

, (30)

where we have taken into account the fact that the HOMnoise contribution is the only one which depends on thetime delay δ. The noise in Eq. (21) can be rewritten interms of the average time t̄ = (t + t′)/2 and of the timedifference τ = t − t′ as (adapting the definition to thecase of a periodic drive53)

SHOM = −(evF )2RT∫ T

2

−T2

dt̄

T

∫ +∞−∞

dτ∆Q(t̄+

τ

2, t̄− τ

2

).

(31)These integrals are performed analytically in AppendixA by introducing the Fourier series

e−ie∫ t0V (t′)dt′ = e−iqΩt

+∞∑l=−∞

pl(q) e−ilΩt (32)

where Ω = 2π/T and the photoassisted coefficients pl arelinked to the probability amplitude for photon absorption(l > 0) or emission (l < 0)26.

6

By using this approach, the ratio (30) can be writtenas

R(δ) =+∞∑l=−∞

|Pl(q; δ)|2|Ωl||p̃l,A(q)|2|Ω(l + q)|+ |p̃l,B(q)|2|Ω(l + q)|

,

(33)where p̃l,A/B(q) and Pl(q; δ) are new photoassisted coef-ficients defined in Appendix A (Eqs. (A4) and (A6)) andthey can be expressed as functions of amplitudes pl de-fined in Eq. (32). They are related to the phases ϕA−ϕBand ϕA/B , respectively, and fully take into account theeffects of interactions. It is worth noting that Eq. (33) aswell as all the following results are obtained in the zerotemperature limit, thermal corrections being marginal inrealistic experimental conditions16,20,26,27.

In the next Section, we specify the above general analy-sis to the case of Lorentzian pulses, a particularly relevantdrive in the context of EQO25,26, considering symmetricand asymmetric configurations. Both of them are ana-lyzed by relying on the general expression (33), wherethe proper photoassisted coefficients of Lorentzian pulses[see Eq. (A5)] will be used.

IV. HOM INTERFEROMETRY FOR LEVITONS

In the previous Section, we have proved that the ex-cess noise in a symmetric HOM configuration, is alwayszero for simultaneous injection from the the sources, in-dependently of the shape of the voltage and of the inter-actions occurring along the channels. In this Section wewill focus on a specific form for the voltage drive which isparticularly relevant in the context of experimental EQOand we present the results for the ratio R as a functionof the time delay δ between the two sources. In order toproperly describe realistic experimental configurations,we consider the two sources to be periodically driven intime. The injection of a periodic train of single elec-trons, without hole contributions, is possible by apply-ing properly quantized Lorentzian voltage pulses23,25,26

of the form

V A1,in(t) ≡ V (t) = V0∑j∈Z

τ0τ20 + (t− jT )2

, (34)

where V0 = −2q/e (~ = 1), with q ∈ N. When q = 1, oneelectron per period T is emitted, realizing a train of so-called Levitons26,27. In Eq. (34) τ0 represents the widthin time of each Lorentzian pulse of the periodic train.

A. Symmetric setup

In this Section we analyze what happens to the noiseratio R, in Eq. (33), when identical Lorentzian voltagepulses with unitary charge (q = 1) are applied to bothcontacts. We consider a symmetrical configuration forthe interferometer, meaning that the lengths of the two

interacting regions are equal (LA = LB = L), as wellas the inter-edge interaction strength (θA = θB = θ) inthe two incoming channels. It is worth noticing that inthis situation the photoassisted coefficients p̃l,A(q) andp̃l,B(q) entering in Eq. (33) are equal.

Due to interactions, as the time delay δ between theright and the left moving electrons is varied, we find threecharacteristic features in the noise profile (see Fig. 3). Atδ = 0 a central dip appears while two symmetrical side-dips emerge at positions δsd = ±|τρ − τσ|. The shape ofthese three dips is Lorentzian reflecting the overall formof the applied voltage pulses, while their width dependson the timescale τ0. According to this the dips are morepronounced for a smaller ratio τ0/T .

This interference pattern is interpreted in terms of thedifferent excitations emerging after the interacting re-gion. Indeed, after the injection, the electronic wave-packet fractionalizes into a slow and a fast mode carry-ing different charges. According to Eq. (27), the centraldip, which corresponds to the situation of simultaneousinjection from the two sources, goes exactly to zero be-cause these identical excitations interfere destructively.This is in striking contrast with what has been observedin a HOM experiment at ν = 2 where the injection wasachieved by means of driven mesoscopic capacitors16,20,where the visibility of the central dip is always reducedby interactions19.

The destructive interference is also responsible for theside-dip structures appearing when fractionalized excita-tions with different velocities collide (see Fig. 3). Forinstance, at a delay δsd = τσ − τρ the fast right movingexcitation and the slow left moving one reach the QPCat the same time. Furthermore, Fig. 3 also shows as areference the behavior of the noise ratio in absence of in-teractions (θ = 0) which always reaches zero (at δ = 0)but does not show any side dip because no fractional-ization occurs in this case. Our numerical curve (black)perfectly coincide with the theoretical analytical formula(gray dots) derived for the HOM noise ratioR of collidingLevitons with unitary charge in the absence of interac-tions (θ = 0)32,34,53

R0(δ) =sin2

(π δT)

sinh2(2π τ0T

)+ sin2

(π δT) . (35)

We have used this reference result as a check for the va-lidity of our numerical calculations.

In view of possible future experimental validations ofour theoretical analysis, the plots of R as a function ofδ/T for different values of θ have been obtained by fix-ing the ratio between the pulse width and the period tobe τ0/T = 0.05, compatible with state of the art mea-surements carried out in narrow constrictions26,27, whilethe time of flight of both slow and fast modes are of theorder of 10 ÷ 100 ps, as interaction lengths are L ∼ µmand velocities are vρ/σ ∼ 104 ÷ 105m/s. This makes ourprediction observable in nowadays EQO experiments.

7

-0.4 -0.2 0.0 0.2 0.4-0.20.0

0.2

0.4

0.6

0.8

1.0

FIG. 3. Ratio R in Eq. (33), for Lorentzian pulses, as a func-tion of time delay over period (δ/T ) for a symmetric setup.The HOM noise generated by the collision of periodical trainsof Lorentzian pulses is shown for different interaction param-eters: θ = 0 (black curve), θ = π/6 (blue curve) and θ = π/4(red curve). Gray dots represent the analytical prediction inEq. (35) for the non-interacting case. Other parameters are:τ0/T = 0.05, vρ = 4 · 105m/s and vσ = 1.8 · 105m/s, withLA = LB = 2µm. Notice that the positions of side dips occurat δsd = ±|τρ − τσ| (gray dotted vertical lines).

B. Asymmetric setup

We now examine the HOM noise ratio in Eq. (33) foran asymmetric configuration where the distances betweenthe injection contacts and the QPC are different (LA 6=LB), still assuming the same inter-edge interaction onboth arms (θA = θB = θ). Notice that our general resultin Eq. (33) can be directly used also to investigate thecase θA 6= θB even if this condition is more difficult tobe controlled experimentally. We did not include thissituation in the paper in order to keep the discussionmore focused. In any case, we expect different interactionstrengths to give a similar qualitative behavior as thepresence of different lengths.

Differently from the symmetric case, when LA 6= LBthe photoassisted coefficients in Eq. (33) are no longerequal (p̃l,A 6= p̃l,B) because of the different interactionlengths. A new scenario thus emerges in this case as nowthe right-moving modes and the left-moving ones do nothave the same times of flight even if they have the samevelocities because of the same interaction strengths. Forthis reason we denote the times of flight of right-movingmodes as τAρ,σ = LA/vρ,σ and those of the left-moving

ones as τBρ,σ = LB/vρ,σ.As before, we consider the noise ratio R as a function

of the time delay δ focusing on the strong coupling regime(θ = π/4) and considering different values of the lengthratio LB/LA. From Fig. 4 (upper panel) one can outlinethat the three dips described before are still present, but

now the overall profiles are very different with respect tothe symmetric case. Indeed, here the central dip does notreach anymore zero (loss of visibility) and its position isshifted with respect to the origin by a time delay

δcd =τBσ + τ

Bρ − τAσ − τAρ

2(36)

which increases proportionally to the length ratioLB/LA. This means that the total suppression of HOMnoise is not achieved because the different interactionlengths result in different times of flight (τAρ,σ 6= τBρ,σ)in such a way that the charge and neutral parts of theincoming signals do not reach the QPC at the same time.The distances of the side dips from the central one satisfy

|δcd − δsd| =τBσ − τBρ + τAσ − τAρ

2(37)

clearly showing the effect of the asymmetric lengths ofinteracting regions on the noise ratio R.Notice that in absence of interactions (lower panel ofFig. 4) we recover the same behavior of the symmetriccase in Fig. 3 up to a simple shift in the delay direction.This is a direct consequence of the lack of fractionaliza-tion of the incoming excitations.

An additional comment on Eq. (36) is worthwhile. Inthe symmetric setup, the central dip corresponds to thesituation where the two charged (or dipolar) modes in-coming from the two sources arrive simultaneously at theQPC. In the asymmetric case, at a delay δ1 = τ

Bρ − τAρ

(δ2 = τBσ − τAσ ) the charged (dipolar) modes reach the

QPC at the same time, but the dipolar (charged) onesdo not. As a result, instead of a single central dip as ap-pearing in Fig. 4, two distinct dips located at δ1 andδ2 should be expected (for additional details see Ap-pendix B). However, for realistic values for τ0/T , thesetwo dips are not resolved (because the wavepackets arenot narrow enough) and merge into a broader one, lo-cated at an average delay δcd = (δ1 + δ2)/2.

In Section IV A we have shown that the HOM noisegoes exactly to zero when we are in a symmetric situationand the excitations are injected simultaneously in the QHedge channels. This time one may wonder whether thesignal periodicity affects the visibility of the central dip(R(δcd)) in an asymmetric setup when the lengths ratiois varied. In Fig. 5 we show the behavior of the mini-mum of the HOM ratio (R(δcd)) as a function of LB/LAin the presence of a periodical Lorentzian source (mainplot) and compare it to the single Lorentzian pulse case(inset). Also in this case we focus on the strong cou-pling θ = π/4 regime. The biggest difference betweenthe two situations lies in the fact that for the periodicdrive the red curve goes to zero three times in the con-sidered range of the ratio LB/LA, including the startingpoint (where LB/LA = 1), while for the single pulse noother zero occurs apart the one corresponding to equallengths. Therefore, the occurrence of additional zeros isa remarkable consequence of the periodicity of the drive

8

-0.4 -0.2 0.0 0.2 0.4 0.6-0.10.00.10.20.30.40.50.60.7

FIG. 4. Ratio R in Eq. (33), for Lorentzian pulses, as afunction of time delay over period (δ/T ) for an asymmetricsetup. In the upper panel, the two curves refer to the strongcoupling regime (θ = π/4) for two different lengths ratios:LB/LA = 1.2 (red curve), LB/LA = 1.5 (blue curve). In thelower one, the curves refer to the non-interacting case (θ = 0)for the same lengths ratios as before (red and blue). It isworth noticing that if no interaction occurs the ratio R goesto zero and that the positions of the minima depend on boththe lengths ratios and the velocity propagation of the freefermions along the channel of the injection (here assumed tobe v1 = vρ). Other parameters are: τ0/T = 0.05, vρ =4 · 105m/s and vσ = 1.8 · 105m/s, with LA = 2µm. Noticethat the positions of central dip are indicated respectively bythe red and blue dashed vertical lines.

and can be used to extract information about the inter-action parameter θ.

In order to better understand the behavior in Fig. 5 wemust start from Eq. (27), for a generic case with δ 6= 0.Therein, the phases ϕA and ϕB must be equal in orderto have a perfect superposition of colliding excitationsand a consequent maximal visibility of the HOM centraldip. The expression giving the lengths ratios at which thecentral dip is maximally visible in the case of a periodicalinjection is

LBLA

=2(k − k′)TτAσ − τAρ

+ 1 (38)

with k, k′ ∈ N and k > k′ (see Appendix C for moredetails). The previous relation describes the zero locatedat LB/LA ≈ 7.6 in Fig. 5, for k − k′ = 1. We alsopoint out that the presence of a second zero, located atLB/LA ≈ 5.6, is a direct consequence of the maximalcoupling θ = π/4. Indeed, as shown in Appendix C, inthis condition additional zeros appear for a length ratio

LBLA

=2(k − k′)TτAσ − τAρ

− 1 . (39)

FIG. 5. Behavior of R(δcd) as a function of the lengths ratioLB/LA. The red curve is obtained for a periodic Lorentzianpulse (τ0/T = 0.05) while the green inset shows the case ofa single pulse. Other parameters are: θ = π/4, LA = 2µm,vρ = 1.5 · 105m/s and vσ = 2 · 104m/s. Notice that herewe have chosen propagation velocities different with respectto the other Figures with the only aim of magnifying thefeatures discussed in the main text.

However, as soon as the coupling departs from the max-imal value (θ < π/4) the second zero is lifted and turnsinto a local minimum (see Appendix C for more details).This is a signature of the different weight of charge anddipole contributions to the fractionalized wave-packetand can be use to extract information about the mis-match in the time of flight and consequently about theinter-edge coupling θ.

The possibility for the HOM central dip to reach zeroat different values of the lengths ratio is a direct conse-quence of the periodicity of the applied signal. In termsof electronic density we can think of what is happeningas follows: one Leviton is injected for every period, itcrosses the interacting region where it fractionalizes intotwo modes with different velocities. If the interactingregion has the proper length, the fast mode of a givenperiod will reach the slow mode of the previous one. Byproperly calibrating the ratio between the lengths intothe two arms it is possible to achieve a situation wherethe colliding objects, the fast and slow modes comingfrom both arms, at the QPC are identical leading to avanishing HOM noise.

V. MEASURING INTERACTIONS

Until now, we have considered a setup where the injec-tion takes place on the inner channels of QH bar only. Amore general analysis consists in considering a case wherethe excitations are also injected in the outer channels.This configuration can be achieved for example by fur-ther exploiting an open quantum dot coupled to the outerchannels16. As we will demonstrate, in this case a col-lisional HOM experiment allows to extract information

9

on the interaction strength between the edge channelsas long as it can be assumed as short-range. In SectionIV B we have shown how the dependence of the visibil-ity of the central dip as a function of the lengths ratiocan be used to indirectly estimate interactions. Here, weconsider a more direct way to measure the interactionintensity encoded in the parameter θ.

Let us start by considering two different input voltages

VA/B1/2 at the entrance of the interaction region, where 1

stands for the inner channels and 2 for the outer ones.Without loss of generality we consider the two drives to

be proportional, namely VA/B2,in = αV

A/B1,in . In what fol-

lows we only consider a symmetric configuration for theinterferometer even if similar results can be obtained foran asymmetric case. This implies that Eq. (15) can bewritten as

VA/B1,out (t) = cos

2 θ VA/B1,in (t− τρ) + sin

2 θ VA/B1,in (t− τσ)

+α sin θ cos θ[VA/B1,in (t− τρ)− V

A/B1,in (t− τσ)]

VA/B2,out (t) = sin θ cos θ[V

A/B1,in (t− τρ)− V

A/B1,in (t− τσ)]

+α sin2 θ VA/B1,in (t− τρ) + α cos

2 θ VA/B1,in (t− τσ).

(40)

From the above equation, we can identify two relevantsituations involving two different values of the propor-tionality parameter: α = tan θ and α = − cot θ. Forthese two values (V

A/B1,in , αV

A/B1,in )

T is an eigenvector of

the scattering matrix S in Eq. (12). In the time domain,this results in(

VA/B1,out (t)

VA/B2,out (t)

)=

(VA/B1,in (t− τρ)VA/B2,in (t− τρ)

)for α = tan θ (41)

and(VA/B1,out (t)

VA/B2,out (t)

)=

(VA/B1,in (t− τσ)VA/B2,in (t− τσ)

)for α = − cot θ .

(42)Therefore, for these values of α, the input voltages are

not mixed by interactions and are transferred unaffectedto the output of the interacting region. This featureis quite surprising because it means that, by properlytuning α, one can inject two input excitations which ef-fectively propagate freely along the edge channels with-out undergoing any fractionalization process, despite thepresence of an interacting region in the system. There-fore, it is possible to regard α as a tunable parameterwith which one can switch off interaction effects on theHOM noise ratio R. As a possible experimental way toimplement such kind of voltage configuration one can ap-

ply the same voltage VA/B1,in (t) to both channels, further

adding a voltage (α − 1)V A/B1,in (t) properly synchronizedwith the first one only to channel 2 by means of a quan-tum dot16.

In order to illustrate this effect we compare in Fig.6 the case where the injection only occurs in the inner

FIG. 6. Ratio R as a function of time delay over period(δ/T ). The full lines describe HOM collisions at finite valuesof α. In a) we set θ = π

6and α = tan

(π6

)≈ 0.58 while in b)

θ = π4

and α = tan(π4

)= 1. The dashed lines describe the

corresponding curves for an injection only in inner channels(α = 0) for both a) θ = π

6and b) θ = π

4). The black dots

show the ratio R in the absence of interactions (θ = 0). Otherparameters are: τ0/T = 0.05, vρ = 4 · 105m/s and vσ =1.8 · 105m/s, with LA = LB = 2µm.

channels (α = 0) with the situation when both inner andouter channels are driven (α 6= 0). The former scenario isrepresented by the dashed curves, showing the side dipsstructure already discussed in Section IV A. The lattercase is represented by full lines and clearly shows that,for the particular value α = tan θ, the side dips disappearand one perfectly recovers the same behavior as in theabsence of interactions, described by Eq. (30) and shownby black dots in Fig. 6.

As a final remark we mention that, for α > tan θ (notshown), the side dips become side peaks as a consequenceof the fact that excitations with opposite charge reach theQPC14,19.

According to the above considerations it is clear thatthe study of the evolution of the side dips as a function ofα can be used as a way to estimate the value of the inter-edge interaction. Indeed, by tuning α in such a way toeliminate the side dips in the HOM signal knowing thatthis occurs precisely at α = tan θ, the mixing angle canbe obtained from this relation.

10

VI. CONCLUSIONS

We have theoretically investigated a Hong-Ou-Mandelexperiment where periodic time-dependent voltage drivesare injected with a tunable delay into Quantum Hall edgechannels at filling factor ν = 2 and collide at a Quan-tum Point Contact. In particular, we have focused onLorentzian voltage pulses carrying unitary charge, usu-ally dubbed Levitons. As a consequence of the screenedCoulomb interactions between the edge channels, the in-jected electrons fractionalize leading to an interestingphenomenology. Indeed, the noise measured just out-side the Quantum Point Contact is characterized by theemergence of side dips as a function of the delay in theinjection. Moreover, differently from what happens inthe case of injection using driven mesoscopic capacitors,the visibility of the central dip remains maximal inde-pendently of the interaction for a symmetric device re-gardless of the form of the voltage used for the injection.This fact is a signature of the robustness of voltage driveswith respect to interaction effects. Our results are evenmore interesting in the case of Lorentzian voltage pulsesbecause of their relevance as on-demand single electronsources. Only by inducing an asymmetry in the device,for example by considering different distances betweenthe injectors and the Quantum Point Contact, the vis-ibility can be reduced. In addition, from the peculiardependence of the visibility on the ratio between thesedistances, it is possible to extract information about theintensity of the interaction along the edge. Along thisdirection, we have also proposed a more direct measure-ment of the interaction based on the application of dif-ferent voltages on the two edge channels along each armof the interferometer. In this case, by properly tuningthe ratio between these voltages, it is possible to preventthe fractionalization with a consequent disapperance ofthe side dip in the HOM noise profile. Therefore, thestudy of the evolution of the side dips in this configura-tion can provide a direct measurement of the strength ofinter-edge interaction.

ACKNOWLEDGMENTS

The authors would like to thank T. Martin, T. Jonck-heere, J. Rech, F. Ronetti and J. Splettstoesser for usefuldiscussions. M.A. acknowledges support from the Euro-pean Unions H2020 research and innovation programmeunder grant agreement No 862683.

Appendix A: Photoassisted amplitudes and HOMnoise ratio

In this Appendix we evaluate the photoassisted am-plitudes p̃l,A/B(q) and Pl(q; δ) in terms of amplitudespl(q) and we show how to obtain Eq. (33). The coeffi-cients we want to determine, for a periodic voltage pulse

source V (τ) = V (τ+T ), are defined by the Fourier series(j = A,B)

e−ie∫ t0V j1,out(τ)dτ = e−iqΩt

∑l

p̃l,j(q)e−ilΩt (A1)

where Ω = 2π/T and

e−ie∫ t0 [V

A1,out(τ)−V

B1,out(τ)]dτ =

∑l

Pl(q; δ)e−ilΩt . (A2)

The explicit expressions of these coefficients in terms ofpl are then obtained by inverting the previous relations.Let us start with p̃l,j(q). From Eq. (A1) we have

p̃l,j(q) =

∫ T0

dt

Tei(l+q)Ωte−ie

∫ t0V j1,out(τ)dτ . (A3)

Next, with the help of Eqs. (29) and (40), the voltages

V j1,out(t) are expressed in terms of the source drive V (t)

given in Eq. (34). In doing that, four different phase fac-tors involving V (t) are obtained, each of which containinga time shift and being differently weighted due to inter-actions. It is then possible to repeatedly use Eq. (32) toexpress these factors as Fourier series involving the pho-toassisted coefficients pl. Finally, after performing thetime integration in Eq. (A3) we obtain the result (ne-glecting unimportant phases)

p̃l,j(q) =∑nrs

pl−n−r+s(q1)pn(q2)pr(q3)p∗s(q3)

× eiΩτjρ(l−n+s)eiΩτ

jσ(n−s),

(A4)

where q1 = cos2 θ, q2 = sin

2 θ and q3 = α sin θ cos θ.Thus, the coefficients p̃l,j(q) are completely specified oncethe expression of pl(q) is known. For a Lorentzian drive,it is given by26,32,39,63

pl(q) = q

+∞∑s=0

(−1)sΓ(q + l + s)e−2πτ0(2s+l)/T

Γ(q + 1− s)Γ(1 + s)Γ(1 + l + s). (A5)

Once p̃l,j are known, it is easy to obtain the photoassistedcoefficients Pl(q; δ) that take into account the time delayδ between the two sources. Indeed, by inverting Eq. (A2)and using Eq. (A1) to express the phase factors involving

the voltages V j1,out(t), we readily arrive at the expression

Pl(q; δ) =∑m

p̃l+m,A(q)p̃∗m,B(q) e

imΩδ, (A6)

where, again, unimportant phases have been neglected.Now we have all the ingredients to evaluate the HOM

ratio defined in Eq. (30). Recall that SHOM is obtainedby evaluating Eq. (31) and the contributions SHBT,j areparticular cases when one of the two sources is switchedoff. The first step is to express the phases ϕj(t, t

′) by

11

relying on the photoassisted coefficients we have deter-mined in this Appendix. For instance,

e−iϕA(t̄+τ2 ,t̄−

τ2 ) =

(∑l

p̃l,A(q) e−ilΩ(t̄+ τ2 )e−iqΩ(t̄+

τ2 )

)×(∑

l′

p̃∗l′,A(q) eil′Ω(t̄− τ2 )eiqΩ(t̄−

τ2 )

)=∑ll′

p̃l,A(q)p̃∗l′,A(q) e

iΩt̄(l′−l)e−iΩτ2 (l+l

′+2q)

(A7)

and similarly for e−iϕB , where the time delay δ has tobe taken into account. This expression is then used intoEq. (27) to obtain the function ∆Q(t + τ/2, t − τ/2).Finally, the two time integrations in Eq. (27) can be per-formed yielding (in the limit of zero temperature)

SHOM = −(evF )2RT(π∑l

|Pl(q; δ)|2|Ωl|)

(A8)

for the general HOM case and

SHBT,j = −(evF )2RT(π∑l

|p̃l,j(q)|2|Ω(l + q)|)

(A9)

for the HBT contributions. From these expressions, thenoise ratio R in Eq. (33) follows straightforwardly. Allthe above (infinite) sums are convergent and their valuehas been obtained numerically by summing over a finitenumber of coefficients until the desired precision is ob-tained.

Appendix B: Central dip resolution in theasymmetric length case

In this Appendix we want to comment the resolutionof the central dip in the asymmetric case θA = θB andLA 6= LB .In principle when the interferometer is asymmetric, oneshould expect four dips, whose positions would be indeeddetermined by two delay times only. At δ1 = τ

Bρ −τAρ , the

two charged modes arrive at the QPC simultaneously, butthe dipolar ones do not. Similarly, at δ2 = τ

Bσ − τAσ , the

two dipolar modes arrive at the QPC at the same timewhile the charged ones do not. However, the proper visu-alization of those dips requires enough resolution, namelythe wave-packets have to be narrow enough. This featureis simply not resolved in Fig. 4 (upper panel) where wehave chosen a value for τ0 (width of the Lorentzian volt-age) and T (period of the source) in the typical rangeaccessible for the experiments. As a result, the two ex-pected dips at δ1 and δ2 merge into a broader one, locatedat δcd (Eq. (36)), which is the average delay between theprevious two: δcd = (δ1 + δ2)/2. In Fig. 7 we show howthe ratio R should be if we consider unrealistically nar-row pulses (for a period T = 100τ0). Here the two dips

-0.10 -0.05 0.00 0.05 0.10 0.15-0.10.00.10.20.30.40.50.6

FIG. 7. Ratio R in Eq. (36), for Lorentzian pulses, as afunction of time delay over period (δ/T ) for an asymmetricsetup. The lengths ratio of the two arms is LB/LB = 1.5,the channel coupling strength is maximal (θ = π/4) and onefix T = 100τ0. Vertical solid lines correspond respectively toδ1 = τ

Bρ − τAρ and δ2 = τBσ − τAσ while the dotted vertical

one to δcd (in Eq. (36)). Other parameters are: τ0/T = 0.05,vρ = 4 · 105m/s and vσ = 1.8 · 105m/s, with LA = 2µm.

are well resolved respectively at positions δ1 and δ2 sym-metrically with respect to δcd, bringing the number ofobserved dips in the HOM ratio from three (as in Fig. 4)to four. Concerning the sideband dips, they are locatedat δ = τBσ − τAρ (coincidence between the dipolar modeincoming from B and the charged one incoming from A)and δ = τBρ −τAσ (coincidence between the charged modeincoming from B and the dipolar one incoming from A).Therefore their positions are determined by two timesonly. In Eq. (37) we have expressed the location of theseside dips relative to the broad central one.It is worth noticing that in the symmetric situation whenLA = LB we have δ1 = δ2 = δcd = 0 as expected.

Appendix C: Central dip visibility in theasymmetric length case

The result in Eq. (38), relating the lengths ratio tothe periodicity of the signal used for explanation of theminimum value for R(δcd) in Fig. 5, is obtained startingfrom the equality in Eq. (29), that is still true also for thephases ϕj(t). The phases are (considering the injectiononly in one channel)

ϕA(t) = cos2 θ ϕ(t− τAρ ) + sin2 θ ϕ(t− τAσ )

ϕB(t+ δ) = cos2 θϕ(t− τBρ + δ) + sin2 θϕ(t− τBσ + δ)

(C1)

12

1 2 3 4 5 6 7 80.0

0.2

0.4

0.6

0.8

FIG. 8. Behavior of R(δcd) with respect to lengths ratioLB/LA for three different fixed value of the interaction angle:θ = π

4(red curve), θ = π

6(cyan curve) and π

8(brown curve).

Other parameters are: τ0/T = 0.05, LA = 2µm vρ = 1.5 ·105m/s and vσ = 2 · 104m/s.

where ϕ(t) =∑k∈Z arctan

(t−kTτ0

)for Lorentzian peri-

odic pulses. In order to solve ϕA(t) = ϕB(t + δ) wehave to specify the interaction angle. Firstly considering0 < θ < π4 (Fig. 8) we know that sin θ 6= cos θ and thismeans that Eq. (29) is satisfied when

ϕ(t− τAρ/σ) = ϕ(t− τBρ/σ + δ) (C2)

These two conditions lead to the same result, thereforewe can focus on the first one

∑k

tan−1

[t− kT − τAρ

τ0

]=∑k′

tan−1

[t− k′T − τBρ + δ

τ0

](C3)

which yields

(k′ − k)T = τAρ − τBρ + δ. (C4)

Because we want to study how the minimum of centraldip varies, the delay δ has to be fixed by

δcd =τBρ + τ

Bσ − τAρ − τAσ

2. (C5)

By substituting this expression into Eq. (C4) and recall-ing that τBρ/σ = τ

Aρ/σ LB/LA, we arrive at the result in

Eq. (38).An interesting additional feature forR(δcd) is obtained

when θ = π/4. Due to the fact that in this case sin θ =

cos θ, there is another possibility to fulfill ϕA(t) = ϕB(t+δ), namely

ϕ(t− τAρ/σ) = ϕ(t− τBσ/ρ + δ). (C6)

FIG. 9. Visualization of single electron charge density prop-agation along an edge channel. In a) a Leviton enter theinteracting region x > 0, in b) it is clear how the Coulombcoupling mechanism works: the fast mode and the slow onestart separating. In d) the green curve describes the case whena second Leviton enter the interacting region: its fast modereaches the slow mode of the previous period, recreating apurely electronic wave-packet. By repeating this mechanismfor all the periods one can justify Eq. (38) and the relationsbetween the lengths ratio and the periodic windows.

Again, this two conditions lead to the same result, whichreads

LBLA

=2(k′ − k)TτAσ − τAρ

− 1. (C7)

This analysis clearly shows that the zeros in R(δcd) de-scribed by Eq. (38) are stable with respect to the changeof the interaction strength, while those described byEq. (C7) are only present at maximal coupling (see alsoFig. 8). In order to further characterize the physics be-hind this phenomenology we have reported in Fig. 9 somesnapshots of the evolution of the particle density, show-ing how Levitons emitted in different periods of the drivecan recombine due to the interaction-induced fractional-ization process.

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Collisional interferometry of Levitons in quantum Hall edge channels at =2AbstractI IntroductionII ModelIII General aspects of HOM interferometryIV HOM interferometry for LevitonsA Symmetric setupB Asymmetric setup

V Measuring interactionsVI Conclusions AcknowledgmentsA Photoassisted amplitudes and HOM noise ratioB Central dip resolution in the asymmetric length caseC Central dip visibility in the asymmetric length case References