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Introduction

I would like to give a few examples of the use of digital audio effects in my computer music pieces. The effects used include mixing with delays and transpositions, reverberation, 3D-sound, use of tuned resonant filters, time-frequency scaling, cross-synthesis. Rather than emphasizing on the technical implementation, I shall emphasize the musical purpose of the effects. Thus, these effects will not be presented in any systematic order.

Simple time-domain processing (Inharmonique, Songes)

In many of my computer music works, and specially the earliest ones (Little Boy, Mutations, Dialogues, Passages, Contours), I have only used sound synthesis. However, I began to introduce some computer-processed sounds in pieces such as in Inharmonique, Songes, and L'autre face. In these pieces, I only used fairly simple processing, mostly confined in the time domain.

In Inharmonique, a piece for soprano and computergenerated tape, realized in 1977 at IRCAM, I generated the tape entirely with the MUSICV sound synthesis program: most of the sounds are synthetic. However, I recorded a melodic fragment sung by the soprano Irène Jarsky, and I processed it with MUSICV to spatialize it. MUSICV was originally designed by Max Mathews so as to enable and to perform the processing of digitized sounds as well as synthesis, but a special unit generator had to be added by John Gardner to actually input the sound from a sound file into the program. The sound file could be read with increments different from unity, which yields sound transposition. In Inharmonique, no transposition was used, only a mere echolike mixing of the recorded melody to itself (this can be heard between 13mn42s and 13mn54s on track # 5 on the INA C1003 recording). This gives the impression that the voice spreads into space.

In Songes, realized at IRCAM in 1979, I recorded simple melodic motives played separately by soloists of the Ensemble Intercontemporain (violin, viola, cello, flute, oboe, clarinet, horn, harp). I transposed some recordings by two semi-tones. I used MUSICV as a (non real-time) mixing program, so that the beginning of the piece sounds like a chamber ensemble -an illusory one, insofar as the performers never actually played together (this can be heard on track 1 of the Wergo 2013-50 CD recording, between 0 and 37s and between 48s and 1mn28s). I also used later in the piece some "spatialized" trills or melodies that seem to run across space, using the technique developed by John Chowning, except for the Doppler effect (this can be heard on track 1 of the Wergo 2013-50 CD recording: moving trills at 1mn29s, flying flute at 1mn43s, trajectories after 7mn05s, suggesting imaginary birds, with glissandi possibly hinting Doppler-like frequency shifts). The processing is intended to take the listener from an instrumental "reality" into a dream-like world, where material constraints dissolve into fluid textures and flying sounds.

Processing to merge natural and synthetic sounds (Sud )

In 1983, François Bayle, then director of the Groupe de Recherches Musicales of Paris (INA-GRM), invited me to

Examples of the Musical Use of Digital Audio Effects

Jean-Claude Risset Laboratoire de Mécanique et d'Acoustique, CNRS, Marseille, France compose a piece. GRM was founded by Pierre Schaeffer, the inventor of musique concrète. My own activity had mostly been in dealing with synthetic sounds: so I decided to build this piece around the encounter of synthetic and natural sounds. These initially markedly different sound worlds would undergo transformations making them merge gradually. At GRM, the late Benedict Mailliard had developed efficient programs for processing sounds, and these programs, together with Mathews's MUSICV -often used internally in Mailliard's routines -afforded possibilities to transform sounds in ways that could be appropriate to my project of merging together the world of natural and synthetic sounds.

I assembled a few soundscapes recorded along the shore and in the country near Marseille: the ebb and flow of the tide; roars of a mistral tempest; waves beating against the rocks; remote birds and voices; insect buzzes; seeds cracking in the sun; the stridulation of crickets at night (the name of the piece is Sud). I also collected a few other sound sequences: a short cello note; the banging sound from wood chimes and metal chimes; a fast arpeggio gesture played on the piano; short sequences of tropical and equatorial birds songs from a recording found in Singapore. Some of the above soundscapes appear in the piece as "phonographies," akin to photographs, with only minor modifications or no modification at all, for instance the sea at the beginning, or the insect sounds toward the end of the first movement. Fifteen rather short recordings, lasting between 1 and 20 seconds, were selected as a kind of sonic kernel: these were subsequently submitted to several generations of processing, as described below.

In Marseille, I also generated some synthetic sounds with the MUSICV program. I chose a defective major-minor pitch scale, which does not exactly octaviate (G natural -B -E -F sharp -G sharp -B -E -F sharp -G natural -B -E -F sharp -G sharp. . . .), as the trade mark of synthetic sounds in the piece. Most of the synthetic sounds follow this scale. Although they were not generated in real-time, these synthetic sounds were conceived as gestures. For instance, I have specified ample arpeggioes in terms of curves abstracted from wave contours and quantified to the steps of the majorminor scale. I also generated tones which appear to fly in open space like birds: sine waves modulated in frequency by complex, supple, almost visible contours (I, 5mn50s, 6mn29s, 8mn50s; III, 56s, 4mn41s, 5mn45s). In contrast, the natural sounds I gathered -for instance, songs of birds or crickets -originally comprise no pitches intended by the composer; they may even have no distinctive pitch, such as sea sounds.

Next I modified and proliferated my original soundsmost recorded, some synthetic -by submitting them to the transformations implemented by Mailliard in studio 123 at GRM. I hereafter describe various kinds of transformations and the way I used them for my project.

-First, sound recording is akin to photography: one must select the view point and the focus. Using sharp microphones as microscopes, I zoomed into the insect world to get close-ups of humming bees and cracking seeds in the summer heat (I, 5mn46s to 6mn and 8mn26s to 8mn59s).

One also must select among the recorded material. Somewhat arbitrarily, I choose a found object -the sound of a wave, recorded on a certain morning in June, 1984 -to serve as a seminal cell used throughout the piece.

-With the GRM program BRASSAGE, one can shuffle sonic elements, i.e., control the division of sounds into short time fragments and the reassembling of these fragments in various ways. I used this "sound shuffling" to perform sonic developments from short sounds, such as wood chimes (I, 3mn20s to 4mn36s) and to produce a "stuttering" piano, further processed by ring modulation (I, 4mn30s, 5mn; III, 2mn16s to 2mn38s). Starting from "found objects" such as birds songs, I rearranged them compositionally to obtain first a pointillistic rendering, then a stretto-like episode (I, 1mn42s to 2mn47s).

-A few sounds were spatialized. The piece is originally in four tracks, but it is often treated as dual stereo -back and front. Spatialization was performed on recording of birds to give the illusion of a controlled and somewhat (purposely) artificial movement of rotation. Thirty seconds after the beginning of the piece, a relatively slow movement is heard behind the listener; then higher tones seem to rotate faster in front. Spatialization was also performed in the course of synthesis on "bird-like" tones modulated in frequency by supple and complex curves (I, 5mn50s, 6mn29s, 8mn50s; III, 56s, 4mn41s, 5mn45s).

-I used reverberation to smooth certain processed sounds (for instance, shuffled wood chimes), but I avoided it for recorded soundscapes, so as to suggest to the listener the actual spaces when they were captured. (This can be heard specially in the first three minutes or at the end of the first movement). I did add a little background reverb to reinforce the depth in illusory rotations.

-By mixing the sound to itself with a short time delay, I induced a pitch coloration: the pitch heard corresponds to a frequency which is the inverse of the delay. One can increase the degree of coloration by adding more copies shifted in time. I used this at the beginning of the third movement to introduce the color of a G sharp -an important pitch in the major-minor scale permeating the piece (III, 0s to 39s).

-By phasing -mixing a sound with transpositions of itself with a minute frequency difference (say a twentieth of a Hertz), one can turn steady periodic tones into a pattern where the harmonics of the tone wax and wane at different rates, proportional to the rank of the harmonic. This happens because the small frequency differences cause phase differences to slowly go from 0 to 360°: thus beats appear, with a period of 20 seconds for the first harmonic, but of only 2 seconds for the 10th harmonic, and of 20/n seconds for the nth harmonic. This produces an unexpected effect. It requires the precision of digital processing, and it is reinforced by the addition of more transpositions with the same frequency difference. I have used this trick before to animate synthetic sounds in a number of pieces: Little Boy, Inharmonique, Songes. Through phasing, I produced spectral scanning of cello tones, (III, 39s, 1mn24s).

I tried to submit non-periodic sounds to phasing: I applied the process to sea roars recorded on a day of a strong North wind -"a mistral:" heavy sounds with wide spectra. The result came to me as a surprise: added to the sea sounds was a powerful glissando downward, starting shrill and deepening in spectrum until it finally got lost into low depths -not to reappear again. The explanation is not simple, but the effect is quite strong -a "found effect," just as I used sonic "found objects." My encounter with it made me decide to change my plan for the piece: the second movement would be a metaphoric tempest, scanned by these impressive descents, and with a catastrophic ending -a metaphor of a wreckage.

-Applying time reversal permitted me to obtain ascending curves from the preceding descending ones: an upward scanning cannot be obtained in real-time, since the descent corresponds to the entropic loss of order, while an ascent would be a statistically impossible route from chaos to coherence.

-I also resorted to mixing with enveloping and tranposition. I used this to shape synthetic inharmonic tones by energy fluxes of the wind and the waves of the sea (for instance, at the beginning of the second movement). I used stylized amplitude contours extracted from actual dynamic curves found in recorded sounds, with frequency transposition intervals also calculated in accordance to these dynamic curves, but quantized to the steps of the major-minor scale (III, 3mn14s to 3mn30s and 4mn21s).

I also used transposition to exert control over certain pitches of the natural sounds. For instance, shortly before the end of the first movement, one can hear cricket-like chirps on the following pitches: G sharp, G sharp, F sharp, E, G natural, B. The crickets I recorded stridulated on F sharp: transposition made them follow the steps of the chosen scale (I, 9mn10s). In the third movement, low and slightly coloured sea sounds are presented successively so as to be heard as G natural, B, E, F sharp, G sharp (III, 2mn51s).

filtering. I extensively resorted to filters implemented by Mailliard. I used about 25 resonant filters tuned to the steps of the minor-major scale. Their impulse response increases when the bandwidth narrows. Very sharp filters are in effect oscillators: an excitation will make them resonate at the frequencies they are tuned to, and the long resonance will completely blur the temporal behavior of the excitation. One can view the array of resonant filters as a huge piano where the only raised dampers correspond to the notes of the majorminor scale. On the contrary, wider filters will do better in preserving the original identity of the excitation, but they will be weaker in the imposition of the scale.

As the reader may have gathered, real-world sounds and synthetic sounds are made to come closer principally through two kinds of transformation (in addition to cross synthesis described below): (1) external morphologies from real-world sounds are imprinted onto synthetic sounds; (2) the majorminor pitch scale is gradually imposed to real-world sounds.

In addition to transposition, the use of resonant filters tuned to the steps of the minor-major scale played a major role in the latter operation.

I applied such resonant filtering to various sounds: for instance, the seminal wave contours which opens the piece can be heard filtered in the third movement, a kind of liquidly singing and ringing to the scale. In the first part of the third movement, birds songs and caws are followed by their resonance. Later in this movement, I let a "bird raga" flourish by filtering a selected bird song through a set of relatively wide resonant filters: then other pitches than those of the scale can come through, which are heard as a kind of appogiatura preceding the E (III, 4mn06s to 6mn19s).

In the second movement of Sud, the pseudo bell-buoy and the sea sounds resonate in the manner of an organ. I obtained this by applying some of the above-mentioned transformations: resonant filtering creating a harmonic skeleton corresponding to the steps of the scale, and mixing several octave transpositions of motives (II, 36s, 57s, 3mn50s to 5mn).

modulation or cross-synthesis. Starting from two sounds, the process of modulation or cross-synthesis produces an hybrid which combines certain characteristics of both original sounds. In the digital domain, ring-modulation is a simple multiplication of waves: from sine waves of frequencies f1 and f2, it produces additional components of frequencies f1 + f2 and f1 -f2. I have modulated piano arpeggioes to make them inharmonic; they were then submitted to resonant filtering (III, 2mn16s to 2mn38s).

Linear predictive coding (LPC) is more elaborate. LPC analysis separates a sound into two parts: one primarily characterizes the excitation, the varying, dynamic, agogic aspects, whereas the other determines the response, the fixed, spectral, structural aspects. I thus crossbred sounds of different origins -birds with metallic plate or cello sounds; sea waves with wood chimes or synthetic "harpsichords." The former yields intriguing "chimeras" -bird-gong, bird-cello (III, 2mn04s, 5mn24s). The latter produces "waves" of wood chimes, waves of harpsichord (III, 1mn43s), thus imprinting on recognizible sounds the powerful energy profile of other sounds. (Note that the effect of a cross synthesis of this nature is much stronger that the mere imposition of an amplitude envelope taken from one sound onto another sound). My purpose was mainly to endow certain sounds with energy fluxes from other sounds, injecting specific qualities without necessarily confusing the identities -like Cézanne wanted to merge "feminine curves with hill shoulders."

When one performs several generations of crosssynthesis, spectra can become very thick and dark. I resorted to the hybridation of sea sounds with wood and metal chime sounds for the catastrophical ending of the second movement. Even though the origin of the sounds gets confused, one still gets convincing suggestions of energy: for instance, the excitation by metallic percussions is transformed, but its effect is still felt as powerful shocks (this can be heard at the end of the second movement).

As I already mentioned, I had selected a found objectthe ebb and flow of a wave on rough beach sand -as a seminal cell used throughout the piece. It appears at the beginning, and it is later repeated in different forms; its dynamic flux permeates the whole piece. It is imprinted onto other sounds, real or synthetic. It is even reflected in the form of the work, where many structures are build around the idea of the wave.

The formal scenario tells the story of the encounter between natural and synthetic sounds, first presented separately, and the growth of their mutual interactions -merging nature and artifice. The defective major-minor scale initially characterizes the artificial sounds, but it is gradually woven into the natural ones. The energy contours from natural sounds progressively print their mark over the synthetic ones. The compositional logic is not one of pitch: the pitch structure does not evolve; it is merely animated by dynamic fluxes.

Processing in pieces for instruments and tape

In Voilements, for tenor saxophone and tape, the soloist plays against his sonic shadow. The tape contains mostly saxophone motives recorded by Daniel Kientzy and processed digitally. The tape first echoes the soloist, multiplying his sound by means of delayed and transposed echoes -here harmonizer-like transformations were performed with the SYTER real-time audio-processor (cf. for instance on CD INA 2000, track 7, 1mn22s, 1mn45s, or 3mn49s to 4mn48s). This rivalry becomes more and more troublesome for the player: it alters the way of playing, it warps it, as a wheel which does not go round (the word "Voilements" alludes to a veil or a sail, but is also means "buckles" or "warps"). The equal temperament tuning is eroded, the tension increases, up to a fight and a genuine crisis: melodic lines get twisted into loops (track 8, 12s). Then, as if there were a zoom backwards, the perspective changes (track 8, 1mn05s); the relation between the soloist and the tape is altered: the timbral range of the tape sounds expands, and the tape becomes a remote and passive background for the gestures of the soloist.

Similar processes are used in Lurai, for celtic harp and tape, dedicated to Denise Mégevand, and Attracteurs Etranges, for clarinet and tape. The tape for this latter piece mostly comprises of clarinet sounds recorded by Michel Portal and processed by the computer in several waysnotably to achieve spatialization and time stretching. The piece stages dialogues and close encounters between the live clarinet player and the tape sounds. It tries to illustrate metaphorically the idea of attractors as geometrical descriptors of dynamic systems: punctual attractors, which correspond to equilibrium positions, and strange attractors -with a fractal structure -which correspond to chaotic systems with a destiny highly sensitive to initial conditions. The wind instruments occasionnally resort to turbulent flows and multiphonics, which are instances of chaos. The unfolding of the piece uses polar pitches, cycles or quasi-cycles and bifurcations.

Processed voice: Invisible

The title of the piece refers to Italo Calvino's book Invisible Cities. The principal sonic material is the recorded voice of Irène Jarsky speaking or singing texts -mostly French translations of quotes by the Tao poet and philosopher Tchouangtseu, but also fragments of poems by Basho ¯, Dante, Heine, Longfellow, and others, read in their original languages. I also used sounds of natural origin: rain, thunder, wind, and birds. These natural sounds often intervene in a simple way: but they are also transformed and combined with other sounds, natural or synthetic.

I used Daniel Arfib's program sound mutations to perform intimate modifications of sounds. Thus, in order to echo Tchouang-tseu's evocation of breath as void capable of modeling itself to external objects, I have performed cross synthesis of wind sounds with speech, replacing in effect the vocal chords of the soprano by the blowing of the wind (7mn36s to 7mn45s). I have also slowed down the sound of the word "langsam" (German for "slow") without frequency transposition, to echo Heine's evocation of the sluggishness of time (9mn43s to 9mn56s; cf. also at 30s and 8mn28s the stretching of "die Zeit"). The rate of time expansion can be higher than 100: however the consonants are not expanded, so that the spoken words remain intelligible. The name "Chloé" is also slowed down at a variable rate, reckoned to follow a composed melodic contour (10mn40s). I performed similar transformations on the spoken expression "une sensation de vide" (a sensation of void or vacuum) to get a musical development similar to variations on a theme (5mn15s to 6mn20s). To illustrate "la voix et le son s'harmonisent" (voice and sound harmonize each other), I imprinted the same harmonic pattern onto both synthetic "bells" and the soprano's voice (11mn05s to 11mn50s). In the previous cases, computer processing permits to preserve the identity of the sound origin while transfiguring its time scale, hence its dynamic behavior, as well as its inner harmony. Here composition specifies time within sound objects, rather than setting sound objects in time.

I synthesized bell-like tones to compose them as minor chords, as normal bells, but also as major chords -to echo a poetic suggestion by Tchouang-tseu: zephyr produces a minor harmony, squalls a major harmony (6mn50s to 7mn10s). In several sections, I also controlled the synthesic acoustic environment (3mn29s to 3mn40s). Sonic transformations bring the voice in fictitious, virtual acoustics, which do not come from the bouncing of sound waves against solid obstacles: but of course we interpret these acoustics by assuming a space where sounds travel (2mn25s to 2mn32s).

Specifying the acoustics and their transformations is part of the composition; the space which the listener is lead to imagine has connotations that must be taken into account.

Resorting to digital synthesis and processing, therefore permits to implement immaterial processes, to produce illusory voices, to set the sounds in imagined spaces similar to Calvino's invisible cities -even though Calvino's imagination is more agile and varied than the sound simulacra I was able to produce.

Electroacoustic pieces: Avel, Elementa

In my latest pieces, Avel (1997) and Elementa (1998), realized at GRM, Paris, I used most of the processes described above. Avel is a study about the wind. The piece begins with a "phonography" (a sound photography): then it evokes eolian sounds -flute, harp -and goes up and down while stylizing the whistles, whines and howls of the wind. Swaying harmonies are activated by these energy fluxes.

Elementa was completed in 1998 for the fiftieth anniversary of "musique concrète." The piece includes sounds synthesized with the MUSICV program and the Synclavier synthesizer, but the sound material mostly consists of recordings. The nature of the sound sources is not hidden: the evocation of the four elements -Terra, Aqua, Aer, Focus (earth, water, air, fire) -takes advantage of recognizable sound recordings. The composition relies upon the connotations and the symbolic implications of these recordings.

Elementa was realized using Pro Tools and GRM Tools. GRM Tools serve as "plugs-in" for the Pro Tools program, permitting to control audio effects such as time shuffling, spatialization or filtering. For instance, in the second movement, Focus, recordings of fire sounds are spatialized to suggest elaborate movements of the fire around the listeners; also comb filters react to the explosions in the sound to generate complex percussive rhythmical patterns that mirror the fire's internal rhythm. The recorded sounds are weaved into figures, phrases, developments and sections, together with synthetic figures with related "gaits": a compositional processing, careful to preserve the autonomy of organic sound objects and their dynamics of flux, duration and energy.

Conclusions

It seems to me that the increasing refinement of available audio effects bridges the gap between the resources of synthesis, with the precise and ductile control it affords, and those of processing, with its inexhaustible variety of content.

Although I fully realize the interest of real-time and reactivity, I must stress that most of my compositional work in this field implies freeing the elaboration of the pieces from the constraints of real-time.


Claude Depollier

Zine El Abidine Fellah

Mohamed Fellah
email: fellah1@yahoo.com

Propagation of Transient Acoustic Waves in Layered Porous Media: Fractional Equations for the Scattering Operators

Keywords:

Acoustic waves scattering from a rigid air-saturated porous medium is studied in the time domain. The medium is one dimensional and its physical parameters are depth dependent, i.e., the medium is layered. The loss and dispersion properties of the medium are due to the fluid-structure interaction induced by wave propagation. They are modeled by generalized susceptibility functions which express the memory effects in the propagation process. The wave equation is then a fractional telegraphist's equation. The two relevant quantities are the scattering operators -transmission and reflection operators -which give the scattered fields from the incident wave. They are obtained from Volterra equations which are fractional equations for the scattering operators.

Introduction

When a sound wave travels in a rigid air-saturated porous medium, the dispersive effects are due to the frequency dependence of the complex functions of the physical parameters of the medium thereafter referred to as the generalized susceptibilities which describe the fluid-structure interactions. The analysis of the propagation of transient waves in such media raises two different problems: the direct and the inverse scattering problems. The direct scattering problem is that of determining the scattered fields that arise when a known incident field impinges on a scattering medium with known physical properties. The inverse scattering problem consists of the problem of determining the physical properties from the scattered fields produced by a known incident wave. In most cases, the determination of medium parameters is done in the frequency domain. The reflection and transmission coefficients which contain the information on the physical parameters of the medium are obtained from the Fourier transform of the scattered signals divided by the Fourier transform of the incident signal. Unfortunately, the coefficients of the wave equation are nonlinear functions of the parameters of the medium which are to be solved to achieve the right values of the generalized susceptibilities. Such solution obtained from the Fourier transform procedure might not reproduce important qualitative features of the solutions and produce results which may be difficult to use in applications. So, the frequency domain methods are limited in probing the inhomogeneous media when parameters are depth dependent.

In the present paper a time domain method is presented for finding the dispersion relation from a time domain experiment. The advantages of time domain methods over frequency domain methods are their ability: (i) to solve the inverse scattering problem; (ii) to give causal solution to the direct scattering problem.

In time domain, a dispersive medium is characterized by susceptibility kernels which are the Fourier transforms of the generalized susceptibilities. They act as memory functions on modeling the relaxation processes which develop in the medium: the viscous and the thermal effects.

The time domain method presented in this paper is only approximate [START_REF] Fellah | Transient acoustic wave propagation in rigid porous media: A time-domain approach[END_REF]. In the time domain, the full wave equation in a dispersive and lossy medium is difficult to be solved. So, we deal with wave equations simpler than the full one that retain the most significant features of the underlaying physics and remain analytically tractable.

Twoof us (Z.E.A. F. and C.D.) have developed a model of memory functions which describe the loss and the dispersion properties of the medium on a relevant frequency interval [START_REF] Fellah | Transient acoustic wave propagation in rigid porous media: A time-domain approach[END_REF]. In high frequency range, dispersion and attenuation of waves are well described by a fractional power of the frequency. So, by Fourier transform, we have shown that the behavior of the waves is determined by a Fractional Telegraph Equation (FTE). To investigate the wave scattering by a layered medium, the relevant quantities are the operators which map the incident wave in the reflected and transmitted ones. Reflection and transmission operators are defined by their kernels which are the solutions of differential equations. In a inhomogeneous porous medium, we show that the scattering kernels satisfy coupled fractional differential equations. The paper is organized as follows. In Section 2 the basic elements of the physical derivation of the FTE are presented. The fluid equivalent model of the propagation in porous media is introduced and the FTE is derived. Section 3 describes the layered porous medium and the wave splitting. In Section 4 the scattering operators are defined and their fractional differential equations are worked out.

Derivation of the FTE

THE FLUID EQUIVALENT MODEL OF POROUS MEDIA

We consider a slab of homogeneous porous material which occupies the region between x = 0 and x = l (x is the Cartesian coordinate in the direction perpendicular to the slab interfaces). In our model the slab is characterized by constant parameters. The incident wave is planar and normally incident. It propagates in the positive x direction and impinges the slab at x = 0.

In air saturated porous media, the structure is assumed to be motionless: the acoustic waves travel only in the fluid filling the pores. The wave propagation is described by the equivalent fluid model which is a particular case of the Biot's theory. In this model, the interactions between the fluid and the structure are taken into account in two frequency dependent response factors which are the generalized susceptibilities: the dynamical tortuosity of the medium α(ω) [START_REF] Johnson | Recent developpments in the acoustic properties of porous media[END_REF] and the dynamical compressibility of the air included in the medium β(ω) [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF]. These two response factors are complex functions which heavily depend on the frequency f = ω/2π. These functions represent the deviation from the behavior of the fluid in the free space as the frequency increases. Their theoretical expressions are given by Johnson [START_REF] Johnson | Recent developpments in the acoustic properties of porous media[END_REF], Allard [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] and Lafarge [START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF]:

α(ω) = α ∞   1 + ηφ jωα ∞ ρ f k 0 1 + j 4α 2 ∞ k 2 0 ρ f ω η 2 φ 2   , (1)
β(ω) = γ -(γ -1)   1 + ηφ jωρ f k ′ 0 Pr 1 + j 4k ′2 0 ρ f ω Pr ηφ 2 ′2   -1/2 , (2)
where j 2 =-1, γ represents the adiabatic constant, Pr the Prandtl number, α ∞ the tortuosity, k 0 the static permeability, k ′ 0 the thermal permeability and and ′ the viscous and thermal characteristic lengths [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF][START_REF] Champoux | Dynamic tortuosity and bulk modulus in air-saturated porous media[END_REF]. This model was initially developed by Johnson, Koplik and Dashen [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF], and completed by Champoux and Allard by adding the description of thermal effects [START_REF] Champoux | Dynamic tortuosity and bulk modulus in air-saturated porous media[END_REF]. Later on, Lafarge has introduced the parameter k ′ 0 which describes the additional damping of sound waves due to the thermal exchanges between fluid and structure at the surface of the pores [START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF].

The functions α(ω) and β(ω)e xpress the viscous and thermal exchanges between the air and the structure which are responsible of the sound damping in acoustic materials. These exchanges are due on the one hand to the fluid-structure relative motion and on the other hand to the air compressionsdilatations produced by the wave motion. The part of the fluid affected by these exchanges can be estimated by the ratio of a microscopic characteristic length of the media, as for example the sizes of the pores, to the viscous and thermal skin depth thickness δ = (2η/ωρ 0 ) 1/2 and δ ′ = (2η/ωρ 0 P r ) 1/2 .Forthe viscous effects this domain corresponds to the region of the fluid in which the velocity distribution is perturbed by the frictional forces at the interface between the viscous fluid and the motionless structure. For the thermal effects, it is the fluid volume affected by the heat exchanges between the two phases of the porous medium, the solid skeleton being seen as a thermostat.

THE FRACTIONAL TELEGRAPH EQUATION

The high frequency approximations of α(ω) and β(ω)(ω →∞) are given by the relations:

α(ω) ≈ α ∞ 1 + 2 η jωρ f 1/2 , (3)
β(ω) ≈ 1 + 2(γ -1) ′ η jω Prρ f 1/2 . ( 4
)
In these equations, the factor ( jω) -1/2 ,e xpresses in the frequency domain the semiintegral operator defined as [START_REF] Samko | Fractional Integrals and Derivatives[END_REF]]

I 1/2 ψ(x) = 1 √ π t a ψ(τ ) √ t -τ dτ, ( 5
)
where a is real valued In the time domain, the susceptibilities are operators [START_REF] Samko | Fractional Integrals and Derivatives[END_REF] and their asymptotic expressions are given in [START_REF] Fellah | Transient acoustic wave propagation in rigid porous media: A time-domain approach[END_REF] as:

α(t) = α ∞ δ(t) + 2α ∞ η ρ f 1/2 t -1/2 , (6)
β(t) = δ(t) + 2(γ -1) ′ η Prρ f 1/2 t -1/2 . ( 7
)
In each of these equations the first term in the right hand side is the instantaneous response of the medium while the second term is the memory function. In electromagnetism, the instantaneous response is called optical response. It describes all the processes which cannot be resolved by the signal. In this framework, the basic equations of the acoustic waves propagation along the ox axis are:

ρ f α(t) * ∂v x ∂t =- ∂ p ∂ x (8) β(t) K a * ∂ p ∂t =- ∂v x ∂ x , (9)
where K a is the bulk modulus of the air, p is the acoustic pressure and v the particle velocity. * is the shorthand notation for the time convolution

( f * g)(t) = t 0 f (t -t ′ )g(t ′ )dt ′ . ( 10
)
Foracausal function v(x, t), the wave equation is deduced from these equations as the FTE:

∂ 2 v ∂ x 2 - 1 c 2 ∂ 2 v ∂t 2 + A t 0 ∂ 2 v/∂t 2 √ t -t ′ dt ′ -B ∂v ∂t = 0, (11)
where A and B are the following constants:

A = 2α ∞ K a ρ r η π 1 + γ -1 √ Pr ′ (12)
B = 4α ∞ (γ -1)η K a ′ √ Pr ( 13
)
c is the velocity c = c 0 / √ α ∞ in the porous medium, and c 0 = ρ f /K a is the wave velocity in the free space. The other coefficients govern the spreading and the attenuation of the waves. During the last decade, FTE has been investigated by many authors [START_REF] Hanyga | Wave propagation in micro-heterogeneous porous media: A model based on an integro-differential wave equation[END_REF][START_REF] Cascaval | Fractional telegraph equations[END_REF]. Fractional differential equations appear more and more frequently in different research areas and engineering applications. They are derived by the generalization of classical equations and they seem to be an essential complementary tool in the description of a lot of phenomena.

Layered Slab of Porous Material

THE BASIC EQUATION

We consider now a slab of inhomogeneous porous material between x = 0 and x = l. The slab parameters are x-dependent, i.e., the medium is layered. However, in 0 < x < l,itisassumed that the parameters of the medium are continuous to avoid jump discontinuities in the operators kernels. In a layered medium, the coefficients A, B and c of Equation [START_REF] Courant | Methods of Mathematical Physics,V ol. 2[END_REF] are no more constant. They are written as A = a(x), B = b(x) and c = c(x). Then the FTE has the following form

∂ 2 x -c(x) -2 ∂ 2 t -b(x)∂ t -a(x)∂ 3/2 t v(x, t) = 0, (14)
where ∂ x v(x, t) = ∂v(x, t)/∂ x ...

WAVE SPLITTING

To split the wave, we put the Equation ( 14) in the matrix form:

∂ x v(x, t) ∂ x v(x, t) = 01 c -2 (x)∂ 2 t 0 v(x, t) ∂ x v(x, t) + 00 b(x)∂ t + a(x)∂ 3/2 t 0 v(x, t) ∂ x v(x, t) (15)
Equation ( 15) describes the evolution of vector (v, ∂ x v) t ( t denotes the transpose) when x varies on the direction of propagation. Since v(x, t) and ∂ x v(x, t) are linearly independent quantities, they form a basis of a two-dimensional space from which any propagating wave can be generated. In the wave splitting, quantities of interest are two independent linear combinations of v(x, t) and ∂ x v(x, t): i) which form a new basis of the space; ii)which are convenient for physical interpretations. A particular choice is {v + ,v -} defined by the transformation:

v + (x, t) v -(x, t) = S v(x, t) ∂ x v(x, t) where S = 1 -c -1 (x)∂ -1 t 1 c -1 (x)∂ -1 t . v ± (x, t) are the split fields which satisfy the relation v(x, t) = v + (x, t) + v -(x, t).
The dynamical equations for these fields are deduced from Equation ( 15) in the following form:

∂ x v + (x, t) v -(x, t) = M v + (x, t) v -(x, t) , (16)
where

M = SMS -1 -S(∂ x S -1 ) + SNS -1 , (17)
M = 01 c -2 (x)∂ 2 t 0 , (18)
N = 00 b(x)∂ t + a(x) √ π∂ 3/2 t 0 . ( 19
)
The wave splitting converts the matrix M into diagonal form. So, during their travel through the material, the split fields are mixed by the S(∂ x S -1 ) and SNS -1 matrices i.e., by the losses and dispersion processes:

SMS -1 = ∂ c∂t -10 01 (20) S(∂ x S -1 ) =-S -1 (∂ x S) = 1 2 c ′ (x) c(x) 1 -1 -11 (21) SNS -1 = c(x) 2 b(x) + a(x) √ π∂ 1/2 t -1 -1 11 (22)
So, the dynamic equation of the split fields is

∂ ∂ x v + v -=     -∂ t c(x) -1 2 c(x)a(x) √ π∂ 1/2 t -1 2 c(x)a(x) √ π∂ 1/2 t 1 2 c(x)a(x) √ π∂ 1/2 t ∂ t c(x) + 1 2 c(x)a(x) √ π∂ 1/2 t   + δ 11 δ 12 δ 21 δ 22   v + v -, (23)
with

δ 11 =-δ 21 =- 1 2 c ′ (x) c(x) - b(x)c(x) 2 , ( 24
)
δ 12 =-δ 22 = 1 2 c ′ (x) c(x) - b(x)c(x) 2 . ( 25
)

Scattering Operators

SCATTERING REPRESENTATION

To describe the scattering properties of the slab of porous material, the scattering operators are introduced [START_REF] Kristensson | Direct and inverse scattering in the time domain for a dissipative wave equation: I. Scattering operators[END_REF]. For a specific incidence from x < 0, the incident wave v i (x, t) which propagates in the positive x direction, is of v + type. In the region x < 0, there is furthermore a reflected wave v r (x, t) which propagates in the negative x direction, i.e., which is of the v -type. v r may be interpreted as a response of the slab to the incident wave v i . The Duhamel's principle establishes that there exist a linear relation between v r and v i namely v r (0, t) = R(0, l, t)v i (0, t)inthe form [START_REF] Courant | Methods of Mathematical Physics,V ol. 2[END_REF].

v r (0, t) = t -∞ R(0, l, t -t ′ )v i (0, t ′ )dt ′ . (26)
R(0, l, t)isthe reflection kernel which refers to the slab between x = 0 and x = l.Inthe same way we can introduce the transmission operator T (l, t) which relates the transmitted wave v t and the incident wave v i :

v t (l, t) = v i (0, t) + t -∞ T (0, l, t -t ′ )v i (0, t ′ )dt ′ , (27)
where T (0, l, t)isthe kernel of T and v t (l, t)isav + type wave such that

v t (l, t) = v + (l, t + τ (0, l)), (28)
with τ (x 1 , x 2 )isthe time it takes for the wave to travel from x = x 1 to x = x 2 :

τ (x 1 , x 2 ) = x 2 x 1 dx ′ c(x ′ ) . ( 29
)
The influence of the slab 0 < x < l is expressed through the arguments (0, l ...)ofthe kernels R and T . They indicate that these kernels depend of the whole slab between the two faces x = 0 and x = l. They are interpreted as the responses of the slab to a δ-function excitation:

v i (x, t) = δ(x -c 0 t)

IMBEDDING EQUATIONS FOR THE SCATTERING OPERATORS

When the slab is layered, we cannot consider just the global scattering kernels. By global kernels we understand functions which depend of on the whole slab. We must also determine the variations of the scattering kernels through the slab. For that we define the local scattering kernels R(x 1 , x 2 , t) and T (x 1 , x 2 , t) relative to the subslab between x = x 1 and x = x 2 .It is assumed that the subslab is embedded between two half-spaces specified by their velocities c(x 1 ) and c(x 2 ). The global kernels are obtained as the limits of the local kernels when x 1 and x 2 approach, respectively 0 and l. Stokes was the first to develop this approach [START_REF] Stokes | On the intensity of the light reflected and transmitted through a pile of plates[END_REF] referred as invariant imbedding method.

Reflection Operator

The local reflection operator is defined by the equation

v -(x, t) = R(x, t) * v + (x, t). ( 30
)
Applying ∂ x to (30) leads to:

∂ x v -(x, t) = ∂ x R(x, t) * v + (x, t) + R(x, t) * ∂ x v + (x, t). (31)
Thus, ∂ x v ± are reported from the dynamic equation of the split fields to give:

0 = 1 2 a(x)c(x) √ π∂ 1/2 t [v + (x, t) + v -(x, t)] + 1 c(x) ∂ t v -(x, t) + δ 21 v + (x, t) + δ 22 v -(x, t) -∂ x R(x, t) * v + (x, t) -δ 11 R(x, t) * v + (x, t) -δ 12 R(x, t) * v -(x, t) + 1 2 a(x)c(x) √ π R(x, t) * ∂ 1/2 t [v + (x, t) + v -(x, t)] + 1 c(x) (R(x, t) * ∂ t v + (x, t)). ( 32
)
Using the following relations:

R(x, t) * ∂ t v + (x, t) = R(x, 0)v + (x, t) + ∂ t R(x, t) * v + (x, t) (33) ∂ t v -(x, t) = 2R(x, t) * v + (x, t) + R(x, 0)v + (x, t) (34)
Equation (32) becomes:

0 = 1 2 a(x)c(x) √ π∂ 1/2 t [v + (x, t) + R(x, t) * v + (x, t)] + 2 c(x) ∂ t R(x, t) * v -(x, t) + 1 c(x) R(x, 0)v + (x, t) + δ 21 v + (x, t) + δ 22 v -(x, t) -∂ x R(x, t) * v + (x, t) + 1 c(x) R(x, 0) * v + (x, t) + 1 c(x) ∂ t R(x, t) * v + (x, t) + 1 2 a(x)c(x) √ π R(x, t) * ∂ 1/2 t [v + (x, t) + R(x, t) * v + (x, t)] -δ 11 R(x, t) * v + (x, t) -δ 12 R(x, t) * R(x, t) * v + (x, t). ( 35
)
In these equations the fractional derivatives ∂ t 1/2 v ± (x, t) can be written as the following convolutions:

∂ 1/2 t v + (x, t) =- 1 2 √ π 1 t 3/2 * v + (x, t) (36) ∂ 1/2 t v -(x, t) =- 1 2 √ π 1 t 3/2 * R(x, t) * v + (x, t). ( 37
)
A term by term identification of (35) gives:

-from the constant terms:

R(x, 0) =- 1 4 (c ′ (x) + b(x)c 2 (x)) (38)
-from the convolution terms

∂ x R(x, t) - 3 c(x) ∂ t R(x, t) =- 1 2 c ′ (x) c(x) -a(x)c(x) R(x, t) * R(x, t) - 1 4 a(x)c(x) 1 t 3/2 + 2 t 3/2 * R(x, t) + 1 t 3/2 * R(x, t) * R(x, t) + b(x)c(x)R(x, t), (39)
with the conditions:

R(x, 0) =- 1 a [c ′ (x) + b(x)c 2 (x)] and R(l, t) = 0. (40)

Transmission Operator

The transmission operator is defined by the equation

v + (x, t + τ (L , x)) = a(l, x)v + (x, t) + T (x, t) * v + (x, t), ( 41
)
where a(l, x)isthe attenuation factor. The differential equation for transmission operator is determined in the same way that the one of the reflection operator. After a straightforward calculation and the term by term identification one finds the following equations:

-from the constant terms

∂ x a(l, x) =-δ 11 a(l, x), ( 42
) or a(l, x) = c(x) c(l) exp - 1 2 l x b(x ′ )c(x ′ )dx ′ ; (43)
-from the coefficient of ∂ t v + (x, t) with the condition T (0, t) = 0. Note that Equations ( 26) and ( 46) which contains R and T are coupled nonlinear differential equations. Equation (26) contains only the unknown R(x, t). They both contain fractional derivatives which model the memory effects which develop in the porous medium when the waves propagate in it.

∂ x τ (l, x) =- 1 c(x) , (44)
These equations may be solved analytically when the profiles of parameters variations are tractable [13] or by numerical methods [START_REF] Kirsh | Two methods for solving the inverse acoustic scattering problem[END_REF] as for example finite difference method or the method of characteristics [START_REF] Weng | Waves and Fields in Inhomogeneous Media[END_REF].

Conclusion

In this paper the fractional differential equations for the scattering operators have been established. However, they are deduced from the fractional telegraph equation which describes the transient acoustics signals propagation in a layered porous medium. This model is well adapted to investigate the propagation of ultrasonic pulses in acoustics porous materials. The model is restricted to a one-dimensional problem to reduce the inverse scattering problem but more general situations like obliquely incident wave may be tackled in the same way.

  (x, t) =-δ 11 T (x, t)δ 12 {a(l, x)R(x, t) + T (x, t) * R(x, t)}

	-from the convolution terms
	∂ x T (x, t) +	1 c(x)	∂ t T -	1 4	a(l, x)	1 t 3/2 + a(l, x)	1 t 3/2 * R(x, t)
						+	1 t 3/2 * R(x, t) +	1 t 3/2 * R(x, t) * T (x, t)	(46)
	leading to
	τ (l, x) =	x	l	dx ′ c(x ′ )	;		(45)

Acknowledgements

We would like to thank the anonymous referees for their invaluable suggestions.


F Martins

S Faniel

B Rosenow

H Sellier

S Huant

M G Pala

L Desplanque

X Wallart

V Bayot

B Hackens
email: benoit.hackens@uclouvain.be.

Coherent tunnelling across a quantum point contact in the quantum Hall regime

Keywords: numbers: 73.21.La, 73.23.Ad, 03.65.Yz, 85.35.Ds

The unique properties of quantum Hall devices arise from the ideal one-dimensional edge states that form in a two-dimensional electron system at high magnetic field. Tunnelling between edge states across a quantum point contact (QPC) has already revealed rich physics, like fractionally charged excitations, or chiral Luttinger liquid. Thanks to scanning gate microscopy, we show that a single QPC can turn into an interferometer for specific potential landscapes. Spectroscopy, magnetic field and temperature dependences of electron transport reveal a quantitatively consistent interferometric behavior of the studied QPC. To explain this unexpected behavior, we put forward a new model which relies on the presence of a quantum Hall island at the centre of the constriction as well as on different tunnelling paths surrounding the island, thereby creating a new type of interferometer. This work sets the ground for new device concepts based on coherent tunnelling.

Electron phase coherence is the cornerstone of quantum devices and computation [1,2]. In that perspective, quantum Hall (QH) devices are particularly attractive in view of their large coherence times [3]. Quantum Hall edge states (ES) formed by Landau levels (LL) crossing the Fermi energy near sample borders are ideal one-dimensional (1D) systems in which scattering vanishes exponentially at low temperature T [1,4]. Edge state loops surrounding potential hills or wells, referred to as localized states or quantum Hall islands (QHIs), then form unique zero-dimensional (0D) systems [START_REF] Ilani | The microscopic nature of localization in the quantum Hall effect[END_REF]. The last few years witnessed great progresses in the transport spectroscopy of model QH localized states created by patterning quantum dots [START_REF] Altimiras | Non-equilibrium edge-channel spectroscopy in the integer quantum Hall regime[END_REF] or antidots [START_REF] Sim | Electron interactions in an antidot in the integer quantum Hall regime[END_REF][START_REF] Goldman | Resonant tunnelling in the quantum Hall regime: measurement of fractional charge[END_REF][START_REF] Maasilta | Energetics of quantum antidot states in the quantum Hall regime[END_REF] in a two-dimensional electron system (2DES).

In parallel, new tools were developed to probe the microscopic structure of confined electron systems in the QH regime. In particular, scanning gate microscopy [START_REF] Topinka | Imaging coherent electron flow from a quantum point contact[END_REF][START_REF] Crook | Imaging cyclotron orbits and scattering sites in a high-mobility two-dimensional electron gas[END_REF][START_REF] Pioda | Spatially resolved manipulation of single electrons in quantum dots using a scanned probe[END_REF][START_REF] Hackens | Imaging and controlling electron transport inside a quantum ring[END_REF][START_REF] Martins | Imaging electron wave functions inside open quantum rings[END_REF][START_REF] Pala | Local density of states in mesoscopic samples from scanning gate microscopy[END_REF][START_REF] Pala | Scanning gate microscopy of quantum rings: effects of an external magnetic field and of charged defects[END_REF] (SGM) makes use of a movable metallic tip, which is voltage-biased, to finely tune the electrons' confining potential in its vicinity. This way, the geometry of propagating edge states and localized states can be modified at will [START_REF] Paradiso | Spatially resolved analysis of edge-channel equilibration in quantum Hall circuits[END_REF]. Very recently, SGM allowed us to locate active QHIs in a QH interferometer [START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF]. Importantly, it appeared that QHIs do not only form around antidots, but potential inhomogeneities also induce QHIs in the arms or near the constrictions connecting a quantum ring to source and drain reservoirs [START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF]. Therefore, lateral confinement, e.g. in Quantum Point Contacts (QPCs), offers the possibility to connect a QHI to ES through tunnel junctions, and thus form a new class of 1D-0D-1D QH devices (Fig. 1). In this case, the 0D island is characterized by a weak coupling (σ << e 2 /h) and a large charging energy (E c = e 2 /C >> k B T ) (C is the island capacitance), which induce Coulomb blockade (CB) [1]. In such devices, Aharonov-Bohm (AB) like oscillations of the resistance can be explained by Coulomb coupling between fully occupied LLs and confined states in the QHI [START_REF] Sim | Electron interactions in an antidot in the integer quantum Hall regime[END_REF][START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF][START_REF] Taylor | Aharonov-Bohm oscillations in the Coulomb blockade regime[END_REF][START_REF] Kataoka | Detection of Coulomb charging around an antidot in the quantum Hall regime[END_REF][START_REF] Rosenow | Influence of interactions on flux and back-gate period of quantum Hall interferometers[END_REF][START_REF] Zhang | Distinct signatures for Coulomb blockade and Aharonov-Bohm interference in electronic Fabry-Pérot interferometers[END_REF]. It was also suggested that AB oscillations reported on a QPC [START_REF] Van Loosdrecht | Aharonov-Bohm effect in a singly connected point contact[END_REF] could be attributed to tunnelling paths around the saddle point [START_REF] Jain | Model tunneling problems in a high magnetic-field[END_REF]. In contrast, transport through QH devices, but in the strong coupling limit (σ >> e 2 /h), revealed coherent effects analog to those observed in optical Mach-Zehnder [3,[START_REF] Ji | An electronic Mach-Zehnder interferometer[END_REF][START_REF] Neder | Unexpected behavior in a two-path electron interferometer[END_REF] or Fabry-Pérot [START_REF] Zhang | Distinct signatures for Coulomb blockade and Aharonov-Bohm interference in electronic Fabry-Pérot interferometers[END_REF][START_REF] Sivan | Aharonov-Bohm and quantum Hall effects in singly connected quantum dots[END_REF][START_REF] Van Wees | Observation of zero-dimensional states in a one-dimensional electron interferometer[END_REF][START_REF] Mcclure | Edge-state velocity and coherence in a quantum Hall Fabry-Pérot interferometer[END_REF][START_REF] Yamauchi | Universality of bias-and temperature-induced dephasing in ballistic electronic interferometers[END_REF][START_REF] Ofek | Role of interactions in an electronic Fabry-Pérot interferometer operating in the quantum Hall effect regime[END_REF][START_REF] Halperin | Theory of the Fabry-Pérot quantum Hall interferometer[END_REF] interferometers.

Here, we examine an unexplored regime of transport across a QPC where QH edge states are weakly coupled, but phase coherence is preserved. The SGM tip is used as a nanogate to tune the potential landscape and hence edge states' pattern and coupling. At first sight, one expects that transport should be driven by tunnelling, and possibly by Coulomb block-

RESULTS

Our sample is a QPC etched in an InGaAs/InAlAs heterostructure holding a 2DES 25 nm below the surface. The QPC lithographic width is 300 nm. All the experiments were performed at temperature between 4.2 K and 100 mK, in a dilution refrigerator. Here, the perpendicular magnetic field B ∼ 9.5 T, which corresponds to a LL filling factor ν ∼ 6 in the 2DES. The SGM experiment is schematically depicted in Fig. 1. It consists in scanning Using Eq. ( 1) for the two consecutive fringes highlighted with the white dashed lines in (b), we calculate in (c) the diameter of the QHI as the tip-QHI distance δ x is varied.

a metallic atomic force microscope tip, polarized at voltage V tip , along a plane parallel to the 2DES at a tip-2DES distance of 50 nm while recording a map of the device resistance R [START_REF] Hackens | Imaging and controlling electron transport inside a quantum ring[END_REF][START_REF] Martins | Imaging electron wave functions inside open quantum rings[END_REF]. The QPC resistance is defined as R = dV /dI, where V and I are the voltage and the current through the device, respectively.

The 2DES being on a quantized Hall plateau, whenever some current tunnels between opposite edge channels, R deviates from the zero value expected in QH systems at very low T [2,4,[START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF]. In our case, the SGM resistance map recorded at B = 9.5 T, V tip = -4 V and T = 4.2 K and presented in Fig. 2(a) reveals concentric fringes superimposed on a slowly varying background. The origin of the background, related to reflection of ES at the QPC, is discussed in the supplementary information. The fringe pattern can easily be understood in the presence of a QHI surrounding a potential hill, close to the saddle point of the QPC and tunnel-coupled to the propagating ES (Fig. 1). Indeed, approaching the polarized tip gradually changes the potential of the QHI, and hence its area A, defined as the surface enclosed by the "looping" ES. The enclosed magnetic flux φ varies accordingly and the tip generates iso-φ lines when circling around the QHI. Since adding one flux quantum φ 0 corresponds to trapping one electron per populated LL in the island, CB oscillations are generated whenever B or A are varied [START_REF] Rosenow | Influence of interactions on flux and back-gate period of quantum Hall interferometers[END_REF], thereby producing AB-like oscillations [START_REF] Sim | Electron interactions in an antidot in the integer quantum Hall regime[END_REF][START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF][START_REF] Taylor | Aharonov-Bohm oscillations in the Coulomb blockade regime[END_REF][START_REF] Kataoka | Detection of Coulomb charging around an antidot in the quantum Hall regime[END_REF][START_REF] Zhang | Distinct signatures for Coulomb blockade and Aharonov-Bohm interference in electronic Fabry-Pérot interferometers[END_REF]. Isoresistance lines visible on Fig. 2(a) are, therefore, iso-φ lines that are crossed as the tip-island distance is varied [START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF]. Consequently, the center of concentric fringes in Fig. 2(a) indicates the position of the active QHI, which connects opposite propagating edge channels through tunnel junctions (Fig. 1).

In the framework of this model, the area of the QHI can be determined thanks to the B-dependence of AB-like oscillations [START_REF] Rosenow | Influence of interactions on flux and back-gate period of quantum Hall interferometers[END_REF]:

∆B = (φ 0 /A)/N ( 1
)
where N is the number of completely filled LL in the bulk (here N = 6). The combined effect of moving the tip along the dashed line in Fig. 2(a) and changing B is illustrated in Fig.

for V tip = -6 V. Along the B-axis, AB-like oscillations are highlighted with the white dashed lines. The negatively polarized tip approaching the QHI raises its potential, which increases its area A, and hence reduces the magnetic field that separates two resistance peaks ∆B. This is illustrated in Fig. 2(c), where we assume that the QHI has a surface equivalent to that of a disk with diameter d obtained from Eq. ( 1): d is found to increase from ∼65 nm to ∼95 nm as the tip-island distance δ x decreases from 1300 nm to 300 nm, respectively. Noteworthy, as expected for Coulomb dominated transport in a QH interferometer, increasing B is equivalent to applying a more negative V tip , yielding a positive dV tip /dB for isoresistance stripes [START_REF] Zhang | Distinct signatures for Coulomb blockade and Aharonov-Bohm interference in electronic Fabry-Pérot interferometers[END_REF][START_REF] Ofek | Role of interactions in an electronic Fabry-Pérot interferometer operating in the quantum Hall effect regime[END_REF][START_REF] Halperin | Theory of the Fabry-Pérot quantum Hall interferometer[END_REF]. Since approaching the negatively charged tip has the same effect as decreasing V tip , Fig. 2(b) seems consistent with the Coulomb dominated transport.

But, surprisingly, the temperature dependence of fringes amplitude (δR, measured on SGM maps), shown on Fig. 3, reveals a peculiar behaviour: it clearly does not follow the T -1 dependence expected in the quantum regime of CB [START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF][START_REF] Yacoby | Coherence and phase sensitive measurements in a quantum dot[END_REF][START_REF] Kouwenhoven | Electron transport in quantum dots[END_REF] (data from ref. [START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF] are presented for comparison in Fig. 3). Instead, δR deceases very slowly from 100 mK to 4.2 K.

Indeed, for coherent transport through a Fabry-Pérot geometry, thermal smearing of interference gives rise to a temperature dependence δR(T ) ∼exp(-T /T 0 ) in the low temperature regime. In contrast, for transport processes involving a weakly coupled Coulomb island, this form for δR(T ) is expected only for temperatures larger than the charging energy [START_REF] Halperin | Theory of the Fabry-Pérot quantum Hall interferometer[END_REF]. In

R [Ω] T [K]
ref [START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF] FIG. 3: Temperature dependence : Coulomb blockade vs coherent transport. δR vs T obtained from SGM maps with V tip = -1 V (circles) and from data in ref. [START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF] (squares). The dashed line corresponds to a T -1 dependence. The gray region corresponds to an exponential dependence exp(-T /T 0 ) with 9.5 K < T 0 < 19.1 K, consistent with magnetoresistance data and edge state velocity estimate along Ref. [START_REF] Mcclure | Edge-state velocity and coherence in a quantum Hall Fabry-Pérot interferometer[END_REF]. The solid line corresponds to T 0 = 16.2 K, consistent with the spectroscopy data (see text).

the Fabry-Pérot situation, T 0 is linked to the excited states level spacing ∆E Ex according to the relation T 0 = ∆E Ex /k B = 2 v/(dk B ) where v is the local edge state velocity, related to the gradient of the confining potential. From experimental data measured in a GaAs QH Fabry-Pérot interferometer [START_REF] Mcclure | Edge-state velocity and coherence in a quantum Hall Fabry-Pérot interferometer[END_REF], one can infer that, in our sample, 5 × 10 4 m/s < v < 10 5 m/s. Given this range for v, and d ∼ 80 nm (from Fig. 2(c), taking into account that the T -dependence data were measured at δ x ∼ 630 nm), we obtain the range of T -dependence represented as a gray region in Fig. 3, which reproduces quite well the behaviour observed experimentally. The corresponding range of 9.5 K < T 0 < 19.1 K is consistent with the low temperature limit and hence with a Fabry-Pérot behaviour. Earlier experiments already evidenced such an exponential decay with temperature, but only in Mach-Zehnder and ballistic devices, which are known to be coherent [START_REF] Yamauchi | Universality of bias-and temperature-induced dephasing in ballistic electronic interferometers[END_REF][START_REF] Roulleau | Finite bias visibility of the electronic Mach-Zehnder interferometer[END_REF][START_REF] Litvin | Edge-channel interference controlled by Landau level filling[END_REF]. However, our main observation confirming the preserved electron phase coherence emerges from the analysis of non-linear transport through the QPC. Scanning gate spectroscopy is realized by positioning the tip right above the QHI, and sweeping both V tip and the dc current I through the QPC. The voltage across our tunnel device, i.e. between propagating edge states, is the Hall voltage: V bias = h/(e 2 N * )I [2], where N * is the (integer) number of transmitted ES at the QPC (Fig. S1 -supplementary material). The measurement configuration is indeed identical to that used to perform conventional electrical spectroscopy on isolated quantum dots. If the QHI were weakly tunnel coupled to the propagating edge states, one would expect to observe a "Coulomb diamond" pattern [START_REF] Martins | Scanning gate spectroscopy of transport across a quantum Hall nano-island[END_REF]. Fig. 4(a) shows dR/dV tip as a function of the dc component of V tip and V bias . Instead of Coulomb diamonds, the spectroscopy displays a checkerboard pattern of maxima and minima, indicating that both V tip and V bias tune the interference of transiting electrons. Each bias independently adds a phase shift between interfering paths, so that the transresistance is modulated by a product of cosines and an exponential term accounting for a voltage-dependent dephasing induced by electrons injected at an energy e|V bias | [START_REF] Mcclure | Edge-state velocity and coherence in a quantum Hall Fabry-Pérot interferometer[END_REF][START_REF] Yamauchi | Universality of bias-and temperature-induced dephasing in ballistic electronic interferometers[END_REF][START_REF] Roulleau | Finite bias visibility of the electronic Mach-Zehnder interferometer[END_REF][START_REF] Van Der Wiel | Electromagnetic Aharonov-Bohm effect in a two-dimensional electron gas ring[END_REF]:

dR dV tip = D cos 2π V bias ∆V bias cos 2π V tip ∆V tip + ϕ exp -2πγ V bias ∆V bias n , (2)
where D is the zero-bias visibility of the oscillations, ∆V tip is the oscillation period induced by V tip , ϕ is a constant phase factor, ∆V bias = 4 v/(ed) is the oscillation period along the V bias axis, and γ is directly related to the V bias -dependent dephasing rate: τ -1 ϕ = γ(e|V bias |)/2 [START_REF] Mcclure | Edge-state velocity and coherence in a quantum Hall Fabry-Pérot interferometer[END_REF]. n varies from 1 to 2 according to Ref. [START_REF] Mcclure | Edge-state velocity and coherence in a quantum Hall Fabry-Pérot interferometer[END_REF][START_REF] Yamauchi | Universality of bias-and temperature-induced dephasing in ballistic electronic interferometers[END_REF][START_REF] Roulleau | Finite bias visibility of the electronic Mach-Zehnder interferometer[END_REF][START_REF] Van Der Wiel | Electromagnetic Aharonov-Bohm effect in a two-dimensional electron gas ring[END_REF] and was set to 1 as we could not discriminate from fitting the data. As shown in Figs. 4(b-d), we obtain an excellent fit of the data in Fig. 4(a) using Eq. ( 2) with with a transist time τ t = d/v = 1.7 × 10 -12 s, and a parameter γ = 0.2 in the range found in Ref. [START_REF] Mcclure | Edge-state velocity and coherence in a quantum Hall Fabry-Pérot interferometer[END_REF]. Note that in such a small QHI, τ t turns out to be smaller by at least one order of magnitude than the intrinsic τ ϕ in the same 2DES [START_REF] Hackens | Dwell-time-limited coherence in open quantum dots[END_REF]. This renders coherent resonant tunnelling through the whole QHI device possible.

DISCUSSION

To interpret ∆V tip obtained from the fit, one first notes that R evolves very similarly when changing either V tip or B in the vicinity of B = 9 T (Fig. S2 -supplementary material).

Therefore, one can convert ∆V tip into an equivalent ∆B, through a lever arm ∆B/∆V tip = 0.108 T/V. Hence, ∆V tip = 0.46 V corresponds to ∆B = 50 mT for the AB-like oscillations.

In that range of V tip , N * = 5 (Fig. S1(d)). This means that d = 2 φ 0 /(πN * ∆B) = 145 nm, consistent with data in Fig. 2(c) since d is at a maximum when the tip is above the QHI (δ x = 0). Moreover, given the value of τ t = 1.7 × 10 -12 s found in fitting the spectroscopy data, one obtains v = 8.5 × 10 4 m/s, within the range of values that was expected from data in ref. [START_REF] Mcclure | Edge-state velocity and coherence in a quantum Hall Fabry-Pérot interferometer[END_REF], and in agreement with the exponential temperature dependence in Fig. 3. We therefore have a fully consistent picture that explains all magnetoresistance, temperature dependence and spectroscopy data, and shows that tunnelling across the QHI is indeed coherent. One fundamental question remains: why do we observe two distinct behaviours of transport through apparently similar QH devices, Coulomb blockaded transport in our previous work [START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF], and coherent transport in this one? The qualitative difference cannot be explained by the fact that d is smaller than previously examined QHIs. T 0 and ∆V bias would be reduced proportionally, but not enough to explain the observed T -dependence and spectroscopy. On the other hand, signs of coherent transport through CB quantum dots were only obtained for symmetric tunnel junctions [START_REF] Yacoby | Coherence and phase sensitive measurements in a quantum dot[END_REF] that allow resonant tunnelling instead of sequential tunnelling. In that framework, one might thus ascribe the loss of electron coherence in other QHIs to an asymmetry of tunnel junctions. However, a difference in the transmission coefficients T c of the tunnel barriers may point towards an alternative explanation. In the coherent regime, we find a rather strong coupling between the QHI and propagating ES (0.27 < T c < 0.43), which contrasts with the Coulomb blockade regime where T c << 1 [START_REF] Van Houten | Coulomb Blockade Oscillations in Semiconductor Nanostructures[END_REF]. A similar trend is observed in transport experiments at B = 0 T on carbon nanotubes [START_REF] Biercuk | Electrical transport in singlewall carbon nanotubes Carbon Nanotubes[END_REF]: phase coherence is maintained when electrons tunnel through barriers with a large transmission coefficient, so that interference effects can be observed.

Up to this point, our analysis is based on the presence of a QHI near the QPC, connected to propagating ES on both sides through tunnelling paths (Fig. 1). However, one could imagine the presence of additional tunnelling paths between propagating ES, in the vicinity of the QPC saddle point. The resulting model is presented in Fig. 5(a-b). While the "green" paths occur naturally when propagating ES and the QHI are close enough, the "red" paths may originate from potential anharmonicities (i.e. non parabolicity) on both sides of the saddle point, similar to the fast potential variations suggested in Ref. [START_REF] Jain | Model tunneling problems in a high magnetic-field[END_REF]. In this model, transport depends in principle on the various tunnelling probabilities, denoted T 1,2 and T 3,4 in Fig. 5. However, the presence of the QHI should always induce oscillations in the magnetoresistance and spectroscopy of the QPC, either because it is enclosed in an interferometer, created by the "red" paths and propagating ES, when T 1,2 < T 3,4 , or because tunnelling occurs directly through it (T 1,2 > T 3,4 ) as discussed above (Fig. 1). Therefore, whichever T 1,2 or T 3,4 dominates, transport is still controlled by the flux trapped in the QHI and hence its Coulomb charging, so that the analysis developed above to extract parameters from the magnetoresistance and spectroscopy are still valid. In that case, i.e. T 3,4 > T 1,2 , the amplitude of the fringes leads us to 0.043 < T 3,4 < 0.078 (for details see Supplementary Information).

In summary, we report first evidence for preserved electron phase coherence in tunnelling across a quantum point contact in the quantum Hall regime. We propose a framework that explains all magnetoresistance, temperature dependence and spectroscopy data. This scenario relies on the presence of a potential hill that generates a quantum Hall island near the saddle point of the QPC. Our data therefore provide new signatures of coherent tunnelling in an ultra-small QH device.

METHODS

Device fabrication and 2DES parameters. Our device is fabricated from a In-GaAs/InAlAs heterostructure grown by molecular beam epitaxy where a 2DES is confined 25 nm below the surface (the layer sequence of this heterostructure is detailed in [START_REF] Hackens | Imaging and controlling electron transport inside a quantum ring[END_REF][START_REF] Martins | Imaging electron wave functions inside open quantum rings[END_REF]).

The QPC was patterned using e-beam lithography followed by wet etching. The QPC resistance R is measured in a four-probe configuration: a low-frequency (typically 10 to 20 Hz) oscillating current I is driven between contacts 1 and 2 on Fig. 1, and V is measured between contacts 3 and 4 using a lock-in technique, with V across the QPC always less than k B T /e. Next to the QPC, we patterned a Hall bar where we measured a low-T electron density and mobility of 1.4 × 10 16 m -2 and 4 m 2 /Vs, respectively. SGM and SGS techniques. All the experiments are carried out inside a 3 He/ 4 He dilution refrigerator where a home-made cryogenic atomic force microscope (AFM) was integrated [START_REF] Hackens | Imaging Coulomb islands in a quantum Hall interferometer[END_REF]. The AFM is based on a quartz tuning fork to which a commercial metallized Si cantilever (model CSC17 from MikroMasch) is glued by means of a conductive silver epoxy.

We image the sample topography by imposing a feedback loop on the shift in the tuning fork resonant frequency and using standard dynamic AFM mode of operation. After locating the QPC we perform SGM. It consists of scanning the tip along a plane parallel to the 2DES at constant distance of 25 nm from the surface, i.e. 50 nm from the 2DES, with a bias voltage V tip applied to the tip and recording simultaneously the device resistance R. At the end of a set of SGM experiments, we image the topography of the QPC to ensure that, during that period, the position of the QPC did not change. The SGS is performed by positioning the AFM tip at a fixed position in the vicinity of the QHI and by adding a dc current I to the lock-in ac signal between contacts 1 and 2 (Fig. 1). The voltage between edge states V bias is obtained by multiplying the dc current I by h/(e 2 N * ). The transresistance dR/dV tip is measured with a second lock-in using an ac modulation of V tip .

QHI at the centre of a QPC

We first consider the situation represented in Fig. S 8(a) where a QHI is located at the centre of a QPC. In this case we assume that coherence is maintained during the multiple reflections. For an off-resonance condition and assuming T 1 = T 2 = T c , T c is given by: T c = 2T t (1 + T t ) [4], which implies that 0.27 < T c < 0.42.

Interferometer formed around the QPC saddle point

In the case of an interferometer formed around the saddle point of a QPC, as illustrated in Fig. S 8(b), we compute the reflection coefficient 1 -T t , taking into account interferences between different semiclassical paths for electrons : 1) a direct path along the edge state, which does not include transmission through the tunnel barriers, and 2) paths including multiple transmissions through the tunnel barriers. The reflection coefficient is then given by [4]:

1 -T t = (1 -T 3 )(1 -T 4 ) 1 + T 3 T 4 -2 √ T 3 T 4 cos( ) (3)
where is the phase difference accumulated along the two types of trajectories and T 3 and

1 4 2 FIG. 1 :

 121 FIG. 1: Schematic representation of our model and experimental setup. Tunnelling paths (dotted lines) connect opposite ES through a quantum Hall island (circle). Current-carrying contacts (1-2) and voltage probes (3-4) allow resistance measurements. (only one edge state is represented for the sake of clarity)

FIG. 2 :

 2 FIG. 2: Imaging tunnelling across a QPC. (a) SGM map at B = 9.5 T, T = 4.2 K, and V tip = -4 V. Continuous lines correspond to the edges of the QPC. The black bar represents 1 µm. (b) B-dependence of R-profiles over the region marked with a dashed line in (a), with V tip = -6 V.

FIG. 4 :

 4 FIG. 4: Evidence for coherent transport in spectroscopy. (a) dR/dV tip as a function of the dc component of V tip and V bias at B = 9.5 T and T =100 mK. Voltage modulation of V tip was set to 50 mV. (b) 2D fit of dR/dV tip using Eq. (2). (c-d) Transresistance vs V bias taken along the red (c) and blue (d) dashed lines in (a-b). The circles correspond to the experimental data and the continuous lines to the fit.

FIG. 5 :

 5 FIG. 5: Potential landscape and tunnelling paths across the QPC. (a) Schematic representation of the electrostatic potential in the vicinity of the QPC (in brown), with the ES in yellow and the tunnelling paths connecting opposite ES (red and green). Only one edge state is represented, for the sake of clarity. (b) top view of the three-dimensional figure in (a), with the various tunnelling probabilities T i between edge states.

T 4

 4 are the transmissions at each side of the saddle point. Assuming that T 3 = T 4 = T 3,4 , we obtain: 0.043 < T 3,4 < 0.078.

  FIG. S 6: (a-c) Consecutive SGM images obtained at T = 100 mK, B = 9.5 T and V tip = 0, -2 and -4 V, respectively. The top black bar represents 1 µm. (d) R vs V tip with the tip positioned near the saddle point of the QPC. The brown dashed lines indicate the resistance expected for N * .

5 FIG. S 7 : 1 T 2 T 3 T 4 TFIG. S 8 :

 5712348 FIG. S 7: QPC resistance vs B (top axis) and V tip (bottom axis) at T = 4.2 K.

455-493 (Springer, Berlin, 2008).

Acknowledgements:

The authors are grateful to M. Treffkorn and T. Hyart for helpful discussions. F.M. and B.H. are postdoctoral and associate researchers with the Belgian FRS-FNRS, respectively. This work has been supported by FRFC grants no. 2.4.546.08.F and 2.4503.12, FNRS grant no. 1.5.044.07.F, by the FSR and ARC program "Stresstronics", by BELSPO (Interuniversity Attraction Pole IAP-6/42), and by the PNANO 2007 program of the ANR (MICATEC project). V.B. acknowledges the award of a Chaire d'excellence by the Nanoscience Foundation in Grenoble.

Author contributions:

Additional information:

Supplementary Information accompanies this paper.

Competing financial interests: The authors declare no competing financial interests. Importantly, the SGM pattern around the QPC exhibits a strong variation that adds to the concentric fringes on Figs. S6(b-c). The origin of the stronger contrast is found by positioning the tip near the saddle point of the QPC and continuously decreasing V tip (Fig. S6(d)). The main trend shows step-like increases of the device resistance which can be understood by invoking ES reflections at the constriction. Decreasing V tip raises the energy of the saddle point and decreases the local filling factor ν * near the constriction. Every time ν * passes a half integer value, an ES is totally reflected and the device resistance shifts to the next plateau given by R = h/e 2 (1/N * -1/N ) [1,2] (brown dashed lines in Fig. S6(d)), where N * is the (integer) number of transmitted ES at the constriction. The presence of oscillations superimposed on the first plateau at N * = 5, similar to those around V tip = 0 V, indicate that the QHI is active even when one ES is reflected.

CALCULATION OF TRANSMISSION COEFFICIENTS ACROSS A QHI

Here we summarize the details concerning the determination of the coefficients of transmission across the tunnel barriers defining the QHI in the coherent regime (we assume here that all barriers have equal transmission coefficients). In this work it was found that the peak-to-peak amplitudes in the coherent regime (∆R) were within the following intervals:

where T t is the total transmission through the QHI (i.e. taking into account the two barriers defining the QHI), we conclude that T t is the interval: 0.16 < T t < 0.


G Chaplier
email: gerard.chaplier@ens-paris-saclay.fr

V Loyau

About a possibility for a tabletop fusion device

Keywords: Nuclear Fusion, Sonoluminescence, Z-Pinch, Energy, Cavitation, X ray

About a possibility for a tabletop fusion device

G Chaplier, Vincent Loyau

Introduction

The Increasing need of energy and predicable rarefaction of oil cause economic stress all over the world and this problematic is going to be more significant. At this time, the alternative energy sources seem to be a poor answer to the world necessities. The solar, thermal and photovoltaic, and wind energies are intermittent and ask for energy storing solutions which are not easily affordable. The fission nuclear energy has a lot of drawbacks as nuclear waste, needs of a cold source which is usually river water causing its ecosystem warming and limited Uranium reserves available only for fifty to two hundred years. The fusion in huge plants as tokamak seems not to work before a long time and is a very complicated way. It begins with a low pressure gas to be able to ionize it and get a plasma, compress it with high intensity magnetic fields to increase the density, then warm it with micro waves and when the fusion is supposed to be there, at last put all the stuff in a steam machine.

May be there is a place for a more simple device which can be made at different scale as the electric motor or the combustion engine whose dimensions can be in millimeters to ten meters range. It is the aim of this paper to humbly propose such a device as a possibility.

II-Quest for sonoluminescence fusion device

Around the end of the last century, the light emission produced by a small cavitation bubble created by an acoustic standing wave was studded mainly on its SBSL form (Single Bubble Sono Luminescence). A chemistry glass balloon field with a fluid as water vibrates under the excitation of piezoelectric drivers [START_REF] Putterman | Sonoluminescence: Sound into light[END_REF][START_REF] Putterman | Sonoluminescence: The Star in a Jar[END_REF]. A standing wave induces huge pressure variations at the center of the balloon, and a light emitting bubble can appear.

People got excited by sonoluminescence phenomena because of theorists who published results of calculation (neglecting a lot of parameters) showing temperature inside the cavitation bubble as high as 10 7 K [START_REF] Vi | Sonoluminescence and diffusive transport[END_REF], not so far of the 10 8

K temperature threshold for Deuterium-Tritium fusion. Other simulation papers taking into account lost parameters expected temperature in the range of the 10 5 K, and now some 10 4 K [START_REF] Wu | Shock-wave propagation in a sonoluminescing gas bubble[END_REF]. Measurements have shown, by fitting with black body radiation, temperatures from 25 000 to 50 000K [START_REF] Hiller | Spectrum of synchronous picosecond sonoluminescence[END_REF]. The temperature elevation reached by the bubble is due to the compression, its size passing from 50 µm diameter to 5 µm and even to 0.5 µm, involving compression factor from one thousand to one million. At those temperatures the gas inside the bubble is certainly a plasma as claimed in [START_REF] David | Plasma formation and temperature measurement during single-bubble cavitation[END_REF] where temperature of 15 200K were measured and very high energy excited states in Argon were seen. In this situation, the gas is an electric conducting medium. In 2002 was published a paper from Rusi Taleyarkhan (and al.) [START_REF] Taleyarkhan | Evidence for Nuclear Emissions During Acoustic Cavitation[END_REF] of Oak Ridge National Laboratory who claimed have seen nuclear fusion in a sonoluminescence experiment. Unfortunately, nobody seems could not have reproduce successfully this experiment. An overview of the field can be found in [START_REF] Michael | Single-bubble sonoluminescence[END_REF].

III-Z-pinch fusion

Another way to explore nuclear fusion is Z-pinch experiments in every of its representative, from the very large Sandia facility [START_REF] Sinars | Review of pulsed power-driven high energy density physics research on Z at Sandia[END_REF], to the more human size ZaP flow Z-pinch [START_REF] Den | The ZaP Flow Z-Pinch: Plasma Flow Shear and Stability[END_REF], Magnetized Liner Inertial Fusion (MagLIF) [START_REF] Gomez | Performance Scaling in Magnetized Liner Inertial Fusion Experiments[END_REF], Sheared Flow Stabilized Z-Pinch [START_REF] Forbes | Progress Towards a Compact Fusion Reactor using the Sheared-Flow Stabilized Z-Pinch[END_REF], and others. Sandia published extremely high temperatures of more than two billion degrees (2 x 10 9 K !!!) induced by the vaporization of a wire-array where 18 MA flew during 100 ns [START_REF] Haines | Ion Viscous Heating in a Magnetohydrodynamically Unstable Z Pinch at Over 2 x 10 9 Kelvin[END_REF].

The physical principle is the compression of a gas discharge under the force developed by the current inside the discharge under the magnetic field produce by itself. Each charge carrier, electron or ion, is moving under the effect of an electric field and is subject to the magnetic field produced by the motion of all the other charge carriers as shows fig. 1. The force developed in this magnetic field causes an electromagnetic pressure counteracted by the increase of the thermodynamic pressure due to the density and temperature augmentation. The resulting equilibrium is described by the Bennet relation [START_REF] Bennett | Magnetically Self-Focusing Streams[END_REF]. This relation establish the dependence between the current inside the plasma and the density temperature product as 8πNk(ZTe+Ti)=μ 0 I 2 [START_REF] Haines | A Review of Dense Z-Pinch[END_REF].The resulting increase of density and temperature can allow reaching the fusion conditions stated by the Lawson criterion which joints the density n, temperature T and confinement time t E in the following relation:

n T t E > 10 21 (keV m -3 s).
Z-Pinch has been proposed since a long time as a possible way for fusion reactor design [START_REF] Hartman | A conceptual Fusion Reactor Based On The High Plasma Density Z-pinch, Towards a Compact Fusion Reactor using the Sheared-Flow Stabilized Z-Pinch, ion[END_REF]. Many tentative of realization exist as Staged Z-Pinch [START_REF] Wessel | Fusion in a Staged Z-Pinch[END_REF], Sheared Flow Stabilized Z-Pinch [START_REF] Zhang | Sustained Neutron Production from a Sheared-Flow Stabilized Z Pinch[END_REF], Gas Puff [START_REF] Klir | Ion acceleration and neutron production in hybrid gas-puff zpinches on the GIT-12 and HAWK generators[END_REF]… Fusion was witnessed by neutron production measurements [START_REF] Zhang | Sustained Neutron Production from a Sheared-Flow Stabilized Z Pinch[END_REF][START_REF] Klir | Ion acceleration and neutron production in hybrid gas-puff zpinches on the GIT-12 and HAWK generators[END_REF][START_REF] Ruiz | Production of Thermonuclear Neutrons from Deuterium-Filled Capsule Implosions Driven by Z-Pinch Dynamic Hohlraums[END_REF][START_REF] Coverdale | Neutron production and implosion characteristics of a deuterium gas-puff Zpinch[END_REF], but it seems the difficulty consists to obtain a stable operation and neutron emission with an interesting energetic yield.

IV-X-Pinch

Instead of making a cylinder discharge, one other way explored is the try to obtain a punctual one. It is the x-pinch where the discharge is made between to crossing wires. It has been used as a high quality X rays source. A review can be found in [22].

V-Pinch triggered by Sonoluminescence

The first goal of these considerations was the quest for a photon source since the sonoluminescence is supposed to produce some light over a large continuous range of wavelengths from IR to hard X ray [START_REF] Burnett | Modeling a sonoluminescing bubble as a plasma[END_REF]. Furthermore, the radiation is pulsed, allowing temporal experiments as permitted by synchrotron radiation (but not with the same photon flux). In this perspective, it is necessary to think about an adapted geometry which avoids the photon absorption by the used fluid (usually water, acetone or sulfuric acid). One possibility, depicted fig. 2, corresponds to a cut of the classical SBSL balloon and the insertion of two cones.

The sound excitation can come from a current in a coil wrapped around the partly spherical vessel where a magnetic field is imposed. When an alternative current is supply to the coil, the vessel vibrates and this move leads to a wave focused at the center of the device. Very high pressure variations are then generated at the center of the sphere. The photons were supposed to be caught through a diamond window (fig. 3) for the isolation between the fluid and the vacuum (truly, over a small solid angle…). The excitation frequency, if the dimensions are the same as the usual SBSL device, is close to 25 kHz. It is to note that, starting from the SBSL balloon, it is natural to consider a partly spherical vessel to get a partially spherical wave, may be other shapes should have good performances, as cylinder in [START_REF] Taleyarkhan | Evidence for Nuclear Emissions During Acoustic Cavitation[END_REF].

Comparing with z-pinch problematic where people seek for high density and temperature gas in respect to the Lawson criteria, but start with very low gas pressure we considered the complementarities of those two worlds, sonoluminescence and z-pinch (fig. 2).

Assuming an electrically insulating fluid, if a high voltage is set between the two cones, a discharge can occur at the apex of the cones when the gas inside the bubble, which should be Deuterium-Tritium for instance, turns to plasma by effect of cavitation, and so become a conductive media. One problem evocated inside the literature about the z-pinch is the high voltage commutation speed. This problem could be avoided in the proposed configuration because the passage of the insulating state to the conductive one is supposed to be very fast in the bubble, in the sub nanosecond range as the variation speed of temperature let expect [START_REF] Burnett | Modeling a sonoluminescing bubble as a plasma[END_REF],

commutation speed which is very difficult to obtain by other technical ways. At this time, we can hope to obtain similar effects than in classical z-pinch which were only tested at this time, as we know, starting with a low pressure and temperature gas whereas the bubble offers high temperature and high density.

When the discharge is terminated, the switch SW can be set ON to charge the capacitor C for the next cycle. It is to note that the switch SW uses only at the capacitor C charge and can be replaced by a resistor at the cost of more energy dissipation. In the case where the fluid is not a good insulator, to avoid conductive losses the cones must be covered with an insulating layer, except at their apex where the current must flows.

Considering very naïve thermodynamic consideration, supposing that the model is still valid when very high energy density is involved, we can estimate the energy in the bubble at the room pressure and temperature and when the spherical acoustical wave have compressed it at its maximum. For a quite small bubble of 50 x 10 -6 m diameter, at T = 300K and P = 1atm, the number of mole of gas is n = 2.92 x 10 -12 . The thermal energy stocked in the bubble is U = 3/2 nRT = 1.09 x 10 -8 J.

At 30 000K the energy rises to U = 1.09 x 10 -6 J and at T = 10 8 K the energy stored reaches U = 0.00364 J. By comparison with the energy stored inside a capacitor of 1 µF under 100V is Uc = 0.005 J. Since the capacitor can be augmented up to thousands of µF and the applied voltage increased up to kV or more we can see there is a possibility to raise the electric energy communicates to the bubble by a factor of tens to hundreds of thousands, even taking into account all the losses. Considering all those things, we can reasonably hope far exceed the Lawson criteria and get a working tabletop fusion device.

The gas of the bubble can come from the fluid outgasing or by a feeding system installed on the axis of one cone, involving a needle or just a small hole (fig. 3) to let in the gas at the low pressure part of the cycle.

If working, the energy released by fusion could induce a pressure wave inside the fluid which propagates to the vessel. If the coil of wire length L around the vessel is forced to move at the speed v inside the space where is the magnetic field B, a voltage E = v.L.B is generated at the ends of the coil. If the system can be set in a resonant mode, may be it can acts as a direct fusion conversion apparatus and the electric energy harvested directly on the vessel coil, which doesn't have to be use any more for the acoustic excitation but for energy extraction. Due to the small volume of gas inside the bubble, such a device could afford energy in the range of kW or tens of kW, instead of tens of MW as in classical z-pinch realizations, which is enough to supply a house or a vehicle.

Such a device, if working, has to include safety protections about a large sound level, possibly fast neutron emission and heat production. A solution could be its installation in a tank, filled with water or any suitable fluid, set at the proper dimensions and shape according to the working wavelength.

An important point to have a spherical wave is the choice of the cones material which must have specific acoustic impedance far higher than the used fluid. This condition excludes the polymers and can allow the use of metal or ceramic parts.

At the contrary, mechanical spacers set to hold the cones together should use PMMA or any matter with acoustic impedance close to those of the fluid. At the apex of the cone, it also has to withstand electric sparks, and so, intend to select refractory metal as Tungsten or Tantalum.

If the bubble diameter is small regarding the distance between the cones, the high voltage between them must be just under the disruptive voltage in the fluid, which is for pure water around 130kV/cm for instance [START_REF] Gerasimov | Water as an Insulator in Pulsed Facilities[END_REF]. If the gas is in touch of both cones due to a filling system tuning, this voltage should be lower, because when the gas turns to plasma, it can conduct all over the interspaces.

If fusion can be induced by the way describe here, we can imagine other geometries as for instance the push-pull configuration shown fig. 4. Anyway, in all realizable configurations, except for electrical measurements, current and voltage, instrumentation will be quite a challenge and shall require a lot of imagination.

VI-Conclusion

Starting from a state closer to the Lawson criteria, can we hope to reach the fusion conditions more easily? Only the experiments can answer this question.

For the future, we will perform some multiphysic simulation to determine a suitable geometry before trying a realization. In the sake of finding a so desirable energy source, we hope this proposition is realist and other groups can be interested in it. The system should also be investigated as a pulsed X ray source.

Fig. 1 a

 1 Fig. 1 a) Each of two conductors is subject to a force F due to the magnetic field induce by the current inside the other. b) Each charge carrier, ion or electron, is subject of a force F i or F e induced by the magnetic field generated by current of all the others charge carriers.

Fig. 2 13 Fig. 4

 2134 Fig. 2 Sketch of the proposed device


Vandellia Cirrhosa

Peter Henderson

Elmer Martinez

/ Afp

Adina Lazar

Marianne Chouteau

Jean-Philippe Ferrière

Thierry Moyaux

Claire Barrès

Corinne Dorel

Stéphane Frénot

Guillaume Beslon

Bastien Humbert

Clarisse Masse

Diana Martin De Argenta

Paul Saada

Hugo Paris

Laurence Dupont

Aedes albopictus

Keywords: Cryptosporidium, Giardia, Entamoeba, Plasmodium, Trypanosoma

Axinotarsus pulicarius

Qu'est-ce que le vivant ?

u

  Fig. 13 : Biodiversité́et santé́humaine. Reproduit d'après « Connecting global priorities : Biodiversity and human health », Organisation mondiale de la santé(OMS) et Secrétariat de la Convention sur la diversité́biologique (CDB), Copyright (2015) WHO/CBD (2015).

uu

  Bien-être, épanouissement, sport, spiritualité (lien avec la santé et la nutrition) Tous les services écosystémiques ne sont pas quantifiables (notamment pour les peuples autochtones)Olios argelasiusAlexandre Lacroix, Devant la beauté de la nature (2018) Eds Allary. Enjeux n°6 : éthiques u En finir avec la vision écopaternaliste de la protection de la nature u Apprendre à s'émerveiller de la nature pour mieux la comprendre et y trouver sa place Loroglossum hircinum u Peut-on éliminer les baleines, les éléphants, les tigres, les rhinocéros, les sabots de vénus, les orchis vanilles ? Conclusions : des leviers d'actions 34 Aphis rumicis Echelle mondiale : les scénarios de l'initiative Bending the curve (GIEC, IPBES) Rapport du WWF Planète vivante 2020 (voir aussi en annexe un exemple de recommandation de l'IPBES) supérieur, doit jouer son rôle sur ces 4 piliers par une politique volontariste qui est actuellement beaucoup trop timide sur les enjeux du vivant. u L'INSA souffre-t-il du syndrome de myopie au désastre* ? A l'échelle individuelle pour un ingénieur citoyen u Changer nos modes de vie et de consommations (cf. actions pour le climat) u Prendre conscience de notre place et de notre dépendance aux écosystèmes pour toutes nos activités afin d'accepter d'en payer le vrai prix u Sortir de la posture du solutionnisme et de ses propres champs de compétences pour appréhender les problèmes de façon systémique en intégrant les enjeux du vivants u Développer un humanisme qui dépasse le dualisme nature-culture (quête de sens) u « Raviver les braises du vivant » (B. Morizot) Biodiversité : un néologisme récent connoté par la conservation 43 Ludovic Vievard. Biodiversité : usages et représentations. Rapp. pour la Dir. la Prospect. du Dialogue Public Gd. LYON. 2011. Corvus_corone

Source : B. Locatelli et al., Analyser des services écosystémiques pour gérer des territoires (diapo Equipe DDRS FIMI)

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