Benjamin Vincent

Olivier Chadebec

Jean-Luc Schanen

Carlos Antonio

França Sartori

Laurent Krähenbühl

Ronan Perrussel

Kévin Berger

New robust coil sensors for near field characterization

Keywords: Coils electromagnetic sensors, near field measurements, multipolar expansion, radiated EMC, radiated electromagnetic sources characterization

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

With the increase of power density in modern power systems, interactions between various elements become a key point to be addressed. This problem can be handled at various scales: interactions between two variable speed drives or power transformers in the same room ("indoor EMC"), interactions between power and the control board, or between different devices on the same board. On this latter example, magnetic couplings between magnetic components (inductor, transformer) and/or wires are often encountered and may lead to dysfunctions or a bad filtering.

Addressing these magnetic couplings implies to have the ability to characterize the magnetic field close to the sources. The whole knowledge of this near field can be achieved using simulation or experimentation.

The first idea starts from a simple mapping of the near magnetic field, using some magnetic sensors and sampling the space by moving the sensors around the system, by moving the system itself, by increasing the number of sensors, or by using any appropriate combination of all previous solutions. After this step, the magnetic field is characterized by a set of thousands of measurements. These data must therefore be summarized into a simpler equivalent source in order to make possible the study of interactions.

One method to describe equivalent sources is the multipolar expansion, which has previously

New robust coil sensors for near field characterization

shown its ability to synthesize any magnetic source into known standardized sources (dipole, quadrupole, etc.). For instance, it has been shown that a set of several thousands of data points can be summarized into 15 terms, with a large validity region [START_REF] Keradec | Nano-teslameter for characterization of the fields emitted and received by wound electronic components[END_REF].

However, the post-processing of punctual measurements to obtain the coefficients of the multipolar expansion leads to inaccuracies, either due to measurement errors or due to mechanical positioning errors [START_REF] Rouve | Optimal magnetic sensor location for spherical harmonic identification applied to radiated electrical devices[END_REF], [START_REF] Kildishev | Measurement of the Spacecraft Main Magnetic Parameters[END_REF].

The present work tackles particular sensors, which are sensitive to a specific component of the multipolar expansion. The principle is based on the measurement of the magnetic flux through large coils placed around the device. These coils are used to achieve a spatial integration and consequently to reduce positioning inaccuracies [START_REF] Kildishev | Measurement of the Spacecraft Main Magnetic Parameters[END_REF]- [START_REF] Savi | Developing a Van Veen & Bergervoet Antenna[END_REF]. As an example, to measure the first order of the decomposition (dipole), the Standard CISPR 16-1 [6], proposes using three orthogonal loops (Fig. 1).

This standard applies for the medium frequency range: 9 kHz to 30 MHz.

The problem is that no source is composed of pure dipoles. Therefore, even the standard method leads to errors. For the sensor design, we propose to take into account higher order terms in the expansion, and to improve the selectivity property, as proposed initially in [START_REF] Pasmooij | A 'Helmholtz' large loop antenna system for improved magnetic field measurements[END_REF].

II. MULTIPOLAR EXPANSION

The multipolar expansion is a classical tool used for electromagnetic field representation [START_REF]Classical Electrodynamics[END_REF], [START_REF] Wikswo | Scalar multipole expansions and their dipole equivalents[END_REF]. It allows decomposing any field into an infinite sum of simple terms. The first order of the decomposition is the well known dipole. Several mathematical basis functions can be used for the decomposition. For near field studies, the quasi-static approximation is suitable i.e. displacement currents can be neglected. Therefore, the basis does not include propagation terms [START_REF]Classical Electrodynamics[END_REF]. Near field area is limited by /6 meters. According to the maximum frequency, quasi static approximation is available up to 1.7 meter.

A.Time and space harmonic decompositions

Outside a sphere including all radiation sources, the magnetic field can be completely described by a magnetic scalar potential (t). This potential is solution of the Laplace equation:

0 ) (   t  (1)
We consider the harmonic decomposition of time varying sources. Each term of the harmonic decomposition of the time varying solution of (1) can be expressed as follows:

                                   1 1 1 ) , , ( ) ( ) , , ( , 1 )
, , (

n n n m nm n n n m nm n nm r r Y r A r B B grad B         (2)
where:

-Ynm are the spherical harmonics of order n, -r, the distance between the center of the decomposition and the point where the field is expressed, -Bnm, the elementary magnetic fields of the decomposition, -Anm, the coefficients of the decomposition, i.e. the unknowns.

B.Relevance

The choice of this multipolar expansion is mainly motivated by the r -(n+1) decrease of the terms of order n. This insures a hierarchy between each order of the decomposition. The larger the distance to the source is, the fewer are the terms required to reconstruct the field.

For interaction studies, we have decided to limit the expansion to the two first orders of the decomposition. This also corresponds to the physical behavior of most sources in electrical systems, where disturbances often originate from current loops. Therefore, the number of unknowns is limited to only 8 (from (2), there are 2n+1 coefficients for each order n; thus, 3 unknowns for order 1 and 5 for order 2).

The aim of the identification is to deduce these 8 unknowns from measurements achieved around the radiation source. For this purpose, one possible solution is to take advantage of the fast decrease of the high order terms with the distance: far from the source, the sensors are only sensitive to the dipole. This property has been used in Standard CISPR16-1 [4]- [START_REF] Pasmooij | A 'Helmholtz' large loop antenna system for improved magnetic field measurements[END_REF]. This leads to large measurement systems, and furthermore, the accuracy is poor: far from the Device Under Test (DUT), the field is low, and the signal is small. Furthermore, the size implies frequency limitations: it is difficult to reach 30 MHz without accounting for propagation effects. Instead of taking advantage of the r -(n+1) behavior only, our design strategy, introduced in the next section, is to use the geometrical properties of the spherical harmonic functions Ynm.

III. SENSOR DESIGN

A. Preliminaries

Each component sensor, dedicated to a specific component of the multipolar expansion is constituted of several coils. They are located on a sphere SM, with a radius r0, including the validity sphere region. An example is shown in Fig. 2. The principle consists in measuring the magnetic flux through the coils. As a component sensor is constituted of several coils, it is possible to compose all flux (sum or difference). The following representation will be used in this paper: a red coil will correspond to a positive outgoing flux, whereas a blue one will correspond to a negative outgoing flux (see Fig. 2

for example).

Any set of coils on the sphere SM can then be represented by a "sensor function" M defined by:  M(, )= 1 for a portion of SM delimited by a red coil,  M(,) = -1 for a portion of SM delimited by a blue coil,  M(,) = 0 for the remaining SM area.

The measurement corresponds to:

dS r B M dS r M Flux M M S r S      ) , , ( ) , ( ) , , ( ) , ( 0 0         n B ( 3
)
where n is the unit outgoing normal of SM and Br(r0,,) the radial component of the induction on SM, defined by:

          1 2 0 0 ) , ( ) 1 ( ) , , ( n n n m nm nm n r Y A r n r B     (4)
The ideal sensor would be sensitive to a single component of (4). We will show that it cannot be realized but, if the decomposition in ( 4) is truncated to n = 4, solutions exist. Consequently, the most complex multipolar field created by the source is limited to the 4 th order (24 components) in our study. It has been considered sufficient for most of practical power electronic radiating systems.

In the following, let n and m denote the indices of the component Bnm to identify and Mnm the "sensor function" dedicated to Bnm. Indices n' and m' will be used for the other components which can create a parasitic flux. This disturbance flux will have to be cancelled by sensor function Mnm. The problem can be written as:

                      M M S m n nm m n n n n m n S n n n m m n m n n nm nm dS Y M A r n dS Y A r n M Flux ) , ( ) , ( ) 1 ' ( ) , ( ) 1 ' ( ) , ( ' ' ' ' 4 1 ' ' ' ' 2 ' 0 4 1 ' ' ' ' ' ' ' ' 2 ' 0             M S nm nm nm n dS Y M A r n ) , ( ) , ( ) 1 ( 2 0     (5)
Thus, we have to design, for each component, a function Mnm such as:

) , ( ) ' , ' ( , 0 ) , ( ) , ( ' ' m n m n dS Y M M S m n nm         (6)
In other words, the goal is to accomplish a spatial filtering of Br (24 components) thanks to functions sensors Mnm (8 sensors for the first 8 components). Equation [START_REF] Savi | Developing a Van Veen & Bergervoet Antenna[END_REF] shows us that the measurement of flux allows an efficient identification of coefficients Anm. Moreover, we can see that the field distribution on sphere SM is directly linked to the characteristics of functions Ynm. Their study is therefore sufficient for finding Mnm.

B. Behavior of functions Ynm

The real spherical harmonic function of order n and degree m is given by ( 7):

      m nm nm Y    .
where,

 

)) (cos( )! (

)! (   m n nm P m n m n              ) sin( 2 1 ) cos( 2    m m m if 0 0 0    m m m ( 7)
The Pnm are the Legendre polynomials defined thanks to Rodrigues formulae [START_REF] Nedelec | Acoustic and electromagnetic equations: integral representations for harmonic problem[END_REF].

From [START_REF] Pasmooij | A 'Helmholtz' large loop antenna system for improved magnetic field measurements[END_REF], we can deduce that:

) 2 / , ( ) , ( , m Y Y nm m n          (8)
The only difference between Ynm and Yn,-m is a rotation of /2m around z axis. The study can therefore be limited, for each order n, to the cases 0  m . The 8 sensor shapes can be reduced to 5 (2 for n = 1 and 3 for n = 2). Functions Ynm are presented on Fig. 3 for the four first orders. To simplify the coil design process, their shapes will be chosen with constant  or  angles, i.e. these coils will follow only meridians or parallels of the sphere. This design constraint is suggested by [START_REF] Pasmooij | A 'Helmholtz' large loop antenna system for improved magnetic field measurements[END_REF] where the function Ynm depends separately on  and φ variables. Therefore, function Mnm can be written as:

            nm nm nm M M M . ,  (9)
Thus, the design method can separately study geometrical characteristics of functions n'm' and m'.

Indeed, according to [START_REF] Pasmooij | A 'Helmholtz' large loop antenna system for improved magnetic field measurements[END_REF] and ( 9), ( 5) is equivalent to:

                                         d M d M A r n Flux m nm m n nm nm n n n m n nm 2 0 ' 0 ' ' 4 1 ' ' ' ' 2 ' 0 ) ( ). ( ) sin( ) ( ). ( ) 1 ' (                                 d M d M A m nm nm nm nm 2 0 0 ) ( ). ( ) sin( ) ( ). ( (10)
Last, according to [START_REF] Nedelec | Acoustic and electromagnetic equations: integral representations for harmonic problem[END_REF] and ( 6), the problem can be summarized by: 0

) ( ). ( ) sin( ) ( ). ( 2 0 0                                 d M d M m nm nm nm (11a) ) , ( ) ' , ' ( , 0 ) ( ). ( ) sin( ) ( ). ( 2 0 ' 0 ' ' m n m n d M d M m nm m n nm                                   (11b)
Each function Mnm has to maximize (11a) with respect to (11b). In (11b), one of the two integrals has to be zero to fulfill the condition.

1) Behavior of n'm'

Let's suppose that the second integral in (11b) is not equal to zero. The study of the properties of n'm' allows canceling the first integral for some cases. Indeed, when the sum n'+m' is even, functions Yn'm' are symmetrical with respect to the equator plane (they are labeled with a black dot on Fig. 3).

The others are anti-symmetrical. The sensor function Mnm should therefore have the same properties than the corresponding function Ynm. Indeed, if it presents a symmetry with respect to the z = 0 plane, anti-symmetrical Bn'm' components will not create any disturbance flux, and vice versa. In the following, this principle will be chosen. Fig. 4 shows the case of component Y21 and the associated sensor geometry. At this step, there is no constraint on the opening angle S. Thus, it is chosen as maximal (S =  /2 in M21 case on Fig. 4), what allows maximizing the signal for the flux created by

B21.

Fig. 4. Component Y21 and a first design for the associated detection coils shape M21.

2) Behavior of m'

The goal is now to cancel the second integral in (11b), the first one being possibly not equal to zero.

According to the previous step, let us remember that for symmetrical sensors only symmetrical components have to be studied and inversely for anti-symmetrical ones.

The m' functions are m'-periodical. Therefore, functions  nm M will also have this property as shown on Fig. 4. On this example, the opening angle S of each coil is limited to the /m maximal value. It remains now to evaluate the disturbance flux created in such coils by others components of the decomposition.

According to (11.b) and ( 7), we have:

         2 0 ). ' cos( ). ( d m M Flux nm nm (12)
where the variations of 

3) Influences

The previous results allow determining the table I. It shows elements Ynm to be identified in blue, and those, which disturb their identification in red. At this step, constraints imposed on functions sensors are:

 the symmetry (on the right hand side) or the anti-symmetry (on the left hand side) in comparison with the equator plan z = 0,  the periodicity linked to the degree of the searched component.

Fluxes have to be maximized; therefore, all opening angles S and s are chosen to get a maximal value by respecting these constraints. For example, concerning Y11 detection, only elements with an even (n+m) can disturb it; this corresponds to all those appearing in the left part of the table I. But, according to the M11 periodicity, only the elements of degree m' = 3 remains. It corresponds to components Y31 and Y33.

TABLE I. TABLE OF INFLUENCES FOR THE FIVE DIFFERENT

SHAPES n=1 Y 1,0 Y 1,1 n=2 Y 2,0 Y 2,1 Y 2,2 n=3 Y 3,0 Y 3,1 Y 3,2 Y 3,3 n=4 Y 4,0 Y 4,1 Y 4,2 Y 4,3 Y 4,4 m = 0 1 2 3 4 0 1 2 3 4 Y 1,0 Y 1,1 Y 2,0 Y 2,1 Y 2,2 Y 3,0 Y 3,1 Y 3,2 Y 3,3 Y 4,0 Y 4,1 Y 4,2 Y 4,3 Y 4,4 n=1 Y 1,0 Y 1,1 n=2 Y 2,0 Y 2,1 Y 2,2 n=3 Y 3,0 Y 3,1 Y 3,2 Y 3,3 n=4 Y 4,0 Y 4,1 Y 4,2 Y 4,3 Y 4,4 m = 0 1 2 3 4 0 1 2 3 4 Y 1,0 Y 1,1 Y 2,0 Y 2,1 Y 2,2 Y 3,0 Y 3,1 Y 3,2 Y 3,3 Y 4,0 Y 4,1 Y 4,2 Y 4,3 Y 4, 4

4) How to remove the last cross effects?

The last step consists in reducing the opening angles S and S in order to remove the last cross effects. However, it will lead to a decrease of the measured flux.

a)

According to S Equation (13a) points out that it is possible to cancel the parasitic components whose index is a multiple of 3 by choosing S = 2/3. So, influences of Y33 on Y11 and of Y43 on Y21 disappear. It remains then only a single parasitic signature by aimed component.

b)

According to S This opening angle can have values included between 0 and . Two different cases lead to two different studies. For the case where functions Mnm are symmetrical, the only way to reduce them is to cut the coils from poles. The value of the angle, which cancels the disturbance component, is given by the solution of (14a). For the anti-symmetrical case, the cut must begin from the equator and the solution is given by (14b). Indeed, in both cases, the cancellation of the first integral in (11b) is achieved by looking when

  1    nm M .   0 ). sin( . 2 / ) ( 2 / ) ( ' '      S S d m n        (1 4a)   0 ). sin( . 0 ' '    S d m n     (1 4b)

C. Example design of sensor M21

To illustrate the proposed method, we focus on the design of function M21. The complete process is shown on Fig. 6. At first, on the left, in accordance with parts B.1.) and B.2.), a first basic shape is drawn. It fits at the same time the two previous constraints: the anti-symmetry with the z = 0 plane and the periodicity on the azimuth of component Y21. This design, with a smart composition, allows maximizing the fluxes in each coil (please take care to the winding way representing by blue and red colors). According to the table I, only elements Y41 and Y43 still disturb identification. In a second step, according to equation (8b), reducing S = 2/3 allows to cancel the effect of Y43. It is shown on the second sphere. Finally, on the right hand side, an angle, solution of equation (14b), exists, which allows to compensate exactly both lobes create by Y41: s = 1.35 rd. According to notations of Fig. 6 (on the right) and after some algebra, (5) gives identification relation: The method is the same for designing all other sensors defined by functions Mnm. The obtained shapes are presented in table II of the appendix.

 

IV. VALIDATION BY THE FINITE ELEMENT METHOD (F.E.M.)

To validate the method, a toroidal inductor has been simulated (5 cm diameter -Fig. 7). It corresponds to the common mode filter of a variable speed drive. The aim is to determine the coefficients of the multipolar expansion. We have included the identification of the first and second order of the decomposition (all 8 sensors). The coefficients A1m and A2m have been identified at a distance of 10 cm from the inductor geometrical center. This distance is sufficient, with respect to the tore radius (2.5 cm), in order to keep valid the approximation of the reference source (limited at the fourth order).

All tests have been carried out in a finite element Software FLUX3D [11], using a scalar magnetic potential formulation coupled with an infinite box transformation accounting for the external air region. The flux generated through the coils is computed using a post-processing surface integration. Fig. 7. Toroidal inductor and the associated variable speed drive.

A. Identification of Anm and validation

After identification, Fig. 8 presents from 0.1 m to 2 m:  the field obtained by the FEM,  the reconstructed field including the first and second order terms,  the reconstruction using the first order only,  the reconstruction using the first order only, obtained by an identification following the Standard CISPR16-1.

The global behavior of all models is similar. However having a look at the relative error -with respect to the FEM solution-it can be seen that the first order is less accurate than the second order identification, close to the source. This was expected by the multipolar expansion theory.

Furthermore, the first order identification obtained from the standard clearly shows that there was some inaccuracy in the dipole determination: it leads to about 13% of field error even far from the source. On the other hand, our sensors have determined the correct value, since far from the source, the field is well reconstructed.

The error in the determination of the dipole using the standard is due to the third order contribution in the sensor, according to table I. It is especially important since the identification has been carried out at a distance of 10 cm, where the high order components of the source have not decreased sufficiently.

The new sensor shape allows thus providing identification closer to the source than the standard (which recommends 1 meter). This is interesting since it brings more signal level and also reduces some propagation issues in the large coils. First order identification error

B. Identification and robustness

First, we check the robustness of the sensor to a change of sensor radius r0: three different sensors for three different measurement distances have been tested, the identification has been found very stable (less then 1% error on the identification). It confirms that it is not necessary to be far from the source to identify the dipole and quadrupole, contrary to the standard solution [START_REF] Bergervoet | A large-loop antenna for magnetic field measurements[END_REF]- [6].

The second verification concerns a mismatch between the inductance center with respect to the decomposition center. This implies a more complex field (i.e. higher orders in the decomposition) [START_REF] Wikswo | Scalar multipole expansions and their dipole equivalents[END_REF].

Only dipole identification is concern by this phase. The results of the FEM simulation show that an error on the dipole determination occurs (Fig. 9). It is due to the contribution of B5m (and higher orders) terms in our sensor. However, the error is reasonable, and lower than using the Standard antenna, since this latter is sensitive starting from B3m.

V. CONCLUSION

The multipolar expansion is attractive to synthesize a magnetic field because it provides accuracy with only few parameters. To identify the components of the decomposition, it seems not convenient to use several punctual measurements. Therefore, we choose to develop dedicated sensors, sensitive to a specific component of the expansion only. These coils have been designed, with special shapes, until the second order of the decomposition. The validation of all sensors has been carried out using a finite element software. Thus, we have shown that Standard sensor could be improved. Indeed, on the one hand, our sensors do not create identification error when the radius decreases. On the other hand, they are more robust when dipole shifts from the system center. Furthermore, we have shown that it could be possible and interesting to design sensors which also identify the quadrupole components by filtering exactly the components of orders 3 and 4.

The construction of the sensors is in progress, and experimental results will be given in future papers. Nevertheless, the construction of the complete system is still complex. We have restrained the sensor design with constant  or  angles. Without these constraints, other geometrical properties of Ynm lead to the same identification with only two different shapes of sensor: future works will expose these developments.

Fig. 1 .

 1 Fig. 1. From Standard CISPR 16-1, the loop-antenna system, consisting in three perpendicular large loop-antennas. The mathematical model of the equivalent sources is first recalled. The design of the sensor geometry can then be detailed and validated by a finite element model of an actual device.

Fig. 2 .

 2 Fig. 2. An example of a component sensor located on sphere SM and centered on the Device Under Test. The sensor is composed of several coils defined by anglesS and S.

Fig. 3 .

 3 Fig. 3. Spherical harmonic functions of the reference source for 4  n and 0  m .

. 5 .

 5 We are in the well-known configuration of Fourier series decomposition where m' is equivalent to the signal pulsation. Thus:  If m' < m, component Yn'm' do not create any disturbance flux. Indeed, the first non zero coefficient of the expansion corresponds to the fundamental of  nm M . If m' = m, the integral in (12) is not zero.  If m' > m, the only components Yn'm' creating a flux have a degree m'=(2k+1)m (k integer).Indeed, for the m = 1 and m = 2 cases, we get: ) and (13d) the integral vanishes independently of sensor angle S.Finally, let us point out that for the m = 0 cases, only the components of same degree (m' = 0) create a parasitic signal.

Fig. 5 .

 5 Fig. 5. nm M functions for m = 1 and m = 2 cases.

Fig. 6 .

 6 Fig. 6. Complete process for the sensor function M21 design.

Fig. 8 .

 8 Fig. 8. Magnetic field decrease (modulus) vs the distance to the source: comparison between the identified dipole/quadrupole with the proposed and Standard sensors and the actual field, simulated by the FEM.

Fig. 9 .

 9 Fig. 9. Comparison between Standard antenna and the proposed sensor for a shifting source: relative error of the dipole identification compared to the actual dipole. The dipole computation is achieved by summing the three components of the dipole: 2 1 1 2 11 2 10    A A A

2 ( 1 )

 21 Same shape in the opposite side,

( 2 )

 2 for m < 0, sensors are rotated by m 2

  Escola Politécnica da Universidade de São Paulo (Lmag/PEA-EPUSP), 05508-900 São Paulo, SP, Brasil (3) Ampère (CNRS UMR5005), Université de Lyon, Ecole Centrale de Lyon, 69134 Ecully CEDEX, France.

	New robust coil sensors for near field
	characterization
	B. Vincent 1 , O. Chadebec 1 , J.-L. Schanen 1 , C.A.F. Sartori 2
	L. Krähenbühl 2, 3 , R. Perrussel 3 , K. Berger 3
	(1) Grenoble Electrical Engineering Lab (G2Elab, CNRS UMR5269)
	INPG-UJF, ENSIEG, BP 46, 38402 Grenoble, France
	(2)

ACKNOWLEDGMENT

This work was supported in part by the CAPES-COFECUB (project 568/07) and the FAPESP (project 2008/53611-9).


François Robert
email: francois.robert@mnhn.fr

Peter Reinhardt

Romain Tartèse

Mass-Independent Fractionation of Titanium Isotopes -Supplementary Material

Keywords:

Samples B. Experimental conditions of NanoSims analyses

Experimental products were gold coated and mounted on 1 inch holders. Analyses were carried out using an O -primary beam current of 15 pA (D1-2, 300 µm diameter aperture), with an accelerating voltage of 16 kV. Each analysis was preceded by 10 min presputtering using a 100 pA primary beam (D1-1, 750 µm diameter aperture) rastered on the sample over 30 µm × 30 µm areas to eliminate any surface contamination. Secondary ionic species of 12 C + , 35 Cl + , 46 Ti + , 47 Ti + , 48 Ti + , 49 Ti + and 50 Ti + were then collected simultaneously on seven electron multipliers in imaging mode. For high spatial resolution analysis, 42 frames were recorded in approximately 20 minutes for a 3µm × 3µm area, divided in 256 × 256 pixels, with a dwell time of 400 µs/pixel. The grain is located in the center of the area. The thickness of the sputtered surface is of the order of a few tens of nanometers. In order to adequately resolve isobaric interferences of Ti hydrides on Ti isotopes (e.g., (m-1) TiH on m Ti), the mass resolving power was set to 8000 (CAMECA definition). All the data were processed off-line using the l'Image software package (L. Nittler,

Experimental samples were imaged with the NanoSims ( 48 Ti images shown as examples) and observed by scanning and transmission electron microscopy (SEM and TEM). No diffraction patters of the Ti-rich material could be observed suggesting a non-crystalline nature of the Ti-bearing phase. The nature of the Ti-bearing phase (TiN or TiC or Ti as an organic heteroelement?) is therefore still uncertain so far. Figure SM1 shows at different scale, the Ti distribution in the carbonaceous grains deposited on the silicon wafer. Analytical details can be found in Reference 1. [START_REF] Buchachenko | Magnetic isotopes as a means to elucidate Earth and environmental chemistry[END_REF] Ti NanoSims ionic images and SEM micrographs. The image on the left shows a field of µm size grains deposited in the carbonaceous layer of the silicon wafer. The NanoSims image at the center shows, at higher magnification, the distribution of Ti in the grain presented in this paper. Images have been smoothed by a moving 3 × 3 average, corresponding to 235 (left image) and 35 nm (central image) spatial resolution. The SEM image on the right is a cross section of the Ti-rich carbonaceous layer (50 nm) deposited on the silicon wafer where the grains appear as bright spots (arrows).

FIG. SM1:

I. INTRODUCTION

The discovery of the mass-independent isotope fractionation (MIF) of oxygen isotopes in ozone [START_REF] Thiemens | The Mass-Independent Fractionation of Oxygen: A Novel Isotope Effect and its Possible Cosmochemical Implications[END_REF][START_REF] Heidenreich | A non-mass-dependent oxygen isotope effect in the production of ozone from molecular oxygen -the role of molecular symmetry in isotope chemistry[END_REF][START_REF] Thiemens | Introduction to Chemistry and Applications in Nature of Mass Indepedendent Isotope Effects Special Feature[END_REF] has drawn considerable theoretical and experimental interest. [START_REF] Anderson | Laboratory measurements of ozone isotopomers by tunable diode-laser absorption-spectroscopy[END_REF][START_REF] Thiemens | Pressure dependency for heavy isotope enhancement in ozone formation[END_REF][START_REF] Morton | Laboratory studies of heavy ozone[END_REF][START_REF] Krankowsky | Atmospheric chemistry -Heavy ozone -A difficult puzzle to solve[END_REF][START_REF] Mauersberger | Ozone isotope enrichment: Isotopomer-specific rate coefficients[END_REF][START_REF] Janssen | Kinetic origin of the ozone isotope effect: a critical analysis of enrichments and rate coefficients[END_REF][START_REF] Gao | Strange and unconventional isotope effects in ozone formation[END_REF][START_REF] Babikov | A quantum symmetry preserving semiclassical method[END_REF][START_REF] Gao | On the theory of the strange and unconventional isotopic effects in ozone formation[END_REF][START_REF] Schinke | Isotope dependence of the lifetime of ozone complexes formed in O+O 2 collisions[END_REF][START_REF] Schinke | The effect of zero-point energy differences on the isotope dependence of the formation of ozone: a classical trajectory study[END_REF][START_REF] Ivanov | On molecular origin of mass-independent fractionation of oxygen isotopes in the ozone forming recombination reaction[END_REF][START_REF] Teplukin | Several levels of theory for description of isotope effects in ozone: Effect of resonance lifetimes and channel couplings[END_REF][START_REF] Ndengué | Ozone photolysis: Strong isotopologue/isotopomer selectivity in the stratosphere[END_REF][START_REF] Ndengué | Ozone Photodissociation: Isotopic and Electronic Branching Ratios for Symmetric and Asymmetric Isotopologues[END_REF][START_REF] Ndengué | Comparison of the Huggins Band for Six Ozone Isotopologues: Vibrational Levels and Absorption Cross Section[END_REF][START_REF] Robert | An experimental test for the massindependent isotopic fractionation mechanism proposed for ozone[END_REF] However, no consensus has been yet reached in the literature about the origin of MIF. Interestingly, a similar isotopic distribution of oxygen isotopes was observed fifty years ago in the refractory minerals of the carbonaceous meteorites (usually referred to as CAIs, Calcium-and Aluminum-rich inclusions). [START_REF] Clayton | A component of primitive nuclear composition in carbonaceous meteorites[END_REF] These minerals are regarded as the first solids condensed at 1200-1600K in the early Solar System (the so called protosolar disk [START_REF] Ebel | Condensation of Rocky Material in Astrophysical Environments[END_REF] ).

For heavier elements such as N, C, Ti, Si etc., sub-micrometer grains isolated from meteorites have shown isotopic compositions having marked departures from mass-dependent fractionation [START_REF] Ireland | Isotopically anomalous Ti in Presolar SiC from the Murchison meteorite[END_REF][START_REF] Hoppe | Carbon, nitrogen, magnesium, silicon and titanium isotopic compositions of single interstellar silicon carbide grains from the Murchison carbonaceous chondrite[END_REF][START_REF] Zinner | Stellar nucleosynthesis and the isotopic composition of presolar grains from primitive meteorites[END_REF][START_REF] Alexander | The galactic evolution of Si, Ti, and O isotopic ratios[END_REF][START_REF] Hoppe | Presolar dust grains from meteorites and their stellar sources[END_REF][START_REF] Zinner | PreNanoSIMS isotopic analysis of small presolar grains: Search for Si3N4 grains from AGB stars and Al and Ti isotopic compositions of rare presolar SiC grains[END_REF] (MDF). These grains are referred to as "presolar grains" and their isotopic compositions attributed to stellar nucleosynthetic signatures. [START_REF] Zinner | Stellar nucleosynthesis and the isotopic composition of presolar grains from primitive meteorites[END_REF] Note that, heavy element MIF originally found in CAIs [START_REF] Niederer | Absolute isotopic abundances of Ti in meteorites[END_REF] were recently resolved at a much lower level of detection between planetary components. [START_REF] Trinquier | Origin of Nucleosynthetic Isotope Heterogeneity in the Solar Protoplanetary Disk[END_REF][START_REF] Leya | Titanium isotopes and the radial heterogeneity of the solar system[END_REF][START_REF] Schiller | Isotopic evolution of the protoplanetary disk and the building blocks of Earth and the Moon[END_REF][START_REF] Dauphas | Mass Fractionation Laws, Mass-Independent Effects, and Isotopic Anomalies[END_REF] These isotopic effects are attributed to nuclear processes, i.e. to grains condensed in the envelope of dying stars [START_REF] Lodders | Titanium and Vanadium chemistry in low-mass Dwarf stars[END_REF] over the age of the galaxy and having survived the gravitational collapse of the protosolar disk. See a review of these data by. [START_REF] Dauphas | Mass Fractionation Laws, Mass-Independent Effects, and Isotopic Anomalies[END_REF][START_REF] Birck | An Overview of Isotopic Anomalies in Extraterrestrial Materials and Their Nucleosynthetic Heritage[END_REF] The recent ozone interpretation entailed the possibility that MIF effects can be also extended to other chemical elements during oxidizing or reducing chemical reactions in plasma. [START_REF] Thiemens | The Mass-Independent Fractionation of Oxygen: A Novel Isotope Effect and its Possible Cosmochemical Implications[END_REF][START_REF] Robert | The common property of isotopic anomalies in meteorites[END_REF][START_REF] Marcus | Mass-independent isotope effect in the earliest processed solids in the solar system: A chemical mechanism[END_REF] Because these chemical conditions could be transposed to those at the origin of the presolar grains, we have developed an experiment where chemical reactions mimic those presumably responsible for the MIF effect in ozone.

Presolar grains are condensed at high temperature from their gaseous state in reducing or oxidizing conditions. Because of the difficulty to reproduce experimentally these hightemperature conditions in the laboratory, we have chosen to study the reduction of the gaseous titanium chlorides during their condensation on grains in a plasma.

The MIF effect is usually illustrated in the diagram where the 2 isotope ratios 17 O/ [START_REF] Teplukin | Several levels of theory for description of isotope effects in ozone: Effect of resonance lifetimes and channel couplings[END_REF] O and [START_REF] Ndengué | Ozone Photodissociation: Isotopic and Electronic Branching Ratios for Symmetric and Asymmetric Isotopologues[END_REF] O/ [START_REF] Teplukin | Several levels of theory for description of isotope effects in ozone: Effect of resonance lifetimes and channel couplings[END_REF] O are expressed in their δ units, that is [START_REF] Teplukin | Several levels of theory for description of isotope effects in ozone: Effect of resonance lifetimes and channel couplings[END_REF] O; "sple" and "std" standing for sample and standard and m for the mass numbers 17 or 18.

δ m O( ) = R sple R std -1 × 1000 with R = m O/
The isotopic fractionation factor α is defined when the standard is the reservoir from which the chemical products are formed:

δ m O( ) = [α -1] × 1000
In the usual theory, the mass-dependent fractionation (MDF) can be illustrated in the In addition to the fact that MIF effects for oxygen isotopes are not restricted to ozone (they have been observed in several atmospheric molecules [START_REF] Thiemens | Atmosphere science -Mass-independent isotope effects in planetary atmospheres and the early solar system[END_REF] ), recent experimental evidence suggest that they can also be extended to other elements. [START_REF] Thiemens | Use of Isotope Effects to Understand the Present and Past of the Atmosphere and Climate and Track the Origin of Life[END_REF] For example, (i) the synthesis of ammonia in the gas phase via the photochemical dissociation of N 2 -has also yielded a [START_REF] Ivanov | On molecular origin of mass-independent fractionation of oxygen isotopes in the ozone forming recombination reaction[END_REF] N selection in much larger excess than predicted by the classical isotope fractionation theory [START_REF] Chakraborty | Massive isotopic effect in vacuum UV photodissociation of N 2 and implications for meteorite data[END_REF] (ii) the condensation of organic matter from a plasma of CH 4 have generated an unexpectedly large heterogeneous distribution of hydrogen isotopes 45 or (iii) several effects where isotopes are selected by their nuclear spins and nuclear magnetic moments. [START_REF] Yasina | Magnetic isotope effect and oxygen isotope selection in oxidation chain reactions[END_REF][START_REF] Buchachenko | Magnetic Isotope Effect: Nuclear Spin Control of Chemical Reactions[END_REF][START_REF] Buchachenko | Magnetic isotopes as a means to elucidate Earth and environmental chemistry[END_REF] A MIF for sulfur isotopes has also been widely observed in the oldest sedimentary rocks on Earth and is interpreted as a signature of ancient atmospheric chemistry. [START_REF] Farquhar | Atmospheric Influence of Earth's Earliest Sulfur Cycle[END_REF] As a whole, these experiments and natural observations have revealed the importance of radical chemistry triggered by molecular dissociation.

δ
The MIF effect in ozone is characterized by two phenomenological parameters: α and η, the MDF and MIF factors, respectively. Oxygen having only 3 stable isotopes, it is not possible to identify individually α and η since the overall isotopic fractionation results from the product α × η. Titanium having more than 3 isotopes, the diagram δ m Ti versus m (m being the masses 46, 47, 48, 49 and 50) allows to remove this indetermination. A heuristic and purposely simplified example is given in Figure 1 where it can be observed that α and η can be defined for elements having 4 isotopes or more. Hence the choice of titanium (5 isotopes) in the present experimental study.

Beside their possible astronomical implications, [START_REF] Robert | Mass-independent fractionation of titanium isotopes and its cosmochemical implications[END_REF] our results reveal that most chemical elements are likely to show MIF effects similar to those observed in ozone. Since MIF is not understood on a quantum-mechanical basis, it seems important to document experimentally these chemical systems. This is also the purpose of the present study.

II. EXPERIMENTAL METHODS

A. Plasma

The formation of titanium nanoparticles from rapidly quenched plasma mixtures has been experimentally observed and we follow these studies. [START_REF] Murphy | Formation of titanium nanoparticles from a titanium tetrachloride plasma[END_REF][START_REF] Lin | An atmospheric pressure microplasma process for continuous synthesis of titanium nitride nanoparticles[END_REF] According to one of the numerous interpretation proposed for ozone, [START_REF] Reinhardt | Mass-independent isotope fractionation in ozone[END_REF][START_REF] Reinhardt | On the mass independent isotopic fractionation in ozone[END_REF] MIF effects take place when a fraction of the metastable molecules (noted [Ti x Cl y ] * ) involved in an isotopic exchange is retrieved by a chemical reaction. Contrary to thermal gases, in plasma, highly reactive radicals greatly favor isotope exchange reactions. Our aim in using a plasma mixture of TiCl 4 / C 5 H 12 (pentane) was to trigger the isotope exchange between Ti and TiCl 4 followed by the retrieval of the metastable [Cl 2 Ti-TiCl 2 ] * via its reduction with the carbon radicals resulting from the dissociation of C 5 H 12 (C x H • y .) [START_REF] Robert | Hydrogen isotope fractionation in methane plasma[END_REF] The formation of organic Ti-rich films and grains was achieved using a hyperfrequence (HF) excitation, forming a "plasma state". [START_REF] Lacoste | Multi-dipolar plasmas for uniform processing: physics, design and performance[END_REF][START_REF] Lieberman | Principles of Plasma Discharges and Materials Processing[END_REF] Experimental conditions were reported by. [START_REF] Robert | Hydrogen isotope fractionation in methane plasma[END_REF][START_REF] Robert | Mass-independent fractionation of titanium isotopes and its cosmochemical implications[END_REF] B. Morphological structures of the deposits Organic deposits in the plasma are formed by two main paths: (i) homogeneous coagulation of grains, which generally generates sub-micrometric particles, and (ii) heterogeneous growth of organic films due to a direct gas/solid condensation on the cold glass walls (60 o C) of the plasma reactor.

Organic matter (OM) was deposited either on the glass walls or on silicon wafers as 10-100 nm thick films (cf. Supplementary Material) located before, right at the center, and after the discharge. The molecular structure of this OM was established in another study. [START_REF] Biron | Toward experimental synthesis of the chondritic insoluble organic matter[END_REF] It consists of 1 to 4 aromatic ring families linked by short aliphatic chains. This structure likely results from the progressive addition of C x H • y radicals causing a progressive chain elongation, its closure in the gas phase and its condensation as aromatic moieties. [START_REF] Derenne | Model of molecular structure of the Insoluble Organic Matter isolated from Murchison meteorite[END_REF] The analytical technique (GCMS) used does not allow to identify the possible speciation of Ti: metallic Ti, Ti chlorides or Ti as a heteroelement.

On the silicon wafers, organic grains up to 2 µm in diameter were sitting on top of the amorphous OM layer. Elemental analyses show that both OM films and carbonaceous grains are Ti-rich. The spatial distribution of titanium isotope ratios in one of these grains was analyzed and is discussed in details in this paper. Bulk analyses on several other but smaller grains are reported in Ref. 50.

C. Isotopic analyses

The Ti isotope composition of experimental products was measured using the CAMECA NanoSIMS 50L at The University of Manchester. This instrument is an ion microprobe optimizing SIMS analysis for high spatial resolution. The surface of the solid sample is sputtered with a focused primary ion beam and the secondary ions ejected from the sample surface are collected by mass spectrometry. Isotope ratios were recorded in a rastered area of 3 × 3 µm in size. A grain was analyzed in details; data are reported in Table I where [START_REF] Gao | On the theory of the strange and unconventional isotopic effects in ozone formation[END_REF] C, [START_REF] Birck | An Overview of Isotopic Anomalies in Extraterrestrial Materials and Their Nucleosynthetic Heritage[END_REF] Cl and [START_REF] Buchachenko | Magnetic isotopes as a means to elucidate Earth and environmental chemistry[END_REF] Ti ion intensities are given in counts per second and the ion image have been divided in 6 zones based on [START_REF] Buchachenko | Magnetic isotopes as a means to elucidate Earth and environmental chemistry[END_REF] Ti threshold. In this image made of 256 × 256 pixels, isotopic variations were resolvable with a spatial resolution of ≈200 nm and with an analytical depth of ≈10 nm. The grain (≈2µm in diameter) is sitting at the center in the ion image (cf. Fig.

1SM, Supplementary Material).

The statistical error (± 1sigma) on isotope ratio is defined as 1/ √ n with n the total number of ions detected by the mass spectrometer. Analytical conditions are reported in Ref. 50 and in Supplementary Material. The isotopic data (Table I) for the grain are reported in the δ m Ti( ) units with m standing for the mass numbers 46 to 50.

A terrestrial organic synthetic powder (cyclopentadienyl-titanium dichloride) was used as a standard. It has been verified by mass spectrometry (ICPMS [START_REF] Deng | Chaussidon: personal communication[END_REF] ) that the standard has the natural isotopic abundance of Ti i.e. δ m Ti = 0±0.1 at all masses. The corresponding isotopic ratios are 0.1119, 0.1009, 0.07339, 0.07027 for 46 Ti/ 48 Ti, 47 Ti/ 48 Ti, [START_REF] Farquhar | Atmospheric Influence of Earth's Earliest Sulfur Cycle[END_REF] Ti/ 48 Ti, [START_REF] Robert | Mass-independent fractionation of titanium isotopes and its cosmochemical implications[END_REF] Ti/ 48 Ti, respectively.

No analytical isotope effect was detectable at the level of our precision, i.e. for reproducibilities of ±5.5, ±4.8, ±5.1 and ±5.7 for δ 46 Ti, δ 47 Ti, δ 49 Ti and δ 50 Ti, respectively (2σ errors, 42 analyses bracketing the samples).

III. RESULTS

A 50 nm layer of organic matter was deposited on the silicon wafer located downstream of the plasma. Numerous carbonaceous grains were observed deposited on top of this layer (cf. the figure in the Supplementary Material). Their bulk analyses reveal a lower Ti concentration than the organic layer but with systematic enrichments or depletions in [START_REF] Buchachenko | Magnetic isotopes as a means to elucidate Earth and environmental chemistry[END_REF] Ti by several per cents. [START_REF] Robert | Mass-independent fractionation of titanium isotopes and its cosmochemical implications[END_REF] In the grains, the [START_REF] Yasina | Magnetic isotope effect and oxygen isotope selection in oxidation chain reactions[END_REF] Ti, [START_REF] Buchachenko | Magnetic Isotope Effect: Nuclear Spin Control of Chemical Reactions[END_REF] Ti, [START_REF] Farquhar | Atmospheric Influence of Earth's Earliest Sulfur Cycle[END_REF] Ti, [START_REF] Robert | Mass-independent fractionation of titanium isotopes and its cosmochemical implications[END_REF] Ti isotopes are not fractionated relative to each other (Table I).

The data for a grain analyzed in detail are also reported in Fig. 2. Error bars combine the statistical errors and the reproducibilities determined on the standard (≈ ±6 at all masses).

Positive and negative MIF effects were observed at all masses and for all concentration ranges. The δ m Ti increases from -250 to +1250 while the Ti concentration decreases from the border to the core of the grains. At the image bulk scale, the mass balance is partially achieved (i.e. δ m Ti( )≈ 0) with a residual fractionation of +33 , +61 , +62 and +92 for m =46, 47, 49 and 50, respectively. In the diagram shown in Fig. 2 all types of known MDF fractionations in natural samples are limited to a few with strictly linear correlations. The slope of these linear correlations gives the MDF factor expressed per mass unit (cf. Fig. 1). Within the precision of the data, no MDF effect is observable between the different patterns.

Without a proper chemical calibration (not done in the present study) elemental ionic analyses with the NanoSims are not quantitative because the emissivity of each element varies with its electronic affinity and with the chemistry of its embedding matrix. Some qualitative information can nevertheless be extracted from our analyses. In the synthetic standard the Ti/C and Ti/Cl atomic ratios are 0.1 and 0.5, respectively. The corresponding [START_REF] Buchachenko | Magnetic isotopes as a means to elucidate Earth and environmental chemistry[END_REF] Ti + / 12 C + and 48 Ti + / 35 Cl + ionic ratios are (8.7±1.5)×10 2 and (4.1±2.1)×10 2 , respectively.

In the grain, the 48 Ti + / 35 Cl + varies from the surface (4 × 10 3 ) to the core (3

× 10 2 )
with no significant variation of the 48 Ti + / 12 C + ratio (≈ 2 × 10 3 ). Thus, the core has a stoichiometry close to TiCl 4±2 while the surface has an excess of Ti atoms. The bulk grain has a stoichiometry close to TiC 5±2 (i.e. an atomic ratio titanium to carbon twice that of the synthetic standard).

IV. INTERPRETATION: ORIGIN AND FORMALISM OF MIF EFFECTS

A. Origin of the effect

As observed for the isotopomers of ozone (up to +500 for 16 O 18 O 18 O) [START_REF] Janssen | Kinetic origin of the ozone isotope effect: a critical analysis of enrichments and rate coefficients[END_REF] ), the isotopic fractionation for Ti MIF is 2-3 orders of magnitude (in ) higher than generally measured in natural or experimental samples. [START_REF] Miller | Isotopic fractionation and the quantification of 17 O anomalies in the oxygen three-isotope system: an appraisal and geochemical significance[END_REF] Two classes of interpretations for the MIF effect in ozone have been proposed in the literature. They involve either (i) the differences in the isotopic exchange rates caused by differences in the symmetry of the metastable molecule [START_REF] Gao | Strange and unconventional isotope effects in ozone formation[END_REF][START_REF] Babikov | A quantum symmetry preserving semiclassical method[END_REF][START_REF] Gao | On the theory of the strange and unconventional isotopic effects in ozone formation[END_REF][START_REF] Ivanov | On molecular origin of mass-independent fractionation of oxygen isotopes in the ozone forming recombination reaction[END_REF][START_REF] Teplukin | Several levels of theory for description of isotope effects in ozone: Effect of resonance lifetimes and channel couplings[END_REF] or (ii) the higher degree of symmetry of scattering cross sections involving indistinguishable isotopes relative to those involving distinguishable isotopes. [START_REF] Reinhardt | Mass-independent isotope fractionation in ozone[END_REF][START_REF] Reinhardt | On the mass independent isotopic fractionation in ozone[END_REF] Both effects yield the same formalism for the MIF model developed in this article. Note that magnetic isotope effects [START_REF] Yasina | Magnetic isotope effect and oxygen isotope selection in oxidation chain reactions[END_REF][START_REF] Buchachenko | Magnetic Isotope Effect: Nuclear Spin Control of Chemical Reactions[END_REF] (MIE) also yield mass-independent fractionation. The present results allow to reject the hypothesis of a nuclear spin control on the chemical reaction because the 4 isotopes 46 Ti, [START_REF] Buchachenko | Magnetic Isotope Effect: Nuclear Spin Control of Chemical Reactions[END_REF] Ti, [START_REF] Farquhar | Atmospheric Influence of Earth's Earliest Sulfur Cycle[END_REF] Ti and 50 Ti show the same MIF factor although [START_REF] Yasina | Magnetic isotope effect and oxygen isotope selection in oxidation chain reactions[END_REF] Ti, [START_REF] Buchachenko | Magnetic isotopes as a means to elucidate Earth and environmental chemistry[END_REF] Ti and [START_REF] Robert | Mass-independent fractionation of titanium isotopes and its cosmochemical implications[END_REF] Ti have zero nuclear spin.

We now summarize the interpretation [START_REF] Reinhardt | Mass-independent isotope fractionation in ozone[END_REF][START_REF] Reinhardt | On the mass independent isotopic fractionation in ozone[END_REF] calling upon the property of the scattering cross sections involving indistinguishable isotopes.

The formation of TiTiM goes through a formation of a metastable complex [Ti 2 M] * (M standing for a chemical bound) with an associated average lifetime τ

TiM + Ti -→ [Ti 2 M] * -→ Ti + TiM (1)
If isotopes are distinguishable as in classical scattering processes, this average lifetime is

τ = ∞ τ min τ f (τ ) dτ ∞ τ min f (τ ) dτ (2)
with some distribution functions f (τ ), and a minimum lifetime τ min . Beyond this minimum lifetime τ min , [Ti 2 M] * is available for a subsequent chemical reaction; below this threshold,

[Ti 2 M] * spontaneously dissociates as in reaction (1). We now look at lifetimes of a complex

[Ti 2 M] * longer than τ min .

We may split τ in two parts, one arising from scattering with isotope exchange (R), and one without (NR):

τ = y τR + (1 -y)τ NR (3)
Thus:

τR = ∞ τ min τ f R (τ ) dτ ∞ τ min f R (τ ) dτ (4)
and, consequently,

y = ∞ τ min f R (τ ) dτ ∞ τ min (f R (τ ) + f NR (τ )) dτ (5)
with a corresponding construction of τ NR . From (classical) scattering simulation we may obtain the distribution function f (τ ) in its two parts, f R and f NR . Suppose further that we can express the dimensionless y as:

y = Y R P R Y R P R + Y NR P NR (6)
Y R = ∞ 0 f R (τ ) dτ ∞ 0 (f R (τ ) + f NR (τ )) dτ (7
τindist. = Y R τR + (1 -Y R )τ NR = (1 -Y NR ) τR + Y NR τNR (8)
showing the symmetry of the two different R and NR processes -we may have started equally well by defining Y NR .

The two processes (between dist-and indistinguishable isotopes) are schematically compared in Figure 3.

If masses of the isotopes were all equal, we obtain from the reaction rates the massindependent fractionation factor as the dist-to indistinguishable lifetime ratio:

η = τ τindist. . (9)
which we apply as well here, bearing in mind the approximation. However, in the case of ozone a reasonable τ min had been determined which, inserted in the described procedure, led to most of the experimental results.

As discussed in the next Section IV B, in the case of TiCl 4 , we should expect a value of η of the order of 4 in order to reproduce the maximum MIF effect observed in the core of the grain. From the insertion reaction of Ti into TiCl 4 this may be a reasonable value, as the isotope exchange via the intermediate formation of Cl 2 Ti-TiCl 2 is quite difficult with respect to the formation of associative complexes Ti-(TiCl 4 ) and since the intermediate complex will probably be formed in an excited triplet state (Ti( 3 F , triplet) + TiCl 4 (singulet)), and not in its singlet ground state.

B. Chemical reactions at the origin of MIFs

The MIF effect observed here is attributed to the reduction of the metastable complex [Cl 2 Ti-TiCl 2 ] * in the presence of the radical C x H • y resulting from the dissociation of pentane.

We may think then of a reaction process as

i Ti + j TiCl 4 → [ i Ti j TiCl 4 ] * (k f ) ( 10
) i Ti j TiCl 4 * → j Ti + i TiCl 4 (k D ) ( 11
) i Ti j TiCl 4 * + n C x H • y → i Ti j TiC nx H ny + 2 Cl 2 (k Chem ) (12)
Equations 10,11 and 12 should be regarded as the simplest reactions able to produce a long lifetime complex. However, kinetic calculations [START_REF] Murphy | Formation of titanium nanoparticles from a titanium tetrachloride plasma[END_REF] show that a variety of titanium chlorides are formed by the dissociation of TiCl 4 and, in our reaction scheme, isotopic exchange (reaction 11) can also occur between these different species. However, Ti 2 Cl 4 being formed by two stable TiCl 2 should have a longer lifetime likely enhancing the isotopic exchange.

In the same spirit, the molecule C x H • y stands for the numerous carbon radicals resulting from the dissociation of pentane and that can all participate to the reduction of the complex [ i Ti j TiCl 4 ] * . [START_REF] Lombardi | Study of an H 2 /CH 4 moderate pressure microwave plasma used for diamond deposition: modelling and IR tuneable diode laser diagnostic[END_REF][START_REF] Lombardi | Diagnostics spectroscopiques d'espèces carbonées et modélisation physicochimique de plasmas micro-ondes dans les mélanges H 2 /CH 4 et Ar/H 2 /CH 4 utilisés pour le dépôt de diamant[END_REF][START_REF] Mohasseb | Etude d'un procédé de dépôt de films de diamant nanocristallin par des plasmas micro-ondes de CH 4 /H 2 /Ar[END_REF] An example of such a complexity is given by. [START_REF] Scapinello | A study on the reaction mechanism of nonoxidative methane coupling in a nanosecond pulsed discharge reactor using isotope analysis[END_REF] In a plasma, the Ti atoms are produced by the electron impact dissociation of TiCl 4 that subsequently reacts with another TiCl 4 molecule, overabundant in the gas phase. Reaction (10) describes this subsequent reaction forming the [ i Ti j TiCl 4 ] * metastable molecule with the rate constant k f (f for formation). The superscripts i and j stand for isotope mass numbers.

Reaction (11) represents the spontaneous dissociation of the metastable molecule where the inverse of the k D (D for dissociation) stands for the lifetime of the molecule: τ = 1/k D .

In the reaction (11), an isotope has been exchanged between Ti and TiCl 4 but the same reaction ( 11) can be written with no exchange. As reported by Murphy et al, [START_REF] Murphy | Formation of titanium nanoparticles from a titanium tetrachloride plasma[END_REF] Ti atoms condense from the gas phase as metallic nanoparticles.

Reaction (12) corresponds to the possible stabilization of [ i Ti j TiCl 4 ] * by a subsequent chemical reaction forming a solid organic product i Ti j Ti-C x H y . Note that, according to the theoretical analysis of Marcus 37 on reaction rate constants, the isotopic exchange reactions (10) and ( 11) take place on the surface of the grain and not in the gas phase if the density of the reactants is too low. In this respect, the reaction (12), contrary to the formation of ozone, is not a three body reaction but a chemical reaction of the metastable molecule

[ i Ti j TiCl 4 ] * .
As mentioned in Section IV A, the MIF is produced by this chemical scheme because two channels are open for the subsequent reactions of the metastable molecule. We now develop a phenomenogical approach to mimic the MIF patterns that are observed in the δ m Ti( ) vs. m diagrams (Fig. 2). In this approach we assume that all isotopic processes involved in isotopic exchanges are mass-dependent.

Neglecting the possible pressure effect (as documented for ozone), the rate constant r (12) for the reaction 12 is:

ij r(12) = ij k f × ij k Chem × ij τ (13)
In this formalism, ij τ is defined for τ > τ min and stands for the averaged lifetime of the metastable isotopomers available for the subsequent chemical reaction via the reaction 12.

For τ < τ min , the metastable isotopomers spontaneously dissociate via the reaction 11.

In our data, isotope ratios are normalized to [START_REF] Buchachenko | Magnetic isotopes as a means to elucidate Earth and environmental chemistry[END_REF] Ti. Hence, the rate constant ratio ij f = ij r(12)/ 48-48 r( 12) is:

ij f = ij k f × ij k Chem × ij τ 48-48 k f × 48-48 k Chem × 48-48 τ I = ij k f 48-48 k f × ij k Chem 48-48 k Chem × ij τ 48-48 τ I = ij α k f × ij η (14)
with ij η = ij τ / τ I and ij α the MIF and MDF factors respectively (the subscript I in τ stands for indistinguishable). We assume that the k f and k Chem ratios are mass-dependent and thus their product is also mass-dependent; hence a unique notation ij α k f standing for the product of the two ratios. It was shown experimentally and theoretically 9,14 that ij η contains also a MDF term; hence the introduction of :

ij η = ij α η × η (15)
The rate constant ratio becomes:

ij f = ij α k f × ij α η × η (16)
Under the form of equation 16, η is (almost) constant 58 (mass-independent) and depends only on the threshold value τ min that dictates the split between the reactions 11 and 12.

The variations of any MDF factor usually reported by the α notation, can be mimicked numerically with a parameter b which dictates the magnitude of the MDF 36 :

ij α = [µ i-j ] b /[µ 48-48 ] b (17)
with µ the reduced masses of the reactants.

In this phenomenological approach, for i = j, η = 1 (since η = τ I /τ I ). When η = 1, ij f is mass-dependent, and mass-independent for η = 1. η varies with τ and η = 1 (τ = τ I ) when the τ 's are integrated over the whole Boltzmann distribution; in this situation, one of the two channels is closed. Note that, contrary to the classical MDF treatment, an additional mass-dependent term ij α η remains in ij f even for η = 1.

In the usual MDF theory, the fractionation factors are defined as the rate constant ratios of isotope exchange reactions. The relative abundances of each isotopes drop out in these ratios. This is not correct for MIF because, in a scrambled situation, all the possible reactions participate to the overall fractionation with different weights when η = 1 or when η = 1.

The isotopic fractionation factor is not anymore the rate constant ratio, but instead the reaction rate ratio.

In the isotopic reaction rate ratio (noted ij R in eq. ( 18)), the density (or partial pressure, noted as concentration [Ti]) ratio of each isotopic species can be replaced by their relative abundance ratio (or molar fraction):

ij R = [ i Ti] × [ j TiCl 4 ] [ 48 Ti] × [ 48 TiCl 4 ] × ij k f × ij k Chem × ij τ 48-48 k f × 48-48 k Chem × 48-48 τ I = x(i) × x(j) x(48) × x(48) × ij f (18)
The overall isotopic fractionation factor is then the sum of all the possible rate ratios ij R, i.e. the sum of the contribution of all the possible isotopomers. Titanium having 5 isotopes, there are 25 possible reactions between the 5 isotopes of titanium. From equation ( 18) and ( 16), the analytical solution for the overall isotopic fractionation factor i-48 α is:

i-48 α = ( i Ti/ 48 Ti) ( i Ti/ 48 Ti) 0 = i-48 α MD [1 + x(i)(η -1 -1)] [1 + x(48)(η -1 -1)] (19)
( i Ti/ 48 Ti) 0 is the isotopic composition of the reservoir from which the products of the reaction (12) are fractionated. The MDF factors ij α k f and ij α η do not appear individually in the formalism of eq. ( 19) but are gathered in i-48 α M D . Under these conditions, equation (19) is exact on a large domain of variations. This equation is also correct for all the chemical elements, provided no other mass-independent effects take place during the formation of metastable molecules such as those reported in Refs. 46-48.

The model reproduces the 4 isotopic compositions (≈ +1250 ) with a single value of the parameter η=4 50 (with α M D =1 i.e. with a MDF fractionation negligible compared with the observed δ m Ti=≈ +1250 ). Such a numerical result reinforces the validity of the model, and thus the MIF effect initially identified in ozone for oxygen isotopes seems as well responsible for the present data on titanium isotopes.

Note that the isotope effect is the same at all masses although the different isotopes of Ti have different nuclear spin values. This is an indirect evidence that the quantum interference terms between P R and P NR are negligible (as assumed by the MIF model 57 and verified by theoretical calculations. 64 ).

Note also that the isotopic abundances are approximately the same for the 4 isotopes [START_REF] Yasina | Magnetic isotope effect and oxygen isotope selection in oxidation chain reactions[END_REF][START_REF] Buchachenko | Magnetic Isotope Effect: Nuclear Spin Control of Chemical Reactions[END_REF][START_REF] Farquhar | Atmospheric Influence of Earth's Earliest Sulfur Cycle[END_REF][START_REF] Robert | Mass-independent fractionation of titanium isotopes and its cosmochemical implications[END_REF] Ti (≈ 5%); hence the close values for the [START_REF] Yasina | Magnetic isotope effect and oxygen isotope selection in oxidation chain reactions[END_REF][START_REF] Buchachenko | Magnetic Isotope Effect: Nuclear Spin Control of Chemical Reactions[END_REF][START_REF] Farquhar | Atmospheric Influence of Earth's Earliest Sulfur Cycle[END_REF] α values. However, this is not the case for other chemical elements, implying that each element should exhibit its own MIF pattern. This dependency on isotopic abundances represents a test for future experiments.

C. Isotopic constraints on the homogeneous condensation process

As theoretically predicted by Marcus for oxygen isotopes, [START_REF] Marcus | Mass-independent isotope effect in the earliest processed solids in the solar system: A chemical mechanism[END_REF] the present results confirm that the homogeneous condensation of grains promotes MIF effects. In addition, using the measured isotopic variations, it is possible to reconstruct the mechanism of grain growth.

As most other grains [START_REF] Robert | Mass-independent fractionation of titanium isotopes and its cosmochemical implications[END_REF] , the grain reported in Table I does not fulfill an exact mass balance (i.e. a δ m Ti = 0 ) with bulk δ 46 Ti= 33.2 ± 4.9, δ 47 Ti= 61.1 ± 4.7, δ 49 Ti=62.9 ± 4.6, δ 50 Ti= 92.4 ±4.6. Contrary to the grain reported in this paper, most grains do not exhibit a bulk value close to 0 ±100. [START_REF] Robert | Mass-independent fractionation of titanium isotopes and its cosmochemical implications[END_REF] It can then be supposed that an important fraction of the grains are mechanically stripped by electron impact during their stay of the silicon wafer.

The variations in δ m Ti( ) from the core to the surface (from positive to negative values) can be accounted for by several models i.e. distillation of a single reservoir, variations in the MIF factors or mixing models.

However, several constraints must be fulfilled by the model, namely: (i) the various correlations between δ m Ti and δ 46 Ti (Fig. 4) (ii) the variations between δ m Ti and the Ti concentration within a single grain (cf. Table I) and (iii) the maximum and minimum δ m Ti values reached in the core and at the surface of the grain, respectively.

Taking at face values the maximum and minimum of the measured δ m Ti as those of the two end-members (i.e. +1250 and -250 for δ [START_REF] Yasina | Magnetic isotope effect and oxygen isotope selection in oxidation chain reactions[END_REF][START_REF] Buchachenko | Magnetic Isotope Effect: Nuclear Spin Control of Chemical Reactions[END_REF][START_REF] Farquhar | Atmospheric Influence of Earth's Earliest Sulfur Cycle[END_REF][START_REF] Robert | Mass-independent fractionation of titanium isotopes and its cosmochemical implications[END_REF] Ti), a simple mixing model reproduces the observations (Fig. 4). In this model it is assumed that the core is progressively embedded by the growth of carbon-rich layers whose titanium isotopic compositions are around -250 . The variations in Fig. 4 are then dictated by the mass balance:

0 = x δ m Ti(> 0 ) + (1 -x) δ m Ti(< 0 ) (20)
The x variations mimic the various correlations between δ m Ti and δ 46 Ti (Fig. 4).

These variations in the x value for each layer of the grain can also be calculated using the Ti concentration in the core (8.32 × 10 2 ) and in the layers ([Ti]) as:

x = (8.32 × 10 2 )/[Ti] (21)
with 8.32 × 10 2 expressed in the units of the counted 48 Ti ions per second (cf. Table I). As shown in Fig. 5, the calculated (20) and measured (21) x values are in reasonable agreement (m = 46 is shown as an example).

No other models have yet been found that satisfy the data. Therefore, we conclude qualitatively that two chemical speciations of Ti exist in the plasma: [START_REF] Buchachenko | Magnetic isotopes as a means to elucidate Earth and environmental chemistry[END_REF] Ti-depleted TiCl x (or metallic Ti + Cl 2 ) and a 48 Ti-rich Ti bound to carbon. It is the mixing of these two phases that produces the observed isotopic variations between grains, and within a single grain. As a consequence, this isotopic partition is likely dictated by the ratio of the two channels written in the reactions (11) and (12).

If correct, ≈20% of the metastable molecules spontaneously dissociate to form TiCl x while ≈80% react with carbon (as 0.2 × (+1250) + 0.8 × (-200) ≈ 0 ).

The suggested large η value (η = 4.0) is still compatible with the effect involving the reaction between indistinguishable isotopes. Although it is not attempted here to calculate the η factor explicitely through classical trajectory simulation for the present experimental conditions, the possible range of η values can be estimated by analogy with the numerical procedures developed for ozone.

Taking eqns. 9, 3 and 8

η = τdist. τindist. = y τR + (1 -y) τNR Y τR + (1 -Y ) τNR = y (τ R /τ N R ) + (1 -y) Y (τ R /τ N R ) + (1 -Y ) (22)
we may estimate the involved parameters. y, the relative contributions of exchange and non-exchange reactions, should be below 0.5 (in total there should be more NR collisions than R, with a limit of 1/2 if the minimum lifetime τ min → ∞). η is observed to be around 4, and the ratio of average complex lifetimes in the collision Ti + TiCl 4 → Ti 2 Cl 4 → Ti + TiCl 4 should be large, reflecting the difficulty of an insertion with subsequent emission of a Ti (not just an elastic collision). Let us take k = τR /τ N R = 15. From the three equations mentioned we obtain

y(η, Y, k) = η -1 k -1 + Y × η > Y × η (23)
We see immediately that Y , the ratio of overall collisions R and N R, should be quite small, not larger than 0.02, which yields y = 0.294 which still seems quite large. Which values to take may be subjet to explicit collision simulations.

V. CONCLUSIONS

The titanium isotopic fractionation observed in a plasma for the reaction TiCl 4 +pentane follows the empirical rules observed for the oxygen MIF observed in ozone. Therefore, this MIF mechanism seems to describe a general behavior of isotopic exchange reactions coupled with a chemical removal of the metastable molecules involved in this isotopic exchange.

This conclusion should, however, be tested with other chemical elements. It may have profound implications on models developed to account for the formation of solids during the formation of the Solar System. A theoretical treatment to reconcile the classical physics approach developed in this paper with more rigorous quantum mechanical principles remains a challenging open issue.

FIG. 1: Schematic example showing that in a 3 isotope system (oxygen as an example), it is not possible to distinguish between the MDF and MIF contributions (α and η, respectively) in the overall isotope effect. This uncertainty is resolved for n > 3 isotope systems (titanium as an example).

  Several questions are raised by these results, namely: (i) what is the physical origin of MIF, (ii) what are the chemical reactions involved in the effect, and (iii) what mechanism accounts for the radial distribution of titanium isotopes in grains.

  with a "probability" P and a "relative abundance" Y , both positive and Y obeying Y R + Y NR = 1. Y R then stands for the fraction of exchange involved in the formation of [Ti 2 M] * :

  ) while y stands for the fraction of exchange reactions taking place during the chemical reaction of [Ti 2 M] * .[START_REF]P R = τ min 0 R dτ ∞ 0 f R dτ[END_REF] If we have now indistinguishable isotopes, we cannot assign any more R and N R, but should deal with a (quantum-mechanical) superposition of both processes. The simplest modification of the classical scattering processes is to average P R and P NR to a common value which then yields y = Y R . The average lifetime τ indist. for [Ti 2 M] * formed by reactions between indistinguishable isotopes becomes:

FIG. 2 :FIG. 3 :FIG. 5 :

 235 FIG. 2: : Distribution of the titanium isotope ratios m Ti/ 48 Ti in an organic grain deposited from a TiCl 4 /pentane plasma. Units are δ m Ti( ) with m standing for 46, 47, 48, 49, and 50 mass units. Error bars (±1σ) are indicated for the highest and lowest isotopic patterns where they are larger than the symbol size used for the data . On these two patterns, error bars are approximately the same for all the masses (cf. Table I for more precise values on error bars). The δ m Ti( ) values measured in the same domain of concentration are connected by broken lines (solid or dashed).These domains of concentrations correspond to a spatial distribution in the grain with the lowest concentrations associated with the higest δ m Ti( ) in the core and vice versa for the surface.

  [START_REF] Ndengué | Ozone photolysis: Strong isotopologue/isotopomer selectivity in the stratosphere[END_REF] O( ) vs. δ 18 O( ) diagram. In this 3-isotope diagram, all isotope exchange reactionsat equilibrium or not -should define a linear correlation having slope of 0.52.[START_REF] Urey | The thermodynamic properties of isotopic substances[END_REF][START_REF] Bigeleisen | The relative reaction velocities of isotopic molecules[END_REF][START_REF] Young | Kinetic and equilibrium mass-dependent isotope fractionation laws in nature and their geochemical and cosmochemical significance[END_REF][START_REF] Miller | Isotopic fractionation and the quantification of 17 O anomalies in the oxygen three-isotope system: an appraisal and geochemical significance[END_REF] At a first approximation, such a behavior is caused by the difference in mass between[START_REF] Ndengué | Ozone Photodissociation: Isotopic and Electronic Branching Ratios for Symmetric and Asymmetric Isotopologues[END_REF] O and[START_REF] Teplukin | Several levels of theory for description of isotope effects in ozone: Effect of resonance lifetimes and channel couplings[END_REF] O which is twice that between[START_REF] Ndengué | Ozone photolysis: Strong isotopologue/isotopomer selectivity in the stratosphere[END_REF] O and[START_REF] Teplukin | Several levels of theory for description of isotope effects in ozone: Effect of resonance lifetimes and channel couplings[END_REF] O. This property is common to all chemical elements

and observed in natural or experimental situations, in liquid or solid phase reactions. On the contrary, MIF effects are characterized by a slope of ≈1.0 in the δ 17 O( ) versus δ 18 O( ) diagram.

* Correspondence to: François Robert (francois.robert@mnhn.fr)

Acknowledgments Funding: F.R. acknowledges support (Overheads) of the ERC Advanced Grant Pa-leoNanoLife (PI: F. Robert; 161764). R.T. acknowledges support from the UK Science and Technology Facilities Council (ST/P005225/1). At The University of Manchester, the NanoSIMS was funded by UK Research Partnership Investment Funding (UKRPIF) Manchester RPIF Round 2, and installation of a Hyperion RF plasma ion source supported by the Henry Royce Institute for Advanced Materials. P.R. acknowledges the federation IP2CT (Sorbonne University) for computer resources.


Batiste Galmès

Kien Phan-Huy

Luca Furfaro

Yanne K Chembo
email: ykchembo@umd.edu

Jean-Marc Merolla
email: jeanmarc.merolla@univ-fcomte.fr

Nine-Frequency-Path Quantum Interferometry over 60 km of Optical Fiber

The archetypal quantum interferometry experiment yields an interference pattern that results from the indistinguishability of two spatiotemporal paths available to a photon or to a pair of entangled photons. A fundamental challenge in quantum interferometry is to perform such experiments with a higher number of paths, and over large distances. We demonstrate that using indistinguishable frequency paths instead of spatiotemporal ones allows for robust, high-dimensional quantum interferometry in optical fibers. In our system, twin-photons from an Einstein-Podolsky-Rosen (EPR) pair are offered up to 9 frequency paths after propagation in long-haul optical fibers, and we show that the multi-path quantum interference patterns can be faithfully restored after the photons travel a total distance of up to 60 km.

I. INTRODUCTION

Quantum information networks require the manipulation and transportation of entangled photons in longhaul optical fiber networks without destroying their nonclassical correlations. The relevance and viability of any prospective quantum network also critically depends on a strong hardware overlap with off-the-shelf components, readily available in the technologically mature sector of optical fiber telecommunications. Fulfilling these constraints while preserving the non-classical correlations of entangled photons when launched in long-haul optical fibers arises as particularly difficult task, and even more so if the entanglement is high-dimensional.

We present here a proof-of-concept quantum interferometry experiment in which high-dimensional quantum entanglement is preserved even after the photons have propagated in long-haul optical fibers. We report a long-distance nonlocal interference experiment based on frequency domain interferometric scheme using electrooptic phase modulators, in which nonlocal dispersion cancellation is performed. The electro-optic phase modulators leading to N frequency paths instead of the two temporal in the typical Franson interferometer. As N -frequency paths are involved, the dispersion strongly affects the interference pattern and leads to radically different observations. We show that the dispersion not only alters the visibility but also disturbs the shape of the interference pattern and we demonstrate how the effect of the dispersion can be cancelled non-locally. This preliminary results confirm also that the high-dimensional quantum entanglement is preserved even after the photons have propagated in long-haul optical fibers.

The paper is divided into three main parts. We first describe our experiment and we present the effects of the dispersion on the N -frequency path interferences. In the second part, we propose a theoretical analysis of the system, and outline the formalism use to quantify the quantum correlations. The last part presents our main experimental result, in which multi-path quantum interference patterns can be faithfully restored after the photons travel a total distance of up to 60 km, in good agreement with the theoretical predictions.

Phase Modulator

in out

|0 J 0 (s)|0 J -1 (s)e -iφ | -1 J -2 (s)e -i2φ | -2 J 1 (s)e iφ |1 J 2 (s)e i2φ |2
FIG. 1: Principle of multiple frequency paths using a phase modulator seeded with a photon of frequency ω0. The modulator is driven by a sinusoidal RF signal of amplitude sVπ, frequency Ω and phase φ. This modulation creates the eigenstates |n ≡ |ω0 + nΩ with n ∈ Z, which are new frequency paths available with probability J 2 n (s) to any incoming photon in the eigenstate |0 . Only 5 eigenstates are represented here for clarity, but the entangled photons explore up to 9 frequency paths in our experimental transmission system. Along with robustness, this high dimensionality is a distinctive advantage of frequency path quantum interferometry in comparison to spatiotemporal schemes.

II. EXPERIMENTAL SYSTEM

Our system is based on the idea of Franson interferometry [1]. We pump a 3 cm-long periodically poled lithium niobate (PPLN) waveguide around 775 nm to generate entangled photons via spontaneous parametric down conversion (SPDC) around λ 0 = 2πc/ω 0 = 1550 nm. The spectral density f (ω) of this source spans over ∼ 2π × 12 THz around ω 0 (∼ 100 nm around λ 0 ), and the quantum state of the twin-photons is

|ψ = +∞ -∞ dωf (ω)|ω 0 -ω |ω 0 + ω . (1)
The entangled photons are then launched into a standard SMF-28 fiber of length L 0 before being separated by a 3 dB coupler in two arms [A] (Alice) and [B] (Bob). Each arm includes another fiber spool of length L A,B , which transform the input state after propagation into

|ψ = +∞ -∞ dωf (ω)e iΦ(ω) |ω 0 + ω A |ω 0 -ω B ( 2
)
where

Φ(ω) = β(ω)[L 0 + L A ] + β(-ω)[L 0 + L B ] (3)
denotes the phase shift resulting from the propagation in the optical fibers L 0,A,B , and β(ω) represents the corresponding fiber dispersion. Whereas Franson interferometry usually involves Mach-Zehnder (MZ) modulators offering two temporal paths [2][3][4][5], our system exploits instead N 2 frequency paths offered by phase modulators (PM) [6][7][8]. The creation of these frequency paths is performed using two phase modulators PM A,B of half-wave voltage V π , modulated at the frequency Ω/2π = 12.5 GHz with signals of normalized amplitudes {a, b} = V {a,b} /V π and phases {ϕ A , ϕ B }.

III. THEORETICAL ANALYSIS

The modulators PM A,B induce the transformations

|n A,B → k∈Z J k ({a, b})e ik({ϕ A ,ϕ B }-π/2) |n + k A,B , (4)
where

|n ≡ |ω 0 + nΩ (5)
are new frequency paths with n ∈ Z. As explained in Fig. 1, every path |n is accessed with probability J 2 n (s) where s ∈ {a, b} is the real-valued modulation index of the modulator. The closure condition

+∞ n=-∞ J 2 n (s) = 1 (6)
is a property of Bessel functions and holds for all s. However, the probabilities J 2 n (s) decay rapidly to 0 as n → ±∞, and only N 2(s + 1) + 1 frequency paths have a non-negligible probability. In our experimental setup, we have s 3 and accordingly, up to N = 9 paths with n ∈ {-4, . . . , 4} can be explored by the twinphotons. We use the two Bragg filters F A,B of bandwidth Ω F /2π = 3 GHz, which are respectively centered to ω 0 + nΩ and ω 0 -nΩ.

Since the modulators sift the angular frequency by Ω, it is convenient to expand following the frequency-bin decomposition

|ψ = +Ω/2 -Ω/2 dω +∞ k=-∞ f (ω + kΩ) ×|ω 0 -ω -kΩ |ω 0 + ω + kΩ . (7)
On the other hand, the phase shift for a photon in a state |ω after propagation in a length L of optical fiber can be explicitly expressed as

|ω out = e iβ(ω)L |ω in . (8)
As a consequence, the probability of coincidence measurement by the avalanche photodetectors D A and D B is therefore

P n = +Ω/2 -Ω/2 dω g n (ω) k∈Z J k (a)J -k (b)e -iΦ(k,n,ω) 2 , (9)
where up to second-order dispersion termes. In the above equation, β 1 and β 2 are respectively the group velocity and the group velocity dispersion in the optical fibers, L = 2L 0 + L A + L B the total distance traveled by the twin-photons, ∆L = L B -L A the distance mismatch, g n (ω) is spectral density of the source after being altered by the Bragg elements and the propagation along the optical fibers, while µ is the quadratic (secondorder) coefficient of the dispersion compensation module. The superposition of indistinguishable paths as given by the summation over k, and squaring the sum gives rise to quantum interferences. The two-photon interference patterns are obtained by scanning the phase difference ∆ϕ = ϕ Bϕ A . Since the filters have a finite bandwidth Ω F , the integration of ω over [-Ω F /2, Ω F /2] results in a continuous superposition of shifted interference patterns leading to a loss of visibility.

Φ(k, n, ω) = k (∆ϕ + β 1 Ω∆L) -k µ + 1 2 β 2 L (ω + nΩ)Ω +k 2 Ω 2 µ + 1 2 β 2 L (10)
The dispersion phase-shift Φ(k, n, ω) is such that the β 1 -induced shift is easily canceled with the matching L A = L B , while the dispersion-induced shift cannot except when the dispersion cancellation is emulated via µ = -1 2 β 2 L. This phenomenon is illustrated in Fig. 3 which shows the evolution of numerically simulated interference patterns as the twin-photons propagate in the optical fibers. It can be seen that dispersion not only alters the visibility of the fringes, but also disturbs the shape of the interference pattern -and thus, the quantum correlations.

It should be noted that this result is essentially different from conventional time-based system using a Mach-Zehnder modulator [21]. In that case, a sine interference pattern is measured because it involves the superposition of two temporal paths. As the dispersion only affects the phase relationship between the two paths, a superposition of shifted sine patterns is obtained leading to a visibility loss. On the contrary, our method involves the interference of N paths to produce a Bessel-like in-terference pattern. As the dispersion affects the phase relationship between all the N paths, the interference is strongly altered as shown by Fig. 3. Another consequence is that the time-based MZ experiment is positively sensitive to source filtering. As reported in ref. [21], if one uses a narrowband filter to turn the propagation from "impulse" to "continuous-wave", it will isolate one sine interference pattern from the superposition, and reduce effectively the dispersion effects. In our frequency-path system, one can not reduce the source bandwidth to limit the effect of dispersion, and furthermore, the use of N 2 frequency paths leads to a strong deformation of the interference pattern that adds to the visibility loss. This is why an efficient dispersion cancellation strategy is an absolute requirement for this frequency-path interferometric approach.

Indeed, unlike in the classical case, the problem of dispersion cancellation in the context of quantum interferometry is not conceptually trivial, as dispersion can be canceled either locally [9][10][11][12][13][14][15][16][17] or nonlocally [18][19][20][21]. Research is still on-going about the classical or quantum nature of the non-local dispersion approach [22,23].

Dispersion cancellation in two-photon experiments was simultaneously introduced in 1992 by Steinberg et al. [9] and Franson [18]. Steinberg et al. proposed a dispersion cancellation scheme based on a Hong-Ou-Mandel interferometer in which photons are recombined through a beam splitter before detection [24]. Conversely, Franson proposed a nonlocal dispersion scheme in which separated photons experienced independently dispersion effects of opposite-signs without recombination before detection. These seminal papers introduced a debate about the classical or quantum nature of the dispersion cancellation [22,25] stimulating the development of classical experiments [20,26] to describe all of the relevant aspects of nonlocal dispersion [19]. Different experimental set-ups have been proposed to explore nonlocal dispersion cancellation such as nonlocal interferometry or even non-interferometric architectures as suggested by Franson [19]. The sources in that case can be based on spectral [17] or angularmomentum entanglement [16].

Our method requires a minimum bandwidth to allow the N -frequency paths: it is therefore not only sensitive to dispersion by nature, but it actually needs the dispersion to be nonlocal. We show here that the dispersion can be canceled non-locally, thereby restoring the original interference pattern after the twin-photons are propagated in long-haul optical fibers. When L A = L B , the dispersion shift Φ(k, n, ω) depends only the total propagation distance L = 2L 0 +L A +L B traveled by the twin-photons. In fact, the dispersion effect is the same whether both photons co-propagate along the fiber (L 0 ), or only along one of the two arms (L A,B ). This has two major consequences. First, the whole propagation and dispersion can actually be experienced by only one photon and still lead to the same measurements. Second, a negative dispersion applied anywhere cancels the effect. To perform nonlocal dispersion cancellation, a dispersion-compensation module (DCM) is inserted in arm [A], as shown in Fig. 2. This module introduces a negative phase-shift that compensates to the group velocity dispersion phase shift. The coincidence probability of Eq. ( 9) becomes (11) so that the initial Bessel-like interference pattern is fully recovered [5].

P n ∝ |J 0 (c)| 2 with c = [a 2 + b 2 + 2ab cos(∆ϕ)] 1 2 ,

IV. RESULTS

The experimental results are presented in Fig. 4. The acquisition of the interference patterns required stable continuous operation for several days. In particular, each interference pattern corresponds to continuous data acquisition for a duration of 48 h. Photon counting was performed using LynXéa photon counters from Aurea Technology with onboard time-correlator. The filtering was performed using fiber-Bragg gratings (FBGs) with a full width at half-maximum (FWHM) bandwidth of 2.6 GHz, with more than 30 dB isolation at 5.6 GHz offset. This FBG filters are cascaded with circulators, and yielded overall losses around 1.5 dB. The bias voltages of the modulators are set with a precision of 100 µV .

Although full automation provides strong performances and reliability, large distances propagation features specific technical challenges. One of the most important difficulty comes from the phase-shift β 1 ∆L that shifts the interference pattern in Φ(k, n, ω), since ∆L cannot be strictly set to 0 when L A,B is km-long. Indeed, for large distances, this term is very sensitive to temperature as 1 K induces a 2π phase-shift per kilometer. A simple way to overcome this difficulty is to propagate most of the distance without separating the photons, split them and cancel the whole dispersion afterward. Consequently, ∆L is then nulled with the best precision possible, and the temperature fluctuations automatically are compensated for being the same for the twin-photon pair. The expected results are identical to the case in which the photons are first separated and then propagate most of the distance in two identical fibers L A L B . However, this solution may rise questions with regard to the actual separation of the photons and the "nonlocality" of the experiment. For this reason, two sets of experiments were performed.

We first let the photons propagate together in the same fiber L 0 = 2 km with negligible propagation in the arms [A] and [B] (with L A = L B 0). The results are then compared to the experiment in which the photons are separated right after the source (L 0 0), while propagating over L A = L B = 2 km. In both cases, the total distance traveled by the twin photons is L = 4 km. A Finisar WaveShaper 4000S with 10 GHz resolution is used on arm [A] to compensate for the dispersion, and Figs. 4(a) and (b) show how the Bessel interference pattern is restored for both experiments, in excellent agreement with the simulations. This confirms the fact that co-propagation (along L 0 ) or split propagation (along L A and L B ) of the photon pair yield the same results, and as predicted by the theory, the visibility of the interference pattern is also restored.

Since co-propagation or split propagation eventually results in the same outcome, a second set of experiments was performed. Here, in order to show our ability to cancel the dispersion over large distances, both photons travel together within the same fiber of length L 0 = 30 km while L A = L B 0, corresponding to a total distance L = 60 km for the twin-photons. The Finisar DCM is replaced by a DCMX from Teraxion with fixed dispersion cancellation for 60 km. The results are shown in Figure 4(c). One can note a slight dissymmetry between the two Bessel side-lobes in the experimental data, and it originates from the DCM limited bandwidth. However, once the DCM response is accounted for in our simulation, an excellent agreement is achieved for the case of long-haul fiber propagation.

V. CONCLUSION

In conclusion, we have demonstrated robust, highdimensional (N = 9) frequency-path quantum interferometry in long-haul optical fibers, in which twin-photons travel a distance up to 60 km. We expect such systems to play a major role in future quantum information networks, particularly for quantum-key distribution frequencybased systems [6,27]. It is also known that ultra-high Q whispering-gallery mode resonators are ideal platforms to achieve both high-capacity optical fiber communications [28] and quantum entanglement via the nonlinearity of the bulk material [29][30][31], and consequently, we expect as well our interferometric results to be relevant in the context of resonator-based quantum frequency combs [32,33].

FIG. 2 :

 2 FIG. 2: Schematic representation of the experimental setup. (1) Broadband twin-photon source. (2) 3 dB splitter towards arms [A] and [B]. (3) Phase modulators providing the frequency paths. (4) Bragg filters to select the output photon pairs. (5) Detection and correlation. (6) Dispersion compensation system.

FIG. 3 :

 3 FIG. 3: Numerical simulation of the interference patterns with respect to the total distance L traveled by the twin-photons, using Eq. (9). The parameters are a = b = 2.75, ∆L = 0, and the filters are positioned at n = 0. (a) Ideal case with no dispersion (β2 = 0) and perfectly monochromatic filters (ΩF = 0). The interference pattern [which corresponds to the continuous black line in Fig. 3(d)] remains unaffected by the propagation. (b) Case where dispersion is accounted for (β2 = -22 ps 2 /km) but the filters are still perfectly monochromatic. The interference pattern evolves periodically, with a period Lp corresponding to length necessary for the accumulated dispersion phase β2Ω 2 Lp/2 to amount to 2π. (c) Realistic case where both the dispersion and the bandwidth of the Bragg filters (ΩF /2π = 3 GHz) are accounted for. The interference pattern is irreversibly altered during propagation and impedes long-haul quantum interferometry. (d) Quantum interference patterns from Fig. 3(c) at L = 0 km (continuous black), L = 4 km (dotted blue), and L = 60 km (dashed red).

FIG. 4 :

 4 FIG. 4: Experimental measurements of the normalized coincidences as a function of the phase difference ∆ϕ, for moderate (4 km) and long (60 km) distance propagation. The upper row (a)-(c) corresponds to experimental results without any dispersion compensation (DC), while the lower row (d)-(f) corresponds to measurements performed when the nonlocal dispersion compensation scheme is implemented. The theoretical Bessel-like patterns are the continuous lines, while the experimental results are indicated with the symbols. (a) and (d): Interference patterns for 2 km split propagation in LA and LB (L = LA + LB 4 km) (b) and (e): Interference patterns for 2 km co-propagation in L0 (L = 2L0 4 km). (c) and (f): Interference patterns for 30 km co-propagation in L0 (L = 2L0 60 km).

Acknowledgment

The authors would like to acknowledge fruitful interactions with Matthieu Bloch. This research has been funded by the Labex ACTION. It has also been funded by the projects CORPS, IQUINS, and PHYFA of the Région Bourgogne Franche-Comté in France.