Ulrich Kuhl

Olivier Legrand

Fabrice Mortessagne

Microwave experiments using open chaotic cavities in the realm of the effective Hamiltonian formalism

Keywords: Scattering matrix, effective Hamiltonian, quantum chaos, classical wave experiments

published or not. The documents may come

Microwave experiments using open chaotic cavities in the realm of the effective Hamiltonian formalism

The measured quantities are reflection and transmission amplitudes through the introduced channels. Thus, all elements of the scattering matrix are experimentally directly accessible including the phases [START_REF] Stöckmann | Quantum Chaos -An Introduction[END_REF]. Also, the systems can be controlled easily down to the sub-wavelength regime. It is the purpose of this review to give an overview of different quantities using only a single description given in section 2. In general the definition of the effective Hamiltonian and the scattering matrix elements differ by factors of π or 2 in coupling constants and the normalization to the mean level spacing enters at different places. From the experimental point of view, the distribution of scattering matrix elements are better accessible than the resonances of the effective Hamiltonian. Often the resonances can only be extracted in the regime of isolated resonances, i.e. assuming only weak coupling. There are only a few results for strongly overlapping resonances, e.g. for the resonance trapping (see Subsec. 3.3 and [7,8]) or for width distribution (see Subsec. 3.2 and [9]). Hopefully it will be possible in the future to extract the resonances reliably even in the strongly overlapping regime and follow them parametrically in the complex plane. Thus one could directly relate for example the already available results presented for the scattering fidelity (see Subsec. 3.4 and [10,11]) or the Poisson kernel (see Subsec. 3.2) to the dynamics of the complex resonances experimentally.

In Sec. 2 we give a short introduction to the effective Hamiltonian. As the effective Hamiltonian cannot be measured directly we introduce the scattering matrix and relate it to the effective Hamiltonian. We fix our notation which will be used throughout the manuscript and might differ from the original papers. Experiments related to spectral properties are described shortly in the corresponding subsections of Sec. 3. We start with the investigation of avoided crossings occurring for level dynamics under parametric changes of the system (Sec. 3.1). The Poisson kernel and the corresponding width distribution are presented in Sec. 3.2. The resonance trapping effect is investigated in Sec. 3.3 by increasing the coupling to the system using a microwave cavity with attached waveguide, and the fidelity created by a variation of coupling is shown in Sec. 3.4. Finally, we also present results on wave functions and measures of openness in Sec. 4 and then summarize the results and give an outlook in Sec. 5.

Effective Hamiltonian and scattering theory

Effective Hamiltonian for chaotic system

The general case of M scattering channels connected to N levels of the closed cavity can be described in terms of the following effective non-Hermitian Hamiltonian

H eff = H -i M c=1 κ c W c W † c . (1)
Here, the internal Hamiltonian H of the closed system is represented by a Hermitian N × N matrix, whereas W c are M vectors of length N containing the information on the coupling of the levels to the continuum through M coupling channels. The W c are assumed to be normalized to one, W † c W c = 1, and κ c is the coupling constant of channel c. Have in mind that there exists a scale induced by the Hamiltonian which is the mean level spacing ∆.

Such an approach was initially developed in nuclear physics [START_REF] Mahaux | Shell-Model Approach to Nuclear Reactions[END_REF][START_REF] Verbaarschot | [END_REF][14] and since then has been successfully applied to study various aspects of open systems, including wave billiards [START_REF] Stöckmann | Quantum Chaos -An Introduction[END_REF][15][16][17]. Usually, the phenomenological coupling constants κ c are considered as real numbers which enter the final expressions via the so-called transmission coefficients T c . However, in the general case of an experiment it might be difficult to vary only the coupling of a channel without perturbing the system otherwise, thus leading to complex coupling coefficients κ c . The only constraint is Re(κ c ) ≥ 0, due to the causality condition on the S-matrix. We note that quite a similar problem of nonzero Im(κ c ) arises in shell-model calculations due to the principal value term of the self-energy operator, cf. [START_REF] Mahaux | Shell-Model Approach to Nuclear Reactions[END_REF] and [START_REF] Verbaarschot | [END_REF].

In this paper we will restrict ourself to open systems, where the classical counterpart of the wave system is fully chaotic and time reversal symmetry is respected. In this case, the Hamiltonian of the closed system is described by a large N × N random Gaussian orthogonal matrix (GOE), in which case, universal fluctuations are expected in the limit N ≫ 1 over a scale of the order of the mean spacing ∆. More specifically, by restricting the study to the center of the spectrum, the level spacing is given by ∆ = Λπ/N where 2Λ is the radius of the semicircle density for GOE [18]. Depending on the quantities of interest, the induced energy dependence on the imaginary part can be incorporated into the statistics of the closed Hamiltonian, i.e.

H ′ = H -∆H im with ∆H im = M c=1 Im(κ c )W c W † c .
One should have in mind that the GOE assumption for H ′ is an approximation as the induced change of energy ∆H im is not described by GOE. In this paper ∆H im is only playing an important role in section 3.4, where we investigate the fidelity of the system under a change of the coupling.

The non-Hermiticity of H eff yields a set of complex eigenvalues {E n } associated to two distinct sets of eigenvectors called left { ψ L n |} and right {|ψ R n } eigenvectors:

H eff |ψ R n = E n |ψ R n , ψ L n |H eff = ψ L n |E n , (2)
where the complex eigenvalue E n = E n -(i/2)Γ n gives respectively the energy E n and the resonance width Γ n > 0 of the nth resonance. The left and right eigenvectors, which describe the resonance states, satisfy the condition of bi-orthogonality, ψ L n |ψ R m = δ nm and completeness,

N n=1 |ψ R n ψ L n | = 1.
In the context of microwave cavities, the validity of model ( 1) has been established in previous works [17,19]. More general models of non-Hermitian Hamiltonian operators may be used to study open quantum systems (see e.g. the topical review written by I. Rotter [1]). This formalism has been successfully applied to wave billiards for which antennas and absorption are respectively described by physical and fictitious coupling channels [19][20][21].

Scattering theory including the effective Hamiltonian

According to the general scattering formalism [START_REF] Mahaux | Shell-Model Approach to Nuclear Reactions[END_REF][START_REF] Verbaarschot | [END_REF]22], the resonance part of the S-matrix at the scattering energy E can be expressed in terms of H eff as follows [11]:

S ab (E) = δ ab -2i Re(κ a )Re(κ b ) W † a 1 E -H eff W b . (3)
In the case where the coupling amplitudes change slowly with the energy, one may safely consider the complex eigenvalues E n of H eff as the only singularities of the S-matrix in the complex energy plane. Using random matrix theory (RMT) for large matrices at E = 0, the transmission coefficients are given in terms of the average S-matrix diagonal elements [11,[START_REF] Verbaarschot | [END_REF]18]

S cc = (1 -κ * c )/(1 + κ c ) by
T c ≡ 1 -| S cc | 2 = 1 - 1 -κ * c 1 + κ c 2 = 4 Re(κ c ) |1 + κ c | 2 , c = 1, . . . , M. (4)
Thus T c is accessible directly from a reflection measurement and κ c can be determined uniquely if Im(κ c ) = 0. It is worth noting that, in the case of real κ c , one gets

T c = 4κc (1+κc) 2 ∈ [0, 1]
, so that T c ≪ 1 and T c = 1 correspond to an almost closed or perfectly open channel c, respectively. In the case of purely imaginary κ c corresponding to perfect reflection, the channel is closed, T c = 0. We will use this in the investigation of the coupling fidelity (see Subsec. 3.4 and [11]), where the effect of adding a channel (i.e. T c = 0 → T c = 1) is investigated in the scattering fidelity.

Typically, the elements of the scattering matrix can be written as a sum of Lorentzians assuming that the coupling constants κ c have only a weak energy dependance within the resonance width, i.e. one is far from energy thresholds

S ab (E) = δ ab -i n a n E -E n + i 2 Γ n . ( 5
)
where a n is the complex amplitude, E n the eigenenergy and Γ n the width of the resonance. Γ n describes in the time domain the exponential decay of the nth eigenstate. In the weak coupling regime (κ ≪ 1, assuming real κ), the resonances do not overlap and the non-Hermitian part of H eff can be treated as a perturbation of the Hermitian part H and one can relate the parameters of the Lorentzians to the closed system. In this Breit-Wigner approximation the elements of the scattering matrix S can be written as

S ab (E) = δ ab -2i n Re(κ a )Re(κ b ) W † a,n W b,n E -E n + i 2 Γ n . (6)
Assuming real κ, to leading order in κ, the real parts of E n are given by the eigenenergies of the closed system, H|n = E n |n , while their widths are given by :

Γ n = 2 M c=1 κ c n|W c W † c |n . ( 7
)
An investigation of the joint distribution of the complex reflection amplitude S 11 and the width Γ n is presented in Subsec. 3.2. Likewise, the right eigenvectors of H eff read :

|ψ R n = |n -i M c=1 κ c m =n (W c W † c ) mn E n -E m |m . (8)
We investigate the eigenvectors, i.e. the eigenfunctions of the system in section 4. Often one can use a 2 × 2 random matrix toy model to obtain analytic results. An example is shown in Subsec. 3.1.

Classical wave experiments on spectral properties

Level dynamics

It is often relevant to study the dynamics of levels when a parameter of the system is varied. For a completely uncorrelated spectrum typical of a system with regular dynamics, levels evolve ignoring the presence of others. It ensues that for certain values of the parameter, levels may become degenerate. On the contrary, in a chaotic system, level repulsion precludes such a possibility. To characterize the parametric dynamics of levels one may consider the statistics of local minima of level spacings, also called avoided crossings. In this section, following [23], we recall how a two-level model with a large number of open decay channels can be used to describe avoided level crossing statistics in open chaotic billiards. This model allows us to describe the fundamental changes of the probability distribution of the avoided level crossings compared with the case of a closed wave system [24]. In particular, it is found that the decay process due to open channels induces a modification of the probability distribution of the avoided level crossings at small spacings due to an attraction of the resonances. In a microwave experiment presented in [25] the spectral properties of a superconducting two-dimensional microwave billiard were investigated, where the geometry of the cavity is varied by rotating a dielectric (Teflon) wedge creating an evolution of the levels (see inset of Fig. 1). The resonator modeled an open quantum billiard whose classical dynamics is chaotic with the antennas acting as single scattering channels [23]. The influence of the flux of microwave power was so weak that it could not be detected through spectral quantities like the nearest neighbor spacing distribution or the spectral rigidity at a fixed value of the parameter. But the distribution of the avoided crossings is more sensitive to small effects showing clear deviations from the GOE result, which could be attributed to the openness of the cavity.

In reference [23], an effective Hamiltonian H eff (µ) (see Eq. ( 1)) which depends on a continuous parameter µ through its Hermitian part H(µ) is used [25]. The Hamiltonian H(µ) of the closed system is modeled by a 2 × 2 random matrix and the coupling to the environment in terms of M equivalently coupled open channels (κ c ≡ κ). The eigenvalues of the effective Hamiltonian are complex and read ǫ ± = E ± -(i/2)Γ ± , where E ± and Γ ± are, respectively, the two eigenenergies and the two spectral widths of the 2-level model. For the study of statistical properties H(µ) is replaced by a Gaussian random matrix [START_REF] Stöckmann | Quantum Chaos -An Introduction[END_REF] and the coupling matrix elements √ κW c,n are chosen to be real Gaussian-distributed with zero mean and variance σ 2 = κ∆/π, where ∆ is the mean level spacing of the closed system [5]. In the eigenbasis of H(µ) (the µ-dependence will be omitted in the following), the effective Hamiltonian is written as

H eff = E 1 0 0 E 2 - i 2 Γ 11 Γ 12 Γ 21 Γ 22 , (9)
where

E 1,2 are the µ-dependent eigenenergies of H (E 2 > E 1 is assumed) and Γ np = κ M c=1 W c,n W c,p .
The complex eigenvalues of the effective Hamiltonian H eff in Eq. ( 9) read

ǫ ± = E 1 + E 2 -i 2 (Γ 11 + Γ 22 ) ± √ D 2 with D = (E 1 -E 2 ) + i 2 (Γ 22 -Γ 11 ) 2 -Γ 12 Γ 21 . ( 10
)
Thus the spacing between the two eigenenergies is given by d = ǫ + -ǫ -= Re( √ D). Considering the limit of a large number of open channels M in the weak coupling regime, we may apply the central limit theorem and replace the random variables depending on the coupling amplitudes by their averages Γ nn = M σ 2 and Γ np Γ pn = M σ 4 . Then the spacing d is given by

d = √ s 2 -M σ 4 if s > √ M σ 2 0 otherwise , (11)
where s = E 2 -E 1 is the spacing of the eigenenergies of the closed system. Note that since M σ 4 = var(Γ), it means that the changes of the spacings due to the openness of the system are related to the fluctuations of the spectral widths [26]. In the limit M → ∞ and σ 2 → 0 with M σ 2 = Γ fixed, var(Γ) → 0 and thus the spacing between eigenenergies of the open system converges to that of the closed system, d → s, in spite of non-vanishing losses. This implies that the values of the parameter µ at the avoided crossings are the same for the closed and the open system. Thus, to obtain the distribution of avoided crossings c of the open system, the spacings s are assumed to be distributed as the avoided crossings of the corresponding closed system. With Eq. ( 11) the probability distribution of the avoided level crossings p(c) is given by

p(c) = δ(c)θ( √ M σ 2 -s) + δ c -s 2 -M σ 4 θ(s - √ M σ 2 ) , ( 12
)
where θ is the Heaviside step function and the triangular brackets denote averaging with respect to the spacing s.

Using the probability distribution of avoided crossings for (closed) chaotic systems with time reversal symmetry calculated by Zakrzewski and Kuś [24], p(s) = 2 πα 2 e -s 2 /(2α 2 ) , where the mean value of s is given by s = α 2/π, and averaging over s yields

p(c) = erf √ M σ 2 √ πα δ(c) + 2 πα 2 c e -(c 2 +Mσ 4 )/(2α 2 ) √ c 2 + M σ 4 , (13)
where α fixes the average of c. Note that the linear behavior of the distribution p(c) at small spacings induces a dip and the local minima of the spacings have a zero-crossings contribution leading to the presence of a δ-peak at the origin.

In the experiment described in [25], only finite spacings could be measured due to the finite frequency resolution implied by the discrete sampling of the data. Therefore, it is more convenient to consider the distribution of nonvanishing avoided crossings (the second term in the right-hand-side of ( 13) with proper normalization) to compare theory and experiment. The experimental results of [25] were obtained using a superconducting microwave cavity, thus minimizing dissipative processes. Three antennas were attached to the cavity: they correspond, in our model, to M = 3 open channels [27]. Absorption into the walls could be safely neglected in the analysis of the experimental data. The analytic result obtained through a ). In red the analytical prediction with κ f it =0.182 (obtained through a least square procedure with M = 3). Inset: schematic representation of the cavity with a rotating Teflon wedge (see [25]). fit using κ as a parameter (continuous curve) is shown in Fig. 1. The 2-level model result follows closely the experimental histogram. This provides a clear confirmation of the interpretation drawn in [25] that the deviation of the avoided-crossings distribution from the predicted GOE result for closed systems was due to the measurement process, i.e. the influence of the three antennas, which couple the cavity modes inside the resonator to the outside.

In other microwave experiments, exceptional points (EP) have been investigated [28,29]. At an EP, two or more eigenvalues and also the associated eigenvectors coalesce [1,30]. To investigate the EP, resonances are moved in the complex plane by varying two parameters. Typically one parameter is mainly inducing a shift of the real part, whereas the other corresponds to an imaginary perturbation inducing a change of resonance width. Let us assume now a typical EP which involves only two eigenvalues. By encircling the EP parametrically once the two resonances are exchanged while only one wave function acquires a sign. Encircling a second time another exchange occurs but the two wavefunctions differ now by a phase of π from the starting situation. Finally the original situation is restored when encircling the EP 4 times [31]. This behavior has been demonstrated using microwave cavities [28,29].

Poisson kernel and width distribution

Often, one cannot extract the resonances and a direct investigation of the effective Hamiltonian is not accessible, but information of the S-Matrix like reflection and transmission amplitudes are. As we are interested not in the individual properties of the system but in statistical properties, the distribution of amplitudes and phases are of interest. In this section we will restrict to a system coupled only via a single channel to the environment and rely on the references [20,21]. The statistical distributions of the S-matrix elements with imperfect coupling have been addressed by several authors [32][33][34][35][36][37][38]. Experimentally not only the direct processes have to be taken into account but also losses or absorption. When absorption is present, S is a subunitary matrix and in the single-channel case the S matrix can be parametrized using the reflection coefficient R as S = √ Re iθ The coupling between the scattering channels and the interior region is given by the transmission coefficient T a = 1 -| S | 2 (see Eq. 4). The subindex a will be used to denote the antenna coupling. The distribution P R,0 (R) of R is known [17,39] and there exists an analytical approximation for GOE (β = 1) which we will use here [21]

P R,0 (R 0 ) = C e -α/(1-R) (1 -R) 2+1/2 Aα -1/2 + B(1 -R) 1/2 , ( 14
)
where α = γ/2, A = α (e α -1), B = (1 + α -e α ), C = α 3/2 /Γ(3/2) and Γ is the upper incomplete Gamma function. γ is the absorption strength, e.g. the exponentiel decay rate of the intensity, inside the The upper, middle rows correspond to experiment and theory, respectively (see [21]). Note the change of scales for R and θ. Below the corresponding normalized width distribution is shown. The black histogram was obtained from the experiment using the harmonic inversion. The solid line corresponds to the theoretical prediction from [17] taking additionally fluctuating absorption into account [41] (see [21,41]).

closed system. We denote a system with absorption strength γ and perfect coupling (T a =1) by S 0 = √ R 0 e iθ0 . Introducing the S-matrix of the antenna [36,40] and using the combination rule of S-matrices, the distribution P S (S) for the system including direct processes was derived as [21]

P S (S) = ∂(R 0 , θ 0 ) ∂(R, θ) P S,0 (S 0 ) = 1 -S 2 |1 -S S | 2 2 1 2π P R,0 (R 0 ). (15)
The experimental realization using microwaves has been performed by a rectangular cavity where a movable half circle is attached to one wall. The microwaves excite the cavity via a single antenna, where the antenna coupling is frequency dependent. Details of the experimental description can be found in Refs. [9,20,21]. In Fig. 2 the joint probability P (S) = P (R, θ) is shown for three different absorption and coupling regimes (columns a-c). In the upper row the experimental distributions and in the middle row the theoretical distributions are shown. A good agreement is found especially if one takes into account that all parameters of the theory can be fixed in advance by average quantities.

To make a more direct relation to the effective Hamiltonian, we will now investigate the resonances, more precisely the distribution of resonance width Γ. From the previous reflection measurements the poles of the S-matrix in the complex plane have been determined [41]. The resonances have been extracted by means of the harmonic inversion [42]. Thus it became possible to resolve the resonances in a regime where the line widths exceed the mean level spacing up to a factor of 10. The obtained experimental distributions of line widths are shown in the lower row of Fig. 2. The distributions are in agreement with predictions from random matrix theory [17] when wall absorption and fluctuations caused by couplings to additional absorbing channels are considered [41].

The joint probability distributions P (R, θ) shown in the upper row of Fig. 2 is induced by the resonances and eigenfunctions of the effective Hamiltonian. The presented width distribution shown in the lower part is only one part necessary to generate the P (R, θ). The other parts are the distribution of the resonances including their correlations to the widths and the value of eigenfunctions at a fixed position r (antenna position). It will be a future challenge to derive all this correlations theoretically and experimentally.

In a recent work the same data has been used to investigate the Wigner surmise of the nearest neighbor spacing for an open system [43]. Using the effective Hamiltonian approach in the 2 × 2 approximation (see Sec. 3.1) the authors were able to derive an analytical formula and an agreement to the experimental nearest neighbor spacing distribution was found [43].

In this subsection we have shown the width distribution for different coupling strength. In the following subsection we will investigate how the resonances, especially their widths, will change when gradually changing the coupling to a single channel.

Resonance trapping

Resonance trapping is a phenomenon appearing in strongly open quantum systems. We will present results from [7,8] using a slightly different notations. For details refer to the original works. Resonance trapping is caused by the interaction of overlapping resonance states via the continuum of scattering states by which some of the states align with the channels by trapping other ones. Therefore the total coupling strength is given by

Γ tot = N n=1 Γ n ≈ Γ align = M n=1 Γ n i.e. Γ trap = N n=M+1 Γ n ≈ 0 , ( 16
)
where N is the total number of states and M states are strongly coupled to the environment. Due to the reordering processes, the other N -M states are only weakly coupled to the environment. Resonance trapping can be observed if the total coupling strength is varied.

Starting from weak coupling where the resonance states are still isolated, we have Breit-Wigner resonances (see Eq. 6) and the Hamiltonian H eff is almost diagonal. Increasing the coupling the resonances start to overlap and a redistribution in the spectrum of the system takes place due to the non-diagonal terms of the non-Hermitian part κW W † of H eff . The Hamiltonian H eff has to be diagonalized,

H eff | ψn = Ẽn - i 2 Γn | ψn . ( 17
)
leading to

S ab (E) = δ ab -2i n Re(κ a )Re(κ b ) W † a,n Wb,n E -Ẽn + i 2 Γn . ( 18
)
where Re(κ a ) W = Re(κ a )W | ψn . W , Ẽn and Γn are generally energy dependent and W is complex. The poles of the S matrix are obtained from the solutions of the fixed-point equations

E n = Ẽn E=En and Γ n = Γn E=En . ( 19
)
which determine the energies E n and widths Γ n of the resonance states. The eigenfunctions ψn of H eff are bi-orthogonal. The resonances are no longer of Breit-Wigner type because of the energy dependencies of W and Γn (see [44,45]). The energy dependencies ensure the unitarity of the S matrix. This effect was demonstrated by a microwave experiment [7,8], where the coupling to the cavity via a waveguide was varied by a slit (see inset of Fig. 3). The reflection was measured via a waveguide of width D = 23.2 mm where a slit was positioned at distance L = 16 mm from the attachment to the cavity. The cavity has a form of a quarter Sinai shape with length a = 285 mm, width b = 200 mm, and radius of the quarter circle of r = 70 mm. The slit opening d was varied in steps of 0.1 mm and the resonance Clearly a trapping effect is visible for several resonances, where the width Γ for the full opening is smaller than the maximum width observed for smaller openings. For the cavity parameters roughly a quarter of all resonances investigated are trapped. The observed frequency shift in Fig. 3 is due to both the real and imaginary part of κ.

It is worthwhile to note that the width distribution once the resonance trapping has happened contains two contributions. One distribution P (Γ trap ) of the trapped states which is similar to using a corresponding weak coupling and another part P (Γ align ) of the broad aligned states. In the previous subsection the width distribution of the trapped states is shown, but in the case of the joint probability distribution P (R, Θ) both contributions are included (see Fig. 2c).

Scattering Fidelity for varying coupling

In this section we want to investigate the effect of opening the system by adding one additional channel, where another channel is used to measure the scattering matrix. We present here only a short theoretical description, which follows [11] (refer to this reference for details of the theory). In contrast to [11] we will compare here the perfectly reflecting to the perfectly absorbing channel, whereas in [11] a variation of the reflection was presented.

The quantity we want to use is the scattering fidelity [10] which is related to the fidelity [46,47]. Both measure the stability of a wave system against perturbations. The fidelity is the time dependence of the overlap of the same initial pulse propagated by a 'slightly' different Hamiltonian, whereas the scattering fidelity amplitude f ab corresponds to the properly normalized spectral cross correlation function of scattering matrix elements S ab for 'slightly' different systems [10]:

f ab (t) = Ŝab (t) Ŝ′ * ab (t) Ŝab (t) Ŝ * ab (t) Ŝ′ ab (t) Ŝ′ * ab (t) . ( 20
)
This definition ensures that f ab (0) = 1. Furthermore, an overall decay of the correlation functions due to absorption or other open channels drops out, provided the decay is the same for the parametric crosscorrelation functions in the nominator and the autocorrelation functions in the denominator. The scattering fidelity itself is

F (t) = |f ab (t)| 2 . ( 21
)
For chaotic systems and weak coupling of the measuring antenna the scattering fidelity approaches the ordinary fidelity [48]. The time t is given in units of the Heisenberg time t H = 2πh/∆, with ∆ being the mean level spacing.

The scattering fidelity presented here will be denoted as coupling fidelity. We present here a sketch of an exact RMT prediction (for details see [11]). The coupling constant of the added channel will be denoted by κ for Ŝab (t) and κ ′ for Ŝ′ * ab (t). The derivation is general but later the reference system is the system without channel, i.e. κ ′ = 0. For the case where κ = κ ′ the parametric correlation functions Ŝab (t) Ŝ′ * ab (t) in Eq. ( 20) reduces to the autocorrelation function, which is given by the famous Verbaarschot-Weidenmüller-Zirnbauer (VWZ) integral [START_REF] Verbaarschot | [END_REF]. For the case of κ = κ ′ the correlator Ŝab (t) Ŝ′ * ab (t) yields [11]

Ŝab (t) Ŝ′ * ab (t) = Ŝeff ab (t) Ŝeff * ab (t) . ( 22
)
Thus the parametric cross correlation function reduces to an autocorrelation function Ŝeff ab (t) Ŝeff * ab (t) given by the VWZ integral, where the transmission coefficient (Eq. ( 4)) of the varied channel has to be replaced by

T eff = 2 (κ + κ ′ * ) (1 + κ) (1 + κ ′ * ) . ( 23
)
One can interpret T eff as an effective transmission coefficient due to a parametric variation of the coupling strength in the varied channel. For κ = κ ′ , the effective transmission coefficient T eff becomes equal to the conventional transmission coefficient (Eq. ( 4)). In contrast to the transmission coefficient in Eq. ( 4) the effective transmission coefficient T eff is generally complex. The subsequent evaluation of the coupling fidelity cannot be done analytically and will be performed numerically. As in our case only one channel with a coupling strength that depends on frequency is added, and the reflection is measured using the only other channel, i.e. via an attached antenna, the theoretical description of the scattering fidelity can be simplified. By denoting "c" the added antenna and "a" the measuring antenna, the effective Hamiltonian H eff may be written

H eff = H a eff -iκ a W a W † a where H a eff is defined as [11] H a eff = H -iκ c W c W † c . (24)
Note that the coupling constant κ a is generally complex and takes into account the effects of both the channel coupling and any reflection and phase shifts coming from the imperfect experimental environment, e.g. slight mismatches of cable analyzer connection etc. Through this description, the 2 × 2 scattering matrix for both channels has thus been reduced to a 1 × 1 scattering matrix for the measuring antenna only

S aa = 1 -iκ * a W † a 1 E -H a eff W a 1 + iκ a W † a 1 E -H a eff W a . ( 25
)
As a basis system for the measurement we use a chaotic quarter Sinai cavity with length l = 472 mm, width w = 200 mm and a quarter-circle of radius r = 70 mm. Additional elements were inserted to reduce the influence of bouncing balls and a rotatable ellipse was introduced close to the center for ensemble averaging. The two antennas are situated such that there is no direct view of each other. The added perturbing antenna is directly attached to a vector network analyzer, mimicking an infinitely long channel. This gives us also the possibility to measure directly the reflection at the added channel S cc and calculate the coupling strength

T c = 1 -| < S cc > | 2
, where the average is taken over different ensembles and a small energy window. Assuming that the coupling κ c has only a real part we can determine κ c uniquely from T c .

In the upper part of Fig. 4 the fidelity introduced in Eq. ( 21) is plotted for two frequency ranges corresponding to two different coupling strengths κ c . Using the experimentally determined κ c here denoted as coupling parameter κ exp c , a good agreement between experimental results (filled symbols) and theoretical curves (dotted line) is found without any fit. A fit of κ c to the experimental curves to the scattering fidelity |f (t)| 2 (the corresponding values are denoted by κ fit c ) which is plotted as dashed line shows only a minor improvement for the correspondence between experiment and theory. In both cases an agreement is found.

In the lower part of Fig. 4 we present the fidelity decay for the frequency range 7.2 -7.7 GHz. Here we find an obvious deviation between the experimental results (filled circles) and the theoretical curve based on the experimental parameter κ exp c (dotted line). To determine the decay of the experimental fidelity decay we tried two possible ways of fitting κ c to the experimental data. The first is fitting |f (t)| 2 using a real valued κ fit c and the second is fitting the complex f (t) using a complex κ fit,im c . While for the fidelity decays both fitting procedures lead to a good agreement between experiment and theory, the corresponding real f R (t) and imaginary f I (t) parts of the fidelity amplitude presented in Fig. 4 is described satisfyingly only by the red dashed curve, which is generated with the complex valued fitting parameter κ fit,im c . Thus in the frequency range from 7.2-7.7 GHz the attached antenna cannot be described solely by an additional coupling but it has a non negligible scattering part inside the billiard. It is worthwhile to note that the real part of κ fit,im c is matching the experimental κ exp c thus showing a consistent description. In case of the two other ranges shown in Fig. 4 the imaginary part is small and Im(κ c ) = 0 is a reasonable assumption. In Fig. 5 we plotted the coupling constant κ c determined from the experimental transmission coefficient T c as κ exp c (black triangles) and from the fit to the experimental fidelity decay κ fit c (red diamonds). For frequencies from about 6 GHz up to 10 GHz the experimentally determined coupling constant and the fitting parameter do not deviate much from each other apart from some significant peaks like the one around 7.5 GHz which results from system specific features, probably antenna resonances. Beyond 11 GHz the way of determining the coupling constant via a reflection measurement at the perturbing antenna c does not lead to convincing results. This might be due to the resonance trapping effect which was observed in similar microwave billiards (see section 3.3 and [7, 8]). If the coupling of the newly attached channel is sufficiently strong, thus leading to a reorientation of the eigenvalues, an imaginary part of the coupling constant has to be taken into account. Typically the coupling constant and antenna transmission T c increase with frequency and above 10 GHz T c > 0.8. Thus in the regime of resonance trapping, taking the complex coupling coefficient into account is crucial for the coupling fidelity.

Classical wave experiments on spatial properties

Wave functions

We now turn to examine the consequences of openness on the spatial properties of the field.

While field statistics of open chaotic systems have been systematically studied for a given energy and considering the energy as a continuous parameter (see [21] and references therein), statistics of resonance states i.e. left and right eigenvectors of H eff are less understood. The impact of the openness is to turn real eigenfunctions into complex internal wavefunctions associated to resonances. This complexness being uniquely related to the presence of currents inside the system [49,50]. In order to quantify the presence of currents, one can use the complexness parameter q 2 n = (Im ψ n ) 2 / (Re ψ n ) 2 introduced by Lobkis and Weaver [51] as the ratio of the variance of the imaginary and real parts of the nth resonance state (after a phase rotation leading to independent real and imaginary components [52]). These authors experimentally obtained the complexness parameter from the distribution of the phase ϕ of the spatial component of the field : ψ = |ψ|e iϕ . By considering that the real and imaginary parts of the field are independant Gaussian random variables, they obtained:

P (ϕ) = q 2π 1 q 2 cos 2 ϕ + sin 2 ϕ . ( 26
)
This distribution is closely related to the Poisson kernel (eq. 15) for a given frequency. This expression is peaked around 0 and π for purely standing waves in a closed cavity. As losses increase, the phase probability distribution broadens corresponding to a growing traveling-wave component of the mode. The Copyright line will be provided by the publisher 28). In the inserts the corresponding modulus of the wave function is shown (see [56]).

latter feature has been recently experimentally verified by means of elastic waves in a chaotic plate [53].

In the context of microwave cavities, a derived distribution had been obtained by J. Barthélemy et al. [54].

A related quantity, namely the phase rigidity |ρ| 2 can be expressed as a function of the complexness parameter:

|ρ| = ψ 2 R -ψ 2 I ψ 2 R + ψ 2 I = 1 -q 2 1 + q 2 . ( 27
)
By considering the phase rigidity as a continuous function of frequency, P. Brouwer studied the effect of openness [55] and derived the theoretical distribution of |ρ| 2 , which has been experimentally verified by Kim et al. in an open microwave cavity [50,56].

For fixed phase rigidity |ρ| 2 the distribution of wave function intensities is given by [55,57]

P ρ (I) = 1 1 -|ρ| 2 exp - I 1 -|ρ| 2 I 0 |ρ|I 1 -|ρ| 2 . ( 28
)
For |ρ| 2 → 1, P ρ is approaching a Porter-Thomas, whereas it is an exponential distribution for |ρ| 2 = 0. Porter-Thomas and single exponential distributions are χ 2 distributions for one and two degrees of freedom, reflecting the fact that in closed systems the wave function is real, whereas in fully open systems it contains a real and an imaginary part of comparable magnitude. In Fig. 6 the distribution of intensities for three different wave functions of an open microwave cavity is shown with different phase rigidity values ranging from nearly closed (|ρ| 2 ≈ 1) to nearly completely open (|ρ| 2 ≈ 0). A good agreement is seen, especially as the only parameter, the phase rigidity |ρ| 2 , has been fixed beforehand (for details see [50,56]). The complexness parameter q n has been investigated both experimentally [53,54] and theoretically [5,26] through the scope of its relationship to the resonance width. In particular, in Ref. [5] the authors have established that the complexness parameter is also a measure of the non-orthogonality of eigenfunctions in the case of inhomogeneous losses.

A linear relationship between q n and Γ n was first noticed by Barthélemy et al. [54] analyzing hundreds of resonance states of a 2D chaotic microwave cavity at room temperature. This result was then confirmed using the effective Hamiltonian formalism in the limit M ≫ 1, relevant in the experiment [5]. A linear relationship between q n and Γ n was also verified in an elastodynamics experiment for a given resonance when a spatially extended coupling is varied [53]. The complexness parameter was also investigated at arbitrary M by means of its probability distribution in the regime of weak coupling [26]. There, it was shown that the average value of q 2 n is directly proportional to the variance of Γ n , which constitutes the natural measure of the fluctuations of the widths.

In reference [6], the authors confronted theoretical predictions presented in [26] to numerical solutions of the Maxwell equations in a 2D chaotic microwave cavity with lossy boundaries subject to Ohmic dissipation. In the regime of weak coupling, which corresponds to the condition var(Γ) ≪ ∆, where ∆ is the mean level spacing, the anti-Hermitian part of H eff is small compared to the Hermitian part and perturbation theory can be applied. We define the rescaled widths γ = πΓ/(2κ∆) with γ = M where we assume a single and real coupling constant κ for the M fictitious channels accounting for absorption. Considering that the widths are given by a sum of M squared independent Gaussian random variables, they follow a χ 2 distribution with M degrees of freedom:

χ 2 M (γ) = 1 2 M/2 Γ(M/2) γ M/2-1 e -γ/2 , (29)
where Γ(•) is the Euler gamma function.

The distribution of the rescaled complexness parameter X n ≡ q 2 n /κ 2 , was obtained by making use of group integral methods [58] and reads:

P GOE M (X) = M 6X 2 1 + (3 + M )X -1 (1 + X -1 ) M/2+2 , (30)
where strong mode-to-mode fluctuations clearly appear, embodied in the power law tail 1/X 2 of the distribution.

The chaotic cavity considered in [6] has a quarter of a stadium shape with a radius of R = 1 m and a length of l = 2 m (see Fig. 7). In order to reduce the bouncing ball modes between the two parallel sides, an oblique cut is performed on one side and a movable perfectly reflecting half disk of diameter d = 0.3 m is placed on the opposite side. The absorbing boundary condition is imposed on an adjustable part of the boundary of the cavity. The number of channels M is related to the effective absorbing length l abs along the boundary and the wavelength λ through:

M = l abs λ/2 . (31)
Note that M is also related to the first two moments of the width distribution (29):

M = 2 γ 2 / γ 2 -1 . (32)
Fig. 8 shows the distributions of the complexness parameter X compared to the theoretical distribution (30). Three different sets of values of the absorbing lengths, conductivities and frequency ranges are used (see caption). For each frequency interval, the number M of channels used for the comparison corresponds to the nearest integer value of relation (31) computed with the median value of the wavelength in each interval (the estimates (31) and (32) agree within a few percents). The excellent agreement, even in the tail of the distribution (as shown in inset for M = 38), confirms that the prediction (30), obtained within the perturbation theory, contains the essential features to account for the non-orthogonality of the resonance states due to spatially continuously distributed losses.

Note that in the width distributions presented in Subsec. 3.2 in the lower row of Fig. 2, the above presented distributions of the complexness parameter correspond to the additional fluctuating absorption, which must be taken into account to obtain a good agreement between theory and experiment [41]. and from the 700 th to the 800th resonances; (c) l abs = π/2, σc = 1000 Sm -1 and from the 700 th to the 800th resonances. The histograms show the numerical distributions. The solid lines correspond to the theoretical prediction (30) where the number M of channels corresponds to the nearest integer value of relation (31) computed with the median value of the wavelength in each frequency interval (see [6]).

Summary and Outlook

We have shown that classical wave experiments are perfectly suited to investigate systems which can be described by the effective Hamiltonian approach. They permit easy access to the full complex scattering matrix, thus giving the possibility to fix relevant parameters beforehand. Additionally the system can be controlled in various ways, like boundary variations on subwavelength scale or coupling variations. This allows to verify experimentally various predictions given by the effective Hamiltonian approach. Since absorption often plays a crucial role for classical waves, it has to be taken into account properly in the theory. Thus, classical experiments in the realm of effective Hamiltonians are a perfect tool to test theoretical predictions and, additionally, are also a driving part in the further development and understanding of effects, induced by the coupling to the environment, i.e. by an effective Hamiltonian. On the theoretical side, a continuous progress of statistical description of new quantities is going on, where microwave experiments will be one of the major tools to verify these predictions. Recently for example the distribution of 'width velocity' has been predicted by Fyodorov and Savin [59] which should be accessible by microwave experiments. On the other hand experimental findings will trigger further theoretical investigations as it has been the case for including effects of absorption, for local level dynamics [60] and for the coupling fidelity [11]. Experimentally challenging is still the reliable extraction of resonances in the strongly overlapping regime. This would give access to follow the resonances far in the complex plane and investigate nearest neighbor distribution in the complex plane, resonance correlations etc. One could also verify predictions for the fractal Weyl law [61][62][63] which relates the average number of resonances to the fractal dimension of the repeller of the classical dynamics.

Fig. 1

 1 Fig.1Probability distribution of non vanishing avoided crossings. In blue, the experimental histogram ([25]). In red the analytical prediction with κ f it =0.182 (obtained through a least square procedure with M = 3). Inset: schematic representation of the cavity with a rotating Teflon wedge (see[25]).

Fig. 2

 2 Fig.2Joint distribution P (R, θ) of the S-matrix for different coupling and absorption regimes: (a) γ = 0.56, Ta = 0.116 (weak absorption-weak coupling) (b) γ = 2.42, Ta = 0.754, and (c) γ = 8.40, Ta = 0.989 (stronger absorption-nearly perfect coupling). The upper, middle rows correspond to experiment and theory, respectively (see[21]). Note the change of scales for R and θ. Below the corresponding normalized width distribution is shown. The black histogram was obtained from the experiment using the harmonic inversion. The solid line corresponds to the theoretical prediction from[17] taking additionally fluctuating absorption into account[41] (see[21,41]).

Fig. 3

 3 Fig. 3 Eigenvalues, obtained by fitting the experimental reflection data to Lorentzians, for slit position L = 16 mm. The open circles denote the eigenvalues by full opening (d = D). The inset shows a sketch of the billiard with attached waveguide and slit within the waveguide (a = 285 mm, b = 200 mm, r = 70 mm, L0 = 204 mm, D = 23.2 mm, L = 16 mm) (see[8]).

Fig. 4

 4 Fig. 4 Fidelity |f (t)| 2 , real fR(t), and imaginary part fI (t) of the fidelity amplitude. Filled symbols show the experimental results and the theoretical curves are dotted for experimental parameter and dashed for fitting parameter. In the upper row the chosen frequency ranges are: 8.7 -9.2 GHz (black triangles) with κ exp c = 0.24, κ fit c = 0.21 and 9.8 -10.3 GHz (green diamonds) with κ exp c = 0.43, κ fit c = 0.46. In the lower row the frequency range is 7.2 -7.7 GHz. Filled circles show the experimental results. The theoretical curves are dotted for experimental parameter κ exp c = 0.19; dashed black for real valued fitting parameter κ fit c = 0.37 and dashed red for complex valued fitting parameter κ fit,im c

Fig. 5

 5 Fig.5 Coupling constant κc as a function of the mean frequency ν; determined from the experimental transmission coefficient Tc (black triangles) and as real valued fitting parameter to the experimental fidelity decay (red diamonds).

Fig. 6

 6 Fig. 6 Distribution of intensities for three different wave functions at frequencies ν = 8.0 (a), 16.9 (b), and 15.4 GHz (c) are shown with different phase rigidities |ρ| 2 . The solid line represents the distribution from eq. (28). In the inserts the corresponding modulus of the wave function is shown (see[56]).

Fig. 7

 7 Fig.7Real (on the left) and imaginary (on the right) component of the 500th resonance state obtained by means of the Finite Element Method (see[6]).

Fig. 8

 8 Fig.8 Distributions of the complexness parameter for different absorbing lengths, conductivities and frequency ranges: (a) l abs = π/18, σc = 80 Sm -1 and from the 300 th to the 400th resonances; (b) l abs = π/6, σc = 400 Sm -1 and from the 700 th to the 800th resonances; (c) l abs = π/2, σc = 1000 Sm -1 and from the 700 th to the 800th resonances. The histograms show the numerical distributions. The solid lines correspond to the theoretical prediction(30) where the number M of channels corresponds to the nearest integer value of relation(31) computed with the median value of the wavelength in each frequency interval (see[6]).

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We would like to acknowledge J. Barthélemy, P. Brouwer, B. Dietz, R. Höhmann, B. Köber, G. Luna-Acosta, R. A. Méndez-Sánchez, E. Persson, C. Poli, I. Rotter, D. Savin, P. Sebbah, H.-J. Stöckmann, and O. Xeridat for discussion, data acquisition, data evaluation, etc.


Robin Cressent

Vincent Idasiak

Frederic Kratz

Reusing FIDES knowledge in the MéDISIS Dysfunctional Behavior Database

Keywords: Model-Based System Engineering, SysML, FIDES guide, Reliability database

For 3 years our research team has applied on different industrial projects our MéDISIS methodology. The MéDISIS methodology aims to ease the integration of dependability analysis through the system engineering process. MéDISIS represents now a complete framework described through various publications. Currently, the use of MéDISIS relies on a functional system model in SysML, the Dysfunctional Behavior Database and processes of translation from SysML to FMEA, Altarica DataFlow, AADL and Simulink to perform dependability studies. Until recently, the DBD was a tool dedicated to MéDISIS. Consequently, the DBD was only designed to be updated through dependability analysis of project implementing the MéDISIS methodology. We describe in this article how we can complete the DBD with knowledge from other sources such as reliability repository. The main source we have studied is the FIDES guide. We highlight how the meta-model of our DBD and the model of the dependability database allows to connect them. We describe the process leading to failure law definition and failure rate calculation in the FIDES guide. It matches perfectly several concepts that are associated to the description of failure modes in the DBD meta-model.

INTRODUCTION

In the current highly competitive context, designing and producing systems needs to rely more and more on optimized processes to master complexity in a restrained time, while validating system performances. In every project but particularly when designing critical systems, dependability analysis needs to be integrated in the design process to benefit from it since the early specification phases. Moreover it has been proven that it was cost-effective to detect potential failures as soon as possible, since it permits to avoid reengineering costs [START_REF] David | Supporting ISO 26262 with SysML, Benefits and Limits[END_REF][2][3].

Design processes exchanging data between system engineering activities and dependability studies were described in [START_REF] Cressent | Dependability analysis activities merged with system engineering, a real case study feedback[END_REF] and in various industrial dependability standards [2,3]. Our work also described various tools to ease the exchanges between system engineers and dependability experts. These tools are organized in a framework supporting the MéDISIS methodology. It relies on the Model-Based System Engineering (MBSE) [START_REF] Friedenthal | A Practical Guide to SysML : The Systems Modeling Language[END_REF] approach and the use of SysML (System Modeling Language) [START_REF]Systems Modeling Language V1[END_REF] as the modeling language for system engineering activities. MéDISIS then provides means of extracting information from SysML models to perform dependability studies. Currently, MéDISIS counts four processes that translate SysML models into target languages: the FMEA generation process and the Altarica DF translation process were described in [START_REF] David | Reliability study of complex physical systems using SysML[END_REF], the SysML to AADL translation process and the process to Simulink were described in [START_REF] Cressent | Increasing Reliability of Embedded Systems in a SysML Centered MBSE Process: Application to the LEA Project. 1st M-BED workshop[END_REF][START_REF] Cressent | Mastering safety and reliability in a Model Based process[END_REF].

To support this translation processes the framework also embeds the Dysfunctional Behavior Database (DBD). The DBD grants the possibility to import and perpetuate feedback gained on previous studies concerning dependability features of components. The DBD is necessary because system engineering activities don't deal with dysfunctional behavior.

Traditional reliability databases are necessary to manage dependability aspects during a project to perform reliability analysis and match failure rate requirements. Reliability databases evolved since the first dependability information collection: "the Martin Titan Handbook" [START_REF]Procedure and Data for Estimating Reliability and Maintainability[END_REF]. This handbook consisted in gathering failure rates for electrical and mechanical component in a uniform way used during a large missile design project. This evolution transcribed by [START_REF] Fragola | Reliability and risk analysis data base development: an historical perspective[END_REF] explains how upon time reliability databases took the benefits of supporting computer systems and then introduced the environmental conditions and the reasons of failure in failure rates computation.

Yet, the current databases are not al not model-oriented, forcing a manua of reliability and risk analysis databa Such a reliability database would be that currently begin to widely spre contained in these databases are e modeling that requires the faulty mod We can note that the first version o starting point regarding its structure each new step in the evolution of the implies that newer databases must e DBD needs to embed data from curre

In this article we will first present sh of knowledge stored in the FIDES re the concept of FIDES and our DBD.

THE DYSFUNCTIONAL BEH

The DBD is organized around the showing that a given SysML block type of entity we register in our DBD a SysML block stereotyped by "Fai "ComponenType" and "FailureMode inherits "ComponentType" so that a its failure modes. This construction modes definition. Failure modes are statemachine (synchronizing behavio A failure mode is composed of 4 par failure development, a statemachine entities stereotyped as "FailurePara either for triggering, developing or s t already formed to fully integrate MBSE processes ual application of their concepts which point out the bases. be of great help to support the Model-Based Safet pread amongst the safety analysts, see for examp expected to help conducting fault injection ana odeling of components and its failure behavior.

of Dysfunctional Behavior Database (DBD) we g ure and the predicted help it will provide. Howeve the reliability databases, the field of knowledge sto t embed data from former ones. Extending this rea urrent reliability databases such as FIDES [START_REF] Group | Reliability Methodology for electronic systems -FIDES Guide[END_REF] or M t shortly our DBD and its meta-model. Then we wil reliability database. Ultimately, we will highlight t D.

HAVIOR DATABASE

Figure 1 The DBD meta-model e component type. We propose to use the "Comp ck (blocks are used to model modular units of syst BD. Then, each failure mode of this type of compo FailureMode". This is presented in Fig. 1 by the a ode". The black arrow between these two blocks sh t all the behavior and attributes declared for a com ion makes it possible to access the component pa are defined by a set of parameters, operations (sho viors). Diverse stereotypes have been set up to defin parts: entities used to describe the failure triggering ine to synchronize the normal and faulty behavior arameters" which express every parameter involv r synchronizing. We will detail the way triggering i ses. In fact their format is the need for new concepts fety Assessment activities ple [START_REF] Lisagor | Towards a practicable process for automated safety assessment[END_REF][START_REF] Lisagor | Incremental Safety Assessment: Enabling the Comparison of Safety Analysis Results[END_REF]. The models nalyses and failure logic e gave in [START_REF] David | Reliability study of complex physical systems using SysML[END_REF] is a valuable ever, we noticed that with stored increased. That fact reasoning to our case, our Mil-HBDK 217f [START_REF]Reliability Prediction of Electronic Equipment[END_REF].

will study the organization t the connections between mponentType" stereotype ystem description [START_REF]Systems Modeling Language V1[END_REF]) is a ponent is registered using e association between the show that "FailureMode" omponent are reusable by parameters for its failure showing its behavior) and efine these elements.

ing, entities to describe the ior of the component, and olved in the failure mode g is modeled since it is the part that is most compatible wit development and synchronization are F

The failure trigger is modeled by a S which indicates the type of trigge Lognormal or Systematic). This "Tri 2; SysML constraints are modeling parameters) stereotyped by "EvalTr "FailureType" attribute of the "Trigg a random failure or to a systematic fa

In the example of modeling a random the trigger (e.g. Exponential, Weib specific to the dysfunctional aspect o the diverse indicators on component will appear in our DBD as "FailureP reliability databases: Failure probabi to supply the information needed for Concerning the modeling of dysfunc faulty, it may execute differently its functions are modeled by SysML op these operations are allocated to S faulty function in detail. The last statemachine that synchronizes the stereotyped "DysBehavior".

THE FIDES GUIDE

The core methodology

Concerning the reliability databases, analysis taking into account variou currently only accounts for electron and components. The method we pre our DBD the possibilities of the FID ith former reliability databases and quickly s are modeled.

Figure 2 DBD meta-model allocations a SysML operation stereotyped as "Trigger" that po ggering law followed by the failure mode (i.e.

Trigger" operation is expressed by its allocated Sys ling constructs that express mathematical express lTrigger" and uses parameters from the various "F igger" operation permits to define if the failure mod c failure as it is done in the IEC 61508 [3].

om failure, the "EvalTrigger" constraint represents eibull etc.). Some of the parameters participating ct of the component behavior, like the famous 1 use ent technology and manufacturer used in databases reParameters". This structure is used to reflect wha abilities, their associated parameters and the way to for failure behavior quantification. nctional behavior description, we assume that when its functions or execute new functions not specifie operations stereotyped by "DysOperation". As it w SysML constraints stereotyped by "EvalDysOpe st specific stereotype used in the DBD is made he normal and faulty behavior of the componen es, we decided to use FIDES [START_REF] Group | Reliability Methodology for electronic systems -FIDES Guide[END_REF] as an example sin rious impacting factors and is recognized in the onic components but allow high level of details w present in this article aims at connecting our DBD IDES guide in terms of reliability analysis. nts the probabilistic law of ng to this failure law are 1 sed in exponential laws or ses as FIDES. All of them hat is available in current to compute them; but also hen a component becomes ified in its mission. These it was done for "Trigger", peration" to describe the e to declare the attached ent. This statemachine is since it embeds reliability the industry [START_REF] Charpenel | The right way to assess electronic system reliability: FIDES[END_REF]. FIDES when describing systems D and FIDES to embed in

The FIDES guide 2009 is composed of three parts. The first describes the methodology and the philosophy behind FIDES. The second part contains the actual data for computing components failure rates. The last part details how to control and audit the reliability engineering process. The core methodology of FIDES relies on the principle of calculating the failure rate of the component (1) for its whole life cycle. This 1 will depend on the life phases and various parameters reflecting the impact of the design process, of the manufacturing process, of the manufacturer, etc… This core methodology represents the generic formulas common to every component FIDES deals with. Then, for every family of components, or even for specific components, FIDES extends its core methodology with more precise formulas. Before analyzing the equations that composes the FIDES core methodology, we need to define two terms used within the FIDES guide to define the various granularity levels:

1 Product: This refers to the assembled entity for which reliability is being studied.

1 Item: In this guide, an item refers to an elementary entity, not broken down, for which the reliability can be studied. 1 Within FIDES, an actual item can be a very specific component such as a resistor or a capacitor. The main equation of the FIDES methodology is to determine the failure rate of the product through its whole life, 1 product :

1 = item item product 1 1 (1)
The 1 item is composed of 3 factors:

Process PM physical item 2 2 1 1 × × = (2)
1 1 physical describes to the physical contributions of the failure rate. 1 2 PM (PM = Part Manufacturing) represents the item quality. The evaluation method may vary depending on the nature of the item considered. 1 2 process represents the quality and technical control over the development, manufacturing and usage process for the product containing the item.

As it is defined by FIDES, 2 PM and 2 process are multiplication terms that mitigates the purely physicals contributions represented by 1 physical . The term 2 PM symbolizes the quality of the item manufacturing. The term 2 process symbolizes the quality and technical control over reliability in the product design process. In complete FIDES analysis, 2 process is calculated based on results of a very detailed audit and 2 PM is calculated by evaluating the manufacturer's process. FIDES also plans to simply assign a constant value to each of them based on experience in case of a lack of sufficiently detailed data. For example, 2 process is by default set to 4 and it is advised to set 2 PM to 1.7 for active components and to 1.6 for other components and COTS. We won't describe in details how to determine these multiplication terms because it is part of the component specific part of the FIDES guide [START_REF] Group | Reliability Methodology for electronic systems -FIDES Guide[END_REF], and focus on the failure rate.

To determine 1 Physical we will combine 2 other formulas given by the FIDES guide: (3)

( ) induced ons contributi on accelerati 0 physical 2 2 1 1 × 8 9 A B C D × = 1 (4)
The formula (3) expresses the impact of the life profile on the failure rate for a one year period of time. The parameter Annual_time is the time spent in a specific phase and the 1 phase i is the failure rate in the conditions of a specific phase. Formula (4) expresses the impact of various stresses on the basic constant failure rate of the item 1 0 . The 2 induced translates the sensitivity to usage conditions. It is composed of various factor for each family of physical stresses (thermal, electrical, mechanical, chemical, humidity, temperature cycling). The 2 induced represents the overstresses not usually listed as such: item placement in the equipment and usage environment but also parameters such as ruggedizing and sensitivity to overstresses. At first sight we have a contradiction since there is two definition of how to calculate 1 physical . However FIDES describes 2 induced as a factor that depends on the life phases, 2 accelleration depends on usage conditions and even 1 0 depends on physical stresses. In fact, formula (4) is only relevant in a specific phase. Finally we can say that in formula (4), it is not 1 physical that is calculated but the 1 phase i .

The two most important aspects of the FIDES guide are:

1 Both physical and process contributions to risk are calculated, 1 The physical contributions are highly dependent on the life cycle definition. Even without detailing the formulas used to compute 2 accelleration or 2 induced that rely on the description of life phases, we can say that adjusting parameters describing the life phases of a product accurately is the reliability expert most important job during FIDES analysis.

To make it easier to analyze this core methodology, we use a SysML model to highlight the connection between these formulas and extract the list of parameters needed to compute the failure rate of a product. In SysML, each formula will be modeled as a constraint property. We will then be able to connect those constraints within a parametric diagram (Fig 3).

Figure 3 Parametric diagram of the FIDES methodology

To ease the understanding of Figure 3, we will summarize a few rules of SysML parametric diagram. There are 3 types of entities:

1 Constraints properties, materialized by round cornered rectangle. It is an instance of a Constraint Block. Constraint block defines a generic formula and the constraint property defines its use in an actual system/context and connected eventually with other constraint properties. For example, "Ohm Law" would be a constraint block, the matching constraint property in a system could embed the specific formula { U1 = (R1 + R2) × I1 }. 1 Constraint parameters, shown as squares on the edge of constraint properties. These represent the parameters involved in the formula of the constraint property. Following with the previous example, U1, R1, R2, and I1 are constraint parameters. 1 Value properties, presented by rectangles. These are the properties of components of the system.

They are used to show how components properties are linked with each other through constraint properties. For example, U1 could be linked to the property "voltage generated" of a voltage generator.

On this Figure 3, we identify the formulas 1 to 4 as the constraints properties: "Product Failure rate", "Item Failure rate", "Impact of life profile" and "Physical contributions". The other constraints properties represent how to calculate 2 PM , 2 process , and 2 induced .

2 PM is calculated from four criteria (where higher number means better results in terms of reliability):

1 QA manufacturer is the manufacturer's quality assurance criteria. It is a note between 0 and 3 based on the qualification of the manufacturer. 1 QA item is the item quality assurance criteria. It is a note between 0 and 3 based on the certification of the item itself. 1 RA item is the item reliability assurance. The FIDES guide defines RA item as relevant only for integrated circuits, discrete semiconductor, LED and optocouplers. It is a note between 0 and 3 based on the tests the item took. 1 Epsilon is the experience factor. It represents the component purchaser's experience and trust with the manufacturer. It is in general specific to a manufacturer for all its components but it can be component specific if relevant. Epsilon is a note between 1 and 4.

2 process is calculated from the Process Grade. This process grade is obtained through the FIDES audit we mentioned earlier.

The 1 physical of a phase is calculated with: 1 1 0 item which is the intrinsic failure rate of the item relatively to a type of physical stress (thermal, humidity, mechanical,…). It is a value that only depends on the type of the studied item. 1 2 acceleration which represents the sensitivity of an item to a type of physical stress. It is item specific but is often computed using more detailed parameters of the item and the life phase considered). 1 2 induced which model the contributions of overstresses on the failure rate.

The factor 2 induced itself is computed using:

1 C sensitivity which represents the coefficient of sensitivity to overstresses inherent to the item technology considered. This value only depends on the type of the item. 1 2 placement which expresses the influence of the position of the item in the product studied (particularly whether or not it is interfaced). 1 2 application which represents the influence of the usage of an item. This criterion is evaluated by answering 8 questions concerning the usage in the considered phase. 1 2 ruggedising which is a factor describing the policy of taking account of overstresses in the product development. This criterion is evaluated by answering questions on the ruggedizing policy.

CONNECTING FIDES AND THE DBD

Through our description of the FIDES guide, we presented how to calculate failure rate. This failure rate associated to an exponential failure law corresponds to the operation "Trigger" in our DBD that represents the triggering of a failure. Furthermore, the "EvalTrigger" is defined in the DBD as the constraint property that allows to compute the "Trigger". The "EvalTrigger" constraint property must connect all "Failure Parameters" to handle the calculation of the failure rate 1. In Fig 3, we present the FIDES methodology as a constraint property detailed with a parametric diagram. This parametric diagram connects various value properties with each other to finally compute the failure rate. Ideally, every parameter needed by the FIDES methodology should be stored in the DBD with "Failure Parameters". However Failure Parameters as described within the DBD should be specific to a single type of component but value properties of the FIDES methodology sometimes depends on product design or life profile. As a result, not every needed piece of information can be stored in our DBD. Besides, the core methodology is represented on Fig. 3 FailureParameter (1)

Csensitivity (C sensitivity )

FailureParameter Pplacement (product design) (2 placement )

FailureParameter (1)

Pruggedising (product design) (2 ruggedising )

FailureParameter (1)

Papplication(phase, product design) (2 application )

FailureParameter (1)

QAmanufacturer (QA manufacturer ) FailureParameter QAitem (QA item ) FailureParameter RAitem (RA item ) FailureParameter Epsilon FailureParameter Process_grade FailureParameter (2) (1)
: Failure Parameters refined through additional constraint properties depending on component type. Eventually, some of the sub-parameters can be project specific.

(2) : Failure Parameters inherited from the company level point of view. NULL: No correspondent entities.

In Table 3 we can see two types of exceptions that require specific adjustments:

1 There are parameters specific to the company used in the FIDES methodology. Parameters such as the process grade should be implemented on the DBD at a company level and instantiate in a project useful version of the DBD. This mechanism of instantiation between company and project levels is not yet formalized. In the meantime the process grade is a Failure Parameter used for every component. 1 There are parameters specific to a single project, such as the duration of the phases, the thermal environment in each phases, etc. These parameters should not be stored in the DBD since these data are not reliability parameters. However, they are determined through system engineering when describing the life cycle of the designed system, meaning that these parameters are deductible from a functional model of the system.

Finally, by matching the concept of the DBD and the FIDES reliability database, we point out that system engineering activities lead to defining major aspects of the FIDES analysis (Life profile, Design process, …). We will illustrate that fact in the following example by analyzing how to use both SysML and the DBD to perform FIDES reliability studies. To illustrate the connection of our DBD and the FIDES guide, we realized a FIDES analysis of a simple electric circuit (extract of a radio product Fig. 4) and extracted useful data from the DBD. Our example is an electric circuit from an FM radio. It is composed of 4 transistors, 1 resistor and 1 capacitor as showed on Fig. 4. We suppose that every item is purchased from the same manufacturer and system engineering activities have already been realized resulting in a SysML model (extract on Fig. 4 and Fig. 5). From the item depicted in our SysML internal block diagram on Fig. 4, we are able to search our DBD to get reliability parameters for FIDES analysis. Recent works showed how the system functional model of a system can contain life cycle information [START_REF]Systems Modeling Language V1[END_REF][START_REF] Bassi | A SysML-Based Methodology for Manufacturing Machinery Modeling and Design[END_REF].

CASE STUDY: RELIABILITY STUDY OF A SIMPLE PRODUCT

In our case, a context diagram in SysML permits to describe life phases and their associated requirements, constraints and properties (Fig. 5). The system context and life cycle as modeled during system engineering activities can be instantiated as the life profile needed for FIDES analysis (Table 1). During system engineering activities two phases were considered: On and Off. On Fig. 5, phase properties are described in the blocks Phase_On and Phase_Off. Phase specific constraints and requirement are also present in SysML. The resulting FIDES life profile is summarized on Table 1. We also attributed the mark for 2 application in each phase. Then we can sum up the reliability parameters needed by FIDES that are present in the DBD. The non physical contributions of the risk are composed of the Part Manufacturing and the Design Process. Concerning our design process, we assume we did not had an audit yet and we set 2 process = 4. The manufacturer of the items is certified ISO 9000 which correspond to QA manufacturer = 1. The items are qualified internally by the manufacturer which correspond to QA component = 1 (The scale for this parameters depends on component type family but internal qualification is equivalent to 1 for resistor, capacitor and transistor). Concerning resistors and capacitors, RA component isn't relevant and set to 0 (if the software used for computing needs a valor). For our transistors, RA component = 1 since it passed designed tests (High Temperature Reverse Bias,…). In our case, the manufacturer is recognized but their processes weren't analyzed which leads to Epsilon = 3. To compute the physical contribution to the risk, two parameters are fully available in the DBD: C sensisivity , and 1 0 item . The 1 0 item for each type of stress and the C sensisivity of each item are summarizeded in Table 1. The remaining parameters are deducted from the system engineering activities. The FM radio mixer is not placed as interface in the considered system, resulting in 2 placement = 1.0 for every item. 2 ruggedising is set to its default value (1.7) because no study was realized to evaluate it in detail. Finally, 2 acceleration are computed using both DBD constraint properties and life profile definition parameters. The resulting failure rate calculated is 8.9 FIT (FIT: Failure In Time, 1 FIT = 1 failure per 10 9 hours).

CONCLUSION

In this paper we described both our proposition for a model based dysfunctional database and the methodology behind the FIDES guide which is designed as a reliability database. We showed how it is possible to fully benefit from feedback formalized in FiDES within our DBD by connecting the two. The FIDES guide is still a work in progress since new type of components are periodically added to it, making the reach of the FIDES methodology greater. By matching the concept of the DBD and the FIDES reliability database, we also point out that system engineering activities lead to defining major aspects of the FIDES analysis (Life profile, Design process, …). All these make FIDES a worthy target candidate for a process that would extract data from a SysML functional model and use data from the DBD. Such a process would feed one of the software implementing FIDES analysis (Windchill Prediction, RAM Commander, Reliability, Care,… [START_REF]Windchill Prediction[END_REF]). This process would add up to the MéDISIS framework that aims to ease the interconnection of system engineering and dependability domains.

Currently, MéDISIS and the DBD are applied in the LEA project [START_REF] Cressent | Dependability analysis activities merged with system engineering, a real case study feedback[END_REF][START_REF] Falempin | French Flight Testing Program LEA Status in[END_REF], from the project activities planning to the product design phase. The LEA project consists in testing a scram jet motor in real flight conditions which implies a high level of reliability. To address reliability concerns, FIDES analysis will certainly be conducted for critical subsystems of the LEA vehicle. Furthermore, this case study permits to quantify the benefits brought by the use a model-based approach combined with MéDISIS and the DBD.

One of the remaining issues that we need to address is the definition of the different levels of our DBD. As we saw, parameters such as the process grade are not specific to a component but to the entire company. In fact, the information contained in the DBD, as we are foreseeing it, belongs to at least two different levels: the Company DBD (common to every project) and the Project DBD (completed through the project progresses). The process to handle the storage of information, from one level to another, needs to be defined in details and the benefits of each level should be identified.

Finally, addressing the relation between FIDES and our DBD leads us to address the challenges that represent the use of Components Off The Shelf (COTS) in a complex system with high dependability requirements [START_REF] Redmill | Analysis of the COTS debate[END_REF]. FIDES currently possess methods to analyze COTS impact on system dependability. These mechanisms will be studied in details to assure that the DBD is fitted to make good use of them.

  summarize how failure possesses a "FailureType" .e. Exponential, Weibull, ysML constraint (see Fig. essions shared by several "FailureParameters". The ode corresponds either to

  As a result, Fig 3 permits to identify the properties of components (value properties) that are needed to compute the failure rate of a product with the FIDES methodology. The next step to assure the compatibility between our DBD and FIDES is to link the value properties identified in Fig 3 with the concept of our DBD described in Fig 1.

Figure 4 FM

 4 Figure 4 FM Radio Mixer circuit: Electronic schema and SysML internal block diagram

Figure 5

 5 Figure 5 Context Diagram in SysML of the FM radio mixer circuit

Table 1

 1 but it may vary since several parameters (such as 2 acceleration ) may need refinement through additional parametric diagrams depending on the type of component considered. Nevertheless, this doesn't constitute an issue because it only means that EvalTriggers from different component type may differ which is allowed by the DBD definition. Table3resumes the conceivable reconnection of the FIDES methodology and our DBD meta-model: Correspondence between FIDES and DBD entities

	FIDES entities	DBD entities
	Constraint blocks and properties	Used within EvalTrigger
	Product Failure rate (1 product )	Used within EvalTrigger
		Parameter of Trigger
	Life Cycle (Annual Time phase)	NULL
	Lambda_0_item(stress) (1 0 item )	FailureParameter
	Pacceleration(stress) (2 acceleration )

Table 1

 1 Life profile description

	Phase	t annual-phase	Thermal T ambient	Humidity RH	Mechanical G RMS-phase	Induced 2 application
	Off	6 570h	25 °C	70%	0,0 Grms	2
	On	2 190h	25 °C	20%	0,1 Grms	4

Table 2

 2 Physical contributions parameters

	transistor	capacitor	resistor