Dario Ferraro
email: dario.ferraro@cpt.univ-mrs.fr

Thibaut Jonckheere

J Ér Ôme Rech

Thierry Martin

Electronic quantum optics beyond the integer quantum Hall effect

Keywords: Nanophysics, quantum noise, single electron sources, integer/spin quantum Hall effect, Andreev reflection

The analog of two seminal quantum optics experiments are considered in a condensed matter setting with single electron sources injecting electronic wave packets on edge states coupled through a quantum point contact. When only one electron is injected, the measurement of noise correlations at the output of the quantum point contact corresponds to the Hanbury-Brown and Twiss setup. When two electrons are injected on opposite edges, the equivalent of the Hong-Ou-Mandel collision is achieved, exhibiting a dip as in the coincidence measurements of quantum optics. The Landauer-Büttiker scattering theory is used to first review these phenomena in the integer quantum Hall effect, next, to focus on two more exotic systems: edge states of two dimensional topological insulators, where new physics emerges from time reversal symmetry and three electron collisions can be achieved; and edges states of a hybrid Hall/superconducting device, which allow to perform electron quantum optics experiments with Bogoliubov quasiparticles.

 imposed on a two dimensional electron gas. The edge states in the IQH effect, which are exempt of backscattering, play the role of such wave guides and a QPC placed downstream is equivalent to a half-silvered mirror, as it partitions electrons which are either reflected by or transmitted through the QPC. Electrons differ from photons in many aspects: they obey fermionic statistics, they are charged particles which interact among themselves and with their environment and, finally, they are always accompanied by a Fermi sea close to which electron

hole pairs may be easily generated. Moreover, while in quantum optics experiments the coincidence rate is measured at the two outputs, here the noise cross-correlation signal at zero frequency is typically computed or measured [2].

One of the main achievement of EQO has been the realization of the electronic equivalent of the Hanbury-Brown and Twiss [10] (HBT) experiment where a single electron source injects electrons on a QPC, followed by the Hong-Ou-Mandel [11] (HOM) experiment, where two electrons incident from two independent sources collide at the QPC. These scenarios represent fundamental tests of quantum mechanics for both photons and electrons, as they probe both their statistics and the form of the injected WPs. In the electronic HBT interferometer the Pauli principle leads to the anti-bunching between the injected electrons and the thermal ones incoming from the second channel [12]. In the HOM fermionic setup, when the emissions of the two colliding electrons are perfectly synchronized, one expects a suppression of the noise due again to the Pauli principle because the electrons are forced to emerge on opposite sides of the QPC. Conversely, for a long enough delay, twice the HBT signal is obtained for the noise in the Two chiral edge channels meet at a QPC. Each one is coupled to a SES in the optimal regime of emission of electrons (holes). Current cross-correlations at the two outputs are measured at zero frequency as a function of the delay between the electron emissions. Picture taken from [13].

HOM setup as the two sources are independent [13]. Experiments [14] do validate the presence of a Pauli dip in the noise correlations, although this dip does not fall to zero at coincident injection: this is attributed to decoherence effects because of the presence of two or more interacting channels [15].

In this paper, we wish to point out that EQO is not limited to the IQH regime. In addition to presenting the basic physics of HOM interferometry for IQH at filling factor ν = 1, we explore two different situations where the paradigms of EQO are also present. First, we consider the situation of the quantum spin Hall (QSH) effect in a two dimensional (2D) topological insulator [16,17], where two counter-propagating edge states carrying electrons with opposite spin appear on each side of the Hall bar [18]. In this situation (with or without spin flip processes at the QPC) it is possible to study the interplay between Fermi statistics and time reversal symmetry (TRS) using HOM interferometry, with two or even three SES. Next, we study the interplay between the IQH effect and superconductivity, as Andreev reflections convert an injected electron into a coherent superposition of electrons and holes -a Bogoliubov quasiparticle -which can subsequently be used as an input quasiparticle for an HBT or an HOM interferometric device [19]. Such Bogoliubov quasiparticles, although generated from electrons of finite energy above the Fermi sea, have recently been identified as possible realization of Majorana fermions in a condensed matter system [20,21,22]. For simplicity, in both of these extensions, we work solely with scattering theory and do not include interactions between the edges or with the electromagnetic environment surrounding the device.

2 Hong-Ou-Mandel electron collisions in the integer quantum Hall effect. We start by considering the case of standard IQHE in the QPC geometry [23] (see Fig. 1). A SES injects electrons on each incoming edge state, and current cross-correlations are detected at the two outputs of the QPC, exhibiting a dependence on the delay between injections. In experiments, the voltage drive is typically a periodic square wave, which results in the controlled emission of a regular train of single electrons and holes [3,4,24].

We consider the outgoing current cross-correlations at zero frequency:

S out 12 = dt dt I out 1 (t)I out 2 (t ) c (1)
where I out 1 (t) and I out 2 (t) are the currents outgoing from the QPC (see Fig. 1) and where we defined the connected correlator AB c = AB -A B . We can safely assume a linear dispersion for the electrons along the edge and chirality (from now on we assume the Fermi velocity v F = 1, reintroducing it only where needed). We compute the cross-correlations at the output of the QPC, using conventionally a x-axis pointing in the propagation direction of each edge. A scattering matrix with transmission (reflection) probability T (R = 1 -T ) characterizes the QPC. The currents outgoing from the QPC can be written in terms of the incoming field operators as

I out 1 = T I 1 + RI 2 + ie √ RT (Ψ † 1 Ψ 2 -Ψ † 2 Ψ 1 ) I out 2 = RI 1 + T I 2 -ie √ RT (Ψ † 1 Ψ 2 -Ψ † 2 Ψ 1 ),
with Ψ l (l = 1, 2) the annihilation operator for an electron on edge l and where we neglected the time dependence for notational convenience. Replacing these expressions into Eq. ( 1) allows to express the outgoing noise as [25] S out 12 = RT (S 11

+ S 22 + Q) . (2)
The last term encodes quantum interference contributions:

Q = -e 2 dtdt Ψ 1 (t)Ψ † 1 (t ) Ψ † 2 (t)Ψ 2 (t ) + Ψ † 1 (t)Ψ 1 (t ) Ψ 2 (t)Ψ † 2 (t ) , (3)
while the first two terms are the incoming auto-correlation noise associated respectively to I 1 and I 2 . Notice that the averages are evaluated with respect to the initial state |ϕ and correspond to the first order electronic coherence functions defined in [25]. Calculations are performed in the simple case in which a single electron, with a given WP, is injected into each edge. This allows to obtain analytical expressions for the HOM dip, which corresponds to the cross-correlated noise when two WPs collide from opposite sides of the QPC.

The state corresponding to an electron injected with WP ϕ 1,2 (x) on edge 1, 2 reads:

|ϕ 1,2 = dx ϕ 1,2 (x) Ψ † 1,2 (x) |F (4)
where |F indicates the edge state Fermi sea at finite temperature T . For identical WPs ϕ 1,2 = ϕ(x), reaching the QPC with a delay δt, the noise normalized by twice the HBT noise (only one source emitting [12]) is:

S HOM (δt) 2S HBT = 1 - ∞ 0 dk| φ(k)| 2 e -ikδt (1 -f k ) 2 ∞ 0 dk| φ(k)| 2 (1 -f k ) 2 2 , (5)
where

f k = 1 1 + e (k-k F )/T (6) is the Fermi distribution (k F the Fermi momentum), φ(k)
is the WP in momentum space, and

S HBT = -e 2 RT ∞ 0 dk| φ(k)| 2 (1 -f k ) 2 ∞ 0 dk| φ(k)| 2 (1 -f k ) 2 . (7)
Eq. ( 5) shows immediately that S HOM (0)/2S HBT = 0, as a consequence of Fermi statistics. On the opposite, when δt is much larger than the inverse width of φ(k), S HOM (∞) is the sum of the HBT noise of the two electrons taken independently, and S HOM /(2S HBT ) = 1. At low temperature, when the injected WP does not overlap with the Fermi sea, one has:

S HOM (δt) 2S HBT = 1 - dτ ϕ(τ ) ϕ * (τ + δt) 2 , (8)
which is similar to the case of optics [11] as the profile of the HOM dip corresponds to the auto-convolution of the WP. The functional forms of the HOM dip can be obtained analytically for various types of WPs. The SES of Ref. [3] is believed to generate Lorentzian WPs of the form:

φ(k) = √ Γ √ 2π 1 (k -k 0 ) + i Γ 2 (9)
which corresponds to Breit-Wigner resonance associated with the emission by the discrete level of a quantum dot of width Γ/2 at energy k 0 , with a half exponential (see Fig 1)

real space profile ϕ(x) = √ Γ e ik0x e Γ 2 x θ(-x) (θ(x)
is the Heaviside function). At zero temperature, the noise corresponding to two such WPs, centered at the same energy k 0 but with different widths Γ 1,2 reads:

S HOM (δt) 2S HBT = 1- 4Γ 1 Γ 2 (Γ 1 + Γ 2 ) 2 θ(δt)e -Γ1δt +θ(-δt)e Γ2δt .
(10) This HOM dip lacks mirror symmetry: its exponential behavior is characterized by the time constants Γ -1 Other parameters are: T 0 = 400 (in units of h/∆) the period of the applied voltage, and T = 0.01∆ the temperature. Picture taken from [13].

or Γ -1 2 respectively depending on the sign of δt. The dip does not reach zero as for Eq. ( 5). The reduced contrast (less than unity) reflects the distinguishability of the injected electrons. This asymmetry is clearly not present for WPs generated by a Lorentzian voltage applied directly to the edge channels [7,9,26] or when considering the adiabatic limit for the emission of a SES [23,27,28].

We compare our formulas with the numerical results of a Floquet calculation [29,30,31] properly modeling the emission process from a realistic periodic source [6,24]. This emitter consists of the mesoscopic capacitor [3,4,12]: a quantum dot with discrete levels connected through a QPC to the edge state which is driven by a gate applying a periodic square drive V (t). The highest occupied state is first positioned above the Fermi level, causing the tunneling of a dot electron to the edge; this (now) empty level is next placed below the Fermi sea, causing the emission of a hole. Periodic square voltage with an amplitude identical to the dot level spacing ∆ yield optimal emission. The Floquet approach allows to evaluate numerically current and noise [24]. Fig. 2 shows the comparison between the numerical results for the HOM dip with the analytical formula in Eq. (10). The upper panel shows a symmetric profile, due to the fact that the two SES are identical. The dips (with minimum value 0 at zero delay) are broader for lower emitter transparencies. This is a consequence of the fact that electrons take a longer time to exit the dot leading to a broader WP. In the optimal regime, the electron emission time corresponds to [24,32].

τ = 2π ∆ 1 D - 1 2 . ( 11
)
This value (with Γ = τ -1 ) is chosen to plot the analytical predictions from Eq. ( 10) (dotted-dashed curves).

The agreement is excellent, especially at low enough transparency, where the single electron emission is properly achieved [4,25]. For what it concerns the asymmetric profile (lower panel), the contrast is smaller than 1. Also in this case both the numerical and the analytical approaches agree very well.

The HOM interferometry with fermions is thus characterized by a dip in the zero-frequency current crosscorrelations in collisions between two electrons, whose shape depends on the injected WPs. This same setup also offers the interesting possibility (not shown) to achieve electron hole collisions, which produce an HOM peak at large enough temperature [13].

3 Electronic quantum optics with 2D topological insulators. Such materials exhibit the QSH effect [16,17]. The first experimental observations of this peculiar state of matter have been carried out in CdTe/HgTe [33,34] and InAs/GaSb [35,36,37] quantum wells. QSH effect manifests itself through a gapped bulk and metallic edge states [38] where electrons with opposite spin propagate in opposite directions along the edges as a consequence of spin-orbit interaction. The topologically protected helical edge states of QSH effect, with their spin-momentum locking properties, suggests that they could be studied in a EQO context [18] especially given the recent proposals for an electron source and beam-splitter. Indeed, the characterization of the SES has already been provided [39,40]. This pair electron source (PES) are predicted to trigger the injection into the helical edge states of a pair of electrons (holes) with opposite spin per period. However, the experimental realization of a QPC in the QSH regime still represents a challenge due to the same Klein mechanism which prevents confinement of massless Dirac fermions in graphene [41]. Still, new generation heterostructures [36,37] give reasonable hopes of possible applications to EQO.

Model.

The Hamiltonian of the system, in the presence of a QPC, is given by [40,42,43,44] .

H = H 0 + H sp + H sf (12) Σ R, ↑ L, ↓ R, ↓ L, ↑ P ES1 P ES2 P ES3
with

H 0 = -ih α=R,L σ=↑,↓ +∞ -∞ dxξ α : Ψ † α,σ (x)∂ x Ψ α,σ (x) : (13)
the free Dirac Hamiltonian of the one-dimensional helical edge channels. Here Ψ α,σ (x) extends the previous definition for the electronic annihilation operator by taking into account the chirality (α = R, L) and spin (σ =↑, ↓) degrees of freedom. Moreover, ξ R/L = ±1 represents the chirality index and : ... : indicates the conventional normal ordering. Assuming a local QPC, one obtains two additional contributions:

H sp = 2h σ=↑,↓ γ sp Ψ † R,σ (0)Ψ L,σ (0) + h.c. ( 14
)
the spin-preserving and

H sf = 2h α=R,L ξ α γ sf Ψ † α,↑ (0)Ψ α,↓ (0) + h.c. (15)
the spin-flipping tunneling Hamiltonian. TRS of the total Hamiltonian H is guaranteed as long as γ sp and γ sf are real numbers [42,45] (γ sp > γ sf is generally assumed). The evolution of the fermionic field operators is specified by the Heisenberg equation of motion in Dirac form. This allows to specify the incoming/outgoing scattering states/operators, which are related by a 4 × 4 scattering matrix describing the QPC [42,46], whose structure reflects the TRS. The scattering matrix elements are then parametrized by λ pb , λ f f , λ pf [42,46], namely the amplitude probabilities of spin-preserving backscattering, spin-flipping forward scattering and spin-preserving forward scattering processes (respectively orange, magenta and green dotted lines in Fig. 3) [40]. They naturally satisfy the constraint

|λ f f | 2 + λ 2 pf + |λ pb | 2 = 1 (16)
imposed by charge conservation.

Auto-correlated noise.

We want to investigate now the auto-correlated outgoing noise. We will focus for simplicity on the (R, ↑) channel, the expressions for the other possible cases can be derived proceeding in the same way. It reads

S out R↑,R↑ = dtdt I out R↑ (t)I out R↑ (t ) ρ,c (17)
where the notation indicates that the connected correlator is calculated over a density matrix of the form ρ = |ϕ ϕ|.

In terms of the incoming signals, it can be written as

S out R↑,R↑ = |λ f f | 4 S R↓,R↓ + λ 4 pf S R↑,R↑ + |λ pd | 4 S L↑,L↑ + Q. ( 18
)
Notice that the interesting physics is encoded in the last term of Eq. ( 18) which extends what we already evaluated in the IQH case, while the others are the auto-correlations of the incoming currents, which vanish when taking the average over one period and computing their zero frequency Fourier component [4,12]. Its explicit form here is

Q = (A + B + C)Q (F S) + (A + B)Q (HBT ) R↓ +(A + C)Q (HBT ) R↑ + (B + C)Q (HBT ) L↑ +AQ (HOM ) R↓,R↑ + BQ (HOM ) R↓,L↑ + CQ (HOM ) R↑,L↑ (19)
where

A = |λ f f | 2 λ 2 pf , B = |λ f f | 2 |λ pb | 2 , C = λ 2 pf |λ pb | 2 (
these can be tuned by modifying the QPC parameters [44,47]). The Fermi sea [48,[START_REF] Martin | Noise in mesoscopic physics[END_REF], the HBT [12], and the HOM [13,14] noise contributions read respectively:

Q (F S) = e 2 π d tdξf ξ (1 -f ξ ) (20)
Q (HBT ) a = e 2 2π d tdξ∆W (e) a ( t, ξ) (1 -2f ξ ) (21)
Q (HOM ) a,b = - e 2 π d tdξ∆W (e) a ( t, ξ)∆W b ( t + δ, ξ) (22)
with a and b the channels of injection and

∆W (e) a ( t, ξ) = dτ e iξτ ∆G (e) a t + τ 2 , t - τ 2 (23)
the Wigner function [START_REF] Ferraro | [END_REF] obtained as a partial Fourier transform of the excess first order coherence [25] ∆G

(e) a (t, t ) = Ψ † a (t )Ψ a (t) ρ -F |Ψ † a (t )Ψ a (t)|F . ( 24
)
In the absence of spin-flip (A = B = 0), C becomes the product of the transmission and reflection probability of the QPC and one recovers what is observed in the IQH case. On a more general ground, when one of the scattering amplitudes is zero only one of the A, B, C parameters survives and we recover an equivalent result to that of the IQH situation.

In experiments, one subtracts the Fermi sea contributions to define the excess noise:

∆Q = Q -(A + B + C)Q (F S) . ( 25
)
When considering the emission of a pair of identical WPs in the form of Eq. ( 9) from a PES, the HBT contributions reduce to Q (HBT ) a ≈ e 2 while the HOM contributions [13] read

Q (HOM ) a,b
(δt) ≈ -2e 2 exp (-Γ |δt|). Notice that, for the sake of simplicity, we have neglected the overlap of the injected electron WP with the Fermi distribution of the other channels (emission high above the Fermi sea and at very low temperature [12]).

Two-electron collision.

We consider the injection of electrons into the (R, ↑) and the (L, ↑) incoming channels. Here, only P ES1 and P ES3 (respectively green and orange circles in Fig. 3) are "on". This process is therefore the QSH equivalent of the IQH case (equal spin injection). The more relevant physical quantity to look at is the ratio between the HOM noise (two sources emitting together with finite delay δt) and the sum of the HBT noises associated with the same sources:

q (2) R↑,L↑ (δt) ≈ 1 -Ie -Γ |δt| (26)
where I = 2C/(A + B + 2C) is the visibility (see Fig. 4). Eq. ( 26) predicts a dip in the noise for electrons reaching the QPC with a delay such as Γ |δt| < 1. Moreover, the exponential form of the dip is reminiscent of the WP profile.

As in the IQH case discussed above, this Pauli dip is due to the fermionic statistics of the electrons [14], while the reduced visibility (compared to [13]) is due to the presence of additional channels coupled at the QPC [51]. Indeed, more outgoing channels lead to more partitioning at the QPC and a consequent enhancement of the HBT noise contribution. This effect can be very small or conversely quite important depending on the visibility I (and consequently of the intensity of γ sp and γ sf ) as shown in Fig. 5. Notice that, for γ sf = 0 (absence of spin-flipping) one has I = 1 and we recover the result of the IQH. It is worth mentioning that the suppression of the visibility discussed here has a different physical origin with respect to the one observed in the IQH effect at filling factor ν = 2 [14,15]. Indeed, even if in this case several scattering channels are present, the QPC can be easily experimentally tuned in a region where this effect is absent (partial transmission of the outer channel and total reflection of the others). Moreover, additional checks allows to unequivocally identify the inter-channel interaction as the dominant cause for the loss of contrast in the IQH case [52]. Due to spin flip processes occurring at the QPC, opposite spin electron interferometry is also possible for this kind of device. For electrons of the same chirality we have: (2)

q (2) R↓,R↑ (δt) ≈ 1 -J e -Γ |δt| (27)
R↑,L↑ (δt) as a function of δt (in units of Γ -1 ) for different values of spinflipping and spin-preserving amplitudes: γ sp = 2, γ sf = 0 (full black curve), γ sp = 2, γ sf = 1.5 (dashed green curve) and γ sp = 1, γ sf = 0.3 (dotted blue curve). Picture taken from [18].

when P ES1 and P ES2 are "on" (green and magenta circles in Fig. 3). Notice that we have defined a new visibility factor J = 2A/(2A + B + C) (see Fig. 4). It is clear that the interference process between (R, ↑) and (L, ↑) electrons can be mapped into the one involving (R, ↑) and (R, ↓) electrons by exchanging the spin-preserving and spin-flipping contributions. Notice that at γ sp = 0 (B = C = 0) we achieve the maximum visibility (J = 1).

The present setup offers also the novel possibility to realize the interference of electrons with opposite spin and opposite chirality:

q (2) R↓,L↑ (δt) ≈ 1 -Ke -Γ |δt| (28)
with P ES2 and P ES3 turned "on" (magenta and orange circles in Fig. 3) and K = 2B/(A + 2B + C). Maximum visibility is reached when A = C = 0, namely in a circle of radius 1 in the (γ sp , γ sf ) plane (see Fig. 4).

δt q

(2)

R↓,L↑ (δt)

q (2)
R↓,R↑ (δt)

Figure 6 (Color on-line) Behavior of q

(2)

R↓,R↑ (δt) (full black) q

(2) R↓,L↑ (δt) (dashed green) as a function of the δt (in units of Γ -1 ). Parameters are γ sp = γ sf = 2. Picture taken from [18].

In the three different two-electron collision configurations (Eqs. ( 26), ( 27) and ( 28)) the maximal visibility occurs when one of the scattering amplitudes (respectively λ f f , λ pb or λ pf ) is zero. Indeed, in this case, only two outgoing channels are available for the two electrons and we recover a zero noise as in the IQH case.

However, the noise suppression observed for collision of electrons with opposite spin is by far not trivial and is due to the constraints imposed by TRS and charge conservation in the QSH system (see Ref. [53] for a similar discussion in the case of a continuous current). This phenomenon, known as Z 2 dip, has also been discussed in Ref. [40] for a specific range of parameters (γ sp = 0, γ sf = 0), a particular case of the more general analysis reported here.

Three-electron collision.

Unique to the QSH effect is the possibility to translate three-photon HOM experiments [54] to EQO. Here, the three PES of the setup in Fig. 3 are operating. The delays in the electron emission are respectively δt 1 between (R, ↓) and (R, ↑); δt 2 between (R, ↓) and (L, ↑). Consequently, (δt 2δt 1 ) rep-resents the delay between (R, ↑) and (L, ↑). As previously we obtain a normalized noise:

q (3) (δt 1 , δt 2 ) ≈ 1 - A A + B + C e -Γ |δt1| - B A + B + C e -Γ |δt2| - C A + B + C e -Γ |δt2-δt1| . ( 29
)
Remarkably enough, a perfect synchronization between the PES leads to q (3) (δt 1 = 0, δt 2 = 0) = 0 for an arbitrary QPC. This total noise suppression is a consequence of the interplay between the fermionic statistics and the TRS in the QSH systems. This can be understood by considering the input state a † R↑ a † R↓ a † L↑ |F (creation of three electrons simultaneously, on in each input channel). In terms of the scattering matrix, one can rewrite this as:

a † R↑ a † R↓ a † L↑ |F = λ * pb b † L↑ + λ pf b † R↑ + λ * f f b † R↓ × λ * pb b † L↓ + λ * f f b † R↑ + λ pf b † R↓ × λ pf b † L↑ + λ f f b † L↓ + λ * pb b † R↑ |F . ( 30
)
TRS guarantees that each incoming operator a † ασ can be expressed as a linear combination of only 3 out of the 4 outgoing operators b † ασ . Therefore, exploiting the unitarity of the scattering matrix and the Pauli principle one obtains

a † R↑ a † R↓ a † L↑ |F = λ pb b † L↑ b † L↓ + λ pf b † R↓ b † L↑ +λ f f b † R↓ b † L↓ b † R↑ |F . ( 31
)
As it is easy to note, this leads to the superposition of three outgoing states, each involving the creation of an electron in the (R, ↑) outgoing channel. Consequently this channel is always populated (no current fluctuations) independently of the final outcome of the scattering process. The partition noise in this channel vanishes, which constitutes a direct consequence of the interplay between TRS and Fermi statistics.

In the absence of synchronized injections the phenomenology is even richer and crucially depends on the QPC parameters (γ sp and γ sf ). Situations may occur where the noise suppression is dominated by the (R, ↓ |L, ↑) interference channel (see top panel of Fig. 7), or oppositely equal spin injection (R, ↑ |L, ↑) can be favored (see bottom panel of Fig. 7). According to this, the present setup allows to investigate different interference configurations, by changing the function of the QPC. Alternatively, HOM interferometry can be seen as a way to quantify the characteristics of the QPC, and in particular to assess the relative weight between the spin-preserving and the spin-flipping tunneling amplitudes.

To summarize, 2D topological insulators exhibiting the QSH effect, are characterized by a very rich physics related to the peculiar connection between spin and momentum of the electrons propagating along the edges. Equal spin, as well as opposite spin interference are allowed here as a consequence of TRS. Moreover, three-electron injection/interference phenomena similar to the ones observed for photons in the conventional quantum optics can be investigated in this setup, differently from what happens in the IQH case [18].

Non-local interference and Hong-Ou-Mandel collisions of individual Bogoliubov quasiparticles.

EQO scenarios are now revisited using a SES in the IQH regime put in proximity with a superconductor (SC) (see Fig. 8). The electrons which are injected can perform several Andreev reflections [55] and can thus be converted partially or totally into holes. Collisions between two Bogoliubov quasiparticles at the location of a QPC can be obtained in the framework of HOM interferometry. Ref. [22], considered this setup first, with electron "beams" (DC voltage imposed between the two opposite edges) impinging on the QPC rather than single quasiparticle excitations. It has been argued [21,22] that since Bogoliubov quasiparticle creation operators are related by a unitary transformation to their annihilation counterpart, these excitations qualify as Majorana fermionic excitations. This constitutes an alternative proposal for Majorana fermions [20] compared to their topological superconductor counterparts [56,57] which give rise to a zero bias anomaly in tunneling experiments [58,59]. The DC proposal [22] SES SC W Figure 8 (Color online) Schematic view of a single electron source (SES) injecting electron and hole WPs into a IQH edge state at filling factor ν = 2 coupled with a SC contact of length W . Picture taken from [19]. failed to address the single shot creation and collision of two Bogoliubov quasiparticles, leading in principle to the annihilation of single Majorana excitations at the QPC. Moreover, this proposal requires the measurement of high frequency noise [60] in a normal metal/SC device. More than a decade ago the finite frequency noise of a normal metal/SC junction was computed [61], but it has so far eluded experimental observation. Single particle/quasiparticle injection, presents the great advantage that only zero frequency noise needs to be measured [4]. Here, the injection process is first characterized in terms of current and noise, exhibiting the non-conservation of the charge and the conservation of the excitation number. Next, EQO interferometric configurations with a QPC are studied, and it is shown that the averaged current (first order coherence [1]) is independent of the SC phase difference, while oscillations dependent on the SC phase difference appear in the noise (second order coherence [62,63]), which constitutes a clear signature of non local phenomena [19] together with the annihilation of Majorana excitations at the QPC.

4.1 Source of Bogoliubov quasiparticles. Two edge channels at ν = 2 (Zeeman splitting and inter-channel interaction are ignored [64]) are coupled to a SES and to a SC contact of length W (see Fig. 8). The SES injects into the channels an electron (a hole) with well defined WPs. A spin-singlet coupling between the Hall channels and the SC contact can for instance be realized in graphene [65,66].

The action of the SC on incoming electrons with energy below the induced SC gap ∆ s is described in terms of an energy dependent 4 × 4 transfer matrix M, constrained by unitarity and particle-hole symmetry [22,67]:

M(ξ) = (τ x ⊗ I) M * (-ξ) (τ x ⊗ I) . ( 32
)
In the above expression I is the identity in spin space, while from now on we indicate with τ i (i = x, y, z) the Pauli matrices acting on the electron-hole space and with σ i (i = x, y, z) the ones related to the spin degree of freedom. According to this, the transfer matrix M is applied to a 4-component spinor state [22]:

c(ξ) =      c e,↑ (ξ) c e,↓ (ξ) c h,↑ (ξ) c h,↓ (ξ)      , (33)
where e (h) indicates the electron (hole) state and ↑ (↓) the up (down) spin direction and ξ is the energy of the incoming excitation. The particle-hole symmetry in Eq. ( 32) leads to the constraint c(ξ

) = τ x ⊗ I c † (-ξ) T [57].
The parameters which enter the transfer matrix are: δ = W/v F the time required for the excitation to cross the SC region; α ≈ W/l s (l s = hv/∆ s the proximity-induced coherence length) and β ≈ W/l m with l m = (h/eB) 1 2 the magnetic length of the Hall system (B the magnetic field); φ the phase of the SC; γ and γ the relative phase shifts of electrons and holes in presence of the magnetic field [22,67,68].

After some algebra, the explicit form of this transfer matrix which takes into account Andreev reflection reads [22]:

M(ξ) = e iξδ e iΓ τz U( θ, φ)e iΓ τz (34)
where we introduced the phase shifts

Γ = γ + Ω, Γ = γ + Ω, with Ω = arctan β tan α 2 + β 2 / α 2 + β 2
and the matrix

U( θ, φ) = exp i θσ y ⊗ (τ x cos φ + τ y sin φ) , (35)
with θ an angle such that

sin θ = α sin( α 2 + β 2 )/ α 2 + β 2 . (36)
By comparing the expression for the upper critical magnetic field in a Type II SC in terms of the coherence length B c = Φ 0 /(2πl 2 s ) and the definition of the magnetic length l m , the condition l s l m (and consequently α β) must be enforced to preserve the superconductivity.

We consider now a SES injecting a spin up electron with the same WP as in Eq. ( 9) into the SC region. For the sake of simplicity we will investigate the zero temperature case. The outgoing state from the SC is a Bogoliubov quasiparticle (a coherent electron/hole superposition with opposite spin):

|B = W e |e, ↑ + W h |h, ↓ = cos θ|e, ↑ + sin θe -iΦ |h, ↓ (37)
with Φ = 2Γφ and |e, ↑ , |h, ↓ a notation for electron/hole outgoing states from the SC region.

The averaged total current and particle density outgoing from the SC region are defined as:

ϕ|I|ϕ = -e ϕ| : Ψ † τ z Ψ : |ϕ (38) ϕ|ρ|ϕ = ϕ| : Ψ † Ψ : |ϕ (39)
where -e < 0 and we have omitted the time dependence for notational convenience. Note that, in the above expressions, the definition Ψ (t) = (4π) 1/2 +∞ -∞ dξe -iξt M(ξ)c(ξ) for the outgoing spinor is required to avoid double counting [22].

Applying Wick's theorem and considering well localized WPs in the positive energy domain, the current reduces to:

ϕ|I(t)|ϕ = -e cos(2 θ)ϕ(t -δ)ϕ * (t -δ). ( 40
)
The outgoing electronic current of Eq. ( 40) differs from the incoming one [1,[START_REF] Ferraro | [END_REF]

is not conserved as a consequence of the creation/destruction of Cooper pairs in the SC. Conversely, due to the unitarity of the scattering matrix, the outgoing excitation density is given by:

ϕ|ρ(t)|ϕ = ϕ(t -δ)ϕ * (t -δ) = ϕ|ρ in (t -δ)|ϕ , (42)
which implies a mere time delay δ with respect to the incoming one. The prefactor is given by

|W e | 2 + |W h | 2 = 1,
which illustrates the conservation of the number of injected excitations:

N = +∞ -∞ dt ϕ|ρ in (t -δ)|ϕ = 1 . (43)
Both the non-conservation of the charge and the conservation of the excitation number are encoded in the Bogoliubov-de Gennes Hamiltonian.

We also consider the current noise at the output of the SC contact:

S source = +∞ -∞ dtdt ϕ|I(t)I(t )|ϕ c = e 2 sin 2 (2 θ). (44)
The above quantity is proportional to P e •P h : it vanishes in the absence of a SC contact ( θ = 0), as expected [4,14], and when the incoming electron is completely converted into a hole ( θ = π/2). Its maximum is reached for θ = π/4, when the outgoing averaged current is zero.

Cross-correlated noise in a QPC geometry.

We now investigate the outgoing cross-correlated noise in a QPC geometry where one or two SES (SES1 and SES2) inject electronic WPs with spin up in the vicinity of one or two SC regions (see Fig. 9). (t) (written in terms of the operators a j ). Picture taken from [19].

Σ SES1 SES2 SC1 SC2 c 1 M 1 c 1 c 2 M 2 c 2 a 1 a 2 I out 1 (t) I out 2 (t)
The annihilation spinors outgoing from the QPC are related to the ones emitted by the two SES through the scattering matrix, which is parametrized by transmission (reflection) coefficients T (R). The zero frequency crosscorrelated noise outgoing from the QPC is again given by Eq. (1).

Hanbury-Brown-Twiss contribution.

When only one of the two SES (labeled j) is "on" we obtain the HBT contribution to the noise. At zero temperature, the injected excitations crossing the SC contact are converted into Bogoliubov quasiparticles which reach the QPC and get partitioned [12]. This contribution to the noise is:

S (j) HBT = -e 2 RT cos 2 (2 θj ). (45)
This represents the shot noise associated with a WP carrying charge Q (see Eq. ( 41)) and is therefore proportional to Q 2 . In the absence of SC ( θ = 0), S HBT = Copyright line will be provided by the publisher -e 2 RT as expected (see Eq. (7) in the zero temperature limit). For θj = π/4, the state which reaches the QPC (a balanced superposition of electrons and holes) generates no noise at all. This is because this individual zero charged quasiparticle excitation presents a non trivial internal structure which cannot be simply described in terms of an incoherent mixture of electrons and holes [69,70]. For θj = π/2, the electron is completely converted into a hole, and the noise is the same as for θj = 0. 4.2.2 Hong-Ou-Mandel contribution. If both the SES are "on" we obtain the HOM noise signal [13,14] S HOM = ∆S HOM + S (1) HBT + S

(2) HBT (46) with

∆S HOM /S 0 = A(δ 1 -δ 2 -η) 1 + cos(2 θ1 ) cos(2 θ2 ) -cos(Φ 12 ) sin(2 θ1 ) sin(2 θ2 ) , (47)
S 0 = e 2 RT , η the time delay in the emission between the two SES, Φ jk = 2Γ j -2Γ kφ j + φ k , and the squared overlap

A(τ ) = +∞ -∞ dtϕ * (t -τ )ϕ(t) 2
between identical WPs with a delay τ . This constitutes a general, central analytical result, as it addresses the HOM collision of two unsynchronized Bogoliubov quasiparticles.

When the two SC regions only differ in their order parameter phase and the two SES are properly synchronized (η = 0) one obtains the simplified expression:

S 2SC HOM = e 2 RT sin 2 (2 θ) [1 -cos(φ 1 -φ 2 )] (48)
which clearly shows a non-local dependence on the difference of the SC order parameter phases as already pointed out in Ref. [22] in the DC regime. The device shows no dependence on the SC phase at the level of the averaged current (first order coherence), but presents an oscillatory modulation in the noise (second order coherence). This is a clear demonstration of the fact that noise measurements in this kind of devices allow to access purely twoquasiparticle effect in analogy to what was discussed in the framework of IQH effect for the interferometers of Refs. [71,72] or the revisitation of the Franson interferometer [73] proposed in Refs. [27,63]. If in Eq. ( 48) φ 1φ 2 = 0 (mod. 2π), the noise vanishes only when two electrons or two holes reach the QPC at the same time ( θ = 0 or θ = π/2) as a consequence of the Pauli principle [13]. Remarkably, the noise reaches its maximum for θ = π/4. To explain this, look at the structure of the ∆S HOM term in Eq. (47). By considering two Bogoliubov excitations in the form of Eq. (37) simultaneously reaching the QPC, this contribution to the noise is proportional to:

|W 1 e W 2 e * -W 1 h W 2 h * | 2 = | cos θ1 cos θ2 -sin θ1 sin θ2 e -i(Φ1-Φ2) | 2 . ( 49
)
It thus corresponds to the difference between the product of electron and hole probability amplitudes. In particular for θ1 = θ2 = π/4 the Bogoliubov quasiparticles carry zero charge and zero shot noise, but are by far not trivial excitations with a complex structure given by the coherent superposition of electrons and holes which can be detected only at the level of the two quasiparticle interferometry. The above argument is also useful in order to understand why the HOM contribution to the noise in Eq. ( 48) is zero for φ 1 -φ 2 = 0 (mod. 2π). Under this condition indeed, the ∆S HOM term exactly compensates the two (equal) S HBT contributions in analogy to what occurs for the IQH case in absence of interaction.

The peculiar structure of the HOM noise contribution for two synchronized Bogoliubov excitations directly reflects into the divergences associated with the ratio:

R 2SC = S 2SC HOM S (1)
HBT + S

(2)

HBT = - 1 2 tan 2 (2 θ) [1 -cos(φ 1 -φ 2 )] . (50)
We can also achieve collisions between single electrons and Bogoliubov quasiparticles: it is now shown that the electron colliding with the Bogoliubov quasiparticle allows to probe the content of the latter. Starting from Eq. ( 47) the corresponding noise becomes:

S 1SC HOM = e 2 RT 1 + cos(2 θ) A(δ 1 -η) -cos 2 (2 θ) -1 (51)
with a maximum WP overlap A = 1, the reference electron interferes with: a) another electron ( θ = 0) leading to a zero noise (Pauli principle); b) with a hole ( θ = π/2) with a consequent minimum of the noise [13]; c) a more general Bogoliubov quasiparticle. In the latter case the cross-correlated noise assumes positive values when the electron component of the Bogoliubov excitation dominates over the hole one (|W e | 2 > |W e | 2 and consequently for 0 < θ < π/4). By decreasing the WP overlap, the ∆S HOM contribution to the noise is suppressed. However, even away from perfect synchronization, it is possible to observe positive and negative regions from which we can extract the dominant contribution to the Bogoliubov quasiparticle. The situation in the case of a WP exponential in time (see Eq. ( 9)), where A(τ ) = e -Γ |τ | , is illustrated by the density plot in the upper panel of Fig. 10. This represents an extremely useful tool to extract information about the structure of the Bogoliubov excitations through interferometric experiments with a known source (electronic WP). These observations indicate that the considered setup offers richer possibilities to implement a tomographic protocol by means of HOM interferometry with respect to what was proposed in the electronic case [74].

Copyright line will be provided by the publisher HOM in units of S 0 = e 2 RT . "Blue" identifies negative regions where the hole contribution dominates over the electron one, while the little red area represents the positive noise. Lower panel. Density plot of R 1SC as a function of δ 1η and θ. The blue area corresponds to negative values of the ratio which cannot be reached in conventional electronic quantum optics experiments. Picture taken from [19].

Another relevant quantity to explore is given by the ratio:

R 1SC = S 1SC HOM /(S (1)
HBT + S

(2)

HBT ) = 1 - 1 + cos(2 θ) 1 + cos 2 (2 θ) A(δ 1 -η). (52)
This also contains negative regions (blue areas in the lower panel of Fig. 10) which are forbidden in conventional electron quantum optics situations ( θ = 0) [13] due to the constraints imposed by the charge conservation.

5 Conclusions. Scattering theory thus allows to study a number of EQO interferometric situations. In IQH regime, theory has already been confronted with experiment [14]. The prediction of the HOM dip is solid, but its visibility does not correspond to what was expected [13]. Experiments cannot easily be achieved at filling factor one and when several channels are present, interactions be-tween them lead to charge fractionalization thus decoherence effects accompanied by a reduced visibility [15]. Two new directions for EQO have been proposed: the study of HOM collisions in the QSH effect in 2D topological insulators [18] and the possibility of observing non-local phenomena when colliding two Bogoliubov quasiparticles at the location of a QPC [19].

In these two setups, finite temperature effects were discarded. This should not change dramatically our predictions as long as the electrons WPs have a well defined energy above the Fermi sea and, for the former case, well below the gap. However they are likely to be affected when considering electron-hole interferometry [13] where the overlap between electron and hole WPs within an energy window set by the temperature is crucial.

Further work should however address the issue of interchannel interactions as in Ref. [15]. For the QSH setup, one suspects that Coulomb interactions between the counterpropagating edges could lead to a further decreasing in the visibilities of the Pauli and the Z 2 dips in the two electron collisions, and prevent a maximal visibility for the three electron dip. For Ref. [19], the approach of Ref. [15] could be directly transposed in order to predict the modification of the Bogoliubov quasiparticle impinging on the QPC, and the modification of the HOM signal.

Finally, EQO could also be studied in intrinsically interacting systems, which represent a strongly correlated state of matter, such as the fractional quantum Hall effect, in order to study HBT and HOM interferometry of electrons/Laughlin quasiparticles. In this direction, the characterization of a single quasiparticle source was recently proposed [75], and the HBT setup was studied in Ref. [76] for Lorentzian WPs.

Figure 1 (

 1 Figure 1 (Color online) HOM setup in the IQH regime.Two chiral edge channels meet at a QPC. Each one is coupled to a SES in the optimal regime of emission of electrons (holes). Current cross-correlations at the two outputs are measured at zero frequency as a function of the delay between the electron emissions. Picture taken from[13].

Figure 2 (

 2 Figure2(Color online) Normalized HOM noise as a function of the delay in the electron (hole) emission δt. Full curves represent the result obtained within the Floquet scattering matrix formalism in the optimal emission regime of the SES. Dotted-dashed curves are the analytical predictions of Eq. (10) for exponential WPs. Upper panel: symmetric profile, with emitter transparencies D = 0.2 (red curve), 0.5 (green curve) and 0.8 (blue curve). Inset: dips for D = 0.2 on two periods of the applied voltage. Lower panel: asymmetric profile, with transparencies D 1 = 0.2, D 2 = 0.5 (bottom red curve) and D 1 = 0.1, D 2 = 0.8 (top blue curve). Other parameters are: T 0 = 400 (in units of h/∆) the period of the applied voltage, and T = 0.01∆ the temperature. Picture taken from[13].

Figure 3 (

 3 Figure 3 (Color on line) Schematic view of a QSH bar with spin up (full arrows) and spin down (dashed arrows) electrons. Dotted lines indicate the scattering processes, compatible with TRS, which affect the current in the R, ↑ outgoing channel (see main text). Pair-electrons sources (PES) are represented with colored circles. Picture taken from [18].

Figure 4 (Figure 5 (

 45 Figure 4 (Color on-line) Density plot of I (left), J (middle) and K (right) as a function of γ sp and γ sf . Picture taken from [18].

2 Figure 7 (

 27 Figure7(Color on-line) Density plot of q (3) (δt 1 , δt 2 ) as a function of δt 1 and δt 2 (in units of Γ -1 ) for γ sp = 1, γ sf = 0.8 (top) and γ sp = 2, γ sf = 1.5 (bottom). Picture taken from[18].

  ϕ|I in (t)|ϕ ≡ -eϕ(t)ϕ * (t), by a time delay δ and by a factor cos(2 θ), which takes into account the conversion of electrons into holes via Andreev reflections. This is simply the difference between the probability |W e | 2 = cos 2 θ for the incoming electron to emerge as an electron and |W h | 2 = sin 2 θ to be converted into a hole. For θ = 0 (|W e | 2 = 1 and |W h | 2 = 0) the SC contact only induces a delay, while for θ = π/2 (|W e | 2 = 0 and |W h | 2 = 1) the incoming electron is completely converted into a hole and a Cooper pair enters into the SC. More importantly, for θ = π/4 (|W e | 2 = |W h | 2 = 1/2) the electron and hole contributions compensate and no averaged current flows out. Nevertheless, this zero averaged current still bears fluctuations. The charge outgoing from the SC contact, Q = +∞ -∞ dt ϕ|I(t)|ϕ = -e cos(2 θ)

Figure 9 (

 9 Figure 9 (Color online) QPC geometry for Bogoliubov quasiparticles. The individual electron sources SES1 and SES2 inject electrons (described by the operators c j (j = 1, 2)), into the two SC contacts SC1 and SC2. The outgoing excitation (M j c j ) reach the QPC and are partitioned according to the scattering matrix Σ. We consider the cross-correlated noise between the outgoing currents I out j

Figure 10 (

 10 Figure 10 (Color online). Upper panel. Density plot of S 1SCHOM in units of S 0 = e 2 RT . "Blue" identifies negative regions where the hole contribution dominates over the electron one, while the little red area represents the positive noise. Lower panel. Density plot of R 1SC as a function of δ 1η and θ. The blue area corresponds to negative values of the ratio which cannot be reached in conventional electronic quantum optics experiments. Picture taken from[19].

Copyright line will be provided by the publisher

Acknowledgements

The support of Grant No. ANR-2010-BLANC-0412 ("1 shot") and of ANR-2014-BLANC "one shot reloaded" is acknowledged. This work was carried out in the framework of Labex ARCHIMEDE Grant No. ANR-11-LABX-0033 and of A*MIDEX project Grant No. ANR-11-IDEX-0001-02, funded by the "investissements d'avenir" French Government program managed by the French National Research Agency (ANR).


D Ferraro

E Sukhorukov

Interaction effects in a multi-channel Fabry-Pérot interferometer in the Aharonov-Bohm regime

We investigate a Fabry-Pérot interferometer in the integer Hall regime in which only one edge channel is transmitted and n channels are trapped into the interferometer loop. Addressing recent experimental observations, we assume that Coulomb blockade effects are completely suppressed due to screening, while keeping track of a residual strong short range electron-electron interaction between the co-propagating edge channels trapped into the interferometer loop. This kind of interaction can be completely described in the framework of the edgemagnetoplasmon scattering matrix theory allowing us to evaluate the backscattering current and the associated differential conductance as a function of the bias voltage. The remarkable features of these quantities are discussed as a function of the number of trapped channels. The developed formalism reveals very general and provides also a simple way to model the experimentally relevant geometry in which some of the trapped channels are absorbed into an Ohmic contact, leading to energy dissipation.

(2) int 17 A.2 W (1) and W (2) 17 A.3 Y In the last few years various accurate experimental observations shed new light on the remarkable physics associated to the Fabry-Pérot interferometer (FPI) of integer and fractional quantum Hall edge channels [START_REF] Ofek | Role of interactions in an electronic Fabry-Pérot interferometer operating in the quantum Hall effect regime[END_REF][START_REF] Zhang | Distinct signatures for Coulomb blockade and Aharonov-Bohm interference in electronic Fabry-Perot interferometers[END_REF][START_REF] Kou | Coulomb Oscillations in Antidots in the Integer and Fractional Quantum Hall Regimes[END_REF]. Depending on the size of the interference loop two distinct regimes have been achieved. For a small enough loop area, intra-edge interaction plays an essential role and features related to Coulomb blockade occur. In the opposite regime of large central area, the expected Aharonov-Bohm physics of free electrons is recovered in the integer quantum Hall regime. A consistent theoretical interpretation of these results as well as a characterization of the crossover between these two limits has been proposed [START_REF] Rosenow | Influence of Interactions on Flux and Back-Gate Period of Quantum Hall Interferometers[END_REF][START_REF] Halperin | Theory of the Fabry-Pérot quantum Hall interferometer[END_REF]. Various experimental techniques, including gates and ohmic contacts, have been introduced in order to enhance the screening of interaction extending as much as possible the domain of validity of the Aharonov-Bohm regime, where the interaction is negligible and the simple free particles picture seems adequate to properly explain the experimental observations. However, very recently, new measurements carried out by Choi et al. [START_REF] Choi | Robust electron pairing in the integer quantum hall effect regime[END_REF] have suggested that a richer phenomenology occurs also in this apparently trivial case. In particular, when only one channel is transmitted throughout the FPI, by increasing the number of channels trapped in the loop (namely the integer filling factor of the system), one moves from the standard Aharonov-Bohm effect of electrons (at filling factor 1 ≤ ν ≤ 2) to a more puzzling situation in which a pair of electrons seems to interfere (at filling factor 3 ≤ ν ≤ 4). This phenomenology has been deduced from both the halving of the periodicity of the conductance with respect to the Aharonov-Bohm flux and the doubling of the effective outgoing charge measured through shot noise. These two independent measurements seem to confirm the robustness of the result, moreover various experimental checks have been carried out in order to rule out other possible effects leading to similar phenomenology like the suppression of odd winding of the interferometer with respect to the even ones, leaving the mutual interaction among the channels as the principal responsible of this peculiar pairing effect.

The aim of this paper is to provide the proper theoretical background for the description of the experimental setup in Ref. [START_REF] Choi | Robust electron pairing in the integer quantum hall effect regime[END_REF] in the framework of the edge-magnetoplasmon scattering matrix formalism [START_REF] Sukhorukov | Resonant Dephasing in the Electronic Mach-Zehnder Interferometer[END_REF][START_REF] Levkivskyi | Dephasing in the electronic Mach-Zehnder interferometer at filling factor ν = 2[END_REF][START_REF] Levkivskyi | Mach-Zehnder interferometry of fractional quantum Hall edge states[END_REF][START_REF] Degiovanni | Plasmon scattering approach to energy exchange and high-frequency noise in ν = 2 quantum Hall edge channels[END_REF][START_REF] Wahl | Interactions and Charge Fractionalization in an Electronic Hong-Ou-Mandel Interferometer[END_REF][START_REF] Ferraro | Real-Time Decoherence of Landau and Levitov Quasiparticles in Quantum Hall Edge Channels[END_REF] where the two points electron Green's function (first order electron coherence [START_REF] Grenier | Single-electron quantum tomography in quantum Hall edge channels[END_REF][START_REF] Ferraro | Wigner function approach to single electron coherence in quantum Hall edge channels[END_REF]), crucial ingredient to calculate transport properties like conductance and noise, explicitly depends on the transmission of the bosonic mode across the interferometer. As simplest possible case we will assume a strong screened Coulomb interaction and we will investigate carefully the functional form of the scattering matrix as a function of the number of channels trapped into the FPI loop. We will discuss in detail the case of one trapped channel extrapolating then the behavior in case of more channels into the interfering loop. The consequences of the form of these scattering matrices on the current and the conductance will be then investigated. We derive a very powerful and general formalism. On the one hand it allows us to rule out the simplest academic model of short range strong interaction as a possible way to explain what is observed in experiments, on the other hand it appears suitable to extensions towards more realistic models in which finite length of the interaction and dissipations have been taken into account [START_REF] Bocquillon | Separation of neutral and charge modes in one-dimensional chiral edge channels[END_REF].

The paper is organized as follows. In Section 2 we discuss the edge-magnetoplasmon scattering matrix theory for a FPI as a function of the number of integer Hall edge channels trapped into the loop. We focus in particular on a strong short range (δ-like) screened Coulomb potential. The classical and quantum contributions to the current and the associated differential conductance are derived in Section 3 by means of the Kubo formula. The plots of these quantities, as well as the relevant comments concerning the behavior of the system as a function of the number of trapped channels are reported in Section 4, also in view of a possible interpretation of the experimental observations. In Section 5 we investigate the role played by an ohmic contact absorbing some of the channels, analogous to the one used in realistic setup, in terms of a simple model based only on the energy conservation. Section 6 is devoted to the conclusions, while some technical details of the calculation are discussed in Appendix A.

Model

Edge-magnetoplasmon description of two interacting channels

Let us start by discussing the physics of two edge channels capacitively coupled along a finite region of length L. This problem will be investigated in the framework of the bosonization formalism [16,[START_REF] Giamarchi | Quantum Physics in One Dimension[END_REF]. Due to the chirality of the two channels we can identify the incoming region (1), the interacting region (2) and the outgoing region (3) (see Fig. 1). We will analyze them in detail in the following.

Incoming region (1)

In this region the interaction is absent and the Hamiltonian density can be written in term of the Wen's hydrodynamical model [START_REF] Wen | Topological orders and edge excitations in fractional quantum Hall states[END_REF] ( = 1)

H (1) = v 1 4π (∂ x φ 1 ) 2 + v 2 4π (∂ x φ 2 ) 2 . (1) (1)
(2)

(3) φ1,in (x, ω) φ2,in (x, ω) φ2,out (x, ω) φ1,out (x, ω)
Ŝ(L, ω) According to the chirality one can easily identify the incoming region (1), the interacting region (2) (shaded area) and the outgoing region (3). In regions (1) and (3) the dynamics of the bosonic fields is well described in terms of free equations of motion, while the outgoing fields are connected to the incoming ones through the edge-magnetoplasmon scattering matrix Ŝ(L, ω) which encodes the information of the inter-channel interaction (assumed as strong and short ranged in the main text).

Therefore, one can easily associate a chiral bosonic field to the charge density along each channel according to the conventional prescription [16,[START_REF] Giamarchi | Quantum Physics in One Dimension[END_REF]

ρ i = 1 2π ∂ x φ i i = 1, 2. (2)
These bosonic fields propagate freely according to the equations of motion

∂ t φ i (x, t) = -v i ∂ x φ i (x, t) i = 1, 2 (3)
and we have considered different propagation velocities v 1 and v 2 along the two channels.

Interacting region (2)

In this region we assume a density-density short range (δ-like) interaction in such a way that the Hamiltonian density becomes

H (2) = v 1 4π (∂ x φ 1 ) 2 + v 2 4π (∂ x φ 2 ) 2 + v 12 2π ∂ x φ 1 ∂ x φ 2 (4)
with v 12 interaction strength. Notice that, in spite of the fact that high frequency measurements suggest a relevant role played by a finite range of interaction and dissipation, this approximation reveals good at low enough frequencies [START_REF] Bocquillon | Separation of neutral and charge modes in one-dimensional chiral edge channels[END_REF]. According to this, the bosonic fields φ 1 and φ 2 are no longer eigenstates of the Hamiltonian of the system. The equations of motion are then decoupled in terms of a charged and a neutral mode, indicated respectively with φ ρ and φ σ . They diagonalize the Hamiltonian with associated eigenvelocities v ρ and v σ in such a way that the new equations of motion become

∂ t φ η (x, t) = -v η ∂ x φ η (x, t) η = ρ, σ. (5)
Due to the fact that the incoming fields are co-propagating, the above diagonalizing fields are related to φ 1 and φ 2 through a simple rotation of an angle θ in the field space1 . This becomes more transparent in the frequency space, namely through a partial Fourier transform with respect to time. Here, one has φρ (x, ω) = cos θ φ1 (x, ω) + sin θ φ2 (x, ω)

φσ (x, ω) = -sin θ φ1 (x, ω) + cos θ φ2 (x, ω). (6)
It is worth to note that the angle θ (0 ≤ θ ≤ π/4) provides a direct measurement of the strength of the interaction. In particular, θ = 0 corresponds to the non-interacting case, while for θ = π/4 one recovers the strong interacting limit.

Outgoing region (3)

Analogously to region [START_REF] Ofek | Role of interactions in an electronic Fabry-Pérot interferometer operating in the quantum Hall effect regime[END_REF], also in this case inter-channel interaction is negligible and the equations of motion write as in Eq. (3) (H (1) = H (3) ).

The general solution of the above systems of equations can be easily found in the frequency domain and reads φα (x, ω) = e i ω vα (x-x 0 ) φα (x 0 , ω)

φα (x 0 , ω) being the (possibly frequency dependent) amplitude at the initial condition x 0 and where α = 1, 2 in regions ( 1) and ( 3) or α = ρ, σ in region (2) respectively.

To completely solve the system we need now to impose the continuity of the fields at the boundaries of the three regions, namely at x = 0 and at x = L. Notice that, in the Fourier representation we are considering, this is equivalent to impose the conservation of the current across the boundaries.

Open channels

Before investigating the FPI geometry we are interested in, it is useful to recall the expected results in the case of open channels. Here, after some algebra, we obtain the edgemagnetoplasmon scattering matrix representation

φ1 (L, ω) φ2 (L, ω) = Ŝ(L, ω) φ1 (0, ω) φ2 (0, ω) (8)
with Ŝ = cos 2 θe iωτρ + sin 2 θe iωτσ sin θ cos θ e iωτρ -e iωτσ sin θ cos θ e iωτρ -e iωτσ sin 2 θe iωτρ + cos 2 θe iωτσ

and where we have introduced the short-hand notation

τ α = L/v α (α = ρ, σ).
Notice that this result is in full agreement with what is discussed in literature [START_REF] Sukhorukov | Resonant Dephasing in the Electronic Mach-Zehnder Interferometer[END_REF][START_REF] Levkivskyi | Dephasing in the electronic Mach-Zehnder interferometer at filling factor ν = 2[END_REF][START_REF] Degiovanni | Plasmon scattering approach to energy exchange and high-frequency noise in ν = 2 quantum Hall edge channels[END_REF][START_REF] Wahl | Interactions and Charge Fractionalization in an Electronic Hong-Ou-Mandel Interferometer[END_REF][START_REF] Ferraro | Real-Time Decoherence of Landau and Levitov Quasiparticles in Quantum Hall Edge Channels[END_REF] and satisfies the unitarity condition Ŝ • Ŝ † = I as expected.

One channel trapped in the Fabry-Pérot loop

Thanks to the above results it is now easy to investigate the simplest possible example of the geometry described in Fig. 2, where only one edge channel is trapped into the FPI loop (n = 1, k = 0). At filling factor ν = 2 (only blue and red channels in Fig. 2) this system represents the natural starting point to model the Choi's experiment of Ref. [START_REF] Choi | Robust electron pairing in the integer quantum hall effect regime[END_REF], when only one channel is trapped into the interferometer loop, while the other is transmitted with a tunable amplitude. For sake of generality we will assume two different scattering matrices for the upper ( Ŝ(u) ) and the lower ( Ŝ(d) ) part of the interferometer. By properly taking into

Σ(n) x 1 Φ x 2
with

Σ(1) =    S (u) 11 -S (d) 22 f (u) 1-S (u) 22 S (d) 22 S (u) 12 S (d) 12 1-S (u) 22 S (d) 22 S (u) 12 S (d) 12 1-S (u) 22 S (d) 22 S (d) 11 -S (u) 22 f (d) 1-S (u) 22 S (d) 22    , (11)
where we have introduced the phase factor

f (ω) = e iω(τρ+τσ) . ( 12
)
Notice that the frequency dependence has been omitted for notational convenience.

It is worth to note that the edge-magnetoplasmon scattering matrix Σ(1) inherits the properties of matrices Ŝ(u) and Ŝ(d) and is therefore unitary as expected.

Strong interaction limit

In this limit (θ = π/4) the expression in Eq. ( 11) strongly simplifies and one recovers a symmetric configuration Ŝ(u) = Ŝ(d) = Ŝ. Moreover, we can safely consider the limit v ρ → +∞ (τ ρ → 0) for the charged mode.

We then obtain the simple matrix elements

S 11 = S 22 ≈ 1 2 1 + e iξ (13)
S 12 = S 21 ≈ 1 2 1 -e iξ (14)
with ξ = ωτ σ .

Under this approximation, we can then write

Σ(1) (ξ) ≈ 2 -4 3 h (1) (ξ) -1 + 4 3 h (1) (ξ) -1 + 4 3 h (1) (ξ) 2 -4 3 h (1) (ξ) , (15)
where we have defined

h (1) (ξ) = +∞ n=0 - e iξ 3 n = 3 3 + e iξ . ( 16
)
Notice that Eq. ( 16) calls for the possibility of a simple harmonics expansion of the matrix elements of Σ(1) suitable, as will be clearer in the following, for numerics and useful to easily derive asymptotic behaviors. Moreover, in the zero frequency limit (ξ = 0), one directly obtain Σ(1) (0) = I as required for a purely capacitive coupling.

Two channels trapped in the Fabry-Pérot loop

In the strong interaction limit discussed above also the case at filling factor ν = 3, where two channels are trapped in the interferometer loop and the last one is transmitted with a tunable amplitude (see Fig. 2), can be easily handled. Without entering into the details of the calculation, also in this case we have a charge mode with a velocity v ρ that is greater with respect to the one of the two neutral modes (assumed v σ for both2 ). Proceeding exactly on the same way as before, after some quite tedious algebra, one obtains the edge-magnetoplasmon scattering matrix

Σ(2) (ξ) ≈ 3 -12 5 h (2) (ξ) -2 + 12 5 h (2) (ξ) -2 + 12 5 h (2) (ξ) 3 -12 5 h (2) (ξ) (17)
with

h (2) (ξ) = +∞ n=0 - e iξ 5 n = 5 5 + e iξ . ( 18
)
Also in this case the zero frequency limit (ξ = 0) leads to Σ(2) (0) = I.

Current and conductance

We can now discuss the general expression for the current flowing through the system in the framework of the edge-magnetoplasmon scattering matrix formalism and focusing on the weak backscattering regime for both the QPCs in Fig. 2. We will consider the first perturbative order in the backscattering Hamiltonian

H BS = j=1,2 Λ j Ψ † b (x j )Ψ a (x j ) + H.c. ( 19
)
with Ψ a and Ψ b electronic annihilation operators associated to the two edges and Λ j (j = 1, 2) tunneling amplitudes associated to the two QPCs. Under this assumption the backscattering current represents a small correction with respect to the quantized Hall current between source and drain. Moreover, this allows us to neglect multiple reflections which could complicates the description of the interferometer. For sake of simplicity the tunneling contributions are assumed to be local, even if more general extended tunneling can be also investigated [START_REF] Chevallier | Poissonian tunneling through an extended impurity in the quantum Hall effect[END_REF][START_REF] Dolcetto | Tunneling between helical edge states through extended contacts[END_REF]. The associated backscattering current operator is given by

I B = -e j=1,2 iΛ j Ψ † b (x j )Ψ a (x j ) + H.c. (20)
In order to evaluate the average value of the backscattering current we can use, as usual, the Kubo formula [START_REF] Mahan | Many particle physics[END_REF]

I B = -i t -∞ dt ′ I B (t), H BS (t ′ ) (21)
where operators are written in the interaction picture, namely evolved in time according to the free edge Hamiltonian only. Notice that all the averages discussed above are taken with respect to the ground state of the free bosonic systems in absence of backscattering.

In order to evaluate the backscattering current in Eq. ( 21) as a function of the elements of the edge-magnetoplasmon scattering matrices derived in the previous section we need to recall the fact that the fermionic annihilation operator can be seen as a coherent state of edge-magnetoplasmon in the form

Ψ(x) = 1 √ 2πα e iφ(x) , (22)
α a finite-length cut-off. [16] Because the calculation of the averaged backscattering current naturally involves fourvertex operators it is useful to introduce the general correlator e -iA e iB e -iC e iD = exp -

1 2 A 2 + B 2 + C 2 + D 2 + AB -AC + AD + BC -BD + CD } (23)
based on the Baker-Campbel-Hausdorff formula and the Wick theorem applied to the bosonic fields and valid for arbitrary gaussian fields A, B, C and D those commutation relations lead to complex functions. By properly taking into account the action of the edge-magnetoplasmon scattering matrix and assuming the same propagation velocity for both free channels (v 1 = v 2 = v) we can evaluate explicitly all the contributions to the current. Notice that the effect of a bias difference between the edge b (at voltage V ) and the edge a (grounded) is taken into account through the standard Peierls substitution [START_REF] Martin | Les Houches Session LXXXI[END_REF] Ψ b (x, t) → e -iω 0 t Ψ b (x, t) with ω 0 = eV / and where the field Ψ a (x, t) is left untouched.

The averaged current can be naturally separated into a classical contribution, diagonal in the QPCs action, and a quantum contribution, which is the off diagonal interference term. In the following we will discuss these terms in detail mainly focusing on the zero temperature limit.

Classical contributions to the current

For what it concerns the classical contributions to the current, namely the one diagonal in the QPCs tunneling amplitudes, one obtains

I cl B = e 2π 2 α 2 |Λ 1 | 2 + |Λ 2 | 2 +∞ -∞ dz sin (ω 0 z)ℑ J (l) (z) (24)
with

J (l) (z) = exp -2 +∞ 0 dω ω 1 -ℜ Σ (l) ab (ω) 1 -e -iωz (25)
and where ℜ[...] and ℑ[...] indicate respectively the real and the imaginary part. Notice that the above formula can be applied to both the case of one (l = 1) and two (l = 2) trapped channels in the FPI. The conventional non interacting case is easily recovered by neglecting the term associated to the edge-magnetoplasmon scattering matrix (Σ

ab = 0). It is worth to point out the fact that, due to translational invariance in the time domain, the expression in Eq. ( 24) is indeed independent of time.

As expected, the above term does not depend on the position of the QPCs and on the flux of magnetic field piercing the interferometer. It corresponds to the sum of the contributions associated to two distinct single QPC geometries, proportional respectively to

|Λ 1 | 2 and |Λ 2 | 2 .
By replacing the explicit form of the scattering matrix elements respectively for Σ(1) and Σ(2) (see Eq. ( 15) and Eq. ( 17)) one obtains the factorization

J (l) (z) = J (l) int (z)J f ree (z) l = 1, 2 (26)
with (see Appendix A)

J f ree (z) = exp -2 ∞ 0 dω ω 1 -e -iωz e -ω/ωρ (27) = 1 (1 + iω ρ z) 2 (28)
the free fermion contribution and

J (1) int (z) = exp -2 ∞ 0 dω ω cos ωτ σ -1 5 + 3 cos ωτ σ 1 -e -iωz e -ω/ωρ (29)
J (2) int (z) = exp -8 ∞ 0 dω ω cos ωτ σ -1 13 + 5 cos ωτ σ
1 -e -iωz e -ω/ωρ [START_REF] Carrega | Anomalous Charge Tunneling in Fractional Quantum Hall Edge States at a Filling Factor ν = 5/2[END_REF] the corrections due to the interaction. Notice that, for further convenience, we have introduced the convergence factor e -ω/ωρ with high frequency cut-off ω ρ = v ρ /α in order to have well behaved integrals for ω → +∞.

Quantum contribution to the current

Differently from the classical contribution discussed above, this term depends non locally on the two QPCs amplitudes and encodes information about the Aharonov-Bohm interference at the level of the FPI. It is given by

I q B = e π 2 α 2 +∞ -∞ dzℑ Λ 1 Λ * 2 e iω 0 z W (l) ℑ Y (l) (∆t -z) (31)
with ∆t = (x 1 -x 2 )/v,

W (l) = exp +∞ 0 dω ω Σ (l) ab (ω) + Σ (l) ba (ω) (32)
and

Y (l) (z) = exp - +∞ 0 dω ω 2 -Σ * (l) aa (ω) + Σ (l) bb (ω) e iωz . (33)
In analogy with what discussed for the classical current, also this contribution does not depend explicitly on time.

In terms of the explicit form of the scattering matrix elements for Σ(1) and Σ(2) (Eq. ( 15) and Eq. ( 17)), one has

W (1) = exp 2 ∞ 0 dω ω 1 -e iωτσ 3 + e iωτσ e -ω/ωρ (34)
W (2) = exp 4 ∞ 0 dω ω 1 -e iωτσ 5 + e iωτσ e -ω/ωρ . (35)
Moreover, also in this case we can factorize the interaction contribution in the form

Y (l) (z) = Y (l) int (z)J f ree (-z) with, Y (1) int (z) = exp 2 ∞ 0 dω ω cos ωτ σ -1 5 + 3 cos ωτ σ e iωz e -ω/ωρ (36) Y (2) int (z) = exp 8 ∞ 0 dω ω cos ωτ σ -1 13 + 5 cos ωτ σ e iωz e -ω/ωρ . ( 37
)
According to the Fourier series expansion of the functions h (1) (Eq. ( 16)) and h (2) (Eq. ( 18)) it is possible to derive useful asymptotic limits for the functions J In particular, focusing on the role played by W (l) , Eq. ( 31) can be approximated in a very good way by (j = 0, 1, 2)

I q B (Φ, ω 0 ) ≈ - e|Λ 1 ||Λ 2 | π 2 α 2 (ω ρ τ σ ) a j e -γ j cos (ω 0 ∆t -Φ -θ j ) +∞ -∞ dz sin ω 0 zℑ Y (j)
int (z)J f ree (-z) (38) where we have included the conventional free fermion case j = 0. In the above expression we have introduced the compact notation a 0 = 0, a 1 = 2/3 and a 2 = 4/5; θ 0 = 0, θ 1 = π/3 and θ 2 = 2π/5; γ 0 = 0, γ 1 ≈ 0.129656 and γ 2 ≈ 0.099524 (see Appendix A). Moreover, it is worth to note that in the non interacting case we trivially have Y (0) int = 1. We observe that the more dramatic effects associated to the presence of the trapped channels consists in modifying the scaling of the tunneling amplitude through a cut-off dependent contribution and to add an additional phase shift. Both these contributions crucially depend on the number of trapped channels. Every curve is properly further rescaled with respect to the factor (ω ρ τ σ ) a j e -γ j in order to keep track of the interaction induced renormalization of the tunneling amplitudes for the non interacting case (j = 0, full black curve), the one trapped channel case (j = 1, dashed blue curve) and the two trapped channels case (j = 2, dotted-dashed red curve) as a function of the Josephson frequency ω 0 = eV 0 / . Parameters are:

Results

Classical contribution

I cl B /I 0 G cl B /G 0
τ σ = 1, ω ρ τ σ = 1000, |Λ 1 | = |Λ 2 | = |Λ| and = 1.
In Fig. 3 we show the behavior of the classical contribution to the current and the G q B /G 0 Figure 4: Color on-line. Quantum contribution to the differential conductance in units of G 0 , calculated at ω 0 = 0 as a function of the Aharonov-Bohm phase (Φ). Also in this case every curve is properly further rescaled with respect to the factor (ω ρ τ σ ) a j e -γ j in order to keep track of the interaction induced renormalization of the tunneling amplitudes for the non interacting case (j = 0, full black curve), the one trapped channel case (j = 1, dashed blue curve) and the two trapped channels case (j = 2, dotted-dashed red curve). Parameters are:

τ σ = 1, ω ρ τ σ = 1000, |Λ 1 | = |Λ 2 | = |Λ| and = 1.
associated differential conductance

G cl B ∝ ∂ I cl B ∂ω 0 = e 2π 2 α 2 |Λ 1 | 2 + |Λ 2 | 2 +∞ -∞ dzz cos (ω 0 z)ℑ J (l) (z) . ( 39
)
In order to take into account the renormalization of the tunneling amplitudes induced by the presence of the trapped channels and to better compare the results, the curves are normalized here with respect to the proper cut-off dependent factor (ω ρ τ σ ) a j e -γ j (j = 0, 1, 2).

Concerning the current (top panel of Fig. 3) all curves are linear as a function of the Josephson frequency (and consequently of the voltage V ) for ω 0 ≈ 0 as a consequence of the fact that the edge-magnetoplasmon scattering matrix reduces to the identity at low enough frequency. However, the interacting cases (dashed blue and dotted-dashed red curves) strongly deviate from this linearity at higher voltages (ω 0 τ σ ≈ π) and show a remarkable oscillating behavior. This is even more evident for what it concerns the differential conductance (bottom panel of Fig. 3) which is constant (up to a small deviations due to the high frequency cut-off used in the numerics) in the free case, but oscillates and decay quite fast by increasing the number of trapped channels. Notice that this phenomenology is reminiscent of what derived in literature for a FPI in the integer and fractional quantum Hall regime [START_REF] De Chamon | Two point-contact interferometer for quantum Hall systems[END_REF] or for the topological insulators in presence of interaction [START_REF] Ferraro | Spin current pumping in helical Luttinger liquids[END_REF].

Quantum contribution to the current

Concerning the interference contribution, relevant information about the physics of the systems can be extracted form the conductance calculated at zero voltage (j = 0, 1, 2.)

G q B (Φ, ω 0 = 0) ∝ ∂I q B ∂ω 0 | ω 0 =0 ≈ - e|Λ 1 ||Λ 2 | π 2 α 2 (ω ρ τ σ ) a j e -γ j cos (Φ + θ j ) +∞ -∞ dzzℑ Y (j) int (z)J f ree (-z) (40)
The behavior of this quantity as a function of the Aharonov-Bohm phase Φ is represented in Fig. 4, where the phase shifts θ 1 = π/3 and θ 2 = 2π/5 induced by the presence of a different number of channels trapped into the interferometer loop are clearly visible.

It is worth to note that the various curves have the same amplitude. This can be easily found by comparing the non interacting case (j = 0), where the integral in Eq. ( 40) can be evaluated analytically according to the relation

+∞ -∞ dzzℑ 1 (1 + iω ρ z) 2 = - π ω 2 ρ (41)
due to the second order poles in the complex plane and coincide with what obtained numerically for the interacting cases (j = 1, 2). This is a direct consequence of the zero energy properties of the edge-magnetoplasmon scattering matrix which guarantee the fact that the electrons are effectively free at low enough frequency.

Considerations about multiple electron tunneling

As discussed above, Choi's experiment shows a puzzling and extremely interesting evolution of the effective tunneling charge as a function of the number of trapped channels. In order to shed light on this behavior one can imagine to allow also the tunneling of excitations with m times the charge of an electron3 (m ∈ N) at the QPCs, despite the fact that these multiple excitations are less relevant with respect to the electrons in the renormalisation group sense [START_REF] Kane | Transport in a one-channel Luttinger liquid[END_REF]. The vertex operator associated to them is given by

Ψ (m) (x) ∝ e imϕ(x) (42)
with the corresponding tunneling Hamiltonian

H (m) T = j=1,2 Λ (m) j e imϕ b (x j ) e -imϕa(x j ) + H.c. (43)
In this case, as far as we consider a gaussian model for the edge states dynamics, the expressions for this kind of excitations can be derived directly from the one for the electrons through the substitutions X → [X ] m 2 , being X = J (1,2) , W (1,2) or Y (1,2) and with a consequent replacement of the tunneling amplitudes and the charge associated to the excitations. This leads to a superohmic behavior which reflects on the fact that the conductance at zero bias is identically zero. This fact can be verified analytically in the non interacting case due to the relation

+∞ -∞ dzzℑ 1 (1 + iω ρ z) 2m 2 = 0 for m = 1 (44)
which is a consequence of the presence of higher order poles in the complex plane and numerically in the other cases. Therefore, no signature associated to higher charge carrier are expected in the framework of the proposed model. According to this, the comparison with experimental observations rules out the simple picture based on short range strong interaction usually considered as a valuable work hypothesis and seems to suggests that a more involved description is required. In particular dissipative effects at the level of the edge-magnetoplasmon scattering matrix [START_REF] Bocquillon | Separation of neutral and charge modes in one-dimensional chiral edge channels[END_REF], finite range interaction and non-gaussianity in the particle injection process [START_REF] Levkivskyi | Dephasing in the electronic Mach-Zehnder interferometer at filling factor ν = 2[END_REF] could play a relevant role.

Effects of an ohmic contact in the interferometer loop

In order to be further closer to what has been done in experiments, we can also consider a modified set-up in which an integer number k ≤ n of the trapped channels is absorbed by an ohmic contact. This configuration has been introduced in order to both enhancing the screening of the interaction and inducing dephasing among the trapped channels (see Fig. 1) [START_REF] Choi | Robust electron pairing in the integer quantum hall effect regime[END_REF]. We will study this geometry in the full equilibration limit (τ σ → +∞) where we can neglect the oscillations appearing in the edge-magnetoplasmon scattering matrices (see Eqs. [START_REF] Bocquillon | Separation of neutral and charge modes in one-dimensional chiral edge channels[END_REF] and [START_REF] Giamarchi | Quantum Physics in One Dimension[END_REF]). This leads to a semiclassical approximation based only on the conservation of a unitary energy flow. In full generality it writes

Σ(n,k) ↑ = (n-k+1)(n+1) n 2 (k+2)+3n+1 (n-k+1)n n 2 (k+2)+3n+1 (n-k+1)n n 2 (k+2)+3n+1 (n+1) 2 n 2 (k+2)+3n+1 . ( 45
)
The notation ↑ indicates an ohmic contact placed in the upper part of the interferometer loop (see Fig. 2), as opposite to ↓ for an ohmic contact in the lower part whose scattering matrix elements are obtained through the replacements ↑↔↓, a ↔ b. Notice that the position of the ohmic contact does not affect the results concerning the current flowing across the sample. Depending on the incoming arm of injection (a or b) one can have two different fraction of the unitary energy flux leaking into the ohmic contact, namely

∆E a,↑ = 1 -Σ (n,k) aa,↑ -Σ (n,k) ba,↑ = k(n + 1) 2 n 2 (k + 2) + 3n + 1 ∆E b,↑ = 1 -Σ (n,k) ab,↑ -Σ (n,k) bb,↑ = kn(n + 1) n 2 (k + 2) + 3n + 1 . (46)
In absence of ohmic contact (k = 0) the scattering matrix in Eq. ( 45) reduces to Σ(n,0)

↑ = Σ(n,0) ↓ = Σ(n,0) = n+1 2n+1 n 2n+1 n 2n+1 n+1 2n+1 (47)
with an obvious zero energy leakage (∆E a,↑ = ∆E b,↑ = 0). Two important comments are in order at this point. First of all, we notice that it is possible to obtain the expressions for Σ(1,0) and Σ(2,0) directly from Eq. ( 15) and (17) in the semiclassical limit, where all the oscillating terms are neglected. Eq. ( 47) represents therefore a generalization of what is done above as far as equilibration among the channels comes into play. Moreover, this fact also represents a validation of the renormalisation of the tunneling amplitudes previously derived, due to the fact that the dominant contribution to the integrals comes indeed from a region where the considered approximation holds.

According to the previous considerations, the classical and quantum contribution to the current can be written, in terms of the elements of the scattering matrix and the energy loss, as

I cl B = e 2π 2 α 2 |Λ 1 | 2 + |Λ 2 | 2 R ω 0 , ω ρ , Σ (n,k) aa + Σ (n,k) bb + ∆E a + ∆E a (48)
I q B = - e π 2 α 2 (|Λ 1 ||Λ 2 |) cos (ω 0 ∆t -Φ) exp -(∆E a + ∆E b ) +∞ 0 dω ω e -ω/ωρ × R ω 0 , ω ρ , Σ (n,k) aa + Σ (n,k) bb (49) with R [x, y, α] = 2π Γ(α) |x| α-1 y α e -|x|/y sgn(x) (50)
and Γ(α) the Euler's Gamma function. Notice that we have omitted the indication about the placement of the ohmic contact due to the fact that the results are independent of this because of the symmetries relating Σ(n,k)

I cl B + I q B = e|Λ| 2 π 2 α 2 [1 -cos (ω 0 ∆t -Φ)] R ω 0 , ω ρ , Σ (n,0) aa + Σ (n,0) bb (51)
showing oscillation as a function of the piercing magnetic flux modulating the overall power-law behavior ω Σ (n,0) aa +Σ

(n,0) bb -1 0 reminiscent of the one observed for the fractional quantum Hall effect [START_REF] De Chamon | Two point-contact interferometer for quantum Hall systems[END_REF] and topological insulators in presence of interaction [START_REF] Ferraro | Spin current pumping in helical Luttinger liquids[END_REF]. It is worth to note that the exponents satisfies

1 < Σ (n,0) aa + Σ (n,0) bb < 2 (52)
leading to a sub-ohmic and not diverging current-voltage characteristic. The presence of an ohmic contact dramatically affects the quantum contribution, which is exponentially suppressed according to the vanishing prefactor in Eq. (49). Indeed, the integral at the exponent is formally divergent and have to be regularized by adding an additional low frequency cut-off depending on the typical equilibration length of the interferometer loop. However, this exponential suppression still survives at finite temperature even after having taken care of this formal divergence.

Moreover, the classical and quantum contribution to the current scale in a different way as a function of the voltage, namely

I cl B ∝ ω Σ (n,k) aa +Σ (n,k) bb +∆Ea+∆Ea-1 0 (53) I q B ∝ ω Σ (n,k) aa +Σ (n,k) bb -1 0 . ( 54
)

Conclusion

In the present paper we have discussed the physics of a FPI in the integer quantum Hall regime. Motivated by very recent experiments, we focused on a system at filling factor ν = n + 1 (n ∈ N) where only one edge channel is transmitted across the sample, while the other n are trapped into the interferometer loop. Due to screening, the only residual interaction effect is given by a strong short range inter-channel interaction that we described in terms of the edge-magnetoplasmon scattering matrix formalism. The major effects associated to the presence of interacting trapped channels are: a renormalisation of the tunneling amplitudes affecting in exactly the same way the classical and the quantum contribution to the current, an evident damped and oscillatory behavior of the differential conductance in contrast to the constant (ohmic) behavoir observed in absence of trapped channels and an additional phase shift in the Aharonov-Bohm periodicity that is clearly visible in the differential conductance at zero bias. We have also discussed a simple model for a system in which k (0 ≤ k ≤ n) channels in the loop are absorbed by an ohmic contact, based only on the energy conservation. Here we observed that, as long as the energy is not conserved due to losses into the contact, the quantum contribution to the conductance is strongly suppressed and only the classical contribution survives, showing a non-universal power-law behavior reminiscent of the element of the effective edge-magnetoplasmon scattering matrix. It is worth to note that the comparison between our analysis and the experiments allows to rule out this simple and widely accepted model for the inter-edge channel interaction as the proper description of the considered set-up. According to this, additional physical effects like dissipation or nongaussianity could play a major role. However, the developed formalism is general enough to allow estention towards these directions.

n f -nτ σ , 1 ω ρ -iz + f nτ σ , 1 ω ρ -iz . (61)
The above series converge quite fast and are helpful both in the numeric evaluation of the integrals for the current and in order to obtain asymptotic expressions under the natural condition ω ρ τ σ ≫ 1.

A.1 J A.2 W (1) and W (2) In this case there is no dependence on z, but it is still possible to resum the series in an approximate way obtaining W (1) = (ω ρ τ σ )

2 3 e -i π 3 e -γ 1 (66) W (2) = (ω ρ τ σ )

4 5 e -i 2π 5 e -γ 2 .

(67)

Notice that the appearance of a phase factor that will enter directly as a shift of the magnetic flux into the quantum contribution to the current. (68)

Figure 1 :

 1 Figure 1: Color on-line. Schematic view of a two integer quantum Hall channels system (filling factor ν = 2). According to the chirality one can easily identify the incoming region (1), the interacting region (2) (shaded area) and the outgoing region (3). In regions (1) and (3) the dynamics of the bosonic fields is well described in terms of free equations of motion, while the outgoing fields are connected to the incoming ones through the edge-magnetoplasmon scattering matrix Ŝ(L, ω) which encodes the information of the inter-channel interaction (assumed as strong and short ranged in the main text).

2 Figure 2 :

 22 Figure 2: Color on-line. Schematic view of a FPI with n trapped channels in the interfering loop. The action of the interferometer in presence of interaction (assumed strong and short ranged in main text) can be described in terms of the edge-magnetoplasmon scattering matrix Σ(n) . Tunneling of electrons occurs at two quantum point contacts in x 1 and x 2 , respectively with tunneling amplitude Λ 1 and Λ 2 . The interferometer is also pierced by a flux Φ of magnetic field which is responsible of the Aharonov-Bohm effect. Moreover, in order to be closer to what is done in experiments, we can consider the presence of an ohmic contact which on the one hand further enhances the screening of the interaction, on the other absorbs the energy of k ≤ n trapped channels inducing dephasing. account the periodic boundary conditions associated to this closed channel one can derive the scattering matrix Σ(1) for the whole interferometer in the form φout a φout b

  int , W(l) and Y (l) int (l = 1, 2) which are discussed in detail in Appendix A.

Figure 3 :

 3 Figure3: Color on-line. Classical current in units of I 0 = e|Λ| 2 /(πv ρ τ σ ) 2 (top) and differential conductance in units of G 0 = eI 0 (bottom). Every curve is properly further rescaled with respect to the factor (ω ρ τ σ ) a j e -γ j in order to keep track of the interaction induced renormalization of the tunneling amplitudes for the non interacting case (j = 0, full black curve), the one trapped channel case (j = 1, dashed blue curve) and the two trapped channels case (j = 2, dotted-dashed red curve) as a function of the Josephson frequency ω 0 = eV 0 / . Parameters are: τ σ = 1, ω ρ τ σ = 1000, |Λ 1 | = |Λ 2 | = |Λ| and = 1.

  ohmic contact (∆E a = ∆E b = 0), but still considering the equilibrated limit, one obtains (for|Λ 1 | = |Λ 2 | = |Λ|)

J

  In the limit |z| ≫ nτ σ (n ∈ N) one directly obtainsJ (1) int (|z| → +∞) ≈ (ω ρ τ σ ) int (|z| → +∞) ≈ (ω ρ τ σ )

A. 3 Y 1 )

 31 the limit |z| ≫ nτ σ (n ∈ N) these functions reduce to Y (int (|z| → +∞) = Y

( 2 )

 2 int (|z| → +∞) ≈ 1.

Notice that the velocities vρ and vσ are functions of v1, v2 and v12[START_REF] Braggio | Environmental induced renormalization effects in quantum Hall edge states due to 1/f noise and dissipation[END_REF]. However, in what follows we are not interested in their explicit functional form, but only in the fact that typically on has vρ ≫ vσ[START_REF] Levkivskyi | Dephasing in the electronic Mach-Zehnder interferometer at filling factor ν = 2[END_REF][START_REF] Levkivskyi | Mach-Zehnder interferometry of fractional quantum Hall edge states[END_REF].

Notice that this symmetry is reminiscent of the analogous hidden symmetry observed for the states belonging to the Jain's sequence of the fractional quantum Hall effect[START_REF] Kane | Impurity scattering and transport of fractional quantum Hall edge states[END_REF].

Notice that a similar analysis involving the tunneling of multiple fractionally charged excitation has been proposed in Refs.[START_REF] Ferraro | Relevance of Multiple Quasiparticle Tunneling between Edge States at ν = p/(2np + 1)[END_REF][START_REF] Ferraro | Charge tunneling in fractional edge channels[END_REF][START_REF] Ferraro | Neutral modes' edge state dynamics through quantum point contacts[END_REF][START_REF] Carrega | Anomalous Charge Tunneling in Fractional Quantum Hall Edge States at a Filling Factor ν = 5/2[END_REF] as a possible explanation for the unexpected evolution of the effective charge in a QPC geometry in the composite edge states of the fractional quantum Hall effect[START_REF] Chung | Scattering of Bunched Fractionally Charged Quasiparticles[END_REF][START_REF] Bid | Shot Noise and Charge at the ν = 2/3 Composite Fractional Quantum Hall State[END_REF][START_REF] Dolev | Dependence of the tunneling quasiparticle charge determined via shot noise measurements on the tunneling barrier and energetics[END_REF].

Acknowledgements

We thanks I. P. Levkivskyi, E. Idrisov, A. Borin and A. Goremykina for useful discussions. We acknowledge the financial support of Swiss National Science Foundation. D. F. want to acknowledge the support of Grant No. ANR-2010-BLANC-0412 ("1 shot") and of ANR-2014-BLANC "one shot reloaded" is acknowledged. Part of this work was carried out in the framework of Labex ARCHIMEDE Grant No. ANR-11-LABX-0033 and of A*MIDEX project Grant No. ANR-11-IDEX-0001-02, funded by the "investissements d'avenir" French Government program managed by the French National Research Agency (ANR).

A Consideration about the integrals

Aim of this Appendix is to investigate the asymptotic behavior of the functions J

and recalling the Fourier series expansion of h (1) and h (2) in Eqs. ( 16) and ( 18) one obtains