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Atom-photon interactions in a system of coupled cavities

A. El Amili, 1, * S. Gleyzes, 1, † and C. I. Westbrook We give a theoretical treatment of single atom detection in an compound, optical micro cavity. The cavity consists of a single mode semiconductor waveguide with a gap to allow atoms to interact with the optical field in the cavity. Optical losses, both in the semiconductor and induced by the gap are considered and we give an estimate of the cavity finesse. We also compute the cooperativity parameter and show how it depends on the gap width and cavity length.

Maximization of the cooperativity does not always correspond to maximization of the coupling. c 2012 Optical Society of America OCIS codes: 020.5580,130.3120,230.3120,230.4555,270.5570

Introduction

Recent years have seen significant developments in atom optics because of miniaturization, which has allowed workers to trap and manipulate atomic ensembles near a substrate via magnetic fields generated by currents in microfabricated wires [START_REF] Fortagh | Miniaturized wire trap for neutral atoms[END_REF][START_REF] Reichel | Atomic micromanipulation with magnetic surface traps[END_REF][START_REF] Denschlag | Guiding neutral atoms with a wire[END_REF]. These advances have permitted generation of Bose-Einstein condensates on atom chips, and production of coherent sources of atoms [START_REF] Hänsel | Bose-Einstein condensation on a microelectronic chip[END_REF][START_REF] Ott | Bose-Einstein condensation in a surface microtrap[END_REF][START_REF] Schneider | Bose-Einstein condensation in a simple microtrap[END_REF]. These scalable micro structures promise various applications on quantum information processing and micro-interferometry on a chip [START_REF] Enk | Quantum information processing in cavity-QED[END_REF][START_REF] Wang | Atom Michelson interferometer on a chip using a Bose-Einstein condensate[END_REF][START_REF] Schumm | Matter-wave interferometry in a double well on an atom chip[END_REF]. In both cases, an important step is to implement a local atom detector able to detect low atom numbersideally single atoms. Several detection possibilities have been, theoretically and experimentally, demonstrated using the interaction of atoms with resonant light [START_REF] Horak | Possibility of single-atom detection on a chip[END_REF][START_REF] Poldy | Single-atom detection with optical cavities[END_REF][START_REF] Steinmetz | Stable fiber-based Fabry-Pérot cavity[END_REF][START_REF] Trupke | Atom detection and photon production in a scalable, open, optical microcavity[END_REF][START_REF] Colombe | Strong atom-field coupling for BoseEinstein condensates in an optical cavity on a chip[END_REF][START_REF] Kohnen | An array of integrated atom-photon junctions[END_REF]. For detection in an optical cavity, an important figure of merit is the "cooperativity". The cooperativity C = g 2 /κγ describes the competition between the atom-field coupling g and the damping rates given by the half linewidth of the cavity and the atomic transitions (κ, γ) respectively [START_REF] Kimble | Strong interactions of single atoms and photons in cavity QED[END_REF][START_REF] Horak | Possibility of single-atom detection on a chip[END_REF][START_REF] Poldy | Single-atom detection with optical cavities[END_REF]. In other words, C -1 is the number of atoms needed to significantly disturb the cavity. In a simple Fabry-Pérot cavity, the cooperativity C ∝ F /A depends mainly on the cavity parameters such as the finesse F and the intra-cavity beam cross section A. Even for a low cavity finesse, it is possible to reach useful values of the cooperativity if the intra-cavity cross section is small. In a system of several coupled cavities, as we will analyse here, the cooperativity is not a simple function of the finesse and beam cross section. This is because the electric field in different parts of the cavity can be very different and can depend on other parameters such as the lengths of the various sub-cavities. We therefore undertake in this paper to analyse a specific coupled cavity system with a view towards optimizing the cooperativity. We will show that the optimum operating conditions are not always easy to guess.

In a previous paper [START_REF] Gleyzes | Towards a monolithic optical cavity for atom detection and manipulation[END_REF], we described a cavity consisting of an AlGaAs waveguide with mirrors at both ends. A cross section of the waveguide is shown in Fig. 1 (a). To use this device as an atom detector, we propose to open a small gap in the waveguide. Atoms within the gap can be detected or manipulated via their interaction with the intracavity field. Because of the considerable effective refractive index of AlGaAs (n ≈ 3.1), the gapvacuum interface has a reflectivity of 27 % and the gap thus forms a low finesse cavity within the larger waveguide cavity. The various length scales of the entire system are shown in Fig. 1 (b). In order to avoid excessive atom-surface interactions, it is necessary that the gap width d be at least 2 µm [START_REF] Gleyzes | Towards a monolithic optical cavity for atom detection and manipulation[END_REF]. On the other hand, the minimization of diffraction losses within the gap dictates that it be as narrow as possible. We have therefore concentrated on analyzing the characteristics of this system for gap sizes of about 2 µm. In our earlier work we gave an estimate of the cooperativity in the regime where the finesse is limited by the waveguide propagation losses. In the present paper we discuss a more systematic investigation. First we study numerically and analytically the transmission and reflection of the gap as a function of its width, taking into account the losses due to diffraction. Knowing the transfer matrix of the gap, we estimate in section 3 the finesse of a resonator based on such a opened waveguide. In section 4, we compute the amplitude of the field in each part of the resonator in order to calculate the coupling parameter g of an atom to the light field.

These calculations allow us to calculate the cooperativity and optimize it as a function of the cavity parameters.

We confirm the approximate results of Ref. [START_REF] Gleyzes | Towards a monolithic optical cavity for atom detection and manipulation[END_REF]. However, we believe it should be possible to fabricate a waveguide with sufficiently low loss that the total losses are dominated by the unavoidable loss in the gap. In this case, we show that the optimization of the cavity is very different.

Transmission and reflection of the gap

We begin by giving some more details of the calculations of the losses induced by the gap which allowed us to estimate the cooperativity in our previous paper [START_REF] Gleyzes | Towards a monolithic optical cavity for atom detection and manipulation[END_REF]. To perform our analysis, a key issue is the spatial variation of the eigenmode of the waveguide as it propagates in free-space. Because of the complex geometry of the waveguide (Fig. 1 (a)), the spatial mode profile must be calculated numerically. We consider a field that propagates inside the waveguide for z < 0, and that reaches at z = 0 a waveguide/vacuum interface (Fig. 1 (b)). Using a 3 dimensional fully-vectorial aperiodic-Fourier modal method [START_REF] Hugonin | Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization[END_REF], we compute the complex amplitude E 0 (x, y, z) of the field that propagates in free-space in the half-space corresponding to z > 0. Numerically, we perform the simulation from z = 0 to z = 35 µm. We define the "axis of the field" as the coordinates (x 0 , y 0 ) that correspond to the maximum of |E 0 (x, y, 0)| and set the origin of the coordinates so that x 0 = y 0 = 0.

We define E 0 = E 0 (0, 0, 0), and then rewrite E 0 (x, y, z) = E 0 f (x, y, z), where f is the mode profile during it's propagation along the z-axis. Due to the continuity of the electric field, f (x, y, 0) corresponds to the profile of the eigenmode inside the waveguide. We can therefore compute the overlap between the propagated field and the field inside the waveguide, which is expressed as:

Q(z) = f * (x, y, 0)f (x, y, z)dxdy |f (x, y, 0)| 2 dxdy |f (x, y, z)| 2 dxdy . (1)
For z > 0, the overlap function is |Q(z)| < 1, which expresses the fact that because of the diffraction during the propagation in free-space, the field cannot be totally recoupled into the waveguide on the other side of the gap. The knowledge of f (x, y, z) also allows us to compute the phase and amplitude of the electric field on the axis of the waveguide as the field propagates:

f (0, 0, z) = |f (0, 0, z)| exp i(kz + ϕ 0 (z)), (2)
where ϕ 0 is the correction to the plane wave phase and varies slowly with z on a scale λ.

Once we know the free-space propagation of the eigenmode, we can calculate the transmission and reflection of a gap of width d. We consider the gap as two plane interfaces separated by a distance d. Each interface has a transmission and a reflection given by the Fresnel coefficient of reflection and transmission between a material of refractive index n = 3.15 and the vacuum n 0 = 1. On the left interface, the light coupled into the gap has the spatial profile of the eigenmode of the waveguide. The mode inside the cavity is considered as the superposition of E + p and E - p , the fields corresponding to the 2p th (propagating from left to right) and 2p + 1 th (propagating from right to left) reflections at the gap interfaces. The amplitude E + p (E - p ) can be expressed as a function of the incident field amplitude E i as E + p = tr 2p E i and E - p = tr 2p+1 E i . The Fresnel coefficients t and r correspond respectively to the transmission coefficient for a wave travelling from the waveguide into the vacuum, and the reflection coefficient of the vacuum/waveguide interface. The spatial profile of E + p and E - p is related to f (x, y, z) by:

E + p (x, y, z) = E + p f (x, y, 2pd + z) (3)
E - p (x, y, z) = E - p f (x, y, (2p + 2)d -z). (4)
A fraction t ′ , where t ′ is the transmission coefficient for a wave travelling from the gap into the waveguide, of each component E + p is transmitted through the right interface, but since the light diffracts during its propagation in the free-space of the gap, only a fraction

Q + p = Q(2pd + d)
of the transmitted field is coupled back into the eigenmode of the right waveguide. The amplitude E t of field transmitted through the gap is therefore:

E t = t ′ p Q + p tr 2p E i . (5)
In a similar way, one can write the amplitude of the field reflected by the gap as the sum of the amplitude of the light directly reflected by the waveguide/vacuum interface -rE i and the mode-matched part of E - p that is transmitted back into the left part of the waveguide:

E r = -rE i + t ′ p Q - p tr 2p+1 E i , (6)
where

Q - p = Q(2pd + 2d
). Considering the gap as a single entity, its complex reflection and transmission coefficients for the amplitude are rg = E r /E i and tg = E t /E i , and characterize completely the gap.

The phase and amplitude of rg and tg are shown in Fig. 2. To confirm our results, we can also use the simulation software to directly simulate the propagation through a waveguide -vacuum -waveguide stack. We denote by rg,sim and tg,sim the reflection and transmission coefficients for the amplitude deduced from the numerical simulations. Figures 2 (a) and 2 (b) compare the result of the analytical model to the numerical simulations. We see that the amplitude and phase match very well. We also compared the losses, defined as 1-| tg | 2 -|r g | 2 , calculated using the two methods and they agree within 20 %.

Resonator losses induced by the gap

The losses calculated above only correspond to the losses for a single pass inside the cavity.

The losses induced by the gap in a cavity will depend on the interferences between the two propagating fields that create the standing wave inside the resonator. Therefore, to calculate the maximum finesse of a cavity based on a gapped waveguide, we need to estimate the losses induced by the gap in the presence of the counter-propagating field reflected by a perfect mirror that terminates the cavity (reflectivity r 0 = -1) (Fig. 1 (b)). We denote by E 1 (E 3 ) the amplitude just before (after) the gap of the field propagating in the left (right) waveguide from the left to the right, and E -1 (E -3 ) the amplitude just before (after) the gap of the field in the left (right) waveguide propagating from right to left. Following the notation of section 2, one can write:

E 3 = tg (d)E 1 + rg (d)E -3 (7)
E -1 = tg (d)E -3 + rg (d)E 1 (8) E -3 = r 0 exp(2inkL 3 )E 3 , (9)
where n is the effective refractive index of the eigenmode of the waveguide and k = 2π/λ the wavenumber of the light in vacuum. Solving (7-9), we can express E -1 as a function of

E 1 : E -1 = reff (d, k, L 3 )E 1 , ( 10
)
where reff is the effective reflectivity of the ensemble gap + waveguide + end mirror. Once we have calculated reff , we can calculate the finesse of a resonator with a gap. It is the finesse of the equivalent resonator made of a perfect mirror (reflectivity r 0 = -1) facing a mirror of reflectivity reff , separated by a distance L 1 . If we write reff = r ef f exp(2iφ), we can express the finesse as:

F d = π √ r ef f 1 -r ef f 1 n(L 1 + L 3 ) nL 1 + ∂φ ∂k . (11)
In the following, we assume that L 3 ≫ d. Since we are looking for a resonance of the cavity, we consider the response r ef f for wavenumber k + δk such that δk < ∼ 1/L 3 ≪ 1/d. Therefore we will consider that rg (k + δk) ≈ rg (k) and tg (k + δk) ≈ tg (k) and that the only k-dependence of φ comes from the exp(2inkL 3 ) of Eq. 9. Knowing the effective reflectivity

F d = π √ r ef f 1 -r ef f 1 n(L 1 + L 3 ) nL 1 + nL 3 ∂φ ∂ (nkδL 3 ) , ( 12
)
where δL 3 = L 3 mod λ/2n. Equation 12 clearly shows that F d does not change if we multiply L 1 and L 3 by the same factor, as long as we keep δL 3 constant.

The plots in Fig. 3 show that the finesse globally decreases when the gap width d increases. This is consistent with the result of Fig. 2, and corresponds again to the fact that the diffraction losses increase with d. In addition to the global decrease, we observe oscillations of the finesse as a function of both d and L 3 . To understand these oscillations, we recall that the losses correspond to fields that are transmitted across the gap but are not coupled into the guided mode on the other side. In the presence of a perfect right-end mirror, the amplitude of the field leaking out at each interface is the sum of a contribution

proportional to E 1 (the single pass losses of the incident field) and a contribution proportional to E -3 (the single pass losses of the field reflected by the end mirror). Depending on the relative phase of E 1 and E -3 , the interference between these two contributions can be either constructive or destructive, and one can reduce the losses in the gap and reach a much larger finesse than what could be deduced from the single pass losses.

Atom-field coupling

To estimate the cooperativity of the system, we need to calculate the amplitude of a onephoton field in the center of the gap. Following the same argument as in the section 3, the field in the gap E g (z) is the sum of the multiple reflections of the light coming from the left waveguide and the multiple reflections coming from the right. Consequently, using equation 2, one can rewrite E g (z) as the sum of E + g (z) and E - g (z) where

E + g (z) = E 1 p even tr p |f 0 (pd + z)| exp(ikpd + iϕ 0 (pd + z)) +E -3 p odd tr p |f 0 (pd + z)| exp(ikpd + iϕ 0 (pd + z)) exp(ikz) = Ẽ+ g (z) exp(ikz) (13)
E - g (z) = E 1 p odd tr p |f 0 (pd + d -z)| exp(ikpd + iϕ 0 (pd + d -z)) +E -3 p even tr p |f 0 (pd + d -z)| exp(ikpd + iϕ 0 (pd + d -z)) exp(-ikz) = Ẽ- g (z) exp(-ikz), (14)
where |f 0 (z)| = |f (0, 0, z)| is the amplitude of the electric field on the axis of the waveguide.

Here, Ẽ+ g and Ẽg vary slowly on the scale of λ. As a result, the field inside the gap is the sum of two terms evolving at exp(ikz) and exp(-ikz). The interference of these two terms will not necessarily be constructive at the exact center of the gap z = d/2, but there will be an anti-node at a position z * within λ/2 of the center where the two amplitudes will add up constructively and

|E g (z * )| = | Ẽ+ g (z * )| + | Ẽ- g (z * )| ≈ | Ẽ+ g d 2 | + | Ẽ- g d 2 |. (15)
Up to this point, the amplitude of the field in the cavity has been set by an arbitrary factor E 1 . To calculate the single-photon amplitude, one has to write that the total energy stored in the cavity is equal to hω. Since d ≪ {L 1 , L 3 }, we can neglect the energy stored in the gap. The total energy of the cavity is therefore

hν ≈ 4n 2 ǫ 0 A(L 1 |E 1 | 2 + L 3 |E 3 | 2 ), (16)
where ν = c/λ is the photon frequency, ǫ 0 is the electric permittivity of vacuum, and we have assumed that |E -1 | ≈ |E 1 | since the losses are small. Finally, the single photon amplitude is given by:

E = √ hν |E + g | + |E - g | 4n 2 ǫ 0 A(L 1 |E 1 | 2 + L 3 |E 3 | 2 ) . ( 17
)
Assuming that L 1 ≈ L 3 ≈ L, we can rewrite the maximum atom field coupling g at the center anti-node as g = g 0 g(d, L 1 , L 3 ) with:

g 0 = 3πcγ k 2 n 2 AL , (18) and g
(d, L 1 , L 3 ) = 1 √ 2 |E + g | + |E - g | |E 1 | 2 + |E 3 | 2 . ( 19
)
Here again, L 1 is chosen to ensure that the cavity is resonant at the atomic frequency, and is no longer a free parameter once d and L 3 are fixed. We can therefore plot g(d, L 1 , L 3 ) = g(d, L 3 ) as function of d and L 3 (Fig. 4). The value of g varies between 1 and n. This can be very well explained by the inset of the Fig. 4 showing the electric field distribution inside To get a feeling for the magnitude of the coupling, we use the parameters of the above cavity including the effective mode area A = 9.9 µm 2 (see Ref. [START_REF] Gleyzes | Towards a monolithic optical cavity for atom detection and manipulation[END_REF]). The formula [START_REF] Hugonin | Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization[END_REF] gives g 0 ≈ 2π × 30 MHz and so the maximum value of the coupling is about 2π × 100 MHz.

Cooperativity of the atom-cavity system

The figure of merit of the cavity is given by the cooperativity C = g 2 /κγ. The cavity loss rate κ is inversely proportional to the finesse:

κ = πc 2n(L 1 + L 3 )F , ( 20
)
and the cooperativity is then given by:

C = 2 σ abs F πnA g2 (d, L 3 ), ( 21
)
where σ abs = 6π/k 2 is the resonant atomic cross section.

We can see that the cooperativity is proportional to the product g2 F . Comparing the results of sections II and III shows that optimizing the cooperativity will be a trade off between maximizing F and maximizing g. The total finesse of the cavity is given by For the first case, we consider the losses due to absorption and/ or diffusion in the AlGaAs waveguide as experimentally measured in [START_REF] Gleyzes | Towards a monolithic optical cavity for atom detection and manipulation[END_REF], corresponding to F p ∼ 92. The results are shown Fig. 5 and they confirm what was suggested in [START_REF] Gleyzes | Towards a monolithic optical cavity for atom detection and manipulation[END_REF]. If the propagation losses are high, the total finesse is limited by F p , and the maximum of the cooperativity is a point in the parameter space very close to the values d = 1.95 µm and nL 3 = 1000λ which optimize the coupling g. Still, the calculation shows that the cooperativity can be increased by ∼10 % by slightly decreasing d and increasing L 3 , which slightly increases the finesse without losing too much of the coupling.

F -1 = F -1 d + F -1 p ,
We also performed simulations for a lossless waveguide, that is, in the situation where the only loss is diffraction in the gap (F d ≪ F p ). In this case the situation is completely different (see Fig. 6). Optimizing the cooperativity corresponds to maximizing the finesse rather than to maximizing the coupling. To illustrate this point, consider a gap width d = 1.95 µm. For nL 3 = 1000.25λ, g 2 is n 2 ∼10 times smaller than for nL 3 = 1000λ. However, the finesse is ∼29 times higher for nL 3 = 1000.25λ and this choice clearly results in a higher C. By setting d = 2.05 µm and nL 3 = 1000.23λ, one can further increase the finesse and the coupling, reaching a cooperativity C max ≈ 9.4.

It is also interesting to notice that the maximum of C is very narrow with respect to L 3 , but is much wider with respect to d. In the vicinity of d ≈ 1.95 µm, the C > 9 area is 180 nm wide along the d axis. This is important experimentally, because unlike the waveguide parts of the cavity (L 1 and L 3 ), whose refractive index can be changed by changing the temperature or by injecting a current, the gap's optical length cannot be fine-tuned once the cavity has been fabricated. These calculations show that if the gap is fabricated within 90 nm of the optimal width, it will always be possible to reach C > 95 %C max by slightly tuning the index of refraction of L 3 .

Another important parameter characterizing the atom-cavity system is the saturation photon number [START_REF] Enk | Quantum information processing in cavity-QED[END_REF], the photon occupation number in the cavity necessary to saturate the atomic transition, N sat = (γ/g) 2 [START_REF] Kimble | Strong interactions of single atoms and photons in cavity QED[END_REF]. The calculations in Sec. 4 show that typically the atom field coupling is at least 10 times the natural linewidth of the Rb transition (γ = 2π×3 MHz).

Thus the saturation photon number can vary between 0.01 and 0.001 as the gap width and the total cavity length are varied.

The saturation photon number is useful to characterize the cavity transmission in the presence of an atom. If we adjust the cavity parameters to give maximum cooperativity, the coupling is minimal (g ∼ 2π × 30 MHz) while the finesse is about 1500. In this situation we have κ ∼ 2π ×34 MHz close to the value of g, but much larger than the atomic damping rate.

Without an atom the cavity, the transmission spectrum close to resonance show a peak with a width of order κ. Over a broader spectrum of frequencies, the cavity exhibits a complex structure characteristic of a multimirror Fabry-Pérot interferometer as analyzed in Ref. [START_REF] Van De Stadt | Multimirror Fabry-Perot Interferometers[END_REF].

The insertion of an atom into the cavity causes a dip in the in the absorption spectrum when the light frequency is close to the atomic resonance [START_REF] Poldy | Single-atom detection with optical cavities[END_REF]. The width of this dip is of order γ. Reference [START_REF] Poldy | Single-atom detection with optical cavities[END_REF] discusses the optimization of the signal to noise in such a situation and points out that the optimum signal to noise ratio is achieved when the photon number is equal the saturation photon number. Direct observation of the vacuum Rabi splitting is not possible because of the large cavity damping.

Summary

We have analyzed the cooperativity of a cavity consisting of an optical waveguide in which a gap is opened to let atoms interact with the cavity field. Because of the high index of refraction of the guided parts of the cavity, the Fresnel reflections at the interfaces are not negligible and the gap behaves like a low finesse Fabry-Perot inside the high finesse cavity.

We have separately calculated the losses induced by the diffraction of the light in the gap, and the value of the atom-light coupling. Choosing the cavity parameters such that all subcavities are resonant with the electromagnetic field maximizes the atom field coupling, but this choice also maximizes the diffraction losses introduced by the gap. This choice is optimal as long as the losses are dominated by other losses, such as absorption or diffusion in the waveguide. If the non-diffractive losses are small enough, a higher cooperativity is obtained by choosing the gap size and cavity length to decrease the coupling strength and increase the finesse. Our simulation predicts that a finesse on the order of 1500 and cooperativity C ∼ 9 should be possible. We also showed that the value of the cavity parameters, especially the gap width, are tolerant to small deviations. The possibility of tuning the waveguide lengths means that close to optimum performance can be obtained despite small errors in the gap width. The achievable cooperativity can attain values which should be of interest for the detection of single atoms on chips. Although the fabrication of a gap remains a great technical challenge and has not yet been achieved, the fully integrated nature of the device we have analyzed means that we can envision a system of several such structures fabricated in parallel on the same substrate. This would permit a structure similar to that in reference [START_REF] Kohnen | An array of integrated atom-photon junctions[END_REF] with the additional feature of cavity enhanced atom sensitivity. The waveguides are assumed to be lossless.

  reff , we are able to compute the finesse of such a compound micro cavity. The results are shown in Fig.3for different values of d, L 1 and L 3 . We varied d between 1.5 and 2.5 µm, and L 3 between 1000λ/n and 1001λ/n. Once d and L 3 are fixed, the distance L 1 is no longer a free parameter since it has to be chosen to verify the resonance condition of the cavity. However, we can always choose L 1 within λ/2n of 1000λ/n. Therefore, in the calculations, we assume that L 1 ≈ 1000λ/n, and only plot the finesse as function of d and L 3 . The choice of 1000λ/n is arbitrary and simply ensures the two conditions L 3 ≫ d and L 1 ≫ λ/n, necessary to neglect the dependence of L 1 with L 3 and d in the calculations. Since L 3 only appears in equations (7-9) as exp(2iknL 3 ), one can check that

the cavity. If L 1 ,

 1 L 3 and d all correspond to an integer number of half-wavelength of the field, the interferences of the partial reflections at the air-waveguide interfaces enhance the field in the gap by a factor n, leading to g = n. If d corresponds to a half integer number of half wavelengths in the gap, g = √ 2n/ √ n 2 + 1 is constant. If d is an integer number of half wavelengths, but L 1 and L 3 are equal to λ/4n (mod λ/2n), then the amplitude of the standing wave of the electromagnetic field is constant in the whole cavity and g = 1.

  where F d corresponds to the finesse induced by the diffraction losses (calculated in section 3), and F p corresponds to the other intrinsic losses of the cavity (such as the losses due to the absorption or scattering of the light during its propagation in the waveguide part of the cavity). Using our numerical model, we have computed the cooperativity as a function of d and L 3 in the two regime F d ≫ F p and F d ≪ F p . Each time we vary d between 1.5 and 2.5 µm, and L 3 between 1000λ/n and 1001λ/n keeping L 1 ≈ 1000λ/n in the two different cases.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. (a) Transverse geometry of the waveguide. The waveguide is a rectangular ridge of AlGaAs, 4 µm wide. The false color image shows the result of a numerical calculation of the amplitude of fundamental mode of the waveguide operating at λ =780 nm. The mode area is calculated numerically and we find A = 9.9 µm 2 . (b) Top view of the gapped waveguide. Atoms are to be guided into this gap to interact with the intracavity field. The gap constitutes a low finesse cavity within a larger waveguide cavity which is closed by reflecting surfaces at the both ends of the waveguide.

Fig. 4 .

 4 Fig. 4. Contour plot of the atom-field coupling g as function d and L 3 . Insets show the electric field profile inside the coupled cavity for special cases of d and L 3 with a normalisation |E 1 | + |E -1 | = 1. (a) Electric field profile for d = 1.95 µm and L 3 equal to an integer number of half wavelengths. (b) Electric field profile for d = 1.95 µm and L 3 = λ/4n (mod λ/2n). (c) Electric field profile for d = 1.75 µm and L 3 equal to an integer number of half wavelengths. (d) Electric field profile for d = 2.05 µm and L 3 = 0.23λ (mod λ/2n) (value that optimizes C in the lossless case). 17

Fig. 5 .Fig. 6 .

 56 Fig. 5. Cooperativity of a Rb atom in the coupled cavity system as a function of d and L 3 , including propagation losses in the waveguides. The attenuation coefficient corresponds to that measured in Ref. [17]. The coupling is maximized for d = 1.95 µm and nL 3 = 1000.5λ.

  1, ‡ 1 Laboratoire Charles Fabry, Institut d'Optique, CNRS et Université Paris-Sud, Campus Polytechnique, 91127 Palaiseau Cedex, France.

Acknowledgments

We thank Philippe Lalanne for helpful discussions and for providing us with propagation simulation software. We acknowledge financial support from the Institut Francilien pour la Recherche en Atomes Froids, from the CNANO program of the Ile de France, and from the


Déborah Ferré

Loïc Hervé
email: herve@insa-rennes.fr

James Ledoux
email: james.ledoux@insa-rennes.fr

Regular perturbation of V -geometrically ergodic Markov chains

Keywords: AMS subject classification : 60J05, 47B07 Stability, Spectral method

In this paper, new conditions for the stability of V -geometrically ergodic Markov chains are introduced. The results are based on an extension of the standard perturbation theory formulated by Keller and Liverani. The continuity and higher regularity properties are investigated. As an illustration, an asymptotic expansion of the invariant probability measure for an autoregressive model with i.i.d. noises (with a non-standard probability density function) is obtained.

Introduction and statements

Let {P ε } |ε|<ε 0 be a family of transition kernels on a measurable space (X, X ), where ε reads as a small perturbation parameter. Throughout the paper V : X →[1, +∞) is a fixed function. The (unperturbed) kernel P 0 is assumed to satisfy the classical V -geometrical ergodicity property, called (VG) in our paper, that is: P 0 admits a unique invariant probability measure π 0 , we have π 0 (V ) < ∞, and there exists some constants c ∈ (0, +∞) and κ 1 ∈ (0, 1) such that:

∀x ∈ X, sup |f |≤V E[f (X n ) | X 0 = x] -π 0 (f ) ≤ c κ n 1 V (x). (VG)
or equivalently ∀x ∈ X, sup

|f |≤V P n 0 f (x) -π 0 (f ) ≤ c κ n 1 V (x).
This means that P 0 has a spectral gap on the weighted supremum normed space B V composed of the measurable functions f : X → C such that f V := sup x∈X V (x) -1 |f (x)| < ∞. We are interested in the two following questions. For |ε| small enough, (I) does P ε admit an invariant probability measure, say π ε , and is P ε V -geometrically ergodic? 1 (II) Do we have any control on π 0 -π ε ?

Under some classical aperiodicity and irreducibility conditions, the property (VG) holds true if and only if P 0 satisfies the so-called drift condition based on the notion of small set (see [START_REF] Meyn | Markov chains and stochastic stability[END_REF] for the definition of the drift condition which is not used here). Consequently a natural and efficient way to study (I) is to prove that the perturbed Markov kernel P ε also satisfies the drift condition (e.g. see [START_REF] Roberts | Convergence properties of perturbed Markov chains[END_REF][START_REF] Breyer | A note on geometric ergodicity and floatingpoint roundoff error[END_REF]). However, to the best of our knowledge, the theory of geometrical ergodic Markov chains does not provide any general answer for question (II), except in terms of weak convergence in specific cases (see [START_REF] Roberts | Convergence properties of perturbed Markov chains[END_REF]). On the other hand the standard perturbation theory, which is a natural way to investigate (II), leads to assume the following continuity condition:

P ε -P 0 B V := sup f V ≤1 P ε f -P 0 f V → 0 when ε → 0 (1)
that is the operator norm of P ε -P 0 on B V goes to 0 as ε → 0. In a series of papers, Kartashov has introduced the concept of "strongly stable Markov chain" for a Markov chain with a transition kernel P 0 such that, in some neighborhood of P 0 with respect to • B V , P ε has a unique invariant probability measure π ε with the property

sup f V ≤1 |π ε (f ) -π 0 (f )| → 0 as P ε -P 0 B V → 0 uniformly in this neighborhood.
Strong stability is shown to be equivalent to the convergence

lim n → ∞ sup f V ≤1 1 n n k=1 P k 0 f -π 0 (f )1 X V = 0.
Moreover, if P 0 is V -geometrically ergodic, then given ρ ∈ (0, 1), one can consider N ∈ N * such that cκ N 1 ≤ ρ, and next if P ε is such that

∆ N := P N 0 -P N ε B V < 1 -ρ, then P ε is V -geometrically ergodic and sup f V ≤1 |π(f ) -π ε (f )| = O(∆ N /(1 -ρ -∆ N ))
. We refer to [START_REF] Kartashov | Strong stable Markov chains[END_REF] for an overview of results in this direction and to [START_REF] Mouhoubi | New perturbation bounds for denumerable Markov chains[END_REF] for a related discussion for discrete state spaces. However, as discussed in [SS00, p. 1126] and in Example 1 below, the continuity condition (1) may be restrictive. Similar questions arise in the context of dynamical systems. To overcome the previous difficulty, Keller introduced the more general assumption

lim ε → 0 sup f 0 ≤1 P ε f -P 0 f 1 = 0
involving two norms • 0 and • 1 (instead of a single one) on the space on which P 0 has a spectral gap [START_REF] Keller | Stochastic stability in some chaotic dynamical systems[END_REF]. This approach has been highly enhanced by the Keller-Liverani perturbation theorem [START_REF] Keller | Stability of the Spectrum for Transfer Operators[END_REF][START_REF] Liverani | Invariant measure and their properties. A functional analytic point of view[END_REF], which has proved to be very powerful in studying the behaviour of the Sinai-Ruelle-Bowen measures of certain perturbed dynamical systems (e.g. see [Bal00, Th 2.10] and [GL06, Th. 2.8]).

The goal of this paper is to show that the Keller-Liverani theorem also provides an interesting way to investigate both the questions (I) (II) in the context of geometrical ergodic Markov chains. In this markovian context, the closest work to ours is [SS00] where Keller's approach is used. The results of [START_REF] Shardlow | A perturbation theory for ergodic Markov chains and application to numerical approximations[END_REF] are improved here thanks to the Keller-Liverani perturbation theorem. Furthermore in this paper, higher regularity properties than continuity are investigated in question (II). Mention that the results of [START_REF] Keller | Stability of the Spectrum for Transfer Operators[END_REF][START_REF] Liverani | Invariant measure and their properties. A functional analytic point of view[END_REF] have been already used in [START_REF] Ferré | A parametric first-order Edgeworth expansion for Markov additive functionals[END_REF] to study some stability properties of parametric autoregressive models (for different purposes from those of Proposition 1 below).

Notations. For β ∈ [0, 1], we denote by (B β , • β ) the Banach space composed of the measurable functions f : X → C such that f β := sup x∈X V (x) -β |f (x)| < ∞. Note that B 0 corresponds to the space of bounded measurable functions on X, with f 0 = sup x∈X |f (x)|, and that B 1 = B V . We denote by (L(B β , B β ′ ), • β,β ′ ) the space of all the bounded linear maps from B β to B β ′ , equipped with its usual norm:

T β,β ′ = sup T f β ′ , f ∈ B β , f β ≤ 1 . We write L(B β ) for L(B β , B β
* B ′ β ,B ′ 0 ≤ T * B ′
β from the continuous inclusion B 0 ⊂ B β . Each perturbed Markov kernel P ε is assumed to continuously act on B 1 . The unperturbed kernel P 0 is assumed to satisfy (VG), namely: P 0 admits a unique invariant distribution π 0 on (X, X ), π 0 (V ) < ∞, and ∃κ 1 ∈ (0, 1),

P n 0 -π 0 (•)1 X 1 = O(κ n 1 ). (V 1 )
We also assume that there exist N ∈ N * , L ∈ (0, +∞) and δ ∈ (0, 1) such that

∀ε ∈ (-ε 0 , ε 0 ), P N ε V ≤ δ N V + L 1 X . (D)
Under some classical aperiodicity and irreducibility assumptions, Property (V 1 ) is equivalent to the drift condition (see [MT93, Chapter 4, Th. 16.0.1]). Condition (D) on the family {P ε } |ε|<ε 0 is weaker than the simultaneous geometrical ergodicity condition introduced in [RRS98] since (D) involves no small set.1

Theorem 1 Conditions (V 1 ), (D) and

lim ε → 0 P ε -P 0 0,1 = 0 (2)
are assumed to hold. Then, setting κ := max(κ 1 , δ), the following statements are fulfilled:

1. for each κ ∈ ( κ, 1), there exists ε 1 ∈ (0, ε 0 ] such that, for all ε ∈ (-ε 1 , ε 1 ), P ε has a unique invariant probability measure π ε , with π ε (V ) < ∞, such that sup

|ε|<ε 1 P n ε -π ε (•)1 X 1 = O(κ n ). (3)
2. We have lim

ε → 0 sup f 0 ≤1 π ε (f ) -π 0 (f ) = 0. (4)
Note that, if each probability measure π ε admits a density, say p ε , with respect to a fixed positive measure ψ on (X, X ), then we have

sup f 0 ≤1 |π ε (f ) -π 0 (f )| = R |p ε (x) -p 0 (x)| dψ(x).

Conclusion (

3) means that P ε is V -geometrically ergodic in a uniform way with respect to the perturbation parameter ε. Conclusion (4) means that the total variation norm of π ε -π 0 goes to 0 when ε → 0. Consequently Theorem 1 provides the same theoretical conclusions that in [Kar96, Chap. 3], but under the continuity condition (2) which is weaker than that in (1). Indeed we have: P ε -P 0 0,1 ≤ P ε -P 0 1 . There are several examples (see for instance [START_REF] Shardlow | A perturbation theory for ergodic Markov chains and application to numerical approximations[END_REF]) showing that the continuity condition (1) of [START_REF] Kartashov | Strong stable Markov chains[END_REF] may fail, while (2) (or (9) below) holds true. The case of an AR process is investigated in Example 1 below. Note that the uniform bound (3) can be obtained from [START_REF] Meyn | Computable bounds for geometric convergence rates of Markov chains[END_REF] when the kernels P ε are assumed to satisfy the drift condition with constants and small set which do not depend on ε. To derive (3) and (4) from the weaker assumptions (V 1 ) and (D), some continuity assumption must be assumed on the map ε → P ε .

Typically Condition (V 1 ) is the assumption AI in [START_REF] Shardlow | A perturbation theory for ergodic Markov chains and application to numerical approximations[END_REF] (see the remark following their assumptions). Condition (2) is weaker than the assumption AII in [START_REF] Shardlow | A perturbation theory for ergodic Markov chains and application to numerical approximations[END_REF] involving the sequence (P n ε ) n≥1 (here Assumption AII is only required for n = 1). Mention that, given any r ∈ ( κ, 1) and setting η := 1 -ln r/ ln δ (η ∈ (0, 1)), the proof of Theorem 1 and further results in [START_REF] Keller | Stability of the Spectrum for Transfer Operators[END_REF] ensure the following:

∃D r ∈ (0, +∞), sup f 0 ≤1 π ε (f ) -π 0 (f ) ≤ D r ( P ε -P 0 0,1 ) η . (5)
This provides an alternative statement to [SS00, Th. 3.1], which states an inequality of type (5) in assuming the existence of π ε . The V -geometrical ergodicity of the perturbed kernel P ε was an open question in [START_REF] Shardlow | A perturbation theory for ergodic Markov chains and application to numerical approximations[END_REF]. Actually, under their assumptions, Condition (D) is a quite natural hypothesis for P ε to inherit the V -geometrical ergodicity of the unperturbed kernel P 0 (note that Condition (D) is not so far from the drift condition). In the proof of Theorem 1, Condition (D) is viewed as a Doeblin-Fortet inequality on the dual space of B 1 in order to use the Keller-Liverani theorem.

Example 1 (AR process) Assume that X := R and (X n ) n∈N is the autoregressive model defined by

n ∈ N * , X n = αX n-1 + ϑ n (6)
where X 0 is a real-valued random variable, α ∈ (-1, 1), and (ϑ n ) n≥1 is a sequence of i.i.d. real-valued random variables, independent of X 0 . Assume that ϑ 1 has a Lebesgue probability density function on X, say ν(•), and admits a first moment, |x|ν(x)dx < ∞. We know that (X n ) n∈N is a Markov chain with transition kernel

P α (x, A) = R 1 A (αx + y)ν(y)dy = R 1 A (y)ν(y -αx)dy. (7) Set V (x) := 1 + |x|, x ∈ R.
It is known (e.g. [Wu04, Sec. 8] or [GHL11, Sec. 5.5]) that, for each α ∈ (-1, 1), (X n ) n∈N is V -geometrically ergodic with an invariant distribution π α . Next, given any a 0 ∈ (0, 1), it can be easily checked that the family {P α , α ∈ (-a 0 , a 0 )} satisfies Condition (D) with N = 1 and with any δ ∈ (a 0 , 1). Finally we prove below that, for every α 0 ∈ (-1, 1), the kernels P α 0 +ε satisfy the weak continuity condition (2). The previous facts ensure that Theorem 1 applies to the family (P α 0 +ǫ ) ǫ , so that the total variation norm of π α -π α 0 goes to 0 when α → α 0 . Note that no specific assumptions are required on the density ν, excepted that it admits a first moment. To the best of our knowledge this last result is new. Anyway, it cannot be deduced from a known method as in [START_REF] Kartashov | Strong stable Markov chains[END_REF] since Condition (1) does not hold, as proved afterwards.

First we show that, for any α 0 ∈ (-1, 1), we have lim α → α 0 P α -P α 0 0,1 = 0, that is

∀ε > 0, ∃η > 0, |α -α 0 | < η =⇒ P α -P α 0 0,1 = sup f 0 ≤1 sup x∈R |P α f (x) -P α 0 f (x)| V (x) < ε.
From (7), we have P α f (x) = R f (y)ν(y -αx)dy so that

P α -P α 0 0,1 ≤ sup x∈R ν(• -αx) -ν(• -α 0 x) L 1 (R) V (x) .
Let ε > 0. Since V (x) → +∞ when x → ∞, we can choose a constant A > 0 such that 4/V (x) < ε for any |x| > A. Therefore, using that ν is a probability density function, we obtain

sup |x|>A ν(• -αx) -ν(• -α 0 x) L 1 (R) V (x) ≤ sup |x|>A 2 V (x) < ε. Next, the map from R into L 1 (R) defined by t → ν(• -t) is continuous, so that we can pick η ′ > 0 such that ∀|x| ≤ A, |αx -α 0 x| ≤ |α -α 0 |A < η ′ =⇒ ν(• -αx) -ν(• -α 0 x) L 1 (R) < ε.
Therefore, given that V ≥ 1, there exists η := η ′ /A such that

|α -α 0 | < η =⇒ sup |x|≤A ν(• -αx) -ν(• -α 0 x) L 1 (R) V (x) ≤ sup |x|≤A ν(• -αx) -ν(• -α 0 x) L 1 (R) < ε.
Second, let us check that, whatever the density ν(•) is, the strong continuity condition (1) is never fulfilled. Let us consider a positive real number a such that Such a real number exists for any density ν(•) (if the previous terms coincide for every a > 0, then we obtain a contradiction when a → +∞). Let α 0 ∈ (0, 1) be fixed. For each α ∈ (α 0 , 1), define

x α := a α -α 0 > 0.
Note that -a + α 0 x α < -a + α x α < a + α 0 x α < a + α x α . Next let us introduce the following function

∀y ∈ R, f α (y) := y 1 [a+α 0 xα,a+αxα] (y) -y 1 [-a+α 0 xα,-a+αxα] (y).
From (7), we have

(P α f α )(x α ) = a 0 (y + αx α ) ν(y)dy - -a -2a
(y + αx α ) ν(y)dy

(P α 0 f α )(x α ) = 2a a (y + α 0 x α ) ν(y)dy - 0 -a (y + α 0 x α ) ν(y)dy.
Set J := a -a y ν(y)dy --a -2a y ν(y)dy -2a a y ν(y)dy. We obtain

(P α f α )(x α ) -(P α 0 f α )(x α ) = J + αx α a 0 ν(y)dy - -a -2a ν(y)dy + α 0 x α 0 -a ν(y)dy - 2a a ν(y)dy .
Since V (x α ) = 1 + x α and x α → +∞ as α ↓ α 0 , it follows that

lim α↓α 0 (P α f α )(x α ) -(P α 0 f α )(x α ) V (x α ) = α 0 a -a ν(y)dy - -a -2a
ν(y)dy -2a a ν(y)dy = 0.

Finally, from

P α -P α 0 1 = P α -P α 0 B V = sup |f |≤V sup x∈R (P α f )(x) -(P α 0 f )(x) V (x) ≥ (P α f α )(x α ) -(P α 0 f α )(x α ) V (x α ) ,
we deduce that P α -P α 0 B V does not go to 0 when α ↓ α 0 .

Reinforcing the continuity condition (2) of Theorem 1 allows us to obtain the following refinement.

Corollary 1 Assume that Conditions (V 1 ) and (D) hold and that there exists β ∈ (0, 1] such that2 ∃κ β ∈ (0, 1),

P n 0 -π 0 (•)1 X β = O(κ n β ) (V β )
and lim

ε → 0 P ε -P 0 β,1 = 0. ( 9
)
Then there exist ε 1 ∈ (0, ε 0 ] (from Theorem 1) and C ∈ (0, +∞) such that:

∀ε ∈ (-ε 1 , ε 1 ), sup f β ≤1 π ε (f ) -π 0 (f ) ≤ C P ε -P 0 β,1 .
Remark 1 Let P be any Markov kernel on X. If we have

P N V ≤ δ N V + L 1 X for some N ∈ N * ,
L ∈ (0, +∞) and δ ∈ (0, 1) and if P ℓ is compact from B 0 to B V (for some ℓ ≥ 1), then P is a powerbounded quasi-compact operator on B V . Moreover its essential spectral radius is such that r ess (P ) ≤ δ.

Actually, denoting by δ V (P ) the infinum of the real numbers δ for which the above inequality holds for some N ∈ N * and L ∈ (0, +∞), we have r ess (P ) ≤ δ V (P ), with equality in many cases. This follows from the Doeblin-Fortet inequalities (11) and [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipchitziens[END_REF], since (P * ) ℓ is compact from B ′ V to B ′ 0 . If in addition P satisfies aperiodicity and irreducibility conditions, then P is V -geometrically ergodic. Such results are fully detailed and applied in [START_REF] Guibourg | Quasi-compactness of Markov kernels on weightedsupremum spaces and geometrical ergodicity[END_REF] (without any perturbation issues). Now, if ρ V (P ) denotes the infinum bound of the real numbers κ 1 such that (V 1 ) holds, then we have ρ V (P ) ≥ r ess (P ). If there are eigenvalues λ of P on B V such that r ess (P ) < |λ| < 1, then ρ V (P ) is the maximal modulus of such eigenvalues (which are in finite number from the definition of r ess (P )); if not, we have ρ V (P ) = r ess (P ). For instance, if (X n ) n∈N is defined by (6) and if the probability density function ν(•) of the noise has a moment of order r ∈ [1, +∞), then (X n ) n∈N is (1 + | • |) r -geometrically ergodic with r ess (P ) ≤ |α| r and ρ V (P ) = |α|. See [Wu04, Sec. 8] or [GHL11, Sec. 5] for details. Now we are interested in asymptotic expansions of the perturbed invariant distributions. Under Condition (VG) and the continuity condition (1), expansions related to the generalized potential R = (I -

P 0 + Π 0 ) -1 , with Π 0 (•) = π 0 (•)1 X , are given in [Kar81, Kar96, HH03], namely π ε = π 0 (I -D ε R) -1 = k≥0 π 0 (D ε R) k = π 0 + n k=1 π 0 (D ε R) k + O((D ε R) n+1 ), when D ε = P ε -P 0 is such that D ε R 1 < 1.
In general this is not a Taylor expansion, except for instance when D ε has the form D ε = ǫD with D ∈ L(B 1 ). This special case is discussed in [START_REF] Altman | Perturbation analysis for denumerable Markov chains with application to queueing models[END_REF]. In fact, for general perturbations, even in the case when (1) is fulfilled, obtaining Taylor expansions for π ε causes difficulties when the derivatives of the perturbed kernels P ε (x, •) w.r.t. to ε yield some weights (see the term x k in (15) for order of regularity k ≥ 1). For this question, the derivation procedure (see [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF][START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]) based on the Keller-Liverani theorem is of great interest.

To illustrate this approach, we only consider the special instance of autoregressive model introduced in Example 1. More specifically, we assume that (X n ) n∈N is given by ( 6) and that the probability density function ν(•) of the noise has a moment of order r, |x| r ν(x)dx < ∞, for some r ∈ [1, +∞). Set V (x) := (1 + |x|) r , x ∈ R. Then, for each α ∈ (-1, 1), (X n ) n∈N is V -geometrically ergodic with an invariant distribution π α . Moreover we assume that r is not an integer (ie. r > ⌊r⌋ where ⌊•⌋ stands for the integer part function on R), that ν(•) is positive and ⌊r⌋ + 1-times continuously differentiable on R, with

j = 1, . . . , ⌊r⌋ + 1, sup t∈R |ν (j) (t)| ν(t) < ∞. ( 10
)
Finally suppose that for all x 0 ∈ R, there exist a neighborhood V x 0 of x 0 and a non-negative measurable function q x 0 (•) such that R (1 + |y|) r q x 0 (y) dy < ∞ and: ∀y ∈ R, ∀v ∈ V x 0 , ν(y + v) ≤ q x 0 (y).

Proposition 1 Under the previous assumptions, there exists β ≡ β(r) ∈ (0, 1) such that the map α → π α is ⌊r⌋-times continuously differentiable from (-1, 1) to the dual space B ′ β of B β . In particular, for all α ∈ (-1, 1), there exist ⌊r⌋ signed measures µ α,1 , . . . , µ α,⌊r⌋ on R such that:

∀A ∈ X , π α+ε (A) = π α (A) + ⌊r⌋ j=1 ε j j! µ α,j (A) + ε ⌊r⌋ R ε (A) with lim ε → 0 sup A∈X R ε (A) = 0.
The previous conditions (10) are well suited to densities of the form ν(x) = c(1 + |x|) -γ For other forms of densities, they can be easily adapted in order that Proposition 1 works.

Proofs

Proof of Theorem 1. We shall repeatedly use the fact that a bounded linear operator (between two normed vector spaces) and its adjoint have the same norm. The adjoint of the rank-one projection

Π 0 (f ) = π 0 (f )1 X is defined by: ∀f ′ ∈ B ′ 1 , Π * 0 (f ′ ) := f ′ (1 X ) π 0 . First, Assumption (V 1 ) gives (P * 0 ) n -Π * 0 B ′ 1 = O(κ n 1
). Second, it follows from the continuity assumption (2) of Theorem 1 that

lim ε → 0 P * ε -P * 0 B ′ 1 ,B ′ 0 = 0,
Third, Assumption (D) gives:

∀ε ∈ (-ε 0 , ε 0 ), ∀f ′ ∈ B ′ 1 , P * N ε f ′ B ′ 1 ≤ δ N f ′ B ′ 1 + L f ′ B ′ 0 . (11)
Indeed, recall that B ′ 1 and B ′ 0 are Banach lattices and that, for each

g ′ ∈ B ′ 1 , we have g ′ B ′ 0 = |g ′ | B ′ 0 = |g ′ |, 1 X and g ′ B ′ 1 = |g ′ | B ′ 1 = |g ′ |, V , where •, • stands for the duality brackets in both B ′ 0 × B 0 and B ′ 1 × B 1 . Next, observe that, for every f ′ ∈ B ′ 1 and every f ∈ B 1 such that |f | ≤ V , we have | (P * ) N f ′ , f | ≤ |f ′ |, P N V . Hence (D) gives (11) as claimed.
The three previous facts show that {P * ε } |ε|<ε 0 satisfies the assumptions of [START_REF] Liverani | Invariant measure and their properties. A functional analytic point of view[END_REF] on B ′ 1 (see also [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] for the use of the perturbation Keller-Liverani theorem in a Markov context). Therefore, for all κ ∈ ( κ, 1), there exists ε 1 ∈ (0, ε 0 ] such that, for all ε ∈ (-ε 1 , ε 1 ), the following properties hold: there exist λ ε ∈ C satisfying lim ε → 0 λ ε = 1 and a rank-one projection Π ′ ε on B ′ 1 such that

sup |ε|<ε 1 (P * ε ) n -λ n ε Π ′ ε B ′ 1 = O(κ n ) and lim ε → 0 Π ′ ε -Π * 0 B ′ 1 ,B ′ 0 = 0. (12)
Up to reduce ε 1 , we have λ ε = 1. Indeed one may assume that |λ ε | > κ. Then we deduce from 1 X = P n ε 1 X and (12) that, for any

f ′ ∈ B ′ 1 , we have lim n λ -n ε f ′ , 1 X = lim n λ -n ε (P * ε ) n f ′ , 1 X = Π ′ ε f ′ , 1 X . Thus (λ -n ε ) n∈N converges in C, so that we have, either λ ε = 1, or |λ ε | > 1. Moreover the sequence (P n ε ) n∈N is bounded in L(B V ) from (D). Thus ((P * ε ) n ) n∈N is bounded in L(B ′ V ), and (12) then implies that (λ n ε ) n∈N is bounded in C. Thus λ ε = 1.
Hence ((P * ε ) n ) n∈N is Cauchy, and so is

(P n ε ) n∈N in L(B 1 ). Therefore ∃ Π ε ∈ L(B 1 ), lim n P n ε = Π ε in L(B 1 ). (13)
Thus

(P n ε ) * → Π * ε in L(B ′ 1 )
, and so Π ′ ε = Π * ε . We have obtained that sup

|ε|<ε 1 P n ε -Π ε 1 = O(κ n ) and lim ε → 0 Π ε -Π 0 0,1 = 0.
From (13) it follows that Π ε is a positive projection on B 1 satisfying

Π ε P ε = P ε Π ε = Π ε . Let us prove that Π ε is rank-one. We know that Π ′ ε is rank-one, namely Π ′ ε (•) := φ ε , • a ′ ε for some φ ε = 0 in the dual space of B ′
1 and for some

a ′ ε = 0 in B ′ 1 . From Π * ε = Π ′ ε , we obtain Ker Π ε = Ker a ′ ε . Thus Ker Π ε is of codimension one, so that we have dim Im Π ε = 1 (use B 1 = Ker Π ε ⊕ Im Π ε ). The last fact shows that Π ε (•) := e ′ ε (•) 1 X for some e ′ ε ≥ 0 in B ′ 1 . Since ∀A ∈ X , lim n P n ε (x 0 , A) = e ′
ε (1 A ) (for some fixed x 0 ∈ X) and P n ε (x 0 , •) is a probability measure on (X, X ) for each n ≥ 1, we deduce from the Vitali-Hahn-Saks theorem that A → π ε (A) := e ′ ε (1 A ) a probability measure on (X, X ). Clearly π ε is P ε -invariant. Next π ε and e ′ ε coincide on B 0 : indeed every f ∈ B 0 can be approached uniformly on X by a sequence (f n ) n of simple functions, so that π ε

(f ) = lim n π ε (f n ) = lim n e ′ ε (f n ) = e ′ ε (f ) (the last convergence holds since e ′ ε ∈ B ′ 1 ). We obtain π ε (V 1 [V ≤n] ) = e ′ ε (V 1 [V ≤n] ) ≤ e ′ ε (V ) < ∞ for every n ≥ 1 since e ′ ε ≥ 0. The monotone convergence theorem gives π ε (V ) < ∞. Thus f → π ε (f ) is in B ′ 1 . Since ∀f ∈ B 1 , lim n P n ε f = e ′ ε (f ) 1 X in B 1
, we deduce that e ′ ε = π ε on B 1 from the invariance of π ε . Finally the last assertion of Theorem 1 follows from lim ε → 0 Π ε -Π 0 0,1 = 0.

Proof of Corollary 1. Let κ ∈ (max(κ β , κ 1 ), 1), and let Γ be the oriented circle in C, centered at z = 1 and with radius less than (1-κ)/2. Note that the assumptions of Corollary 1 imply those of Theorem 1. We know from [START_REF] Liverani | Invariant measure and their properties. A functional analytic point of view[END_REF] and the proof of Theorem 1 that M := sup z∈Γ,|ε|<ε 1 (zI -P ε ) -1 1 < ∞. Moreover, the rank-one eigenprojection Π ε is from the standard spectral theory:

Π ε = 1 2iπ Γ (zI -P ε ) -1 dz. (14)
Next (V β ) gives M 0 := sup z∈Γ (zI -P 0 ) -1 β < ∞. Assume that 1 X 1 = 1 (to simplify). Then, from

(zI -P ε ) -1 -(zI -P 0 ) -1 = (zI -P ε ) -1 (P ε -P 0 )(zI -P 0 ) -1 , we obtain that for every f ∈ B β π ε (f ) -π 0 (f ) = Π ε (f ) -Π 0 (f ) 1 ≤ (1 -κ) 2 M P ε -P 0 β,1 M 0 f β .
Proof of Proposition 1. Recall that V (x) := (1 + |x|) r , x ∈ R. Let a 0 ∈ (0, 1). We apply the derivation procedure of [HP10, App. A] to the family {P α , α ∈ (-a 0 , a 0 )} with respect to the Banach spaces B β . Let β ∈ (0, 1]. For any α ∈ (-a 0 , a 0 ), the Markov kernel P α of the autoregressive model (X n ) n∈N satisfies the drift condition and the aperiodicity/irreducibility assumptions of [START_REF] Meyn | Markov chains and stochastic stability[END_REF]. Hence each P α satisfies (V β ). In the same way, {P α } |α|<a 0 satisfies (D), so that it also satisfies (D) with respect to the function V β thanks to Jensen's inequality. Let α ∈ (-a 0 , a 0 ). Then lim ε → 0 P α+ε -P α 0,β = 0 from Lemma 1 below. Hence the perturbation Keller-Liverani theorem can be applied in L(B β ) to the family (P α+ǫ ) ǫ (as seen with β = 1 in the proof of Theorem 1). This gives the following spectral properties: there exists κ α,β ∈ (0, 1) such that, for all κ ∈ (κ α,β , 1), exists ε > 0 such that the resolvents (zI -P α+ε ) -1 are well-defined and uniformly bounded in L(B β ) provided that |ε| ≤ ε and z ∈ C satisfies |z| ≥ κ and |z -1| ≥ (1 -κ)/2. Now let us introduce the formal derivative operators of α → P α :

∀x ∈ R, P k,α f (x) = (-1) k x k R f (y)ν (k) (y -αx) dy. ( 15
) Lemma 1 Let (β, β ′ ) ∈ [0, 1] 2 . (i) If β + k/r < β ′ ≤ 1 for k ∈ {0, . . . , ⌊r⌋}, then α → P k,α is continuous from (-a 0 , a 0 ) to L(B β , B β ′ ). (ii) If β + (k + 1)/r < β ′ ≤ 1 for k ∈ {0, . . . , ⌊r⌋ -1}, then α → P k,α is continuously differentiable from (-a 0 , a 0 ) to L(B β , B β ′ ), with dP k,α dα (α) = P k+1,α .
In a first stage, let us admit that Lemma 1 holds and apply [HP10, App. A]. For β ∈ (0, 1) and σ > 0, we set: T 0 (β) = β + σ/r and T 1 (β) = β + (1 + σ)/r. Lemma 1 gives for j = 1, . . . , ⌊r⌋:

-if T 0 (β) ∈ (0, 1], then the map α → P α is continuous from (-a 0 , a 0 ) to L(B β , B T 0 (β) ), -if T 1 (T 0 T 1 ) j-1 (β) ∈ (0, 1], then the map α → P α is j-times continuously differentiable from (-a 0 , a 0 ) to L(B β , B T 1 (T 0 T 1 ) j-1 (β) ).
Let β r ∈ (0, 1 -⌊r⌋/r), and let σ > 0 such that β r + [(2⌊r⌋ + 1)σ + ⌊r⌋]/r = 1. In other words we have 0 T 1 ) ⌊r⌋ T 0 (β r ) = 1. Let α 0 ∈ (-a 0 , a 0 ). Then it follows from [HP10, Prop. A.1] that there exists κ ∈ (0, 1) such that, for all z ∈ D := {z ∈ C : |z| ≥ κ, |z -1| ≥ (1 -κ)/2}, the map α → (zI -P α ) -1 is ⌊r⌋-times continuously differentiable from some open interval I α 0 centered at 0 into L(B βr , B 1 ). Moreover the derivatives (up to the order ⌊r⌋) of the last map are uniformly bounded in (α, z) ∈ I α 0 × D. Next, let us define Π α (f ) = π α (•)1 R for any f ∈ B βr . From the previous facts and standard spectral calculus (see ( 14)), the map α → Π α (f ) is ⌊r⌋-times continuously differentiable from I α 0 to L(B βr , B 1 ), so is α → π α (•) from I α 0 to B ′ βr . Since α 0 is any element in (-a 0 , a 0 ), with arbitrary a 0 ∈ (-1, 1), this proves the first assertion of Proposition 1.

Finally let α ∈ (-1, 1). For all borel set A of R, we have dπα dα (A) = lim h → 0 (π α+h (A) -π α (A))/h. Since π α+h (•) -π α (•) is a (signed) measure on R, it follows from the Vitali-Hahn-Saks theorem that there exists a (signed) measure µ α,1 on R such that we have: dπα dα (A) = µ α,1 (A). An obvious induction gives the same conclusion for the derivatives of order j = 2, . . . , ⌊r⌋.

Proof of Lemmas 1. Set A k := sup t∈R |ν (k) (t)|/ν(t) and A := max 0≤k≤⌊r⌋+1 A k . Note that there exists B ≡ B(β) such that, for all α ∈ [-a 0 , a 0 ], we have P α V β ≤ BV β .

For α ∈ (-a 0 , a 0 ), β ∈ (0, 1], f ∈ B β and x ∈ R, we denote F k (α) := (P k,α f )(x). From the assumptions on ν and Lebesgue's theorem, for every k = 0, . . . , ⌊r⌋, F k is differentiable on [-a 0 , a 0 ], with ∂F k ∂α (α) = F k+1 (α). Let k ∈ {0, . . . , ⌊r⌋}, (β, β ′ ) ∈ [0, 1] 2 such that β + k/r < β ′ ≤ 1, and let 0 < σ ≤ 1 be such that β + (k + σ)/r = β ′ . For any (α, α ′ ) ∈ (-a 0 , a 0 ) 2 , we obtain (use Taylor expansion of F k for (16b)):

F k (α) -F k (α ′ ) ≤ 2 A k B f β V (x) β+ k r . ( 16a
)
F k (α) -F k (α ′ ) ≤ |α -α ′ | A k+1 B f β V (x) β+ k+1 r . (16b)
Multiplying (16a) (to the power 1 -σ) by (16b) (to the power σ) gives:

|F k (α) -F k (α ′ )| ≤ 2|α -α ′ | σ AB f β V (x) β+ k+σ r ,
which rewrites as:

P k,α f -P k,α ′ f β ′ ≤ 2AB |α -α ′ | σ f β .
The first assertion of Lemma 1 is proved.

Next let k ∈ {0, . . . , ⌊r⌋ -1}, (β, β ′ ) ∈ [0, 1] 2 such that β + (k + 1)/r < β ′ ≤ 1, and let 0 < σ ≤ 1 be such that β + (k + 1 + σ)/r = β ′ . For any (α, α ′ ) ∈ (-a 0 , a 0 ) 2 , we obtain:

F k (α) -F k (α ′ ) -(α -α ′ )F k+1 (α ′ ) ≤ 2 |α -α ′ | A k+1 B f β V (x) β+ k+1 r (17a) F k (α) -F k (α ′ ) -(α -α ′ )F k+1 (α ′ ) ≤ |α -α ′ | 2 2 A k+2 B f β V (x) β+ k+2 r . (17b)
Then, by multiplying (17a) (with the power 1 -σ) and (17b) (with the power σ), we obtain:

|F k (α) -F k (α ′ ) -(α -α ′ )F k+1 (α ′ )| ≤ 2 |α -α ′ | 1+σ AB f β V (x) β+ k+1+σ r ,
which rewrites as:

P k,α f -P k,α ′ f -(α -α ′ )P k+1,α ′ f β ′ ≤ 2AB |α -α ′ | 1+σ f β .

  ) and T β for T β,β which is a slight abuse of notation. Let (B ′ β , • B ′ β ) denote the dual space of B β . If T ∈ L(B β ), then T * stands for the adjoint operator of T . By definition we have T * ∈ L(B ′ β ) with the corresponding operator norm T * B ′ β = T β . Note that T * also defines an element of L(B ′ β , B ′ 0 ) with corresponding operator norm T

In case ε = 0 (i.e. for a single transition kernel P0), the connection between the "best" constants κ1 in (V1) and δ in (D) is discussed in details in[START_REF] Guibourg | Quasi-compactness of Markov kernels on weightedsupremum spaces and geometrical ergodicity[END_REF]. See also Remark 1.

Under the drift condition, (V1) implies (V β ) by Jensen's inequality for β ∈ (0, 1].


Introduction

The slow hydrogen sorption kinetics of Mg and the high dissociation temperature of MgH 2 strongly limit the practical applications [1] and [START_REF]Hydrogen in intermetallic Compounds II[END_REF]. Even so, Mg is one of the most promising candidates for a hydrogen storage material and a lot of research has been undertaken to overcome these drawbacks. The usual way is the addition to Mg of some metal or alloy by the use of high-energy ball milling (mechanical alloying [START_REF] Gaffet | [END_REF]). Over the last 20 years, a large group of heterogeneous systems has been studied such as Mg-Fe [4], Mg-Ni [5], Mg-LaNi 5 [6] and many others [START_REF] Alefed | Hydrogen in Metals[END_REF]. It has also been shown that mechanical grinding in a reactive atmosphere promotes the solid-gas reaction. Such a procedure has been used to produce nitrides, oxides and hydrides [START_REF] Bobet | [END_REF] at room temperature.

In a previous study [9] we found that, under our milling conditions, the Mg-Co system containing 10 wt.% of Co exhibits the best sorption properties. With a view to application as electrodes in Ni-MH batteries, addition of Co is well known to improve the cycling life of the electrode [10]. The aim of this study was to get a better understanding of the influence of reactive mechanical grinding (RMG) and Co addition on both the crystallographic and the hydrogen sorption behavior of the Mg+10 wt.% Co mixture. In particular, our goal was to try to find the driving parameter of the nucleation and diffusion process.

Experimental

Pure Mg powder was mixed with Co (initial particle sizes of 50 μm and 3-7 μm, respectively, total weight 8 g) in a stainless steel container (with 17 hardened steel balls, total weight 72 g) closed in a hermetic way. All handling was performed in a glove box under Ar in order to prevent oxidation. The mill container was then evacuated to primary vacuum (10 -3 MPa) prior to filling with high purity hydrogen gas (≈1.1 MPa). The disc rotation speed was fixed to 200 rpm for all the experiments. The experimental procedure was as follow: 15 min of milling, 5 min of rest, 15 min of milling, refilling with H 2 gas (1.1 MPa). This procedure was repeated until the total milling time expected was reached. The procedure allows (i) prevention of leaks and (ii) increase in the H/M ratio (initially n H /n Mg ≈0.2).

The crystalline structure of as-milled powders was characterized by X-ray diffraction (XRD) analysis on a Phillips PW 1050 diffractometer using Bragg-Brentano geometry (CuKα radiation). These patterns were scanned by steps of 0.02° (2θ) from 25 to 80° with counting time of 10 s. The diffraction patterns were analyzed by a whole pattern fitting procedure (with the Thomson-Cox-Hastings function) using the program fullprof [START_REF] Rodriguez-Carjaval | Powder diffraction, Satellite meeting of the 15th Congress of IUCr[END_REF] in order to determine the lattice parameters, the weight concentration of each phase and both the crystallite size and the internal strain.

The hydriding apparatus has been described previously [START_REF] Bobet | [END_REF]. The sample chamber was evacuated to primary vacuum for 1 h without heating. The sample was then heated up to 350°C under 0.1 MPa of hydrogen gas. When the temperature was stabilized, hydrogen gas was introduced into the sample chamber to a pressure of about 1.6 MPa. The absorbed hydrogen quantity was measured as a function of time by a volumetric method.

Results and discussion

Structural and chemical study

It has already been shown that the RMG (200 rpm and m balls /m powder =9) of Mg+10 wt.% Co result in the formation of a mixture of MgH 2 , Mg and Co [9]. The amount of MgH 2 formed is successfully calculated using Rietveld refinement. However, when the RMG duration was increased beyond 10 h, the crystallinity of the mixtures decreased and the XRD method fails to determine precisely the MgH 2 content. The use of a thermogravimetric method is then used to determine the fraction of Mg transformed into MgH 2 . This method leads to a larger uncertainty for the experiments at 15 and 20 h. In Table 1, we summarize the results of the crystallographic and thermogravimetric experiments. It appears that Mg remained present even for long periods of RMG (up to 20 h). The formation of MgH 2 seems to be rapid during the first 10 h but slow thereafter. The diffraction patterns did not reveal any change in the lattice parameters (see Table 1) indicating that, as expected (i) almost no solid solution of H in Mg exists and (ii) the stoichiometry H/Mg for MgH 2 is strictly equal to 2. No formation of the ternary hydride Mg 2 CoH 5 is noticed as reported for MgH 2 -Ni and MgH 2 -Fe [13], [14] and [15]. However, the starting products for Liang et al. [13] was MgH 2 as opposed to Mg in the present case. MgH 2 that is formed in situ in our case does not exhibit the same microstructure as that used by Liang et al. [13]. Moreover, the milling apparatus and milling conditions are not the same so the resulting products can be different. However, we can expect that after a longer milling time some ternary Mg 2 CoH 5 could be formed (but then it would be formed as an almost amorphous phase).

The XRD patterns show that the peaks related to MgH 2 are broader than those of Mg indicating a difference in the size of the crystallites (Fig. 1). MG (MG under Ar atmosphere) induces a larger reduction of the crystallites size compared with RMG. As the milling conditions (and then the global milling energy) are the same in both cases we concluded that the exothermic formation of MgH 2 provides some 'thermal' energy that increased the crystallites size. Then the decrease of crystallites size by ball milling is reduced by that exothermic reaction. For RMG experiments, increasing the milling duration beyond 10 h does not lead to significant changes in the crystallites size of either Mg or MgH 2 . As we already notice the same phenomenon on MgH 2 formation, we concluded that the efficiency of milling beyond 10 h is low. The process of MgH 2 formation could be assumed to be firstly the adsorption of hydrogen on particles created by crushing (fresh surface) and then diffusion of the adsorbed hydrogen into the metals to form metal hydrides. On the basis that the milling efficiency beyond 10 h is low, no fresh surfaces would be created and the formation of MgH 2 would be considerably reduced.

The partition of Co in the Mg matrix is also important for the hydrogen sorption characteristics. The results of electron probe microanalysis reveals that a milling time of 2 h allows an almost random dispersion to be obtained even if it has been possible to observe again a few cluster of Co. The size of the Co clusters decreases rapidly with increasing milling time.

Hydriding properties

We have already published [9] a hypothesis on the mechanism of the absorption phenomenon. The plot dA/dt=f(F) presented in Fig. 2, where A is the wt.% of hydrogen absorbed (which is equivalent to a graph dF/dt=f(F) where F is the fraction of MgH 2 formed) and t is the time (of hydrogenation), allows a better understanding of the influence of MG atmosphere. The rate of formation is initially steady, indicating that the reaction is controlled by nucleation process and afterwards, the reaction rate decreases rapidly indicating that the reaction is controlled by a diffusion process. The nucleation process exists even for the first cycle but it starts at F≈0.4 because of the in situ hydrogenation that happened during the RMG. The behavior reported for MG products is equivalent but in this case, the nucleation duration (length) is shorter. Moreover, during the first absorption cycle, the reaction rate was very slow and typical of a reaction controlled by nucleation and growth. The two-step behavior is only observed from the second cycle.

We also notice that the nucleation process duration increases slightly with the number of hydrogenation cycles. As reported in Table 2, a direct relationship between specific surface and nucleation length can be established. Such a direct relationship between the specific surface and the nucleation 'length' clearly establishes that the process involved in the first step of the hydrogenation is controlled by the surface (as the nucleation process is). As the number of hydrogenation cycles increases, the specific surface increases too (decrepitation phenomenon) which leads to an increase of the nucleation duration. Increasing the milling time increases the specific surface and then the nucleation length, however, once again, beyond 10 h; the specific surface is not increased further.

To characterize the second step process, we use a simple model (spherical moving boundary model [16]). The rate equation can be written as:

and C=kD H (P 0 1/2 -P eq 1/2 ) where r is the crystallite radius, k the rate constant which depends on temperature and pressure, P o the applied pressure and P eq the equilibrium pressure (calculated according to the thermodynamic properties published by Liang et al. [17] i.e. ln P eq =-74 400/RT+133.5/R).

In order to compare all the mixtures, we take the Mg RMG for 2 h as a reference product and we normalized the C value estimated for the (Mg+10 wt.% Co) mixture by the C value calculated for this reference. Then, as the milling conditions are the same for all experiments and as, at given temperature and pressure, k and (P o 1/2 -P eq 1/2 ) should be constant (even if some dependence on the milling time exists) comparison of the normalized C values gives us the opportunity to compare the diffusion process. The results are summarized in Table 3.

The diffusion is improved by increasing the milling time up to 10 h as expected from the creation of defects. Beyond 10 h, the milling has been reported to be almost ineffective (no more internal stress generated) and so no more improvement of the diffusion constant is observed. Moreover, the Co addition does not improve the diffusion process -that means that the 3d element acts as a catalyst for the dissociation of H 2 as reported elsewhere [4], [9] and [18]. The use of RMG (compared with MG) leads to the larger improvement of the diffusion process. In the case of RMG, we established a linear relationship between the number of crystallites per particles and the normalized coefficient (Fig. 3). This relationship does not exist for the MG product suggesting two different behaviors. To a first approximation, we could conclude that the driving force of diffusion in the case of MG product is the intragrain diffusion as shown by Zaluska et al. [19] as in the case of the RMG product the intergrain diffusion should not be considered as negligible.

The dehydriding process was not extensively studied. However, the results of dehydrogenation as a function of temperature are shown in Fig. 4. The RMG improves significantly the dehydriding kinetics. Due to the strong dependence of the kinetics behavior upon the driving force (i.e. (P 0 -P eq ) [20]), it is not easy to compare our results which some previous ones. Even so, the kinetic behavior reported by Liang et al. [20] is a little better than ours.

Conclusion

It is reported in this paper that a two-step process can describe the hydrogen absorption of a mixture Mg+10 wt.% Co elaborated by RMG. The first one is a nucleation process and the second step is a diffusion process. Both the nucleation and the diffusion processes can be improved by increasing the milling duration (creation of defects for nucleation and decrease of the number of crystallites per particle for diffusion) up to 10 h. When the milling duration increase beyond 10 h the milling is ineffective and the hydrogen sorption properties barely improve.

It is therefore shown that the RMG process is an effective way of improving the sorption properties as it leads to an embrittlement of magnesium (decrease the particles size) without decreasing the crystallites size too much. It also allows the activation procedure (which is generally required prior to the first absorption reaction) to be suppressed. However, results published elsewhere [17], [19] and [20] using MgH 2 as a starting material are always slightly better than our results. Some experiments are underway on the MgH 2 -Co mixture submitted to RMG.


Karën Fort
email: karen.fort@inist.fr

Vincent Claveau
email: vincent.claveau@irisa.fr

Annotating Football Matches: Influence of the Source Medium on Manual Annotation

Keywords: Manual corpus annotation, Inter-annotator agreement, Football, Speech Transcription

In this paper, we present an annotation campaign of football (soccer) matches, from a heterogeneous text corpus of both match minutes and video commentary transcripts, in French. The data, annotations and evaluation process are detailed, and the quality of the annotated corpus is discussed. In particular, we propose a new technique to better estimate the annotator agreement when few elements of a text are to be annotated. Based on that, we show how the source medium influenced the process and the quality.

Introduction

It has become a cliché to state that multi-modal and multimedia documents are now widely spread and part of our every day life. Yet, proposing intelligent processing of these documents or simply accessing the information they contain is still an issue for real-world applications. Moreover, building and annotating resources to develop and test such applications raises issues that are seldom documented. In this paper, we present the development of an annotated text corpus of football 1 matches built from video (speech transcripts) and specialized websites. In that respect, the corpus is not multi-modal per se, as only text is processed, but it contains data from both written and oral sources. This corpus and the associated annotations were created in order to develop and test automatic tools for video summarizing, re-purposing or event extraction from football broadcasts. This application, developed with an industrial partner, is not further developed in this article, but it has an important impact on the elements to be annotated (see Section 3. and [START_REF] Fort | Towards a Methodology for Named Entities Annotation[END_REF]). In this paper, in addition to introduce a new resource, we aim at two other goals. First, we detail the annotation campaign and how good practice rules were implemented for this heterogeneous corpus. Secondly, across the annotation process, we exhibit quantitative and qualitative differences in results between the written and oral sources. To do so, we use inter-annotator and intra-annotator agreement measures and adapt existing ones to 1) acknowledge the quality of the produced annotation, 2) outline the differences in the quality results between the written and oral (from video) sources. The article is structured as follows. After a brief review of related studies in Section 2., we present the corpus, its annotation, and the campaign in Section 3. We then detail agreement measures and their results, used for the resource 1 in the sense of soccer. development, in Section 4., before concluding.

Related Work

A number of publications dealing with football-based applications (Nemrava et al., 2007, for example) refer to a domain annotated corpus. However, to our knowledge, none of them describe in details the manual annotation of the corpus itself. Besides, none of them concern a French corpus. Other studies used football corpora to create more or less detailed monolingual (Nathalie [START_REF] Gasiglia | Pistes méthodologiques pour l'exploration d'un corpus à haut rendement relatif au parler du football, une langue de spécialité de grande diffusion[END_REF] or multilingual [START_REF] Schmidt | The Linguistics of Football (Language in Performance 38), volume 38, chapter The Kicktionary: Combining corpus linguistics and lexical semantics for a multilingual football dictionary[END_REF] lexicons. In these cases, if the associated publications detail the annotation of the corpora that were used, the annotation itself was more linguisticallyoriented than domain-oriented and therefore raised different issues.

Campaign Preparation

Data Preparation

The corpus we used covers 16 European football matches. It is made of 24 transcripts of the video commentaries of the matches (1 per half-time, for 12 matches) and 16 files containing the written minutes of matches (the same 12 matches that are covered by the transcripts and 4 additional matches). The total size of the corpus reaches about 250,000 tokens. As shown by [START_REF] Fort | Corpus linguistics for the annotation manager[END_REF], its main characteristic is to be very heterogeneous, be it from the point of view of the type of matches (French 1 st league matches, international matches, etc), of the files size (from 1,116 tokens per match for minutes to 21,000 tokens per match for the transcripts), or of the sources (different TV channels and commentators). The speech contained in the video was manually transcribed using TRANSCRIBER [START_REF] Barras | Transcriber: a free tool for segmenting, labeling and transcribing speech[END_REF] and its default guidelines. It is worth noting that the transcripts are aligned with the speech. Therefore, it provides us with a precise timestamp for each word and annotation in the transcripts. The minutes also contain time information since each action is preceded by its occurring time in the match. Thus, it makes it possible to map events described in both sources.

Annotations

We decided to decompose the annotation into three steps, corresponding to layers of analysis of growing complexity, that were easy to annotate simultaneously. Thus, the annotators had to first annotate all the units (named entities, time and location), then the actions and finally the relations (see details in table 1 2 ). These labels were selected in three steps: 1) selection from an available, and rather general, football ontology [START_REF] Crampes | Ontologysupported and ontology-driven conceptual navigation on the world wide web[END_REF] from Ecole des Mines d'Alès3 , keeping our application in mind as recommended in [START_REF] Leech | Developing Linguistic Corpora: a Guide to Good Practice, chapter Adding Linguistic An-notation[END_REF], 2) modifications following the training phase, 3) modifications following the pre-campaign. As the corpus is made of two different media, one of which, the transcripts, is ellipses-prone ("Makoun. Et c'est récupéré. Clerc, avec Cris. Boumsong, Makoun." [Makoun. Saved. Clerc with Cris. Boumsong, Makoun.]), we decided not to annotate the actions' predicate, but the actors. The same goes for the relations. This choice was made in order to maintain only one annotation guide and a homogeneous annotation process. However, and this is especially the case in transcripts, the actor of an action does not always appear in the text: "Grand dribble en pivot bien pris" [Large dribble in pivot well stopped] (here, the actor of the dribble is not indicated). The same goes for the relations, in which the source or/and the target actor can be omitted: "Ribéry avec une faute sur Gourcuff" [Ribery with a foul on Gourcuff ] (here, the source actor of the foul is missing). In these cases, we asked the annotators to anchor the annotation on the action predicate and to add a predefined feature (Missing Actor for actions and Missing source/target for relations). Note that for the relations, we had to add a new unit, ActionPourActeurVide (ActionForEmptyActor), used to annotate the predicate, in order to anchor the relation on it.

Methodology

The annotation was performed by two annotators from INIST-CNRS, one man and one woman, both experts in football (regular player and former player). We chose to use GLOZZ [START_REF] Widlöcher | La plate-forme glozz : environnement d'annotation et d'exploration de corpus[END_REF] as annotation tool: it is easy to use and supports the annotation of relations. The files to be annotated were dispatched between the annotators so that they had a similar workload, taking into account the types of files (league 1 matches, international matches, etc), their source (minutes or transcripts) and their size. Apart from the training part, the corpus was automatically pre-annotated for player and coach names, ontologies/soccerV2.0.daml using lists found on specialized websites. Part of the annotation work was therefore about correcting pre-annotations, which proved to help gaining time and quality in at least the annotation of part-of-speech [START_REF] Fort | Influence of Preannotation on POS-tagged Corpus Development[END_REF]. We also advised the annotators to work first on the match minutes before annotating the transcripts (supposedly more ambiguous), when available. It is important to note that we finally decided to annotate the transcripts directly, not using the video, in order to gain time (more than 2 hours per transcript).

Annotators were asked to annotate layer by layer (see section 3.2.), and to track their time for each file and each annotation step within the file, using a freely available on line tool4 . They were also asked to work at least 10 hours a week on the annotation and keep a steady rhythm at it to optimize the learning curve and the quality of the work, but this recommendation was not always followed, due to busy schedules. We also told the annotators not to hesitate to add comments and we added an Uncertainty feature to the annotations that they could use. Annotator 1 used both possibilities while Annotator 2 did not. We used the methodology described by [START_REF] Bonneau-Maynard | Semantic Annotation of the French Media Dialog Corpus[END_REF] and computed the inter-annotator agreement early in the process in order to check for inconsistencies in the annotation model and obvious ambiguities in the tagset to improve the annotation guidelines. We also computed intra-annotator agreements, as recommended by [START_REF] Gut | Measuring the Reliability of Manual Annotations of Speech Corpora[END_REF]. The annotation campaign itself was done in several phases: 1) training: on the smallest match minutes file (not preannotated), using the annotation tool, 2) pre-campaign I: annotation by both annotators of the same corpus sample (of match minutes), computation of the inter-annotator agreement, discussion about disagreements, update of the guidelines, 3) pre-campaign II: annotation by both annotators (together) of one minutes file, new update of the guidelines, 4) pre-campaign III: annotation by both annotators of the same corpus sample (of transcripts), computation of the inter-annotator agreement, discussion about disagreements, new update of the guidelines, 5) campaign: annotation by annotators of the files assigned to them (match by match). Finally, other inter-annotator agreements and intra-annotator agreements were computed at the end of the campaign.

Results and analysis

Agreement measures

Computing intra-and inter-annotator agreements is essential when developing annotated resources: it is used to assess the reliability, hence the quality of the produced annotations, to set an upper bound of the performance of automatic systems, and in our case, to highlight the difficulty of the task according to the source modality. Cohen's [START_REF] Cohen | A Coefficient of Agreement for Nominal Scales[END_REF] or Carletta's [START_REF] Carletta | Assessing Agreement on Classification Tasks: the Kappa Statistic[END_REF] κ are preferred to simpler measure like F-measure since they take into account the chance agreement [START_REF] Artstein | Inter-coder agreement for computational linguistics[END_REF] Table 1: Annotation steps and corresponding labels require to evaluate the number of markables (entities that may require to be annotated). While the number of markables is obvious and known a priori for some tasks (like PoS tagging), it can only be estimated a posteriori for annotation tasks like ours [START_REF] Grouin | Proposal for an extension of traditional named entities: From guidelines to evaluation, an overview[END_REF]. To overcome this issue, we propose an a posteriori estimation based on the expectation-maximization procedure described in Algorithm 1. It iteratively estimates the number of markables δ (maximization step) using the (iteratively estimated) probability γ that all the annotators miss a same markable computed as the product of probability of A j missing a markable (expectation step). In the following subsection, we use this algorithm to estimate the number of markables when computing Cohen's and Carletta's κ. For instance, the intra-and inter-annotator Cohen's κ for the annotation of entities and actions in the minutes, when computed (as usual) by considering all the tokens as markables, respectively reaches 0.9456 and 0.9404. Such high values mask differences that are better emphasized when computing κ with the estimation of markables that we propose (see Sub-Section 4.4.). The two κ, as implemented, are also very strict in the sense that any slight difference between two annotations will be considered as a disagree-ment. Thus, when possible, we also provide the entropy agreement values as defined by [START_REF] Mathet | Une approche holiste et unifiée de l'alignement et de la mesure d'accord inter-annotateurs[END_REF] and implemented in GLOZZ. This measure takes into account partial matches and thus provides less pessimistic agreement values, but it does not apply to relations.

Data on Process

The choice of the annotation tool has an important impact on the annotation campaign. Our data model was designed to comply with GLOZZ's constraints. Thus, we used no relation with more than two actors and marked the actions with a prefix ("A ") to distinguish them from simple units. Also, as GLOZZ does not allow for the direct modification of the source text, the annotators could not correct the typographic errors, the missing whitespaces or the tokenization problems that occurred in the corpus, in particular in the transcripts. Annotator 1, who inserted a lot of comments, noted 94 of them, all in the transcripts (vs. 1 for annotator 2, in a minutes file). These transcription errors also impacted the automatic pre-annotation: annotator 1 noticed that 321 named entities were not pre-annotated due, in particular, to typographic errors. However, we obtain an inter-annotator agreement (using GLOZZ Entropy measure) between the annotators and the automated pre-annotation of more than 0.9 on transcripts and 0.8 on minutes. The total number of annotations added or corrected by the annotators is 37,784, 27,736 of which (i.e. more than 73%) were added or corrected in the transcripts. All the categories were used, but two of them only twice (Tir-erCoupFrancIndirect and TirerPenalty) and only in the minutes, and the annotators found only 6 red cards (Pren-dreCartonRouge) and 9 President. As for the missing actors, the annotators found 586 of them in the actions and 404 in the relations (190 source actors, 173 target actors and 41 source and target actors). The majority of these missing actors appear in the transcripts (304 out of 586). This is consistent with the comments made by the annotators in which they note a lot of doubts on the FairePasse relation (nearly 800) and, more generally, on what is going on in the transcripts (they noted 1,429 uncertainties in the transcripts files out of a total of 1,505).

Annotation time

Table 2 presents the mean annotation times (per 1,000 tokens) for each annotator and source. In order to check if the differences are significant, we ran statistical tests (Welsh two sample t-test, with p=0.05). These tests show that there are no statistically significant differences between the annotation time of the annotators, both for minutes and transcripts. More interestingly, the differences between the modalities are proved statistically significant, for both annotators, when considering the time spent by token. But we also find that no statistically significant difference is found between the annotation time by annotation produced between the written and the oral modality. These two significance results may seem contradictory, but it is simply explained by the (statistically significant) difference of density of annotations (number of annotations given the number of tokens); the mean density for minutes is 0.16 while those of transcripts is 0.08. Indeed, video commentators tend to make small talk or talk about other events during the match, thus diluting interesting information. [START_REF] Artstein | Inter-coder agreement for computational linguistics[END_REF]. Secondly, the 3 different measures show that annotating relations is more error-prone than annotating unary annotations (units and actions).

Computing annotation agreements has become a standard when developing annotated resources, but in this paper, we would like to promote the interest of a finer grain analysis. This is especially important when the elements to annotate belong to different categories, and when these categories comprise very different population, as in our case. Indeed, more detailed results presented in table 4 show that significant disparities between the annotation categories actually exist. They also highlight the need for post-processing for certain categories of actions. Interestingly, the absence of bias between the annotators is also verified at this level, as well as the higher difficulty of processing transcripts. If some categories yield very low agreement measures (eg. PosséderBallon (HaveBall), FaireTentative2Passe (PassAttempt)), events (actions or relations) initiated or validated by an action of the referee are less open to interpretation and thus obtain better results than other events. Similarly, agreement on entities show a high contrast between the actors and the circumstants.

From a qualitative point of view, a closer analysis of the disagreements shows that the annotators rarely disagree on the type of an annotation, but annotate different elements. Last, unsurprisingly, the agreement values (both inter-and intra-annotator) tend to be lower in transcripts than in minutes. It is especially the case with complex annotations like relations. The previously mentioned oral specifics, in particular ellipses, easily explain this difference.

Qualitative analysis

Based on the quantitative results presented in the previous subsections, the principal causes of disagreements were searched for the most error-prone annotation categories, in one file of minutes and two files of transcripts. This analysis was tedious but very useful; it made it possible to build the following typology of the main causes of disagreements:

• errors due to the a misuse of the annotation tool (eg. units annotated 2 times);

• over-annotation or under-annotation of an annotator;

• disagreements on the frontiers of the annotated linguistic unit;

• disagreements on the anchoring of a relation;

• ambiguities, especially in speech transcripts.

In the first three cases, the disagreements are due to an error of one of the annotators. The errors caused by a misuse of the tool are not frequent. On the contrary, the overannotation of some linguistic phenomena is more frequent, but can be controlled by adding recommendation in the annotation guidelines. The under-annotation and forgotten annotations are more difficult to detect and to solve, since they are mainly caused by lapse of concentration.

The last two types of errors are more complex to handle, as they are not errors per se. For instance, concerning the relation anchoring, both annotators often identified the right actor, but not the same occurrence of its name (although the guidelines gave directions to prevent this). This is what happened in the example presented in Figure 1. Last, ambiguities, mainly found in speech, made the annotation of the transcripts tedious, implying to re-read several times the same sentence. Despite those efforts, many doubts on the annotation may still persist. For instance, one could think that the MarquerBut (ScoreGoal) action, which is very important from an applicative point of view, is fairly easy to annotate, but in the example given in Figure 2, the speech ambiguities misled the annotator to indicate that Gouffran scored, while in fact Gourcuff scored. The results of this detailed analysis suggest different ways to improve the quality of this annotated corpus. First, the fusion of the annotations from the two annotators, possibly corrected by one of them, would provide a more complete and stable result. Additionally, the intra-annotator agreement shows that annotator 2 was less coherent with himself than annotator 1 (except on relations, in the transcripts). This justifies that his/her annotations be reviewed and possibly corrected in priority.

Conclusion

This article presents in details the manual annotation process and quality of a football match annotation campaign. The produced annotations are freely available5 as well as the annotation guidelines, in French. At the heart of this annotation process is the evaluation of annotator agreement. We proposed a new and simple way to estimate the number of markables, which is a key element in usual annotator agreement measures like Cohen's κ, and ensures not to obtain over-optimistic results. Different perspectives are foreseen for this work. First, the qualitative analysis of the corpus is still ongoing and will probably lead to another version of the annotations, with corrections. Secondly, from a more multi-modal point of view, the differences of results between the oral and written sources will also be investigated, and should lead to interesting insights both from a linguistic and applicative points of view.

  for a complete description and comparison). Yet, such measures Units actors Joueur (Player), Equipe (Team), Arbitre (Referee), Entraineur (Coach), ArbitreAssistant (AssistantReferee), Président (President) circumstants EspaceSurTerrain (LocationOnField), LieuDuMatch (MatchLocation), TempsDans-Match (TimeInMatch) Actions initiated by referee TirerCoupFrancDirect (DirectFreeKick), TirerCoupFrancIndirect (IndirectFreeKick), TirerCorner (Corner), TirerPenalty (Penalty), FaireFauteDeJeu (Foul), HorsJeu (Offside), MarquerBut (ScoreGoal), PrendreCartonJaune (YellowCard), PrendreCarton-Rouge (RedCard), PrendreRappelALOrdre (Warning) others Centrer (Center), FaireTentative2Centre (CenterAttempt), Dribbler (Dribble), RaterBut (MissGoal), ArreterBut (StopGoal), IntercepterBallon (Interception), PossederBallon (HaveBall), ActionDuPublic (AudienceAction) Relations initiated by referee FaireFauteSurJoueur (FoulOnPlayer), TaclerFaute (FoulTackle), RemplacerJoueur (ReplacePlayer) others FaireCombinaison (Combination), FairePasse (Pass), FaireTentative2Passe (PassAttempt)

Algorithm 1

 1 EM Algorithm Input: {Mj} (sets of marked elements by annotators Aj) δ0 = j Mj for (i=1 ; change in δ ; i++) do expectation:

Figure 1 :

 1 Figure 1: Example of disagreement on the anchoring of a relation (FaireFauteSurJoueur/FoulOnPlayer)

  Table 3 presents the intra-and inter-annotator agreement values with Cohen's κ, Carletta's κ and the GLOZZ Entropy measure for the different layers of annotation. Several figures are noteworthy. First of all, Cohen's κ and Carletta's κ are very close in almost all cases. It means that there is no annotator bias, i.e. the distributions of annotations produced by the annotators are very similar

Table 3 :

 3 Annotation agreement values Minutes Transcriptions Cohen's κ Carletta's κ Cohen's κ Carletta's κ

	Additionally, annotator 1 could

Table 4 :

 4 Annotation agreements by modality and annotation category be used as a corrector, after sufficient training with the updated guidelines.

Note that the grouping of categories into actors and circumstants, then initiated by referee and others presented in table 1 was only defined a posteriori for the evaluation and did not exist in the data model used by the annotators.

3 http://www.lgi2p.ema.fr/ ˜ranwezs/

TIMETRACKER (http://www.formassembly.com/ time-tracker/#).

under LGPL-LR license at http://www.irisa.fr/ texmex/people/claveau/corpora/FootQuaero/.

Acknowledgments

We want to thank the annotators of the campaign, Claire Ris and Alain Zérouki from INIST-CNRS, for their hard work and their rich feedback, as well as Ali-Reza Ebadat, INRIA-INSA, Véronika Lux-Pogodalla, ATILF-CNRS, and Claire-Hélène Demarty, Technicolor Rennes, for helping us to collect and process the data. This work was realized as part of the Quaero Programme 6 , funded by OSEO, French State agency for innovation.