M C Ulrich

Remillieux

M Lott

C Payan

V Garnier

P Y Le Bas

T J Ulrich

M C Remillieux

Three-dimensional modeling and numerical predictions of multimodal nonlinear behavior in damaged concrete blocks

Three-dimensional modeling and numerical predictions of multimodal nonlinear behavior in damaged concrete blocks

I. Introduction

Geomaterials exhibit mechanical softening under dynamic loading, usually combined with hysteresis and slow-dynamics effects [START_REF] Johnson | Resonance and elastic nonlinear phenomena in rock[END_REF]TenCate et al., 2000). These effects are reversible and deterministic. From a micro-structure standpoint, a recent study has established a correlation between micro-crack density and softening effects (Payan et al., 2014a,b). From an energy balance point of view, it means that some potential energy disappears when the material is loaded dynamically and is slowly recovered when the material is at rest. These properties are relevant to various natural processes and industrial applications including the onset of earthquake and avalanches in geophysics [START_REF] Johnson | Nonlinear dynamics, granular media and dynamic earthquake triggering[END_REF], the aging of civil infrastructure [START_REF] Eiras | Dynamic acousto-elastic test using continuous probe wave and transient vibration to investigate material nonlinearity[END_REF][START_REF] Payan | Applying nonlinear resonant ultrasound spectroscopy to improving thermal damage assessment in concrete[END_REF][START_REF] Vu | Concrete cover characterisation using dynamic acousto-elastic testing and rayleigh waves[END_REF], the design of acoustic meta-materials with extreme stiffness [START_REF] Diani | A review on the mullins effect[END_REF][START_REF] Wang | Extreme stiffness systems due to negative stiffness elements[END_REF], or the assessment of bone fragility in the medical field [START_REF] Haupert | Non destructive characterization of cortical bone micro-damage by nonlinear resonant ultrasound spectroscopy[END_REF].

The dynamic response of these materials has been first modeled by [START_REF] Guyer | Hysteresis, discrete memory, and nonlinear wave propagation in rock: A new paradigm[END_REF] using the Preisach formalism, which is a phenomenological description borrowed from the field of electromagnetics. This model has then been refined to include slow-dynamics effects through thermally activated random transitions between the open and closed states of the hysteretic elements of the Preisach system [START_REF] Delsanto | Modeling nonclassical nonlinearity, conditioning, and slow dynamics effects in mesoscopic elastic materials[END_REF][START_REF] Nobili | Temperature effects on the elastic properties of hysteretic elastic media: Modeling and simulations[END_REF].

More recently, [START_REF] Pecorari | A constitutive relationship for mechanical hysteresis of sandstone materials[END_REF] proposed a hysteretic model sharing some features with the Jiles-Atherton model (also borrowed from the field of electromagnetics), which he enhanced to capture the slow-dynamics effects. It is also worth mentioning the soft-ratchet model of slowdynamics originally proposed by [START_REF] Vakhnenko | Strain induced kinetics of intergrain defects as the mechanism of slow dynamics in the nonlinear resonant response of humid sandstone bars[END_REF] and recently modified by [START_REF] Favrie | Fast and slow dynamics in a nonlinear elastic bar excited by longitudinal vibrations[END_REF] to include classical nonlinearity and viscoelasticity. All these models, however, have been derived in the one-dimensional (1D). Three-dimensional (3D) effects are an essential part of the dynamic response observed in geomaterials and cannot be ignored [START_REF] Egle | Measurement of acoustoelastic and third-order elastic constants for rail steel[END_REF][START_REF] Payan | Applying nonlinear resonant ultrasound spectroscopy to improving thermal damage assessment in concrete[END_REF][START_REF] Payan | Determination of third order elastic constants in a complex solid applying coda wave interferometry[END_REF][START_REF] Tournat | Probing weak forces in granular media through nonlinear dynamic dilatancy: clapping contacts and polarization anisotropy[END_REF]. [START_REF] Lyakhovsky | Non-linear damage rheology and wave resonance in rocks[END_REF] proposed a twodimensional continuum damage rheology model capturing hysteretic nonlinearity and some of the features observed in nonlinear resonance experiments. A similar model was used by [START_REF] Hamiel | Brittle deformation and damage-induced seismic wave anisotropy in rocks[END_REF] to describe stress-induced anisotropy in damaged geomaterials. Note that in these models, damage is not recoverable and slow-dynamics effects are not considered. The recent modeling work of [START_REF] Berjamin | Nonlinear waves in solids with slow dynamics: an internal-variable model[END_REF] includes slow-dynamics effects, but although it can be naturally extended to 3D analysis, the simulations are carried out only for a 1D case. Lott,JASA 4 Attempting to develop a predictive tool to link the microstructural properties of the material to the nonlinear elastic response observed experimentally is a challenging task. Despite a high sensitivity to microstructures, amorphous condensed matter arrangement, and environmental conditions (e.g., temperature, pressure) (Lott et al., 2016;[START_REF] Payan | Determination of third order elastic constants in a complex solid applying coda wave interferometry[END_REF]TenCate et al., 2000), the elastic nonlinear parameters seem to depend on the choice of the experimental setup in much of the previous work reported in the literature. For instance, [START_REF] Renaud | Anisotropy of dynamic acoustoelasticity in limestone, influence of conditioning, and comparison with nonlinear resonance spectroscopy[END_REF] showed that material softening effects in Berea sandstone may vary by up to an order of magnitude depending on the probing direction while the material is isotropic. In the work of Payan et al. (2014a,b) on concrete, two techniques were used to estimate a parameter of nonlinearity. The two techniques showed the same trends but the magnitudes of the measured parameter differ by an order of magnitude. These experimental data where derived using the most common 1D approximation. However, a complex strain field leads to a 3D problem from a mechanical point of view and using a 1D approximation to describe this problem may lead to erroneous conclusions.

Recently, Lott et al. (2016[START_REF] Lott | Nonlinear elasticity in rocks: A comprehensive three-dimensional description[END_REF] show that conditioning can be properly accounted in 3D using a single scalar nonlinear parameter into the appropriate set of elasticity equations using uncoupled single mode resonance type of experiments, i.e., shear and compressional. The aim of this paper is to study the validity of this model to nonlinear resonance experiments, involving strong coupling between shear and compressional motion types. Numerical simulations are compared to experimental data from nonlinear resonances in gradually thermally damaged concrete blocks. Using a single scalar nonlinear parameter appropriately introduced in elasticity equations, the validity of the model is demonstrated and discussed.

II. Theoretical framework

In continuum mechanics, the parameters needed to compute the linear dynamic response of an elastic body are the mass density, the linear elastic tensor, and the damping parameters. If the geometry of the sample is complex and/or the material is heterogeneous, numerical techniques (e.g., finite-element method) should be used and the distribution of these parameters should be known for all computation cells discretizing the sample. In this paper, we study consolidated granular materials, which may be described as a disordered network of mesoscopic-sized "hard" Lott, JASA 5 elements (e.g., grains with characteristic lengths ranging from tens to hundreds of microns) cemented together by a "soft" bond system (e.g., amorphous silica, calcite) [START_REF] Guyer | Nonlinear mesoscopic elasticity: Evidence for a new class of materials[END_REF]. Despite the complexity of the structure, the linear elastic response of these materials at macroscopic scale is well captured by the model of a continuum. This continuum may have heterogeneous properties under dynamic loading as a combined result of the complex spatiotemporal distribution of the strain field in the sample and the strain-induced material softening [START_REF] Payan | Applying nonlinear resonant ultrasound spectroscopy to improving thermal damage assessment in concrete[END_REF][START_REF] Remillieux | Resonant ultrasound spectroscopy for materials with high damping and samples of arbitrary geometry[END_REF][START_REF] Renaud | Anisotropy of dynamic acoustoelasticity in limestone, influence of conditioning, and comparison with nonlinear resonance spectroscopy[END_REF]. Material softening is thought to originate from the microscopic-sized defects (e.g., micro-cracks) in the "soft" subsystem and at the interfaces between the "hard" and "soft" subsystems (Payan et al., 2014a). To incorporate these defects in the macroscopic continuum model, we consider a volume dV that is small enough to satisfy the requirements of the numerical schemes but large enough to ensure a constant micro-crack density from one volume to the next. Furthermore, we assume that the defects are randomly distributed and oriented within the volume dV, with a length-scale much smaller than the typical acoustic wavelengths used in this study. With dV being on the order of 1mm 3 , the above criteria are satisfied.

The approach used in this paper is similar to the one used by [START_REF] Zubelewicz | Overall stress and strain rates for crystalline and frictional materials[END_REF] in transient numerical simulations of rock fractures. Each contact within the medium brings softening effects through nonlinear cohesive mechanisms. These effects are then integrated over an elementary volume dV and used for a mesoscopic mechanical implementation within a continuum with possible heterogeneous properties.

Recently, the authors (Lott et al., 2016[START_REF] Lott | Nonlinear elasticity in rocks: A comprehensive three-dimensional description[END_REF] proposed a general formulation to include material-softening effects in the equations of elasticity. This formulation is an extension of that proposed by Hughes and Kelly [START_REF] Hughes | Second-order elastic deformation of solids[END_REF]. The tensorial product between the strain and stress vector bases, δij=ni stress ⊗nj strain , is the natural basis for the elastic tensor and should now include softening effects as,

Λ 𝑖𝑗 = 𝛿 𝑖𝑗 (1 -αΔ𝜀 𝑖𝑗 * ), (1)
or equivalently in matrix form as,

𝚲 = 𝑰 3 -𝛼𝜺 * , (2)
Lott, JASA

6
where ni is the principal strain direction, Δεij * the strain amplitude, the star symbol denotes the basis formed by the principal strain axes, and α is a scalar quantifying the softening effect. The stiffness tensor is then expressed as,

𝐶 𝑖𝑗𝑘𝑙 * = [𝜆 + 2(𝑙 -𝜆 -𝑚)𝑇𝑟(𝜀) + 2(𝜆 + 𝑚)(𝜀 𝑖 * + 𝜀 𝑘 * ) -2𝜇𝜀 𝑖 ]Λ 𝑖𝑗 Λ 𝑘𝑙 + [𝜇 + (𝜆 + 𝑚 -𝜇)𝑇𝑟(𝜀) + 2𝜇(𝜀 𝑖 + 𝜀 𝑗 + 𝜀 𝑙 )](Λ 𝑖𝑘 Λ 𝑗𝑙 + Λ 𝑖𝑙 Λ 𝑗𝑘 ) + 1 2 𝑛 ∑ (Λ 𝑗𝑣𝑘 𝑖𝑣𝑙 + Λ 𝑗𝑣𝑙 𝑖𝑣𝑘 )𝜀 𝑣 𝑣 (3)
where λ and μ are the Lamé constants and l, m, n are the Murnaghan constants [START_REF] Murnaghan | Finite deformations of an elastic solid[END_REF]. Practically, the material-softening law is applied in the basis formed by the principal strain axes. This basis is obtained from an eigen decomposition of the strain field ε measured in the geometric basis. This decomposition, ε = P ε*P -1 , is always possible because the strain tensor is always real and symmetric. Once the softening law has been applied to the stiffness tensor in the basis formed by the principal strain axes, the following transformation is used to express the conditioned stiffness tensor in the geometric basis: Cijkl = PirPjsPktPluC * rstu. This result is equivalent to classical acoustoelasticity theory when the term quantifying softening is equal to the identity matrix.

In this study, the validity of the model will be assessed using nonlinear resonance experiments or more commonly referred to in the literature as nonlinear resonant ultrasound spectroscopy (NRUS). In these experiments, a sample is subjected to sequences of periodic signals at various frequencies around a resonance frequency and at increasing amplitudes while, for each periodic signal, data are recorded when the sample vibrates at or near a steady state. Under these conditions, the effects of the first-order term of nonlinearity in the classical description, i.e., terms involving β in 1D and (l, m, n) in 3D, average to zero over one cycle of the harmonic excitation. This is not the case with the second-order term of nonlinearity δ but the proposed model does not go beyond the first order. As a result, Eq. ( 3) can be simplified by removing the terms involving third-order elastic constants (i.e., only the linear version of the elastic tensor is used to apply the softening law),

𝐶 𝑖𝑗𝑘𝑙 * = 𝜆Λ 𝑖𝑗 Λ 𝑘𝑙 + 𝜇 (Λ 𝑖𝑘 Λ 𝑗𝑙 + Λ 𝑖𝑙 Λ 𝑖𝑗𝑗𝑘 ). (4)
Finally, each cell (dV) of the sample can be conditioned by the strain amplitude at that cell. This means that the stiffness tensor is no longer uniform over the volume of the sample but depends on the local strain tensor. The sample experiences softening with a complex distribution (i.e., that of the strain field), which in turns lowers its resonance frequencies. Because of the heterogeneity, the resonance frequencies have to be computed numerically.

III. Numerical application to experimental data

The validity of the model is examined against experimental data collected during resonance experiments. The samples are those described by [START_REF] Payan | Quantitative linear and nonlinear resonance inspection techniques and analysis for material characterization: Application to concrete thermal damage[END_REF]. The three samples used in this study consist of concrete blocks with dimensions 6 × 10 × 10 cm 3 . One sample is kept intact and used as a reference. The other two samples are thermally damaged at 120 and at 250°C using a slow increase then decrease in temperature to prevent from eventual mechanical damage induced by thermal gradients. Thermal damage allows the density of micro-cracking within the concrete samples to be increased as needed. Below 300° C, microcracks are essentially caused by drying as well as differential thermal dilatation between cement paste and aggregates. In concrete the most brittle zone is the interface between the aggregates and the cement paste. This zone, commonly referred to as Interfacial Transition Zone (ITZ), is the most porous and crystallized region. It is thus quite natural to assume that most of the thermal damage will occur in the ITZ. With an arbitrary grain distribution, thermal damage of concrete can reasonably be considered as isotropic.

A. Experimental protocol and results

The

B. Numerical protocol

The relatively large mass added by the transducer is accounted for in the model by an aluminum cylinder with a perfect contact between the transducer and the sample. The following protocol is applied (Fig. 2) to describe the numerical process. The simulations are carried out using the "Structural Mechanics" module of the commercial finite-element software package Comsol® Multiphysics.

• Step 1 : A first eigenvalue problem is solved to compute the resonance frequencies of the samples with linear elastic properties (before the material-softening law is applied), which were already measured by [START_REF] Payan | Quantitative linear and nonlinear resonance inspection techniques and analysis for material characterization: Application to concrete thermal damage[END_REF] of strain amplitudes in the samples are recovered from the linear model, based on the laser measurement at a single point.

•

Step 2 : In the second step of the numerical protocol, a frequency-domain simulation is conducted by imposing a normal force on the free flat surface where the transducer is mounted. The amplitude of the normal force is tuned to match the particle velocity at the measured position on the sample.

•

Step 3 (Fig. 2) : The strain tensor is extracted at all nodes of the mesh discretizing the sample and used for the conditioning step. As the media is initially considered isotropic and homogeneous, the initial local stiffness is independent of the choice of basis and usually written Cijkl = λ δijδkl + μ (δikδjk + δilδjk). Using Eq. ( 1) and Eq. ( 4), the material softening effect is applied to the stiffness tensor Cijkl * trough the tensors Λij in the eigen basis.

•

Step 4 (Fig. 2) : The conditioned elastic tensor is then transposed back to the "geometrical" basis. After this numerical procedure, the sample, initially homogeneous and isotropic, becomes heterogeneous and anisotropic.

• Step 5 : A final eigenvalue problem is solved to compute the resonance frequencies of the samples with conditioned elastic properties. Steps 2 to 5 are repeated for several amplitudes to provide a curve of the relative shift of the resonance frequency.

IV. Results and discussion

Predicted and measured material softening in NRUS are shown in Fig. 3 for the three samples.

The values of α0 used in the simulations to match the experimental data are reported in Table .     II.

Sample α0

Reference 300

120°C 520 250°C 1000

Table . II. Values of the parameter α0 used in the NRUS simulations for the three concrete samples.

Note that this value increases by more than 300% between the reference sample and the most damaged sample, which seems to be consistent with the evolutions of the nonlinear parameters reported in the literature. To highlight the efficiency of the present model, the "apparent" nonlinear parameters are measured by linear regression of the relative frequency shift as a function of the strain amplitude for the three modes studied. This apparent nonlinear parameter changes with the mode order and the type of strain (e.g., volumetric and deviatoric) used in the analysis. For each mode and each sample, the apparent nonlinear parameters obtained with volumetric and deviatoric strains are reported in Fig. 4. Tying this apparent nonlinear parameter to a 1D description will lead to erroneous conclusions including that the nonlinearity is dispersive (i.e., depends on the order of the mode) or is dependent on the type of strain involved (e.g., shear or compressional components). It is important to stress that the nonclassical nonlinearity should be handled by a local parameter appropriately integrated within the elasticity equations.

Numerical simulations using a single scalar parameter α0 reproduce well the experimental observations, independently from the type of strain used or the mode order employed in the Lott, JASA 11 nonlinear analysis. There are some discrepancies between numerical simulation and experiments for some cases, but it does not seem to be consistent with the type of mode or a particular sample. Discrepancies are most likely due to the complexity of the experiment and the presence of a large transducer mounted at various positions on the sample to excite the various modes. Also, the sample do not have the perfect parallelepiped geometry used in the model. Even if the sample size was precisely measured using a caliper, parallelisms deviations are not accounted for in the simulations.

For the third mode, all the frequency shifts flatten a little bit at increasing amplitude (Fig. 3).

This behavior may be induced by some hidden resonant peaks which could appear at high amplitude. This can happen especially at "high" frequency when the resonance frequency density becomes high. Even if the location of the transducer was chosen so as to favor a given mode shape, other nearby resonances may alter these curves.

More importantly, the simulation results indicate that, in the case of concrete the nonlinear behavior of the material for both compressional and torsional motions is well captured by a single local scalar parameter. These results are in agreement with the fact that the distribution of defects in the thermally damaged concrete samples is isotropic. is also reported for comparison.

V. Conclusion

A tensorial interpretation of the softening effects under dynamic stress was integrated in a numerical scheme using a single scalar parameter. The key feature of the model relies on locally applying a softening law to the elastic tensor in the eigenbasis of the strain field and transforming this tensor back to the geometrical basis once the softening has been applied.

Numerical simulation results were compared to nonlinear multi-modal resonance experiments on thermally damaged concrete samples. In these samples, it was well adapted to predicting nonlinear behavior of their complex mode shapes involving coupling between shear and compressional components. This is in good agreement with the fact that an isotropic distribution of defects is expected in these materials.

It also shown that without accounting for these issues, the derived apparent nonlinear parameter can lead to erroneous interpretations. In this study, one could conclude to frequency dispersion Lott, JASA 14 or strain type dependences of nonlinearity. However, none of these statements are supported by the appropriate description of nonlinearity in the elasticity equations.

Future improvements of the model will aim at accounting for textured type of microcracking using the more general framework of a vectorial nonlinear parameter. Indeed, experimental data reported in the literature (Remillieux et al. 2016;[START_REF] Lott | Nonlinear elasticity in rocks: A comprehensive three-dimensional description[END_REF] show that many sedimentary rocks (e.g., Berea sandstone) exhibiting preferred oriented micro cracking features would require such a description. is also reported for comparison.

  FIG. 1. (Color online) Experimental (a) first, (b) second and (c) third NRUS curves for the 120°C damaged sample. Insets are the corresponding modal shapes.

  FIG. 3. (Color online) Experimental and numerically predicted resonances curves for the whole set of samples and modal shapes.

FIG. 1 .

 1 FIG. 1. (Color online) Experimental (a) first, (b) second and (c) third NRUS curves for the 120°C damaged sample. Insets are the corresponding modal shapes.

FIG. 4

 4 FIG. 4. (Color online) Experimental and numerically derived apparent nonlinear parameters. α0

Table . I

 . FIG. 2. Scheme of the numerical and experimental protocols. . Resonance frequencies obtained experimentally and numerically with linear RUS.

							Lott, JASA
	120°C 9.95 9.96 (+0.1%) 16.86 16.81 (-0.3%) 23.30 23.79 (+2.1%)
	250°C 9.43	9.53 (+1%) 15.97 14.739 (-7.7%) 22.31 22.30 (-0.1%)
			Resonance frequencies f0 (kHz)
	Sample		Mode 1		Mode 2		Mode 3
		Exp	Num	Exp	Num	Exp	Num
	Ref	10.16 10.09 (-0.7%) 17.33 17.36 (+0.2%) 24.02 24.43(+1.7%)
							10

Table .

 . Table. I. Resonance frequencies obtained experimentally and numerically with linear RUS. II. Values of the parameter α0 used in the NRUS simulations for the three concrete samples.

							Lott, JASA
	TABLES FIGURE CAPTION
			Resonance frequencies f0 (kHz)
	Sample		Mode 1		Mode 2		Mode 3
		Exp	Num	Exp	Num	Exp	Num
	Ref	10.16 10.09 (-0.7%) 17.33 17.36 (+0.2%) 24.02 24.43(+1.7%)
	120°C 9.95 9.96 (+0.1%) 16.86 16.81 (-0.3%) 23.30 23.79 (+2.1%)
	250°C 9.43	9.53 (+1%) 15.97 14.739 (-7.7%) 22.31 22.30 (-0.1%)
				Sample	α0
				Reference 300
				120°C	520
				250°C	1000

Acknowledgments

This work was supported by the French National Research Agency through the ENDE program (Grant No. ANR-11 RSNR 0009) and the U.S. Department of Energy through the Fossil Energy program (Grant No. FE-634-15-FY15).


Dario Ferraro
email: ferraro@fisica.unige.it

Flavio Ronetti

Luca Vannucci

Matteo Acciai

Jérôme Rech

Thibaut Jockheere

Thierry Martin

Maura Sassetti

Hong-Ou-Mandel characterization of multiply charged Levitons

We review and develop recent results regarding Leviton excitations generated in topological states of matter -such as integer and fractional quantum Hall edge channels -and carrying a charge multiple of the electronic one. The peculiar features associated to these clean and robust emerging excitations can be detected through current correlation measurements. In particular, relevant information can be extracted from the noise signal in generalized Hong-Ou-Mandel experiments, where Levitons with different charges collide against each other at a quantum point contact. We describe this quantity both in the framework of the photo-assisted noise formalism and in terms of a very interesting and transparent picture based on wave-packet overlap.

Introduction

The on-demand generation of single-electron states in mesoscopic systems has opened the way to the fascinating field of electron quantum optics (EQO), where individual fermionic excitations ballistically propagating in solid state systems are manipulated with methods borrowed from photonic quantum-optical experiments [1,2,3]. The standard recipe of EQO starts from the on-demand generation of coherent single-particle excitations in a solid state device, which can be achieved using the so-called mesoscopic capacitor [4,5] or driving the quantum conductor with carefully-engineered periodic voltage pulses [6,7,8,9]. The dissipationless edge states of the quantum Hall effect are then exploited to ensure ballistic propagation of the wave-packet throughout the sample, and quantum point contacts (QPC) play the role of beam-splitter allowing to conceive efficient interferometric setups.

However, it is well known that solid state systems can be heavily influenced by interactions, differently from what happens in the traditional photonic quantum optics. For instance, the fractional quantum Hall (FQH) effect [10] is a paradigmatic example of the dramatic consequences of electron-electron interactions. Here, a new strongly-correlated phase emerges in the quantum liquid, with quasiparticle excitations carrying a fraction of the electron charge [11,12,13] and whose statistical properties are neither bosonic nor fermionic, but belong to the more general class of anyons [14]. Since FQH systems host topologically protected chiral modes at the edge [15], an extremely exciting generalization of EQO ideas to this strongly interacting regime can be envisaged.

With the present paper we intend to review and extend recent results concerning EQO in a quantum Hall fluid at integer and fractional filling factor. In particular, we focus on a solid state analogue of the celebrated Hong-Ou-Mandel (HOM) experiment [16], where excitations impinge on the opposite sides of the beam splitter. In this experimental configuration it is possible to extract valuable information about the statistical properties of the colliding particles and the form of their wave-packets. The simplest way to generate clean pulses in the input arms of the HOM interferometer is by means of a periodic Lorentzian voltage drive, which was shown to provide on-demand robust multi-electronic excitations both in the integer [17,18] and the fractional regime [19,20]. According to existing literature, throughout this review we will refer to these excitations carrying q elementary charges as q-Levitons [21]. We find that, in strongly-correlated FQH states, the HOM noise generated by the collision involving q-Levitons with q > 1 shows unexpected sub-dips [21] which are consistent with an interaction-induced self-organization of the multi-electronic wavepacket in the time domain, namely a real-time version of the crystallization occurring in Luttinger liquids confined by a potential well [22,23]. This work collects evidence of the prominent role played by HOM collisional experiments as a tool to characterize the behavior of individual excitations ballistically propagating in condensed matter systems [24].

This review is organized as follows. We illustrate the model for a Hall bar in the Laughlin sequence of the FQH effect in presence of the periodic drives in Section 2. Then, in Section 3, we introduce a quantum point contact and evaluate the corresponding shot-noise generated by tunneling between the two edges. In Section 4 we apply such formalism to study HOM interferometry in EQO. We review recent theoretical and experimental results at integer and fractional filling factor, and discuss new unexpected features emerging from generalized HOM collisions in the FQH regime. Later, we also discuss the relation between current noise and the overlap of Levitons wavefunctions in Section 5. Finally, in Section 6, we draw our conclusions.

Model

Let us consider a Hall bar in a four-terminal geometry, such as the one depicted in Figure 1. For a filling factor ν in the Laughlin sequence ν = 1/(2n + 1) [11], with n ∈ N, a single chiral mode propagates at each edge of the sample. Right-moving and left-moving low-energy degrees of freedom are described by bosonic fields Φ R/L satisfying [Φ R/L (x), Φ R/L (y)] = ±iπsign(xy). The effective Hamiltonian for the edge modes reads [15]

( = 1) H R/L = v 4π dx ∂ x Φ R/L (x) 2 , (1)
with v the propagation velocity of the free chiral fields. We assume that the edge modes are capacitively coupled with two oscillating voltage drives V R (x, t) and V L (x, t), acting separately on the right-moving and left-moving excitations respectively. To this end, we introduce the coupling term where the density operators ρ R/L (x) = ∓ √ ν 2π ∂ x Φ R/L (x) have been introduced. To describe the experimentally relevant situation of infinite, homogeneous contacts we factorize the space and time dependence of the voltage sources as V R/L (x, t) = Θ(∓xd)V R/L (t). Under such assumption, the equations of motion (∂

H V = e dxV R (x, t)ρ R (x) + e dxV L (x, t)ρ L (x), (2)
t ± v∂ x )Φ R/L (x, t) = e √ νV R/L (x, t) are solved by Φ R/L (x, t) = φ R/L (x ∓ vt, 0) + e √ ν t∓ x v -d v 0 dt V R/L (t ). (3)
In the above equation, we have indicated with φ R/L the free chiral bosonic fields at zero voltage (equilibrium configuration with V L = V R = 0). The unimportant constant time shift d/v will be omitted in the rest of the paper. Finally, the standard procedure of bosonization allows one to define annihilation fields for Laughlin quasiparticles as [15] Ψ

R/L (x, t) = F R/L √ 2πa e ±ikF(x∓vt) e -i √ νφ R/L (x∓vt,0) e -ie * t∓ x v 0 dt V R/L (t ) , (4)
where a is a short-distance cut-off, k F is the Fermi momentum, F R/L are the Klein factors [25,[START_REF] Martin | Noise in mesoscopic physics[END_REF][START_REF] Dolcetto | [END_REF]] and e * = νe (e the electron charge) indicates the fractional charge of a quasiparticle in the Laughlin sequence [11].

Shot-noise induced by tunneling

A quantum point contact (QPC) placed at x = 0 induces tunneling of Laughlin quasiparticles between opposite edges [28]. This is accounted for by the tunneling Hamiltonian

H tun = ΛΨ † R (0)Ψ L (0) + h.c., (5)
with Λ the constant tunneling amplitude. Operators for the right and left moving currents are readily obtained from continuity equations, and read J R/L (x, t) = -v e √ ν 2π ∂ x Φ R/L (x, t) downstream of the input terminals. Considering the tunneling as a perturbation, we can write the time-dependent current operators up to second order in H tun as

J R/L (x, t) = -v e √ ν 2π ∂ x φ R/L (x ∓ vt, 0) ± e 2 ν 2π V R/L (t ∓ x v )+ -i t -∞ dt H tun (t ), v e √ ν 2π ∂ x φ R/L (x ∓ vt) + + t -∞ dt t -∞ dt H tun (t ), H tun (t ), v e √ ν 2π ∂ x φ R/L (x ∓ vt) + + o(H 2 tun ). (6)
Notice that, according to the analysis carried out in Ref. [29], the present perturbative approach is physically meaningful as long as the backscattering conductance associated to the quasiparticle tunneling at the QPC is smaller than the quantized conductance of the quantum Hall channel. We define the currents flowing into output terminals 2 and 3 as J 3 (t) = J R (d, t), J 2 (t) = -J L (-d, t), so that currents entering the terminals are taken as positive. The central quantity of interest in this work will be the zero frequency cross-correlated current fluctuation averaged over one period T of the drive [30], namely

S 23 = 2 T 0 dt T +∞ -∞ dt [ J 2 (t)J 3 (t ) -J 2 (t) J 3 (t ) ] . (7)
Using the bosonization identity in equation ( 4), it is possible to express S 23 in terms of the bosonic Green's function

G(τ ) = φ R/L (0, τ ) -φ R/L (0, 0) φ R/L (0, 0
) and the voltage signals only. It is worth noticing that G(τ ) does not depend on the spatial variable x due to the assumption of local tunneling at the QPC.

After some algebra the only remaining contribution is found to be

S 23 = -4(e * ) 2 |λ| 2 T 0 dt T +∞ -∞ dt cos e * t t [V R (τ ) -V L (τ )] dτ e 2νG(t -t) (8)
with λ = Λ/(2πa).

Hanbury-Brown and Twiss setup

In the context of EQO, Hanbury-Brown and Twiss (HBT) interferometry is performed by driving one of the input terminals with a periodic signal V R/L (t) while grounding the opposite one [31,32,33]. To obtain the noise in this configuration, it is useful to expand the cosine in equation ( 8) using the Fourier series representation of the periodic function e

-iχ R/L (t) , with χ R/L (t) = t 0 [e * V R/L (t ) -q R/L ω]dt . Physically, χ R/L (t)
is the phase accumulated by a Laughlin quasiparticle under the action of V R/L (t) and q R/L is the total charge injected during one period of the drive. The latter is linked to the DC component of the signal, namely ωq

R/L = e * T 0 dt T V R/L (t)
with ω = 2π/T [19]. Introducing the Fourier series e -iχ R/L (t) = l p l (q R/L )e -ilωt and the Fourier transform Pg (E) = +∞ -∞ dte iEt e gG(t) , with g a generic real parameter, [34] one gets

S HBT R/L (q R/L ) = -2(e * ) 2 |λ| 2 +∞ l=-∞ p l (q R/L ) 2 × × P2ν q R/L + l ω + P2ν -q R/L + l ω . (9)
We will focus on the interesting case of a periodic train of Lorentzian voltage pulses, whose Fourier coefficients are [6,19,20,35]

p l (q R/L ) = q R/L γ l ∞ s=0 (-1) s Γ (l + s + q R/L ) Γ (1 -s + q R/L ) γ 2s s!(s + l)! . (10)
Here γ = e -2πη and η is the ratio between the half width at half maximum of each Lorentzian peak and the period T [36,37,[START_REF] Glattli | Method and device for phase modulation of a carrier wave and application to the detection of multi-level phase-encoded digital signals[END_REF].

Hong-Ou-Mandel setup

A solid-state equivalent of the photonic HOM experiment is reproduced when identical excitations collide at the QPC [7,16,30,39,40]. Contacts 1 and 4 are thus driven with identical periodic voltage signals, the only difference being a tunable time delay t D between them. Here, we will consider a broader class of possible HOM collisions, where the left and right contact are driven with an identical shape in time but different amplitudes, namely

V L (t) = q L q R V R (t + t D )
. This choice allows us to elaborate on previous results regarding symmetric HOM interferometry, while also discussing some original results about generalized asymmetric collisions.

Following the same procedure we used for the HBT signal we immediately get

S HOM (q R , q L , t D ) = -2(e * ) 2 |λ| 2 +∞ l=-∞ |p l (q R , q L , t D )| 2 × × P2ν [(l + q R -q L )ω] + P2ν [-(l + q R -q L )ω] , (11)
with coefficients pl (q R , q

L , t D ) = m p l+m (q R )p * m (q L )e imωt D .
It is customary to normalize the HOM noise with respect to the value expected for the random partitioning of a single source, namely the HBT signal [30,40]. For this purpose we define the ratio

R(q R , q L , t D ) = S HOM (q R , q L , t D ) -S (0) S HBT R (q R ) + S HBT L (q L ) -2S (0) . ( 12
)
Notice that we have subtracted the equilibrium noise S (0) = -4(e * ) 2 |λ| 2 P2ν (0) both to the numerator and the denominator in equation (12) [19,40].

Before getting into the details of the results, it is worth noticing that our HOM interferometer differs substantially from a gauge-transformed HBT setup. Indeed, there is no gauge transformation that is able to map the equation of motion for the HOM configuration into an effective HBT with V R (t) = V R (t) -V L (t) and V L (t) = 0. However, the HOM noise in equation ( 11) looks like the HBT contribution due to the peculiarity of point-like tunneling [41]. This is linked to the fact that the tunneling amplitude for point-like tunneling between opposite edge states is energy independent in the Landauer-Büttiker picture of quantum transport. Any similarity between our HOM setup and an effective HBT would vanish for more complicated tunneling geometries such as multiple QPC or extended contacts [42,43], where the transmission amplitude acquires a dependence upon energy [44,45].

Results

Symmetric HOM collisions

We first look at the HOM ratio R in the case of integer filling factor (ν = 1) and identical drives (q R = q L = q). This case is reported in panels a) and b) of Figure q R(q R = q, q L = q, t D ) q R(q R = q, q L = q, t D ) 2. Here, exact results based on the scattering matrix formalism are available in the literature, both on the theoretical and experimental sides [6,7]. We observe that the noise is totally suppressed when colliding packets arrive simultaneously at the QPC (t D = 0). This is consistent with the anti-bunching effect expected for identical fermionic particles colliding at the QPC, which forces two identical fermions to leave the interferometer on opposite output arms, thus causing a drop in the fluctuations (the so-called Pauli dip) [7,30,40]. However, we point out that a complete suppression of the noise is observed also for non integer values of q, i.e. when colliding packets are formed by several particle-hole pairs instead of a single electron-like excitation [17,18]. We also note that brighter regions in Figure 2a occur in correspondence of integer values of q. This is a signature of the robustness of Levitons at integer charge, namely a reduced contribution of particle-hole pairs. Finally, the qualitative behavior of R as a function of time delay is almost unaffected by variations of q and shows a unique dip. Things are dramatically different in the fractional regime, where new features linked to the strongly-correlated FQH phase come into play [21]. In Figure 2c we report the behavior of the HOM ratio for filling factor ν = 1/3. One still observes a complete dip in the ratio R when a null time delay is tuned between colliding packets, despite the presence of anyonic quasiparticles in the system. Once again, we ascribe this particular behavior to the geometry of our interferometer, which consists of a single QPC connecting the upper and lower edge. Such a tunneling geometry does not allow for closed-loop trajectories of one Laughlin quasiparticle around another, thus preventing any possible effect due to fractional statistics [46]. At the same time, we note that the simple structure consisting of one single dip at t D = 0 is replaced by a much richer phenomenology. Indeed, Figure 2c shows the emergence of sub-dips for values q > 1. This is even more evident in Figure 2d, where we have isolated the behavior of R at integer values of q. We note that the number of these sub-dips is given by 2q -2.

Such an unexpected behavior suggests that the injected wave-packet reorganizes into several sub-peaks at the output of the QPC. Indeed, due to the presence of a single edge channel, one cannot interpret this structure as a consequence of a spincharge separation mechanism [47,48,49]. Moreover, let us notice that there is no coupling with bulk degrees of freedom in the model presented in Section 2. The subdip structure cannot depend on the distance d between the injector and the QPC as well (see Figure 1), since such a parameter only manifests itself as an additional constant phase shift of the coefficients p l and pl , which is clearly washed out in our final results (see equations ( 9) and ( 11)). We can thus conclude that there is strong evidence of a rearrangement of the injected charge density outgoing from the QPC. This phenomenology can be explained in terms of a crystallization of q-Levitons occurring in the time domain, differently from what is typically discussed in condensed matter systems, and induced by the strong correlation characterizing the FQH state [21].

Two additional features are worth noticing. First, the curves for q = 1 at ν = 1 and ν = 1/3 (black lines in Figures 2b and2d) are identical, which means that the Lorentzian peak carrying exactly one electron generates the same HOM ratio at both integer and fractional filling factor. Indeed, it was shown analytically in [19] that a remarkable factorization in the dependence on temperature and filling factor arises for this particular voltage drive, leading to a universal form of R(q R = 1, q L = 1, t D ) for each value of θ and ν. This universality is consistent with the previously discussed picture, showing the impossibility for a single electronic wavepacket to crystallize. Second, the uniqueness of integer Lorentzian pulses is even more evident in the fractional regime when looking at the behavior for long delays between colliding packets (i.e. t D /T ≈ ±0.5). We observe that generic values of q lead to a rather low HOM noise compared with the HBT contribution, which means that composite objects consisting of several particle-hole pairs are colliding at the beam splitter. However, integer Lorentzian pulses generate bright horizontal lines in Figure 2c. This is another striking confirmation of the fact that q-Levitons are minimal excitation states also in the FQH regime [19,20]. Moreover, this is a signature of the fact that, differently from recently discussed periodic sources based on driven antidots [50], a train of Lorentzian voltage pulse cannot be used as an on-demand source of fractionally charged quasiparticles [19].

Asymmetric collisions

We now examine what happens when wave-packets carrying different charges are injected in terminals 1 and 4 and sent to the collider. For simplicity, we inject integer Lorentzian pulses with q R = 1 on the upper right-moving edge, while the opposite contact is driven with a generic Lorentzian drive with tunable q L and t D . Starting from the ν = 1 case, we observe that the total suppression of HOM noise is achieved only for q L = q R = 1, as expected. However, different values of q L still generate a partial dip in the noise whose value is given by

R(q R , q L , t D = 0) = q L -q R q L + q R , (13)
as one can infer by looking at Figure 3a. For instance, by fixing the value of q L we obtain the four curves shown in Figure 3b, which clearly demonstrate that the anti-bunching effect is not complete -although still present -when excitations with different charge collide at the QPC.

In the FQH regime the values of the ratio R at zero delay is slightly modified, while two unexpected sub-dips appear (see Figures 3c and3d). Despite being extremely well visible in the case of integer q L , they are almost totally washed out when non-integer pulses are considered. This is again due to the fact that only an integer Lorentzian drive generates clean states in the conductor. Moreover, the distance between these two side-dips progressively increases by increasing q L . As deduced previously from Figure 2, the q L -Levitons, with q L > 1, are re-arranged in a crystallized structure, while the single Leviton wave-packet maintains a trivial pattern. Therefore, the latter is not suitable to resolve the more complicated crystal of q L -Levitons in the type of experiment presented in Figure 3. As a consequence, the number of side-dips is totally unrelated to the specific value of q L , in contrast to the configuration with identical drives. A better resolution would require the injection of a very narrow Leviton wavepacket from terminal 1, ideally reaching the limit of a delta-like pulse: anyway, the generation of such a narrow pulse is beyond the state-of-art technology. Another possibility would be to increase the value of q R , thus characterizing the crystal of q L -Levitons with a comparably complicated wave-packet: in this case, a total number of 2q R side-dips would appear.

Wave-packet approach

The current-current correlation generated in a collisional HOM experiment can be equally expressed in terms of the overlap between wavefunctions describing the Levitons impinging at the QPC. In this insightful formalism, a periodic train of q-Levitons is described by the following set of wavefunctions [36]

ϕ k (t) = sinh (2πη) 2 sin k-1 π t T -iη sin k π t T + iη , k = 1, ..., q. (14)
They generalize the set of single-electron wave functions introduced for the Lorentzian pulse [35,51,52], and form a complete orthonormal basis, thus satisfying the condition

T 0 dt T ϕ k (t)ϕ * k (t) = δ k,k .
This set of wavefunctions is usually related to quantities called quasiparticle and quasihole correlation functions, defined as

G (qp) R/L (t , t) = Ψ † R/L (0, t )Ψ R/L (0, t) = 1 2πa e -ie * t t dτ V R/L (τ ) e 2νG(t -t) , (15)
G (qh) R/L (t , t) = Ψ R/L (0, t )Ψ † R/L (0, t) = 1 2πa e ie * t t dτ V R/L (τ ) e 2νG(t -t) . (16)
We retained only temporal variables in equation ( 15), since the space-and timedependent field Ψ R/L (x, t) evolves only through the combination x ∓ vt, due to the chirality of Laughlin edge states. Contrarily to ordinary optics, this correlator does not vanish even at equilibrium, due to the different nature of the ground state in fermionic systems. Since one is mainly interested in characterizing deviations from equilibrium introduced by the voltage drives, it is useful to define the excess quasiparticle correlators as

∆G (qp) R/L (t , t) = G (qp) R/L (t , t) -G (0) R/L (t -t), (17)
where

G (0)
R/L (tt) = 1 2πa e 2νG(t -t) corresponds to the correlator when V L = V R = 0. Their expressions in terms of functions in equation (14) read

∆G (qp) R/L (t , t) = -2iG 0 (t -t) sin π(t -t) T q R/L k=1 ϕ k,R (t)ϕ * k,L (t ), (18)
where ϕ k,R (t) = ϕ k (t) and ϕ k,L (t) = ϕ k (t + t D ) and

G 0 (τ ) = 1 2πa πk B θτ sinh (πk B θτ ) 1 + i vτ a ν . (19)
It is instructive to note that, in the ordinary metallic case (ν = 1) and in the limit of infinite period, equation (18) reduces to the so called single-electron coherence function, a crucial tool in the framework of electron quantum optics [35,51]. Excess correlators for quasiholes can be defined similarly as

∆G (qh) R/L (t -t) = G (qh) R/L (t , t) - G 0 (t -t), yielding ∆G (qh) R = 2iG 0 (t -t) sin π t -t T q R/L k=1 ϕ * k,R (t)ϕ k,L (t ). ( 20
)
By using equations (15) and the definitions of excess correlators for quasiparticles and quasiholes into Eqs. ( 8) and ( 12), the ratio R can be expressed directly as (see Appendix A for more details)

R = 1 - q R k=1 q L k =1 +∞ p=1 +∞ p =1 w k pp g k p (t D )g * k p (t D ) 1 2 (v q R + v q L ) , (21)
with

g kp (t D ) = T 0 dt T ϕ k (t + t D )ϕ * p (t)
and where the coefficients w k pp and v q,r , which encode all the dependence on temperature and interaction, read

w k pp = T 0 dt T +∞ -∞ dτ ϕ k (t)ϕ * k (t + τ )ϕ p (t)ϕ * p (t + τ ) sin 2 πτ T G 2 0 (τ ), (22)
v q R = qr k=1 +∞ -∞ dτ sin πτ T g * kk (τ )G 2 0 (τ ). (23)
Equation ( 21) clearly shows the anti-bunching effect which arises when singleelectron excitations as Levitons collide at a QPC. As the overlap between wavefunctions increases the value of R is reduced and eventually vanishes when the overlap between the two states is perfect.

Finally, starting from equation ( 21) one also gets a simplified formula for the integer case at zero temperature, which reads

R = 1 - 2 q L + q R q R k=1 q L k =1 |g k k (t D )| 2 , (24)
in accordance with previous results [30,36].

Conclusions and perspectives

In this work, we thoroughly reviewed the issue of HOM collisions in a FQH system. This can be done in a multi-terminal setup in presence of a QPC, which can be thought as an effective beam splitter for electron quantum optics experiments. By summarizing and further investigating previous results in the context of EQO, we have shown that synchronized collisions of identical excitations at the QPC always generate a total dip in the crossed shot noise, both at integer and fractional filling. However, the scenario for ν = 1/3 is enriched by the presence of several sub-dips in the HOM ratio, which are consistent with the crystallization of the wave-packet outgoing from the QPC. We have also explored a more generic HOM configuration involving Lorentzian pulses with different amplitudes, namely wave-packets carrying a different number of electrons on the opposite sides of the QPC. Here, the suppression of the shot noise is not complete, and two progressively separated side dips emerge in the FQH regime. The generation and control of Levitons and, more in general, of single to few electronic excitations in one-dimensional ballistic conductors have attracted a huge interest in recent years and is actually at the forefront of research in condensed matter physics, and may lead to fascinating applications in quantum communication and quantum information. However, several important issues, including for instance the role of decoherence [49,53] or screening [54] in the dynamics of single-electron excitations, are yet to be fully understood and will certainly deserve a lot of attention in years to come.

Further interesting developments could also involve the investigation of pseudorandom emission of Levitons, as recently proposed for the free fermion case [55]. Besides closing the conceptual gap between time-periodic injection and individual wave-packet emission -Holy Grail of EQO [56] -this newly proposed emission protocol opens important perspectives for the FQH case, as it promises to strongly magnify the side dip features discussed in the present text.

As a final remark, let us notice that first experimental results about photo-assisted noise in the FQH regime are starting to be available [57]. This is of particular importance in a field like mesoscopic physics, where theory and experiments have always fed each other on the pursuit of new exciting discoveries. We believe that this is only the first step towards a full experimental EQO at fractional filling factor.

Eqs. ( 15) and ( 16 By inserting equations ( 17) and (20) into the above equations, these two contributions can be further expressed in terms of the wavefunctions {ϕ k } as

N = T 0 dt T +∞ -∞ dτ q R k=1 q L k =1 ϕ k (t)

Fig. 1 .

 1 Fig. 1. Four-terminal setup for Hong-Ou-Mandel interferometry in the FQH regime. Periodic trains of Levitons (blue peaks) are injected through input terminals 1 and 4, while contact 2 and 3 are the output terminals where the current and the noise are measured.

Fig. 2 .

 2 Fig.2. HOM ratio R for collisions of identical excitations as a function of the delay tD (in units of the period T ) and the number q of injected charges. Panels a) and b) refer to the case of integer filling factor, while panels c) and d) correspond to fractional filling ν = 1/3. Curves in b) and d) are extracted respectively from the density plots in a) and c) by cutting at q = 1, 2, 3 and 4. The temperature is θ = 0.01 ω, the dimensionless width of the pulses is η = 0.04 and the high energy cut-off is v/a = 10 ω.

Fig. 3 .

 3 Fig.3. HOM ratio R for collisions of integer (qR = 1) generic (qL = q) pulses as a function of the delay tD and the charge q injected in the left-moving edge. Panels a) and b) refer to the case of integer filling factor, while panels c) and d) correspond to fractional filling ν = 1/3. Curves in b) and d) are extracted respectively from a) and c) at q = 1, 2, 3 and 4. The temperature is θ = 0.01 ω, the dimensionless width of the pulses is η = 0.04 and the high energy cut-off is v/a = 10ω.

) as 1 SLR

 1 HOM = (νe) 2 |Λ| 2 (t + τ, t) + G (qh) R (t + τ, t)G (qp) L (t + τ, t) . (A.1) It is interesting to recast the current noise directly in terms of the excess correlation functions ∆G (qp) R/L and ∆G (qh) R/L , which encode all the information about Levitons. By using the relations G (qp) R/L = ∆G (qp) R/L + G 0 and G (qh) R/L = ∆G (qh) R/L + G 0 (we omitted here the time dependence for notational convenience), one finds S HOM = -S (0) + S HBT R (t + τ, t)∆G

L

  (t + τ, t) + ∆G

R

  (t + τ, t)∆G

L

  (t + τ, t) ,(A.2)where we have used the expressions of the equilibrium noise S (0) and the HBT noise S HBT R,L in terms of Green's functions, which readS (0) = 2(νe) (t + τ, t) + G (qh) R/L (t + τ, t) G 0 (τ ). (A.4)By using the previous results the HOM ratio becomes R

  τ, t) + ∆G (qh) r (t + τ, t) G 0 (τ ).(A.7)

  ϕ * k (t + τ )ϕ * k (t + t D )ϕ k (t + t D + τ ) + h.c. ×Finally, by introducing the overlap integrals of wavefunctions g kp (t D ) =

		× sin 2 πτ T		G 2 0 (τ ),		(A.8)
	D =	1 2 r=R,L	0	T	dt T	+∞ -∞	dτ i	qr k=1	ϕ k (t)ϕ * k (t + τ ) + h.c. sin	πτ T	G 2 0 (τ ). (A.9)
	t D )ϕ * p (t), the ratio acquires the form given in the main text	T 0	dt T ϕ k (t +
					R = 1 -	q R k=1	q L k =1	+∞ p=1	+∞ p =1 1 2 (v q R + v q L ) w k pp g k p (t D )g * k p (t D )	,	(A.10)
	where the coefficients are
	w k pp =	0	T	dt T	-∞ +∞ dτ ϕ k (t)ϕ * k (t + τ )ϕ p (t)ϕ * p (t + τ ) sin 2 πτ T	G 2 0 (τ ),	(A.11)
	v qr =	qr k=1 -∞ +∞ dτ sin	πτ T	g * kk (τ )G 2 0 (τ ).	(A.12)

For notational convenience, from now on we have omitted the functional dependence of noise contributions introduced in the main text.

Acknowledgements

We are grateful to E. Bocquillon, G. Fève and D. C. Glattli for useful discussions. L.V. acknowledges support from CNR SPIN through Seed project "Electron quantum optics with quantized energy packets". This work was granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (Grant No. ANR-10-EQPX-29-01). It has been carried out in the framework of project "1shot reloaded" (Grant No. ANR-14-CE32-0017) and benefited from the support of the Labex ARCHIMEDE (Grant No. ANR-11-LABX-0033), all funded by the "investissements d'avenir" French Government program managed by the French National Research Agency (ANR). The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University -A*MIDEX, a French Investissements d'Avenir programme.

Authors contributions

All the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript.

A Calculation of the HOM ratio R

In this Appendix, we express the ratio defined in Eq. ( 12) in terms of the set of wavefunctions given in Eq. ( 14). We can start by expressing the HOM current noise in terms of Green's functions in