Pauline Girault

Nathalie Lorrain
email: nathalie.lorrain@univ-rennes1.fr

Luiz Poffo

Mohammed Guendouz

Jonathan Lemaitre

Christiane Carré

Michel Gadonna

Dominique Bosc

Guillaume Vignaud

Integrated polymer micro-ring resonators for optical sensing applications

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

Introduction

Biological and chemical optical sensors are an active research topic for their many potential applications in medical diagnostics, screening of chemical compounds in drug development, food safety, environment monitoring and security defence. 1,2 Among the existing biological and chemical sensors, micro-resonators (MRs) have been demonstrated to be extremely sensitive and offer the advantage of significantly minimizing the device size, which greatly reduces the amount of molecules to be detected. 3,4 MRs based on integrated optical waveguides possess promising performances, particularly thanks to their integration capability with electronic devices as well as photonic devices, and to their easy fabrication with mature semiconductor processing technologies. [5][6][7] Moreover, these sensors allow label-free detection, compatibility with micro-fluidic handling and capability of providing high specificity using surface chemical modifications. 8,9 They can be easily implemented using polymer materials with a low cost of fabrication. 10,11 Most of the MR sensors rely on evanescent wave to detect the presence of molecules whether adsorbed on the sensor surface (surface sensing) or spread in the surrounding medium (homogeneous sensing). 12 The interaction between these molecules and the evanescent waves induces a shift in the specific resonance wavelength ∆λ due to the refractive index (nc) variation of the upper cladding of the resonator waveguide. In this case, the sensitivity (S) of the sensing optical function is given by S = ∆λ/∆nc expressed in nm/RIU (Refractive Index Unit).

The context of this work is to improve optical sensor sensitivity based on easily integrated micro-resonator (MR) component. We present a preliminary study on polymer integrated MR with sub-micronic ring waveguide gaps for sensing applications by taking advantage of our previous experiments on filters based on MRs. 13,[START_REF] Maalouf | Silicon Photonics and Photonic Integrated Circuits conference[END_REF] SU8 polymer as core layer and PMATRIFE polymer as lower cladding layer have been chosen due to their refractive index contrast (∼ 0.17 at 1550 nm). First results of such MR using PMATRIFE as lower and upper cladding layers have been recently published. [START_REF] Girault | SPIE Proceedings, Optical Sensing and Detection III[END_REF] This current work carries out the study of MR structures with the same couple of polymers, but without an upper cladding layer to allow laying of solutions such as glucose aqueous solutions for sensing applications.

I. Principle

The configuration studied in this work is a single side coupled micro-ring resonator represented in Figure 1. Using a broadband source, wavelengths injected through the first waveguide are coupled into the ring. Those, which are in tune with the incident wave (Σλ) after a round-trip in the ring, resonate by constructive interferences in the ring. The spectrum observed at the output of the straight waveguide shows extinction for each resonant wavelength.

In the coupling region considered as symmetrical, τ² is the self-coupling ratio in the straight or the ring waveguides whereas κ² is the cross-coupling ratio as well from the straight waveguide to the ring waveguide at the entrance as from the ring waveguide to the straight waveguide at the exit. In this case of symmetrical and lossless coupling, τ² and κ² verify τ 2 + κ 2 = 1, and the normalized transmitted intensity can be expressed [START_REF] Yariv | [END_REF] by : 1where:

T(λ E E -a 1 -aτ jϕ (
-Ei and Et are respectively the amplitude of the incident and transmitted fields, -ϕ 2πR neff β (2) is the single-pass phase shift of the propagated wave in the ring of radius R, with β the propagation constant and n eff the effective index of the ring waveguide.

a e -2πR α (3) represents the round trip field attenuation constant and the propagation losses per unit length.

The transmission T(λ) has periodic resonances occurring when λres 2πR neff p (4), corresponding to a single pass phase shift equal to an integer (p) times 2$: ϕ = p 2π .

Two consecutive wavelengths extracted from the ring are separated by a FSR (Free Spectral Range), FSR λres 2 2πR ng (5), where n ( is the group index which can be expressed as a function of wavelength and effective index by:

ng(λ neff(λ -λ dneff dλ (6).
Q λres δλ (7) is the quality factor (with δλ the full width at half maximum of the resonance peak)

and the transmission contrast is expressed by the difference (Tmax -Tmin (8) in which Tmax is the maximum of the transmission and Tmin, the minimum of transmission at the resonant wavelength. FSR, Q and transmission contrast are the characteristic values of MRs taken into account in this study. They depend, for a given design, on the coupling ratio between the ring and the straight waveguide and on the round trip losses.

II. Micro-ring implementation

A. Materials

In this paper, we present the fabrication of micro-ring resonators using Epoxy photoresist SU8 as core material and PMATRIFE (Poly (2,2,2 MethAcrylate of TRIFluoro-Ethyle ))

polymer as cladding layer. No upper cladding layer is used unlike in our previous study [START_REF] Girault | SPIE Proceedings, Optical Sensing and Detection III[END_REF] to implement the structure for sensing applications. SU8 is often used in polymer integrated optics for its optical and chemical properties, as well as its biological compatibilities. 17 To ensure a sufficient light confinement in the waveguide ridge core, PMATRIFE polymer, which offers a relatively low refractive index, has been chosen.

Refractive index evolution of these two polymers as a function of wavelength was investigated by spectroscopic ellipsometry, under ambient conditions (i.e. at 25°C and 55% of relative humidity). In each case, a uniform layer of polymer was deposited on a cleaned wafer of silicium by spin-coating at room temperature. Tetrahydrofuran was used as solvent to dissolve PMATRIFE. The solution was spin-cast at rotation speed of 4000 rpm during 10 seconds, to control the thicknesses of the films. After deposition, the samples were further processed by a two-step thermal treatment (10 minutes at 120°C, then 5 hours at 150°C), to evaporate all the solvents and to optimize its mechanical properties. SU8 corresponding to a liquid formulation was used without solvent. The solution was spin-cast first at rotation speed of 500 rpm during 5 seconds and after at 3000 rpm during 20 seconds. Finally, the samples were also processed by a two-step thermal treatment (first 1 minute at 65°C, after the temperature is increased gradually up to 95°C for a total duration of 4 minutes, the sample being at 95°C during 1 minute).

The ellipsometer (Jobin Yvon-Uvisel instrument) was mainly composed of a Xenon source, ranging from long-range ultraviolet to infrared (250-1700 nm), a polarizer, an analyzer and a monochromator handling the dispersion and the selection of the wavelength to a photomultiplier. Each sample was measured at three incident angles 65°, 70° and 75°.

Measurements were fitted with a Cauchy layer model, which is composed of a silicon substrate with a 2-nm-thick native SiO2 layer and a polymer film on top 18 . The fitting parameters were the refractive index of the polymer film, its thickness and the angle of incidence. The obtained thicknesses were 2.15 ± 0.10 µm for PMATRIFE and 1.05 ± 0.05 µm for SU8.

On figure 2 The effective index variation as a function of wavelength has been fitted by a linear function over the studied range (Figure 3). Taking into account the refractive index dispersion (equation ( 6)) mentioned in section I, the group index of the polymer waveguides have been deduced for the three configurations at 1550 nm. For air upper cladding layer: n g = 1.657 ± 0.002 RIU whereas for deionized water cladding layer, n g = 1.627 ± 0.002 RIU and for glucose aqueous solution (100 g/l), n g = 1.625 ± 0.002 RIU.

B. Design and fabrication

Compared to our previous study of PMATRIFE / SU8 / PMATRIFE MR, we used the structure without PMATRIFE upper cladding layer which has been replaced by air or water superstrates for sensor applications. Nevertheless, we have used the same photolithography mask, the design of which is reminded hereafter. Then we have verified thanks to a coupling study that the gap range is suitable to the SU8 / PMATRIFE MR characteristics values taking into account air and water superstrates.

In order to neglect the bend loss propagation in the ring while keeping in mind the optical integration, a radius of 120 µm has been chosen. [START_REF] Maalouf | Proc. SPIE 6996, Silicon Photonics and Photonic Integrated Circuits[END_REF] Using the equation ( 5), the calculated FSR of 2 nm will allow a relatively large detection window to be explored. The gap separation between the straight waveguide and the ring is an important parameter which is linked to the coupling ratio. Gaps varying from 0.5 to 1 µm have been studied regarding the theoretical photolithography resolution limit of 0.4 µm. [START_REF] Maalouf | [END_REF] Such designed waveguides have been realized using photolithography steps. [START_REF] Girault | SPIE Proceedings, Optical Sensing and Detection III[END_REF] In class 100 clean-room, all the polymer solutions used during the process (Figure 3) were filtered at 0.2 µm before the spin-coating steps. A 7 µm-thick layer of PMATRIFE (500 g/l in THF

C. Coupling study

In order to determine the coupling ratio between the straight and the ring waveguides as function of the gap g between them, we have used the Marcatili method 22 considering two different upper cladding layers: air and water. All the calculations were performed using Matlab software. In this configuration, the distance separation in the y direction varying along the z direction can be expressed by the following equation:

y(z g 5 R(1 -cos(θ (9).
where g is the minimal distance between the straight and the ring waveguides, R the ring radius and θ arcsin( z R is varying in this configuration from -9 to 9 .

We note CTE(z) the coupling coefficient for a z abscissa given in Marcatili method 22 for the mode TE :

|; <= (> | 2 ? k x A 3 π C k 1 k z D1-E n 3 n 1 F 2 G 1 2 a H1 -? k x A 3 π C 2 I 1 2 exp J-π y(z A 3 H1 -? k x A 3 π C 2 I 1 2 K (10
where:

-a is the core waveguide width (in the y transverse direction),

-kz is the axial propagation constant which depends on the transverse propagation constants kx and ky, and on the propagation constant k1 in the free space medium of refractive index n1;

k z Lk 1 2 -k x 2 -k y 2 M 1 2 k x π a N1 5 2n 3 2 A 3 πn 1 2 a O P1 k y π b ?1 5 A 2 5 A 3 πb C P1 R S 2$ T U S
The transverse propagation constants kx and ky depend on the refractive indices (figure 5.b.) of the core waveguides (n1), of the lower (n2) and upper cladding layers (n3) and on the coefficients Ai which are defined for i=2,3 by the expression :

V $ Lk 1 2 -k i 2 M S
Then to obtain the coupling coefficient Cw over the whole distance L of the coupling area (figure 5a), we have to integrate the CTE(z) function between the abscissa -L to L : (10) Then the coupling ratio X is equal to [START_REF] Okamoto | Fundamentals of optical waveguides[END_REF] :

CW = ⌡  ⌠ - L 2 L 2 CTE(z)dz
κ 2 = 1 -( cos(CW)) 2 (11)
κ 2 is calculated at lambda = 1550 nm and plotted as a function of the gap g in figure 6. Within the gap range (0.5 µm to 1 µm) that can be obtained from a technological point of view, coupling ratio varies from 0 to 2.45 ± 0.04 % for air superstrate and from 0 to 12.2 ± 0.3 % for water superstrate. The uncertainties on the coupling ratio have been estimated taking into consideration the uncertainties on the refractive index of the materials ( section II.A). For the sensing study, we expect the highest contrast transmission defined by equation (8). The normalized transmission contrast of the MRs as function of the coupling ratio κ 2 for 3 different propagation losses is plotted in figure 7 using equation ( 1). These values of attenuation 1, 3 and 5 dB/cm were chosen in relation to the ring propagation losses of 2.15 dB/cm measured in our previous study [START_REF] Girault | SPIE Proceedings, Optical Sensing and Detection III[END_REF] and to the material propagation losses of 5 dB/cm in SU8 [START_REF] Poon | [END_REF] .

In our model used to calculate the normalized transmission, the contrast only depends on the coupling ratio κ 2 and the propagation losses. It does not depend on the real effective index that we have calculated previously. Then, the normalized transmission contrast is similar for air and water superstrates.

Figure 7 shows that the maximum of contrast is obtained with a coupling ratio of about 5 % for propagation losses of 1 dB/cm and is about 15 % for losses of 5 dB/cm. According to figure 6, we can approach such coupling values with the gap range of the photolithography mask that we have previously used for PMATRIFE / SU8 / PMATRIFE MR [START_REF] Girault | SPIE Proceedings, Optical Sensing and Detection III[END_REF] . Thus this gap range of 0.5 µm to 1 µm is suitable to obtain sufficient SU8 / PMATRIFE MR transmission contrast taking into account air and water superstrates. The etching edges (Figure 8.d) are quite satisfying at this scale. However, an evaluation of the roughness is difficult to obtain yet we estimated the roughness to be about 50 nm.

Moreover, the gap of 0.5 µm has been successfully made (Figure 8.c). Figure 8.e shows a cross section view of the straight ridge waveguide with the suitable dimensions.

B. Extinction responses

In order to characterize the microresonators, we have measured their spectral responses.

A tunable laser (Agilent 8164A with 81680A tunable source) was injected in the input waveguide using a lensed single mode fiber (mode radius of 2.2 µm). At the output, another lensed single mode fiber is used to couple the output of the waveguide to a power meter or a spectrum analyzer (figure 9). For sensing applications, it is suitable to have one selected polarization to optimize their caracteristic values, such as Q and transmission contrast, and to make the analysis of the MR responses easier. Therefore, a polarization controller is inserted between the laser and the input of the lensed fiber.

The measurements of transmission were done on the through port in the wavelength range of 1550-1560 nm with a step of 2.5 pm. The calculated transmission were adjusted to experimental ones using equation (1) defined in section I. The adjustment of resonances shape was performed by modifying the coupling ratio and the propagation losses in the ring whereas the FSR was adjusted with the index group parameter. Adjustment parameters with their uncertainties are given in Table I. The expected values for coupling ratio and group index, already reported respectively in figure 6 and figure 3, are also indicated. The coupling ratio obtained from the Marcatili method is in the two cases lower than the ones obtained from the adjustment of experimental data. The Marcatili method only matches for low refractive index contrasts between the core and the cladding layers. The index contrast of the couple of polymer (SU8/PMATRIFE) is 0.17, value which might be too high to use the Marcatilli method. The limitation of this method for our refractive index contrast values could explain the difference between the coupling ratio obtained by adjustment of the experimental data and the calculated ones. Another reason which could explain this difference could be technological imperfections at nanometer scale that would favor the coupling. From a technological point of view, according to SEM observations, the gap of 0.5 µm is reached with an estimated accuracy of 0.05 µm. If we take into consideration these accuraties on figure 6, the uncertainties of coupling ratio reach nearly 1.5%. This quite high value could thus also explain the difference between the calculated and adjusted coupling ratio.

Estimation of material propagation losses in the ring have been deduced thanks to a fitting of spectral MR response with equation (1).

Material propagation losses around 4 -5 dB/cm have been obtained for the two studied upper superstrates (air or deionized water). This value is in the same order as other losses obtained in micro-ring resonators using SU8 for core layer but OG-125 (n = 1.456) for lower and upper cladding. [START_REF] Poon | [END_REF] If we take this propagation loss value in figure 7, we can observe that the maximum of contrast is obtained for a coupling ratio of 15 %, value which is experimentally reached for deionized water upper cladding. For air upper cladding, taking into consideration the same propagation losses, a contrast of around 0.7 is expected and corresponds to the measured one. On the contrary, the presence of deionized water in the ring rather than air, induces a decrease of the Q factor (2.7 10 4 in air against 1.5 10 4 in deionized water) due to the increase of the coupling ratio. In this case, the contrast nearly reaches the maximum: the measured value is 0.98. For this gap and for the water superstrate case, the coupling is then optimal. We observe an opposite behaviour for group index. The group index obtained from adjustment is lower than the calculated one. The experimental group index is deduced from the equation ( 5) with FSR and λres taken from experimental data and with the ring radius R equal to the expected one (120 µm). The calculated group index just takes into account the wavelength dispersion. First, it is determined from the experimental material refractive indexes and then from simulation by the finite difference method at several wavelengths. However, the curvature of the ring may also affect the group index, which has not been taken into account.

Concerning the resonant wavelength, the presence of deionized water on the ring rather than air, induced a shift of resonant wavelengths. It can be estimated using the effective index of the waveguide that we have determined (see figure 3). For air upper cladding, the resonant wavelength is 1556.4 nm corresponding to an order p of 700, according to equation ( 4), with an effective index of 1.464. For deionized water upper cladding, because of the effective index increase, and for the same integer order p = 700, the resonant wavelength is about 1600.6 nm.

The red shift of resonant wavelength induced by the increase of effective index approaches 44 nm which corresponds to nearly 22 times the value of FSR. Consequently, we did not estimate any sensitivity from these measurements.

IV. Sensing experiment

Our interest in this work was to use the MR structure as an optical transducer for sensing applications. For this objective and in order to evaluate the sensitivity of the structure by homogeneous sensing, glucose aqueous solutions were used with different concentrations. The solutions were introduced into the MR structure using a micro-syringe and were kept at the ambient temperature of the experimental room. The studied range of glucose concentration varied from 0 to 100 g/l with an accuracy of 2.5 g/l. This concentration range was chosen to prevent a shift of the resonant wavelength higher than the FSR.

The experimental transmission spectra obtained with deionized water superstrate and with a glucose aqueous solution using a concentration of 100 g/l, are reported in Figure 11.a.

In this condition, the increase of the refractive index of deionized water due to the presence of solubilized glucose was 0.012, according to Ciminelli et al. 25 . Such a refractive index variation provided a resonant wavelength red-shift of 1.39 nm at around 1558 nm (figure 11.a). In order to estimate the uncertainty on the red-shift, we performed the measurements 3 times by removing the sample from the optical bench. The error on the red shift has been estimated to 0.05 nm. The sensitivity of the MR was, in this case, 116 ± 8 nm /RIU (Refractive Index Unit) corresponding to the wavelength shift variation per upper cladding refractive index variation.

In the uncertainty of 8 nm/RIU, both the uncertainties on the red shift and on the glucose concentration have been taken into consideration. We observed a decrease in the contrast of nearly 1 -2 dB for each studied glucose concentration. The small increase of upper cladding refractive index may induce, for the same gap between the ring and the straight waveguide, a slightly higher coupling ratio. For aqueous solution, the optimal coupling ratio was reached for the gap of 0.5 µm. Therefore, by increasing the upper cladding refractive index, the increase of coupling ratio would provide a decrease in the contrast according to Figure 6.

In MR 25 and higher sensitivity (300 nm/RIU) 6 was obtained with silicon slot-waveguides-based ring resonator on insulator.

Conclusion

This study aimed to implement integrated polymer MRs for sensing applications using SU8 as core material and PMATRIFE as lower confinement material. Different superstrates such as air, deionized water and glucose aqueous solution were studied. With the aim of determining the index group of the structures, parameter necessary for the simulation study, refractive index of the different polymers, used in the fabrication of MRs, are given as a function of wavelength.

To optimize the design of the MRs, a coupling study was performed to link gap and characteristics values of MRs. These preliminary studies allowed us to implement successfully MRs with sub-micronic gaps varying from 0.5 to 1 µm. The best result is obtained for water superstrate. Normalized transmission contrast of 0.98 with a quality factor of nearly 1.5x10 4 was achieved for a gap of 0.5 µm corresponding to a coupling ratio of 14.7 % and ring propagation losses of 5.5 dB/cm according to the theoretical MR responses. The sensing application is tested with different glucose concentrations in water. We have measured a sensitivity of the SU8/PMATRIFE MRs of 115 ± 8 nm/RIU using a couple of polymers which technology is mastered in our lab. Our current work is to increase further the sensitivity by using other core and cladding materials with larger flexibility of their refractive index.

  After theoretical reminders used in the calculations of MRs characteristics, we present the refractive index study of the polymer waveguide. Micro-ring implementation is then described. A coupling study of the design is presented taking into account different upper cladding mediums. Measurements of extinction responses obtained with MRs with different upper cladding layers are presented and adjusted to theoretical ones in order to determine the group index, coupling and loss parameters. Optical results are discussed taking into account SEM observations and the results obtained in the coupling study. Finally first results of homogeneous sensing experiments using glucose aqueous solution are presented.

FIG. 1 .

 1 FIG. 1. Single side coupled micro-ring resonator. The field amplitudes are denoted by Ei (incident) and Et (transmitted). Self-coupling (τ) and cross-coupling (κ) amplitude coefficients are indicated.

19 FIG. 2 .

 192 FIG. 2. Refractive index evolution of SU8 and PMATRIFE polymers as a function of the wavelength.

FIG. 3 .

 3 FIG. 3. Effective index evolution of SU8 / PMATRIFE waveguides as function of the wavelength with a SU8 ridge core dimensions of 1.5 µm x 1.5 µm. Different waveguide upper cladding layers were studied regarding the sensing applications: air, deionized water without and with 100 g/l of glucose.

FIG. 4 .

 4 photo-sensitive resin (~ 400 nm) have been deposited. The thin SiO2 layer allowed the protection of SU8 from the next UV exposure step. Then, patterns were produced through a chrome photolithography mask under UV exposure and developed. To obtain the desired waveguides, an oxygen plasma RIE was performed. The top view and the cross section of the obtained waveguides at the end of the process are schematized in Figure4.

Figure 5 .FIG. 5 .

 55 FIG. 5. Schema of the coupling region of micro-ring resonator: (a) top view and (b) cross section view at z=0.

FIG. 6 .

 6 FIG.6. Coupling ratio calculated as function of the gap between the straight and the curved waveguides considering two cases of upper cladding layers: air or deionized water.

FIG. 7 .

 7 FIG. 7. Normalized transmission contrast plotted in function of the coupling ratio for propagation losses per unit length of 1, 3 or 5 dB/cm at 1550 nm.

Figures 8 .

 8 Figures 8.b, 8.c and 8.d give a top and a titled views of the coupling region.

FIG. 8 .

 8 FIG. 8. SEM images of the Air/SU8/PMATRIFE MR. (a) Top overview, (b) and (c) top view of the coupling region. (d) Tilted view of the etched waveguides in the coupling region. (e) Cross section view of the straight ridge waveguide.

FIG. 9 .

 9 FIG. 9. Overview of the optical MR characterization system.

Figure 10

 10 FIG. 10. Experimental and calculated transmission spectra for one polarization light and for the gap of 0.5 µm: (a) for Air/SU8/PMATRIFE and (b) for deionized water /SU8/PMATRIFE micro-rings.

Figure 11 .

 11 b, the measured red shifts obtained for glucose concentration of 50, 75 and 100 g/l are plotted as function of the refractive index variation of the aqueous solution with dissolved glucose (Figure 11.b). The sensitivity of our SU8/PMATRIFE MR taking into consideration the different glucose aqueous solutions as upper cladding is around 115 ± 8 nm /RIU. Such sensitivity values are reported in literature for silicon on insulator (SOI)

FIG. 11 .

 11 FIG. 11. (a) Experimental transmission spectra for one polarization light for deionized water/SU8/PMATRIFE and glucose aqueous solution (100 g/l)/SU8/PMATRIFE micro rings. Wavelength shift (∆λ) of the resonance associated with the change of refractive index (∆n) of the upper cladding layer. (b) Wavelength shift (∆λ) of the resonance plotted as a function of glucose concentration dissolved in water and as a function of the refractive index change (∆n) of the upper cladding layer.

TABLE I .

 I Adjustment parameters: coupling ratio, propagation losses in the ring and group index and calculated ones.

	WAVEGUIDE	Air/SU8/PMATRIFE	Aqueous
				solution/SU8/PMATRIFE
	Parameters	From adjustment	Calculated	From adjustment	Calculated
	Coupling rate	3.95 ± 0.05	2.45 ± 0.04	14.7 ± 0.2	12.2 ± 0.3
	(%)
	Propagation losses	4.23 ± 0.08	-	5.55 ± 0.05	-
	(dB/cm)
	Group index	1,628825 ± 10 -6	1.657 ± 0.002	1,610250 ± 10 -6	1.627 ± 0.002

Acknowledgments This work is supported by the PONANT project, funded by Lannion Trégor Communauté, Conseil Général des Côtes d'Armor, Région Bretagne, Feder and the French Research Ministry.


K A Ïs

Mourad Choulli
email: mourad.choulli@univ-lorraine.fr

LOGARITHMIC STABILITY IN DETERMINING TWO COEFFICIENTS IN A DISSIPATIVE WAVE EQUATION. EXTENSIONS TO CLAMPED EULER-BERNOULLI BEAM AND HEAT EQUATIONS

Keywords: Damping coefficient, potential, dissipative wave equation, boundary measurements, boundary observability, initial-to-boundary operator 93C25, 93B07, 93C20, 35R30

We are concerned with the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. We establish stability estimates of logarithmic type when the measurements are given by the operator who maps the initial condition to Neumann boundary trace of the solution of the corresponding initial-boundary value problem. We build a method combining an observability inequality together with a spectral decomposition. We also apply this method to a clamped Euler-Bernoulli beam equation. Finally, we indicate how the present approach can be adapted to a heat equation.

Introduction

We consider the following initial-boundary value problem (abbreviated to IBVP in the sequel) for the wave equation:

(1.1)

   ∂ 2 t u -∆u + q(x)u + a(x)∂ t u = 0 in Q = Ω × (0, τ ), u = 0 on Σ = ∂Ω × (0, τ ), u(•, 0) = u 0 , ∂ t u(•, 0) = u 1 ,
where Ω ⊂ R n , n ≥ 1, is a bounded domain with C 2 -smooth boundary ∂Ω and τ > 0.

We assume in this text that the coefficients q and a are real-valued.

Under the assumption that q, a ∈ L ∞ (Ω), for each τ > 0 and u 0 u 1 ∈ H 1 0 (Ω) × L 2 (Ω), the IBVP (1.1) has a unique solution u q,a ∈ C([0, τ ], H 1 0 (Ω)) such that ∂ t u q,a ∈ C([0, τ ], L 2 (Ω)) (e.g. [7, pages 699-702]). On the other hand, by a classical energy estimate, we have u q,a C([0,τ ],H 1 0 (Ω)) + ∂ t u q,a C([0,τ ],L 2 (Ω)) ≤ C( u 0 1,2 + u 1 0 ).

Here and henceforth, • p and • s,p , 1 ≤ p ≤ ∞, s ∈ R, denote respectively the usual L p -norm and the W s,p -norm. We note that the constant C above is a non decreasing function of q ∞ + a ∞ . Now, since u q,a coincides with the solution of the IBVP (1.1) in which -q(x)u q,a -a(x)∂ t u q,a is seen as a right-hand side, we can apply [START_REF] Lasiecka | Non homogeneous boundary value problems for second order hyperbolic operators[END_REF]Theorem 2.1] to get that ∂ ν u q,a , the derivative of the u q,a in the direction of ν, the unit outward normal vector to ∂Ω, belongs to L 2 (Σ). Additionaly, the mapping

u 0 u 1 ∈ H 1 0 (Ω) × L 2 (Ω) -→ ∂ ν u q,a ∈ L 2 (Σ)
defines a bounded operator.

Let Γ be a non empty open subset of ∂Ω and Υ = Γ × (0, τ ). To q, a ∈ L ∞ (Ω), we associate the initial-to-boundary (abbreviated to IB in the following) operator Λ q,a defined by Λ q,a :

u 0 u 1 ∈ H 1 0 (Ω) × L 2 (Ω) -→ ∂ ν u q,a|Υ ∈ L 2 (Υ).
Clearly, from the preceding discussion, Λ q,a ∈ B H 1 0 (Ω) × L 2 (Ω), L 2 (Υ) . We also consider two partial IB operators Λ q and Λq,a which are given by Λ q (u 0 ) = Λ q,0 u 0 0 , Λq,a (u 1 ) = Λ q,a 0 u 1 .

Therefore, Λ q ∈ B H 1 0 (Ω), L 2 (Υ) and Λq,a ∈ B L 2 (Ω), L 2 (Υ) . Next, we see that ∂ t u is the solution of the IBVP (1.1) corresponding to the initial conditions u 1 and ∆u 0 -qu 0 -au 1 . Hence, repeating the preceding analysis with ∂ t u in place of u, we get Λ q,a ∈ B H 1 0 (Ω) ∩ H 2 (Ω) × H 1 0 (Ω), H 1 ((0, τ ), L 2 (Γ)) . Consequently, Λ q ∈ B H 1 0 (Ω) ∩ H 2 (Ω), H 1 ((0, τ ), L 2 (Γ)) , Λq,a ∈ B H 1 0 (Ω), H 1 ((0, τ ), L 2 (Γ)) . We are interested in the stability issue for the inverse problem consisting in the determination of both the potential q and the damping coefficient a, appearing in the IBVP (1.1), from the IB map Λ q,a . We succeed in proving logarithmic stability estimates of determining q from Λ q , a from Λ q,a and (q, a) from Λ q,a .

We introduce the unbounded operators, defined on H 1 0 (Ω) × L 2 (Ω), as follows

A 0 = 0 I ∆ 0 , D(A 0 ) = H 2 (Ω) ∩ H 1 0 (Ω) × H 1 0 (Ω) and A = A q,a = A 0 + B with D(A) = D(A 0 ), where B = B q,a = 0 0 -q -a . Let C : D(A 0 ) → L 2 (Σ) : ϕ ψ -→ ∂ ν ϕ.
Since we deal with the wave equation, it is necessary to make assumptions on Γ and τ in order to guarantee that our system is observable. To this end, we assume that Γ is chosen in such a way that there is τ 0 such that the pair (A, C) is exactly observable for any τ ≥ τ 0 . We formulate the precise definition of exact observability in the next section in an abstract framework.

We give sufficient conditions ensuring that the pair (A, C) is exactly observable. We fix

x 0 ∈ R n \ Ω and we set Γ 0 = {x ∈ ∂Ω; ν(x) • (x -x 0 ) > 0} and d = max x∈Ω |x -x 0 |.
Let us assume that Γ ⊃ Γ 0 . Following [21, Theorem 7.2.3, page 233], (A 0 , C) is exactly observable with τ ≥ τ 0 = 2d. In light of [21, Theorem 7.3.2, page 235] and the remark following it, we conclude that (A, C) is also exactly observable for τ ≥ τ 0 , again with τ 0 = 2d.

We mention that sharp sufficient conditions on Γ and τ 0 were given in a work by Bardos, Lebeau and Rauch [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF].

Unless otherwise stated, for sake of simplicity, all operator norms will denoted by • . Also, B p (resp. B s,p ) denote the unit ball of L p (Ω) (resp. W s,p (Ω)).

We aim to prove in the present work the following theorem.

Theorem 1.1. We assume that (A 0 , C) is exactly observable for τ ≥ τ 0 , for some τ 0 > 0. Let 0 ≤ q 0 ∈ L ∞ (Ω), there is a constant δ > 0 so that

(1.2) q -q 0 2 ≤ C ln C -1 Λ q0 -Λ q -1/2 , q ∈ q 0 + δB 1,∞ ,
and, for any m > 0,

a 2 ≤ C ln C -1 Λq0,a -Λq0,0 -1/2 , a ∈ [δB ∞ ] ∩ [mB 1,2 ] , (1.3) q -q 0 2 + a 0 ≤ C ln C -1 Λ q,a -Λ q0,0 -1/2 , q ∈ q 0 + δB 1,∞ , a ∈ [δB ∞ ] ∩ [mB 1,2 ] . (1.4)
Here, C is a generic constant not depending on q and a.

Theorem 1.1 gives only stability estimates at zero damping coefficient. The difficulty of stability estimates at a non zero damping coefficient is related to the fact that the operator A is not necessarily diagonalizable. The main reason is that, contrary to case where a = 0, this operator is no longer skew-adjoint. We detail the stability estimate at a non zero damping coefficient in a separate section.

The problem of determining the potential in a wave equation from the so-called Dirichlet-to-Neumann (usually abbreviated to DN) map was initiated by Rakesh and Symes [START_REF] Rakesh | Uniqueness for an inverse problems for the wave equation[END_REF] (see also [START_REF] Cardoso | On the hyperbolic Dirichlet to Neumann functional[END_REF] and [START_REF] Isakov | An inverse hyperbolic problem with many boundary measurements[END_REF]). They prove that the potential can be recovered uniquely from the DN map provided that the length of the time interval is larger than the diameter of the space domain. The key point in their method is the construction of special solutions, called beam solutions. A sharp uniqueness result was proved by the so-called boundary control method. More details on this method can be found for instance in [START_REF] Belishev | Boundary control in reconstruction of manifolds and metrics (BC method)[END_REF] and [START_REF] Katchalov | Lassas Inverse boundary spectral problems[END_REF]. Also, Sun [START_REF] Sun | On continous dependence for an inverse initial boundary value problem for the wave equation[END_REF] establishes Hölder stability estimates and, most recently, Bao and Yun [START_REF] Bao | On the stability of an inverse problem for the wave equation[END_REF] improve the result of [START_REF] Sun | On continous dependence for an inverse initial boundary value problem for the wave equation[END_REF]. Specifically, they prove a nearly Lipschitz stability estimate. An extension was obtained by Bellassoued, Choulli and Yamamoto [START_REF] Bellassoued | Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem[END_REF] in the case of a partial DN map by a method built on the quantification of the continuation of the solution of the wave equation from partial Cauchy data. We refer to the introduction of [START_REF] Bellassoued | Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem[END_REF] for a short overview of inverse problems related to the wave equation. We finally quote a very recent paper by Bao and Zhang [START_REF] Bao | Sensitivity analysis of an inverse problem for the wave equation with caustics[END_REF] dealing with sensitivity analysis of an inverse problem for the wave equation with caustics.

It is worthwhile to mention that contrary to hyperbolic inverse problems, for which the stability can be of Lipschitz, Hölder or logarithmic type, elliptic and parabolic inverse problems are always severely ill-posed. That is the corresponding stability estimates are in most cases of logarithmic type. In [START_REF] Alessandrini | Examples of instability in inverse boundary-value problems[END_REF], Alessandrini gives an example in non destructive testing showing that the logarithmic stability is the best possible. This text is organized as follows. We consider in Section 2 the inverse source problem for exactly observable systems in an abstract framework. This material is necessary to establish stability estimates for the determination of the potential and the damping coefficient appearing in the IBVP (1.1). We devote Section 3 to the proof of Theorem 1.1 and we give in Section 4 a sufficient condition which guarantees that A is diagonalizable. The condition that A is diagonalizable is used in an essential way to get a variant of Theorem 1.1 at a non zero damping coefficient. We apply in Section 5 our approach to a clamped Euler-Bernoulli beam equation. The possible adaptation of our method to a heat equation is discussed in Section 6. Due to the fact that a heat equation is not exactly observable but only observable at final time, we obtain a stability estimate only when we perturb the unknown coefficient by a finite dimensional subspace.

An abstract framework for the inverse source problem

Let H be a Hilbert space and A : D(A) ⊂ H → H be the generator of continuous semigroup T (t). An operator C ∈ B(D(A), Y ), Y is a Hilbert space which is identified with its dual space, is called an admissible observation for T (t) if for some (and hence for all) τ > 0, the operator Ψ ∈ B(D(A), L 2 ((0, τ ), Y )) given by

(Ψx)(t) = CT (t)x, t ∈ [0, τ ], x ∈ D(A),
has a bounded extension to H.

We introduce the notion of exact observability for the system z ′ (t) = Az(t), z(0) = x, (2.1)

y(t) = Cz(t), (2.2)
where C is an admissible observation for T (t). Following the usual definition, the pair (A, C) is said exactly observable at time τ > 0 if there is a constant κ such that the solution (z, y) of (2.1) and (2.2) satisfies

τ 0 y(t) 2 Y dt ≥ κ 2 x 2 H , x ∈ D(A).
Or equivalently

(2.3) τ 0 (Ψx)(t) 2 Y dt ≥ κ 2 x 2 H , x ∈ D(A).
We consider the Cauchy problem

(2.4) z ′ (t) = Az(t) + λ(t)x, z(0) = 0,
and we set

(2.5) y(t) = Cz(t), t ∈ [0, τ ].
By Duhamel's formula, we have

(2.6) y(t) = t 0 λ(t -s)CT (s)xds = t 0 λ(t -s)(Ψx)(s)ds. Let H 1 ℓ ((0, τ ), Y ) = u ∈ H 1 ((0, τ ), Y ); u(0) = 0 . We define the operator S : L 2 ((0, τ ), Y ) -→ H 1 ℓ ((0, τ ), Y ) by (2.7) (Sh)(t) = t 0 λ(t -s)h(s)ds. If E = SΨ, then (2.6) takes the form y(t) = (Ex)(t).
Theorem 2.1. We assume that (A, C) is exactly observable for τ ≥ τ 0 , for some

τ 0 > 0. Let λ ∈ H 1 ((0, T )) satisfies λ(0) = 0. Then E is one-to-one from H onto H 1 ℓ ((0, τ ), Y ) and (2.8) κ|λ(0)| √ 2 e -τ λ ′ 2 L 2 ((0,τ )) |λ(0)| 2 x H ≤ Ex H 1 ℓ ((0,τ ),Y ) , x ∈ H.
Proof. First, taking the derivative with respect to t of each side of the integral equation

t 0 λ(t -s)ϕ(s)ds = ψ(t),
we get a Volterra equation of second kind

λ(0)ϕ(t) + t 0 λ ′ (t -s)ϕ(s)ds = ψ ′ (t).
Mimicking the proof of [15, Theorem 2, page 33], we obtain that this integral equation has a unique solution ϕ ∈ L 2 ((0, τ ), Y ) and

ϕ L 2 ((0,τ ),Y ) ≤ C ψ ′ L 2 ((0,τ ),Y ) ≤ C ψ H 1 ℓ ((0,τ ),Y ) . Here C = C(λ) is a constant.
Next, we estimate the constant C above. From the elementary convexity inequality (a + b) 2 ≤ 2(a 2 + b 2 ), we deduce

|λ(0)|ϕ(t) 2 Y ≤ 2 t 0 |λ ′ (t -s) |λ(0)| [|λ(0)| ϕ(s) Y ] ds 2 + 2 ψ ′ (t) 2 Y .
Thus,

|λ(0)| 2 ϕ(t) 2 Y ≤ 2 λ ′ 2 L 2 ((0,τ )) |λ(0)| 2 t 0 |ϕ(0)| 2 ϕ(s) 2 Y ds + 2 ψ ′ (t) 2
Y by the Cauchy-Schwarz's inequality. Therefore, using Gronwall's lemma, we obtain in a straightforward manner that

ϕ L 2 ((0,τ ),Y ) ≤ √ 2 |λ(0)| e τ λ ′ 2 L 2 ((0,τ )) |λ(0)| 2 ψ ′ L 2 ((0,τ ),Y )
and then

ϕ L 2 ((0,τ ),Y ) ≤ √ 2 |λ(0)| e τ λ ′ 2 L 2 ((0,τ )) |λ(0)| 2 Sϕ H 1 ℓ ((0,τ ),Y ) .

In light of (2.3), we end up getting

Ex H 1 ℓ ((0,τ ),Y ) ≥ κ|λ(0)| √ 2 e -τ λ ′ 2 L 2 ((0,τ )) |λ(0)| 2 x H .
We shall need a variant of Theorem 2.1. If (A, C) is as in Theorem 2.1, then it follows from [21, Proposition 6.3.3, page 189] that there is δ > 0 such that for any P ∈ B(H) satisfying P ≤ δ, (A + P, C) is exactly observable with κ(P + A) ≥ κ/2.

We define E P similarly to E by replacing A by A + P .

Theorem 2.2. We assume that (A, C) is exactly observable for τ ≥ τ 0 , for some τ 0 > 0. Let λ ∈ H 1 ((0, T )) satisfies λ(0) = 0. There is δ > 0 such that, for any P ∈ B(H) satisfying P ≤ δ, E P is one-to-one from

H onto H 1 ℓ ((0, τ ), Y ) and (2.9) κ|λ(0)| 2 √ 2 e -τ λ ′ 2 L 2 ((0,τ )) |λ(0)| 2 x H ≤ E P x H 1 ℓ ((0,τ ),Y ) , x ∈ H.
We now apply the preceding theorem to the following IBVP for the wave equation

(2.10)    ∂ 2 t u -∆u + q(x)u + a(x)∂ t u = λ(t)f (x) in Q, u = 0 on Σ, u(•, 0) = 0, ∂ t u(•, 0) = 0.
We recall that

A 0 = 0 I ∆ 0 , D(A 0 ) = H 2 (Ω) ∩ H 1 0 (Ω) × H 1 0 (Ω) and A = A q,a = A 0 + B q,a with D(A) = D(A 0 ), where B q,a = 0 0 -q -a . Also C : D(A 0 ) → L 2 (Γ) : ϕ ψ -→ ∂ ν ϕ.
We fix q 0 , a 0 ∈ L ∞ (Ω) and we assume that (A q0,a0 , C) is exactly observable with constant κ. This is the case when Γ ⊃ Γ 0 = {x ∈ ∂Ω; ν(x) • (x -x 0 ) > 0} (see for instance [12, Theorem 1.2, page 141]1 ).

Corollary 2.1. There is δ > 0 such that, for any q ∈ q 0 + δB 1,∞ and a ∈ a 0 + δB ∞ , we have

f 2 ≤ 2 √ 2 κ|λ(0)| e τ λ ′ 2 L 2 ((0,τ )) |λ(0)| 2 ∂ ν u f H 1 ((0,τ ),L 2 (Γ)) ,
where u f is the solution of the IBVP (2.10). This is nothing else but a Lipschitz stability estimate for the inverse problem of determining the source term f from the boundary data ∂ ν u f |Υ , when λ is supposed to be known.

Proof of Theorem 1.1

Proof of Theorem 1.1. Let (λ k ) and (φ k ) be respectively the sequence of Dirichlet eigenvalues of -∆ + q 0 , counted according to their multiplicity, and the corresponding eigenvectors. We assume that the sequence (φ k ) forms an orthonormal basis of L 2 (Ω).

We recall that according to the min-max principle, the following two-sided estimates hold

(3.1) c -1 k 2/n ≤ λ k ≤ ck 2/n .
Here, the constant c > 1 depends only on Ω and q 0 . Let u q be the solution of the IBVP (1.1) corresponding to q, a = 0, u 0 = φ k and u 1 = 0. Taking into account that u q0 = cos(t √ λ k )φ k is the solution of the IBVP (1.1) corresponding to q = q 0 , a = 0, u 0 = φ k and u 1 = 0, we see that u = u q -u q0 is the solution of the IBVP

(3.2)    ∂ 2 t u -∆u + qu = -(q -q 0 ) cos(t √ λ k )φ k in Q, u = 0 on Σ, u(•, 0) = 0, ∂ t u(•, 0) = 0.
In the remaining part of this proof, C is a generic constant independent on k. Let δ be as in Corollary 2.1. If q ∈ q 0 + δB 1,∞ , we get by applying Corollary 2.1

(q -q 0 )φ k 2 ≤ Ce Cλ k ∂ ν u H 1 ((0,τ ),L 2 (Γ)) .
Since |(q -q 0 , φ k )| ≤ |Ω| 1/2 (q -q 0 )φ k L 2 (Ω) by Cauchy-Schwarz's inequality, the last inequality entails

|(q -q 0 , φ k )| ≤ Ce Cλ k ∂ ν u H 1 ((0,τ ),L 2 (Γ)) . But, ∂ ν u = (Λ q0 -Λ q )φ k . Therefore (3.3) |(q -q 0 , φ k )| 2 ≤ Ce Cλ k Λ q0 -Λ q 2 .
Let λ ≥ λ 1 and N = N (λ) be the smallest integer so that λ N ≤ λ < λ N +1 . Then

q -q 0 2 2 = k |(q -q 0 , φ k )| 2 = k≤N |(q -q 0 , φ k )| 2 + k>N |(q -q 0 , φ k )| 2 ≤ k≤N |(q -q 0 , φ k )| 2 + 1 λ k>N λ k |(q -q 0 , φ k )| 2 ≤ k≤N |(q -q 0 , φ k )| 2 + Cδ 2 λ .
Here we used the fact that

k≥1 (1 + λ k )(•, ϕ k ) 2 1/2
defines an equivalent norm on H 1 (Ω).

In light of (3.3), we get

q -q 0 2 2 ≤ CN e Cλ Λ q0 -Λ q 2 + Cδ 2 λ .
By (3.1), N ≤ Cλ n/2 . Hence

q -q 0 2 2 ≤ Ce Cλ Λ q0 -Λ q 2 + Cδ 2 λ .
Minimizing with respect to λ, we obtain that there is δ 0 > 0 such that if Λ q0 -Λ q ≤ δ 0 , then

q -q 0 2 ≤ C ln(C -1 Λ q0 -Λ q ) -1/2 .
Estimate (1.2) follows then from the continuity of the mapping

q ∈ L ∞ (Ω) → Λ q ∈ B(H 1 0 (Ω) ∩ H 2 (Ω), H 1 ((0, τ ), L 2 (Γ)).
We proceed similarly for proving (1.3). In the actual case we have to replace the previous u q0 by u q0 = λ -1 k sin(t √ λ k )φ k , corresponding to the initial conditions u 0 = 0 and u 1 = φ k . Therefore, we have in place of (3.2)

(3.4)    ∂ 2 t u -∆u + a∂ t u = -a cos(t √ λ k )φ k in Q, u = 0 on Σ, u(•, 0) = 0, ∂ t u(•, 0) = 0.
We continue as in the preceding case by establishing the estimate

|(a, φ k )| 2 ≤ Ce Cλ k Λ q0,a -Λ q0,0
and we complete the proof of (1.3) as above.

We end the proof by showing how we proceed for proving (1.4). Taking into account that the solution corresponding to q = q 0 , a = 0,

u 0 = φ k and u 1 = iλ k φ k is u q0 = e i √ λ k t φ k , then in place of (3.2) we have the following IBVP (3.5)    ∂ 2 t u -∆u + qu + a∂ t u = -[(q -q 0 ) + i √ λ k a]e i √ λ k t φ k , in Q, u = 0 on Σ, u(•, 0) = 0, ∂ t u(•, 0) = 0.
We can argue one more time as in the proof of (1.2). We find

|(ϕ, q -q 0 ) + i λ k (ϕ, a)| 2 ≤ Ce Cλ k Λ q,a -Λ q0,0 2 , entailing |(ϕ, q -q 0 )| 2 ≤ Ce Cλ k Λ q,a -Λ q0,0 2 , |(ϕ, a)| 2 ≤ Ce Cλ k Λ q,a -Λ q0,0 2 .
We end up getting (1.4) by mimicking the rest of the proof of estimate (1.2).

Stability around a non zero damping coefficient

We limit ourselves to the one dimensional case and, for sake of simplicity, we take q identically equal to zero. But the analysis we carry out in the present section is still applicable for any non negative bounded potential.

We assume in the present section that Ω = (0, π). We introduced in the first section the unbounded operators, defined on

H = H 1 0 (Ω) × L 2 (Ω), A 0 = 0 I d 2 dx 2 0 , D(A 0 ) = H 2 (Ω) ∩ H 1 0 (Ω) × H 1 0 (Ω) := H 1 and A a = A 0 + B a with D(A) = D(A 0 )
, where

B a = 0 0 0 -a .
From [21, Proposition 6.2.1, page 180], (A 0 , C) is exactly observable for any τ ≥ 2π when

C : ϕ ψ ∈ D(A 0 ) -→ dϕ dx (0).
On the other hand, it follows from [21, Proposition 3.7.7, page 101] that the skew-adjoint operator A 0 is diagonalizable with eigenvalues λ k = ik, k ∈ Z * , corresponding to the orthonormal basis (g k ), where

g k = 1 √ 2 f k ik f k , k ∈ Z * ,
where (f k ) k∈N is an orthonormal basis of L 2 (Ω) consisting of eigenfunctions of the unbounded operator

A 0 = d 2 dx 2 under Dirichlet boundary condition and f -k = -f k , k ∈ N * . Let H ± be the closure in H of span{g ±k ; k ∈ N * }. Clearly, H = H + ⊕ H -and H ± is invariant under A 0 . Let then A ± 0 : H ± → H ± be the unbounded operator given by A ± 0 = A 0|H ± and D(A ± 0 ) = {u ∈ H ± ; k∈N * k 2 | u, g ±k | 2 < ∞}.
Here •, • is the scalar product in H.

Let A ± a0 = A ± 0 + B a0 and set ̺ = k≥1 1 (2k + 1) 2 and α = 1 2 2(1 + ̺) .
In light of [19, Theorem 2 and Lemma 10], we get Theorem 4.1. Under the assumption ρ := a 0 ∞ < α, the spectrum of ±A ± a0 consists in a sequence (iµ ± k ) such that, for any δ ∈ (0, 1 -ρ 2 /α 2 ), there is an integer

k such that |iµ ± k -ik| ≤ α = α(a 0 ) := ρ 4ρ 2 + δ , k ≥ k.
In addition, H ± admits a Riesz basis

(φ ± k ) = ϕ ± k iµ ± k ϕ ± k k∈N *
, each φ ± k is an eigenfunction corresponding to iµ ± k . We denote by ( φ ± k ) the Riesz basis biorthogonal to (φ ± k ) and define the sequence

(φ k ) k∈Z * (resp. ( φ k ) k∈Z * ) as follows φ -k = -φ - k and φ k = φ + k (resp. φ -k = -φ - k and φ k = φ + k ), k ∈ N * . Set also µ -k = -µ - k and µ k = µ + k , k ∈ N * . Therefore, A a0 φ k = iµ k φ k , k ∈ Z * , and, for any u ∈ H, u = k∈Z u, φ k φ k = k∈Z u, φ k φ k .
Additionally,

(4.1) α u 2 H ≤ k∈Z * | u, φ k | 2 , k∈Z * | u, φ k | 2 ≤ β u 2 H ,
where the constants α and β do not depend on u (see for instance [21, Lemma 252, page 37]).

We pick a 0 as in the preceding theorem. Then it is straightforward to check that u a0 = e iµ k t ϕ k , k ∈ Z * , is the solution of the IBVP (1.1) with q = 0, a = a 0 , u 0 u 1 = φ k . If u a is the solution of the IBVP (1.1), then u = u a -u a0 is the solution of the IBVP (4.2)

   ∂ 2 t u -∆u + a(x)∂ t u = (a 0 -a)iµ k e iµ k t ϕ k in Q, u = 0 on Σ, u(•, 0) = 0, ∂ t u(•, 0) = 0.
We fixe δ as in the statement of Theorem 4.1. Then, for some integer k, e iµ k t ≤ e |iµ k -i|k||t e i|k|t ≤ e ατ , |k| ≥ k,

|µ k | ≤ |k| + α, |k| ≥ k.
These estimates at hand, we can proceed as in the previous section to get, where

ψ k = iµ k ϕ k , (4.3) |(a -a 0 , ψ k )| 2 = 0 a -a 0 , φ k 2 ≤ Ce Ck 2 Λ a -Λ a0 2 .
It follows from (4.1),

(4.4) α a -a 0 2 2 = α 0 a -a 0 2 H ≤ |k|≥1 0 a -a 0 , φ k 2 .
In light of (4.3) and (4.4), we have

α a -a 0 2 2 ≤ CN e Cλ Λ a -Λ a0 2 + 1 λ |k|>N k 2 |(a -a 0 , ψ k )| 2 ≤ CN e Cλ Λ a -Λ a0 2 + 1 λ |k|≥1 k 2 |(a -a 0 , ψ k )| 2 . (4.5)
Here λ ≥ λ 1 and N = N (λ) be the smallest integer satisfying N 2 ≤ λ < (N + 1) 2 .

We note that we cannot pursue the proof similarly to that of (1.2) because (ψ k ) is not necessarily an orthonormal basis of L 2 (Ω). So instead of the boundedness of a -a 0 in H 1 (Ω), we make the assumption, where m > 0 is fixed,

(4.6) |k|≥1 k 2 |(a -a 0 , ψ k )| 2 ≤ m.
Under the assumption (4.6), (4.5) entails

α a -a 0 2 2 ≤ Ce Cλ Λ a -Λ a0 2 + m λ .
where Λ a = Λ 0,a and Λ a0 = Λ 0,a0 . The same minimization argument used in the proof of (1.2) (see Section 3) allows us to prove the following theorem.

Theorem 4.2. There exist two constants C > 0 and δ > 0 so that

a -a 0 2 ≤ C ln C -1 Λ a -Λ a0 -1/2
, a ∈ a 0 + δB ∞ and (4.6) holds. Remark 4.1. Let us explain briefly why the result of this section can not be extended to a higher dimensional case. The main reason is that, even for simple geometries, the eigenvalues of the unperturbed operators A ± 0 do not satisfy a gap condition which is the main assumption in [START_REF] Shkalikov | On the basis property of root vectors of a perturbed self-adjoint operator[END_REF]Theorem 2]. If (ρ k ), ρ k = k, is the sequence of eigenvalues of ±A ± 0 , we used in an essential way that ρ k+1 -ρ k = 1.

When Ω = (0, a) × (0, b), the eigenvalues operator

A + 0 consist in the sequence π 2 k 2 a 2 + ℓ 2 b 2
k,ℓ∈N * . These eigenvalues are simple when a 2 b 2 ∈ Q but can condensate in finite interval and therefore they don't satisfy a gap condition like in the one dimensional case.

An application to clamped Euler-Bernoulli beam

For the same reason as in the preceding section, we limit our analysis to the one dimensional case. So we let Ω = (0, 1).

We introduce the following spaces

H 0 = L 2 (0, 1), H 1/2 = H 2 0 (Ω), H 1 = H 4 (0, 1) ∩ H 2 0 (Ω).
The natural norm of H s will denoted by • s , s ∈ {0, 1/2, 1}. On H = H 1/2 × H 0 , we introduce the unbounded operator A given by

A = 0 I -d 4 dx 4 0 , D(A) = H 1 × H 1/2 := H 1 .
We consider a torque observation at an end point. We define then C :

H 1 → C by C ϕ ψ = d 2 ϕ dx 2 (0).
We are concerned with following IBVP for the clamped Euler-Bernoulli beam equation (5.1)

       ∂ 2 t u + ∂ 4 x u = 0 in Q, u(0, •) = u(1, •) = 0 on (0, τ ), ∂ x u(0, •) = ∂ x u(1, •) = 0 on (0, τ ), u(•, 0) = u 0 , ∂ t u(•, 0) = u 1 .
From [21, Proposition 3.7.6, page 100], A is skew-adjoint and therefore it generates a unitary group on H. Consequently, for any u 0 u 1 ∈ H 1 the IBVP (5.1) has a unique solution u so that (u, u ′ ) ∈ C([0, τ ], H 1 ) ∩ C 1 ([0, τ ], H). Moreover, by [21, Proposition 6.10.1, page 270], (A, C) is exactly observable for any τ > 0 and there is a constant κ such that (5.2)

κ 2 ( u 0 2 1/2 + u 1 2 0 ) ≤ ∂ 2 x u(0, •) 2 L 2 ((0,τ )) .
Here the constant κ is independent on u 0 and u 1 .

Let B a be the operator, where a = a(x),

B a = 0 0 0 -a .
This operator is bounded on H whenever a ∈ L ∞ (Ω). Therefore, bearing in mind that A + B a generates a continuous semigroup, the IBVP (5.3)

       ∂ 2 t u + ∂ 4 x u + a(x)∂ t u = 0 in Q, u(0, •) = u(1, •) = 0 on (0, τ ), ∂ x u(0, •) = ∂ x u(1, •) = 0 on (0, τ ), u(•, 0) = u 0 , ∂ t u(•, 0) = u 1 has a unique solution u = u a (u 0 , u 1 ) satisfying (u, u ′ ) ∈ C([0, T ], H 1 ) ∩ C 1 ([0, T ], H), for any u 0 u 1 ∈ H 1 .
Moreover, the same perturbation argument used in the proof of Theorem 2.2 enables us to show that (A+ B a , C) is exactly observable with constant κ 2 ≥ κ 2 /2 provided the norm of the operator B a is sufficiently small. That is, there is δ > 0 such that for any B a ∈ B(H) with B a ≤ δ, we have

(5.4) (1/2)κ 2 ( u 0 2 1/2 + u 1 2 0 ) ≤ ∂ 2 x u(0, •) 2 L 2 ((0,τ )) .
In light of [21, Lemma 6.10.2, page 218], the spectrum of A consists in a sequence of simple eigenvalues (iρ k ) k∈Z * , where

ρ k = π 2 k - 1 2 2 + a k , k ∈ N * ,
(a k ) a sequence converging exponentially to 0, and

ρ -k = -ρ k , k ∈ N * .
Let A 0 be the unbounded operator on L 2 (Ω) defined by A 0 = d 4 dx 4 and D(A 0 ) = H 4 (Ω) ∩ H 2 0 (Ω). Then A 0 is diagonalizable with eigenvalues (ρ 2 k ) k∈N * . Let (f k ) k∈N * be a basis of eigenfunctions, each f n is an eigenfunction corresponding to ρ 2 k . Let

g k = 1 √ 2 f k iρ k f k , and
g -k = -g k , k ∈ N * .
With the help of [21, Lemma 3.7.7, page 101], we get that (g k ) k∈Z * is an orthonormal basis of A 0 . Define H ± as the closure of span{g ±k ; k ∈ N * }. Then H = H -⊕ H + and H ± is invariant under A 0 . We consider A ± 0 : H ± → H ± the unbounded operator given by A ± 0 = A 0|H ± and

D(A ± 0 ) = {u ∈ H ± ; k∈N * k 2 | u, g ±k | 2 < ∞},
where •, • is the scalar product in H, and we set A ± a0 = A ± a0 + B a0 . Since ρ k+1 -ρ k → +∞ as k → +∞, (ρ k ) k∈N * satisfies the a gap condition. Precisely, there exists d > 0 so that

ρ k+1 -ρ k ≥ d, k ∈ N * . Set α ′ = d 2 2(1 + ̺) ,
where ̺ is as in Section 4.

We have similarly to Theorem 4.1, Theorem 5.1. Under the assumption ρ := a 0 ∞ < α ′ , the spectrum of ±A ± a0 consists in a sequence (iµ ± k ) such that, for any δ ∈ (0, 1 -ρ 2 /(α ′ ) 2 ), there is an integer k such that

|iµ ± k -iρ k | ≤ α = α(a 0 ) := ρd 4ρ 2 + d 2 δ , k ≥ k.
In addition, H ± admits a Riesz basis

(φ ± k ) = ϕ ± k iµ ± k ϕ ± k , each φ ± k is an eigenfunction corresponding to iµ ± k .
We define the IB operator Λa by Λa :

u 1 ∈ H 1/2 -→ ∂ 2 x u a (0, u 1 )(0, •) ∈ L 2 (0, τ ).
One can prove in a straightforward manner that Λa is bounded operator between H 1 and H 1 ((0, τ )) and its norm can be uniformly bounded, with respect to a, by a constant, provided that the L ∞ -norm of a is sufficiently small. We carry out a similar analysis to that after Theorem 4.1 to get the following stability estimate. Theorem 5.2. Given m > 0, there exist constants C > 0 and δ > 0 so that

a -a 0 0 ≤ C ln C -1 Λa -Λa0 -1/4 , if a ∈ a 0 + δB 1,∞ and |k|≥1 λ k |(a -a 0 , ψ k )| 2 ≤ m,
where

ψ ±k = iµ ± k ϕ ± k , k ∈ N * .
We mention that the method used in this section and in the previous one is easily adaptable to a Schrödinger equation.

The case of a heat equation

We consider the following IBVP for the heat equation (6.1) [START_REF] Choulli | Une introduction aux problèmes inverses elliptiques et paraboliques[END_REF]Theorem 1.43, page 27], for any q ∈ L ∞ (Ω) and u 0 ∈ H 1 0 (Ω), the IBVP has a unique solution u q = u q (u 0 ) ∈ H 2,1 (Q) and, for any m > 0,

   ∂ t u -∆u + q(x)u = 0 in Q, u = 0 on Σ, u(•, 0) = u 0 . Let H 2,1 (Q) = L 2 ((0, τ ), H 2 (Ω)) ∩ H 1 ((0, τ ), L 2 (Ω)). From
u q H 2,1 (Q) ≤ C u 0 1,2 ,
where the constant C = C(M ) is independent on q, q ∞ ≤ m.

Let Γ be an arbitrary nonempty open subset of ∂Ω and set Υ = Γ × (0, τ ). Using the trace theorem in [11, page 26], we obtain that the following IB mapping

Λ q : u 0 ∈ H 1 0 (Ω) -→ ∂ ν u q (u 0 ) ∈ L 2 (Υ) is bounded.
The following lemma will be useful in the sequel. Its proof is sketched in Appendix A.

Lemma 6.1. Let q 0 , q ∈ L ∞ (Ω) so that q ∈ q 0 + W 1,∞ (Ω). Then Λ q -Λ q0 defines a bounded operator from H 1 0 (Ω) into H 1 ((0, τ ); L 2 (Γ)). Additionally, for each m > 0, there exits C > 0 so that Λ q -Λ q0 ≤ C, for all q 0 , q ∈ mB ∞ . Here, Λ q -Λ q0 is the norm of Λ q -Λ q0 in B(H 1 0 (Ω); H 1 ((0, τ ); L 2 (Γ))). In the sequel Λ q -Λ q0 is considered as an operator acting from H 1 0 (Ω) into H 1 ((0, τ ); L 2 (Γ)). We assume, without loss of generality, that q ≥ 0. Indeed, substituting u by ue -q ∞t , we see that q in (6.1) is changed to q + q ∞ . So, we fix q 0 ∈ L ∞ (Ω) satisfying 0 ≤ q 0 and we let 0 < λ 1 < λ 2 . . . ≤ λ k → +∞ be the sequence of eigenvalues, counted according to their multiplicity, of -∆ + q 0 under Dirichlet boundary condition. An orthonormal basis consisting in the corresponding eigenfunctions is denoted by (ϕ k ).

Let q ∈ mB ∞ ∩ q 0 + W 1,∞ (Ω) . We pick a positive integer k. Taking into account that u q0 (ϕ k ) = e -λ k t ϕ k , we obtain that u = u q (ϕ k ) -u q0 (ϕ k ) is the solution of the IBVP (6.2)

   ∂ t u -∆u + q(x)u = (q 0 -q)e -λ k t ϕ k in Q, u = 0 on Σ, u(•, 0) = 0.
We set f = (q -q 0 )ϕ k and λ(t) = e -λ k t . Therefore (6.2) becomes (6.3)

   ∂ t u -∆u + q(x)u = λ(t)f (x) in Q, u = 0 on Σ, u(•, 0) = 0. It is straightforward to check that (6.4) u(x, t) = t 0 λ(t -s)v(x, s),
where v is the solution of the IBVP   

∂ t v -∆v + q(x)v = 0 in Q, v = 0 on Σ, v(•, 0) = f.
In light of the Carleman estimate in [START_REF] Choulli | Une introduction aux problèmes inverses elliptiques et paraboliques[END_REF]Theorem 3.4, page 165], we can extend [START_REF] Choulli | Une introduction aux problèmes inverses elliptiques et paraboliques[END_REF]Proposition 3.5,page 170] in order to get the following final time observability inequality (6.5) v(•, τ

) H 1 0 (Ω) ≤ C ∂ ν v L 2 (Υ) .
Here C is a constant depending on m but not on q.

By the continuity of the trace operator w ∈ H 2,1 (Q) → ∂ ν w |Υ ∈ L 2 (Υ), we get from (6.4)

∂ ν u(x, t) |Υ = t 0 λ(t -s)∂ ν v(x, s) |Υ .
We proceed as in the beginning of the proof of Theorem 2.1 to deduce the following estimate (6.6)

∂ ν v L 2 (Υ) ≤ √ 2e τ 2 λ 2 k ∂ ν u H 1 ((0,τ ),L 2 (Γ)) .
On the other hand

(6.7) v(x, t) = ℓ≥1 e -λ ℓ t (f, ϕ ℓ )ϕ ℓ . Hence v(•, τ ) 2 2 = ℓ≥1 e -2λ ℓ τ |(f, ϕ ℓ )| 2 .
Arguing as in Section 3, we get, for any λ ≥ λ 1 and N

= N (λ) satisfying λ N ≤ λ < λ N +1 , (6.8) f 2 2 ≤ e 2λ k τ v(•, τ ) 2 + 1 λ 2 ℓ>N λ 2 ℓ |(f, ϕ ℓ )| 2 .
By Green's formula, we obtain

λ ℓ (f, ϕ ℓ ) = - Ω ∆(q -q 0 )ϕ ℓ ϕ k dx + 2 Ω ∇(q -q 0 ) • ∇ϕ k ϕ ℓ dx + λ k (f, ϕ ℓ ).
Therefore, under the assumption that q ∈ q 0 + W 2,∞ (Ω) and q -q 0 2,∞ ≤ m,

λ ℓ |(f, ϕ ℓ )| ≤ (1 + λ k )m + λ k |(f, ϕ ℓ )|.
This estimate in (6.8) yields

f 2 2 ≤ e 2λτ v(•, τ ) 2 2 + 2(1 + λ k )m 2 + λ 2 k λ 2 ℓ>N |(f, ϕ ℓ )| 2 ≤ e 2λτ v(•, τ ) 2 2 + 2(1 + λ k )m 2 + λ 2 k λ 2 f 2 2 ≤ e 2λτ v(•, τ ) 2 2 + Cλ 2 k λ 2 f 2 2 ≤ e 2λτ v(•, τ ) 2 2 + Cλ 2 k λ 2 .
This inequality together with (6.6) imply

(q -q 0 )ϕ k 2 2 = f 2 2 ≤ √ 2e 2τ 2 λ 2 k +2λτ ∂ ν u 2 H 1 ((0,τ ),L 2 (Γ)) + Cλ 2 k λ 2 . But ∂ ν u H 1 ((0,τ ),L 2 (Γ)) = Λ q (ϕ) -Λ q0 (ϕ k ) H 1 ((0,τ ),L 2 (Γ)) ≤ Λ q -Λ q0 ϕ k H 1 0 (Ω) ≤ λ k Λ q -Λ q0 .
Whence, (6.9) (q -q 0 )ϕ k

2 2 = f 2 2 ≤ √ 2λ k e 2τ 2 λ 2 k +2λτ Λ q -Λ q0 2 + Cλ 2 k λ 2
. Now the usual way consists in minimizing, with respect to λ, the right hand side of the inequality above. By a straightforward computation, one can see that the minimization argument is possible only if

λ k e -2τ 2 λ 2 k Λ q -Λ q0 2 ≫ 1.
But this estimate does not guarantee that Λ q -Λ q0 can be chosen arbitrarily small uniformly in k. However, the minimization argument works if we perturb q 0 by a finite dimensional subspace. That what we will discuss now.

Let I > 0 be a given integer and

E I = span{ϕ 1 , . . . , ϕ I }. Since |(q -q 0 , ϕ k )| 2 ≤ |Ω| (q -q 0 )ϕ k 2
2 by Cauchy-Schwarz's inequality, we get from (6.9)

q -q 0 2 2 = I k=1 |((q -q 0 ), ϕ k )| 2 ≤ C I e 2λτ Λ q -Λ q0 + 1 λ 2 ,
for some constant C I depending on I. We observe that, according to the preceding analysis, C I surely blows-up when I → +∞.

Minimizing with respect to λ the right hand side of the inequality above, we get

q -q 0 2 ≤ C I |ln ( Λ q -Λ q0 )| -1 ,
provided that Λ q -Λ q0 is sufficiently small. By a simple continuity argument, we see that Λ q -Λ q0 is small whenever q -q 0 1 , ∞ is small. If Λ I q = Λ q|E I , we end up getting Theorem 6.1. Under the preceding notations and assumptions, there exist two constants C I and δ I so that q -q 0 2 ≤ C I ln Λ I q -Λ I q0 -1 , if q -q 0 1,∞ ≤ δ I . Remark 6.1. We consider on L 2 (Ω) the following norm, which weaker than its natural norm,

w * =   k≥1 e -3τ 2 λ 2 k |(w, ϕ k )| 2   2
, w ∈ L 2 (Ω).

Then (6.9) yields q -q 0 2 * ≤ √ 2e 2τ λ Λ q -Λ q0 2 + C λ 2 . We get by minimizing the right hand side with respect to λ q -q 0 * ≤ C |ln ( Λ q -Λ q0 )| -1 .

Appendix A

Proof of Lemma 6.1. We start be considering the following homogenous IBVP for the heat equation

(A.1)    ∂ t u -∆u + q(x)u = φ in Q, u = 0 on Σ, u(•, 0) = ψ.
Let φ ∈ L 2 (Q) and ψ ∈ H 1 0 (Ω). We obtain by applying one more time [11, Theorem 1.43, page 27] that the IBVP (A.1) has a unique solution u φ,ψ ∈ H 2,1 (Q) provided that q ∈ L ∞ (Ω). Moreover, for any m > 0, there exists C > 0 so that

(A.2) u φ,ψ H 2,1 (Q) ≤ C ψ H 1 (Ω) + φ L 2 (Q) ,
uniformly in q ∈ mB ∞ .

Let A q be the unbounded operator on L 2 (Ω) defined by A q = -∆ + q, D(A q ) = H 2 (Ω) ∩ H 1 0 (Ω). Then the solution of (A.1) is given by u φ,ψ (•, t) = e -tAq ψ + t 0 e -sAq φ(•, t -s)ds, where e -tAq is the semigroup generated by -A q .

In the sequel, we write u φ for u φ,0 . Let φ ∈ C 1 ([0, τ ]; L 2 (Ω)) so that φ(•, 0) ∈ H 1 0 (Ω). Then In other words, ∂ t u φ = u ∂tφ,φ(•,0) . Thus, estimate (A.2) entails

(A.3) ∂ t u φ H 2,1 (Q) ≤ C φ(•, 0) H 1 (Ω) + ∂ t φ L 2 (Q) .
Next, let φ ∈ H 1 ((0, τ ); L 2 (Ω)) with φ(•, 0) ∈ H 1 0 (Ω). Observing that u φ = u φ + u φ(•,0) , where φ = φ -φ(•, 0), and ∂ t u φ(•,0) = u 0,φ(•,0) , we see that is sufficient to consider the case φ(•, 0) = 0.

By density, there exist a sequence (φ k ) in C ∞ 0 ((0, τ ]; L 2 (Ω)) converging to φ ∈ H 1 ((0, τ ); L 2 (Ω)). Armed with (A.2), we get in a straightforward manner that u φ k and u ∂tφ converge respectively to u φ and u ∂tφ in H 2,1 (Q). But, in light of the smoothness of φ k , ∂ t u φ k = u ∂tφ k . Therefore, we have ∂ t u φ = u ∂tφ and (A.3) holds true for all φ ∈ H 1 ((0, τ ); L 2 (Ω)) with φ(•, 0) ∈ H 1 0 (Ω). Now, let q 0 , q ∈ mB ∞ so that q ∈ q 0 + W 1,∞ (Ω). Let u 0 ∈ H 1 0 (Ω). By an elementary computation, we get that u := u q (u 0 ) -u q0 (u 0 ) = u φ , with φ = (q 0 -q)u q0 (u 0 ). Consequently, from the preceding discussion, u, ∂ t u ∈ H 2,1 (Q) and

(A.4) u H 2,1 (Q) + ∂ t u H 2,1 (Q) ≤ C u 0 H 1 (Ω) .
For some constant C = C(m). Finally, using the continuity of the trace operator w ∈ H 2,1 (Q) → ∂ ν w ∈ L 2 (Υ), we obtain from (A.4) Λ q (u 0 ) -Λ q0 (u 0 ) H 1 ((0,τ );L 2 (Γ)) ≤ C u H 1 (Ω) .

That is, we proved Λ q -Λ q0 ≤ C, where Λ q -Λ q0 is the norm of Λ q -Λ q0 in B(H 1 0 (Ω), H 1 ((0, τ ); L 2 (Γ)).

∂ 0 e

 0 t u φ (•, t) = e -tAq φ(•, 0) + t -sAq ∂ t φ(•, t -s)ds.

We note that from the proof of this theorem it is not possible to extract the dependance of κ on q 0 and a 0 .


Lodewyck

C Shi

J.-L Robyr

U Eismann

M Zawada

L Lorini

R Le Targat

J Lodewyck

Polarizabilities of the 87 Sr Clock Transition

Keywords: numbers: 06.30.Ft, 42.62.Fi, 37.10.Jk

Introduction

Optical lattice clocks are now the most stable frequency references [START_REF] Hinkley | An atomic clock with 10 -18 instability[END_REF][START_REF] Hagemann | Providing 10 -16 short-term stability of a 1.5-µm laser to optical clocks. Instrumentation and Measurement[END_REF][START_REF] Bloom | An optical lattice clock with accuracy and stability at the 10 -18 level[END_REF], and their accuracy is steadily improving towards 10 -17 to 10 -18 [START_REF] Bloom | An optical lattice clock with accuracy and stability at the 10 -18 level[END_REF][START_REF] Falke | A strontium lattice clock with 3 × 10 -17 inaccuracy and its frequency[END_REF][START_REF] Targat | Experimental realization of an optical second with strontium lattice clocks[END_REF][START_REF] Ushijima | Cryogenic optical lattice clocks[END_REF]. In these clocks the systematic effects due to the atomic motion are cancelled by trapping a large number of ultra-cold atoms (typically 10 4 ) in the Lamb-Dicke regime, using an optical lattice formed by a standing wave laser beam. A specificity of these optical clocks is the strong light shift induced by the intense trapping light. At the so-called "magic wavelength", for which the scalar polarizabilities of the fundamental and excited clock states are identical, this light shift is largely suppressed [START_REF] Katori | Ultrastable optical clock with neutral atoms in an engineered light shift trap[END_REF]. However, in order to achieve a high accuracy, the residual polarizationdependent and higher order light shifts have to be evaluated. In reference [START_REF] Westergaard | Lattice-induced frequency shifts in Sr optical lattice clocks at the 10 -17 level[END_REF], we reported exhaustive measurements of these light shifts with two 87 Sr lattice clocks, and showed that they are compatible with an accuracy of 10 -17 at a trapping potential of 100 recoil energies. We reported the first experimental resolution of the vector-, tensor-, and hyper-polarizability and put an upper bound on higher order effects for 87 Sr.

In this paper, we focus on the first-order electric dipole interaction to propose a theoretical study of the coefficient of the decomposition of its Hamiltonian in vector and tensor irreducible operators that can reproduce the experimental results reported in [START_REF] Westergaard | Lattice-induced frequency shifts in Sr optical lattice clocks at the 10 -17 level[END_REF]. The first section introduces the irreducible operator formalism describing the atomic polarizability. In the second section, we make use of this decomposition to theoretically evaluate the vector and tensor polarizability coefficients. The last section details the measurement of these coefficients and show their agreement with the theoretical estimates. Finally, we describe the non-linear dependence of the vector and tensor shifts with the depth of the trapping potential. Furthermore, we report on precise measurements of the magic wavelength and its sensitivity that provides insight in the physical properties of electronic levels of 87 Sr that are useful for characterizing the clock accuracy [START_REF] Middelmann | High accuracy correction of blackbody radiation shift in an optical lattice clock[END_REF].

I. FACTORIZATION OF THE POLARIZABILITY OPERATOR

The Hamiltonian that describes an atomic level |φ = |nJF m in the presence of the electromagnetic field of a trapping light and a static bias magnetic field B s reads:

Ĥ = ĤZ + Ĥeff . (1)
ĤZ is the Hamiltonian representing the Zeeman interaction:

ĤZ = gµ B ˆ F • B s , (2)
where g is the Landé factor and µ B the Bohr magneton. Ĥeff is the effective Hamiltonian representing the the off-resonant electric dipole interaction between the atom and the trapping light of complex amplitude E or power density I = ε 0 c|E| 2 /2 in second-order perturbation theory [START_REF] Cohen-Tannoudji | Experimental study of zeeman light shifts in weak magnetic fields[END_REF]:

Ĥeff = - 1 4 α|E| 2 , (3)
where the polarizability operator α reads:

α = 1 |φ ε * • ˆ d |φ φ | ε • ˆ d ω 0 -ω l + ε • ˆ d |φ φ | ε * • ˆ d ω 0 + ω l , (4)
where the sum runs over all excited states |φ = |n J F m . ˆ d is the electric dipole operator, ω 0 is the energy difference between |φ and |φ , and ω l and ε are the angular frequency and polarization of the trapping light, respectively. Because the two states involved in the clock transition (5s 2 1 S 0 and 5s5p 3 P 0 , as show in figure 1) both have J = 0, their respective polarizabilities do not a priori depend on the magnetic sub-level m considered. However, the hyperfine Hamiltonian describing odd Sr isotopes with a non-zero nuclear spin slightly breaks the rotational invariance and introduces a minute dependence of the polarizabilities of and ε. This dependence can be made analytically explicit by expanding the polarizability operator as the sum of three irreducible operators [START_REF] Romalis | Zeeman frequency shifts in an optical dipole trap used to search for an electric-dipole moment[END_REF][START_REF] V D Ovsyannikov | Polarisation and dispersion properties of light shifts in ultrastable optical frequency standards[END_REF]. In the (2F + 1)dimensional basis |m ≡ {|nJF m , -F ≤ m ≤ F } of the hyperfine manifold of the Hilbert space describing either clock state, the matrix elements of this expansion read:

m 2 |α|m 1 = 2 j=0 K j j M =-j (-1) M C F m2 F m1j -M [ ε ⊗ ε * ] jM , (5)
where the first sum runs over the tensor rank j. In this expression, the dependence in ε is exclusively contained in the tensor product [ ε ⊗ ε * ] jM , the dependence in m in the Clebsch-Gordan coefficient C F m2 F m1j -M and the dependence in |φ in the coefficients:

K j = 1 |n J F k j 1 ω 0 -ω l + (-1) j ω 0 + ω l , (6)
where k j is expressed using Wigner 6j-symbols and reduced dipole elements:

k j = (-1) j-F -F 2j + 1 2F + 1 F F j 1 1 F |(F ||d||F )| 2 .
(7) The canonical scalar α s , vector α v and tensor α t polarizabilities are then defined by rescaling these coefficients:

α s = - 1 √ 3 K 0 , α v = 2F F + 1 K 1 ,
and

α t = 2F (2F -1) 3(F + 1)(2F + 3) K 2 . (8)
These three physical parameters are sufficient to completely explicit the matrix of the polarizability operator given by equation [START_REF] Targat | Experimental realization of an optical second with strontium lattice clocks[END_REF]. The first term of this equation, for j = 0, is the scalar polarizability. It is the main, rotationally invariant contribution to the polarizability. The vector polarizability (j = 1) appears only when F ≥ 1/2 and when the light has a non-linear polarization. It is equivalent to a fictitious magnetic field along the light wave vector. It is an odd function of m, and accordingly vanishes when the light shift is averaged over opposite values of m. The last term (j = 2), appearing if F ≥ 1, is the tensor polarizability that results in a polarization dependent polarizability with an even dependence in m.

The following section is devoted to the theoretical calculation of the three polarizability coefficients for the two atomic levels involved in the 87 Sr clock transition.

II. THEORETICAL CALCULATION OF THE POLARIZABILITIES

In order to further calculate the polarizabilities, the k j coefficients can be related to the |φ → |φ transition rate Γ φφ , or to the radiative lifetime τ of the excited state |φ through:

|(F ||d||F )| 2 2F + 1 = B hfs 3πε 0 c 3 ω 3 0 Γ φφ = B hfs 3πε 0 c 3 ω 3 0 B fs ζ τ , (9)
where the fine and hyperfine branching ratios read:

B fs = (2J + 1)(2L + 1) J 1 J L S L 2 , (10)
B hfs = (2F + 1)(2J + 1) F 1 F J I J 2 , ( 11
)
and ζ is a dimensionless parameter accounting for the fine structure splitting.

When neglecting the hyperfine interaction, the two states of the clock transition with J = 0 do not exhibit vector or tensor polarizabilities, as can be seen by setting F = 0 in equation [START_REF] Katori | Ultrastable optical clock with neutral atoms in an engineered light shift trap[END_REF]. However, the hyperfine splitting of the excited states |φ :

ν hfs = 1 2 AC + B 3 4 C(C + 1) -I(I + 1)J (J + 1) 2I(2I -1)J (2J -1) , (12)
where C = F (F + 1) -I(I + 1) -J (J + 1) and A and B are magnetic dipole and electric quadrupole constants, gives rise to vector and tensor components. The relative magnitude of these latter components is expected to be on the order of the ratio between the hyperfine splitting ν hfs and the frequency of the optical transitions |φ → |φ [START_REF] Katori | Ultrastable optical clock with neutral atoms in an engineered light shift trap[END_REF]:

α v,t α s ≈ ν hfs ν optical . (13)
However, for a state |φ with J = 0, we can show that for any value of I, and with J = 1 according to the electric dipole selection rules:

I+J F =I-J k 2 C = 0 (J = 0, J = 1), (14)
such that the contribution of the magnetic dipole term A to the tensor polarizability is cancelled at first order in ν hfs . Hence, the dominant contribution to the tensor shift is the quadrupole B, or the second order in ν hfs , whichever is the largest. No such cancellation occurs for the vector shift. This property is particularly observed for 87 Sr, as the excited states with the largest hyperfine splitting feature B A, like 5s6s 3 S 1 and 5s5p 3 P 1 . As a consequence, their contribution to the tensor polarizabilities of the clock states is orders of magnitude smaller than what could be expected from [START_REF] Katori | Ultrastable optical clock with neutral atoms in an engineered light shift trap[END_REF].

Using experimental measurements of the lifetimes of excited states, completed by theoretical estimates of the strength of atomic transitions [START_REF] Boyd | High precision spectroscopy of strontium in an optical lattice: Towards a new standard for frequency and time[END_REF], we can calculate the dynamic polarizabilities for various states of 87 Sr at the magic wavelength (ω l = 2π × 368.6 THz).

The scalar, vector and tensor polarizabilities of the fundamental state 5s 2 1 S 0 are derived by summing over the excited states 5s5p 3 P 1 and 5skp 1 P 1 with k = 5 . . . 20. They read, expressed in atomic units (a.u.):

5s 2 1 S 0 α s α v α t 279.8 4.75 × 10 -5 1.57 × 10 -5 (15)
The main contribution to the scalar polarizability comes from the 5s5p 1 P 1 state, while the vector and tensor terms are dominated by the 5s5p 1 P 1 and 5s5p 3 P 1 states. As expected, the ratio between the vector (resp. tensor) polarizability and the scalar polarizability is on the order of 10 -7 , comparable with the ratio between magnetic dipole term A -about 200 MHz (resp. the quadrupole term Babout 50 MHz) and optical frequencies -about 500 THz. The polarizability of the 5s5p 3 P 1 (F = 9/2) is estimated by summing over the 5sks 3 S 1 states with k = 6 . . . 10, the 5p 2 3 P states and the 5skd 3 D states with k = 4 . . . 9:

5s5p 3 P 1 (F = 9/2) α s α v α t 304.4 -50.3 -101.2 (16)
As expected from the fact that J = 0 for 3 P 1 , the vector and tensor polarizabilities are on the same order of magnitude as the scalar polarizability. Figure 2 confronts these numerical values to the experimental spectroscopy of the 5s 2 1 S 0 → 5s5p 3 P 1 (F = 9/2) inter-combination line.

In the case of the excited clock state 5s5p 3 P 0 , a direct calculation is not sufficient because state mixing with the 5s5p 3 P 1 (F = 9/2), 5s5p 3 P 2 (F = 9/2) and 5s5p 1 P 1 (F = 9/2) states alters its physical properties.

FIG. 2:

Spectroscopy of the saturated 5s 2 1 S0 → 5s5p 3 P1(F = 9/2) transition by depletion of 87 Sr atoms in an optical lattice, for a linearly polarized trapping light, no magnetic field, and a trap depth of 1520 ER, where ER = h × 3.47 kHz is the recoil energy associated with the absorption of a lattice photon with frequency ω l . In this configuration, the polarizability operator ( 5) is already diagonal when choosing the quantization axis along the lattice polarization. Its eigen-values are α(m) = αs + αt 3m 2 -F (F +1)

F (2F -1)
. The graph is centred on the light shift free frequency of the transition, and displays the splitting between magnetic sub-levels induced by the scalar and tensor polarizabilities of both states. The arrows indicate the theoretical light shift δν = -1 4 α(m)|E| 2 for each value of |m|, as calculated from equations ( 15) and [START_REF] Blatt | Rabi spectroscopy and excitation inhomogeneity in a one-dimensional optical lattice clock[END_REF].

Since these states have J = 0, their vector and tensor polarizabilities are large and may contribute to the polarizabilities of the 3 P 0 state. This state mixing is written:

|φ = p c p |φ p 0 , (17)
where the coefficients c p are [START_REF] Boyd | Nuclear spin effects in optical lattice clocks[END_REF]:

|φ p 0 5s5p 3 P 0 5s5p 3 P 1 5s5p 3 P 2 5s5p 1 P 1 (F = 9/2) (F = 9/2) (F = 9/2) c p 1. 2.3 × 10 -4 -1.38 × 10 -6 -4.1 × 10 -6
.

(18) When state mixing is involved, expanding the matrix elements of the polarizability operator ( 4) is more involving, but the result is that equations ( 5) through [START_REF] Romalis | Zeeman frequency shifts in an optical dipole trap used to search for an electric-dipole moment[END_REF] remain valid expressions of these matrix elements, provided they are averaged over all (|φ p 0 , |φ q 0 ) combinations with a weight c p c * q , and provided the branching ratios are replaced by: Bfs = (-1) Sp+Sq+2L +Jp+Jq (2J p + 1)(2J q + 1)

× (2L + 1) J 1 J p L S L J 1 J q L S L , (19)
Bhfs = (2F +1)(2J +1) F 1 F J p I J F 1 F J q I J . (20)
Using these expressions, we can show that for J p = 0 and J q = 1, I+J F =|I-J | k 2 = 0 such that this combination has a negligible contribution to the tensor shift, and for J p = 0 and J q = 2, I+J F =|I-J | k 1 = 0 such that this second combination has a negligible contribution to the vector shift.

Among all the (|φ p 0 , |φ q 0 ) combinations involving the states listed in the table of equation ( 18), only a few lead to a non-negligible contribution. They are listed in bold characters in the following table:

5s5p 3 P 0 |φ p 0 |φ q 0 α s α v α t 3 P 0 3 P 0 288.8 9.3 × 10 -3
1.9 × 10 -6 3 P 1 3 P 1 2 × 10 -5 2.7 × 10 -6 -5.4 × 10 -6 3 P 0 3 P 1 2 × 10 -6 1.82 × 10 -1 -1.5 × 10 -6 3 P 0 3 P 2 ≤ 10 -13 -1.8 × 10 -8 3.70 × 10 -4

Total 288.8 1.91 × 10 -1 3.65 × 10 -4 (21) In this table, the (|φ p 0 , |φ q 0 ) and (|φ q 0 , |φ p 0 ) contributions for p = q are grouped together. As expected from the theoretical considerations above, the tensor term of the ( 3 P 0 , 3 P 0 ) configuration is negligible, and the main contribution to the vector and tensor polarizabilities respectively come from the mixing combinations ( 3 P 0 , 3 P 1 ) and ( 3 P 0 , 3 P 2 ).

To express these polarizabilities in units more adapted to experimental purposes, we note U 0 = 1 4 α s |E| 2 the trap depth. At the magic wavelength, α s and hence U 0 are by definition identical for the two clock states. The numerical values of α s shown in equations ( 15) and (21) show that this fact is approximately rendered by our calculation, which also gives an order of magnitude of a few percent for its accuracy, arising from uncertainties on the decay rates from excited states. For the vector and tensor polarizabilities, we introduce the coefficients:

κ v = - α v α s 1 h 1 2F and κ t = - α t α s 1 h 1 2F (2F -1)
.

The following table gathers the theoretical values for κ v and κ t for the two clock states, as well as their difference ∆κ v,t = κ v,t ( 3 P 0 )κ v,t ( 1 S 0 ):

|φ κ v (mHz/E R ) κ t (µHz/E R ) 3 P 0
-255 -60.9 1 S 0 -6.5 × 10 -2 -2.7

3 P 0 -1 S 0 -255 -58.2 (23)

III. EXPERIMENTAL DETERMINATION OF THE POLARIZABILITIES

During the clock operation, a static bias magnetic field is applied to split the magnetic sub-levels. The energy levels are then the eigenvalues of the full Hamiltonian written in equation ( 1). Because ˆ F is a vector operator, the Zeeman Hamiltonian adds up to the vector component of the polarizability operator. Therefore, the vector component of the Hamiltonian is equivalent to the action of an effective magnetic b eff field equal to the vector sum of the static magnetic field B s and the fictitious magnetic field that represents the vector light shift (see figure 3):

b eff = κ v i( ε ∧ ε * )U 0 g + B s , (24)
where g = gµ B /h. If the quantization axis e Z is chosen along the effective magnetic field (i.e e Z = b eff /|| b eff ||), the vector component of the Hamiltonian is diagonal in the basis {|m , -F ≤ m ≤ F } of eigen-vectors of F Z . However, the tensor part of the Hamiltonian is a priori not diagonal in this basis. Yet, since κ v κ t , if we assume that the bias field is large enough for the Zeeman splitting to be much larger than the energy splitting due to the tensor shift (For a trap depth U 0 ∼ 100 E R , this assumption is satisfied if B s 1 µT), the states {|m } remain approximate eigen-states of the full Hamiltonian. Expanding the tensor product in equation ( 5) then yields the energy shift δν of level |m of a given hyperfine manifold:

δν(m) = - U 0 h + mg|| b eff || + κ t βU 0 , (25)
with

β = (3| ε • e Z | 2 -1)(3m 2 -F (F + 1)). (26)
The middle term of equation (25) includes both the vector light shift and the Zeeman shift. It can be isolated from the scalar and tensor terms by measuring the half difference Z s (m) = 1 2 (δν(m)δν(-m)) between opposite magnetic sub-levels:

Z s = mg|| b eff || = m (gB s cos θ + κ v ξU 0 ) 2 + (gB s sin θ) 2 , (27)
where ξ = ||i( ε∧ ε * )|| is the degree of circular polarization of the trapping light, and θ is the angle between the wave vector k and the bias field B s (as shown on figure 3). The Taylor expansion of the later equation to second order in U 0 yields:

Z s mgB s + mκ v ξ cos θ U 0 + m sin 2 θ (κ v ξ) 2 2gB s U 2 0 . ( 28
)
To experimentally evaluate the differential vector polarizability coefficient ∆κ v , we measured ∆Z s = Z s ( 3 P 0 ) -Z s ( 1 S 0 ) for different trapping depths with m = ±9/2 and a circular polarization for the trapping light (|ξ| = 1).

From these data we extrapolated the derivative of the Zeeman shift at zero trap depth ∂∆Z s /∂U 0 (U 0 = 0) = m∆κ v ξ cos θ. The trap depth is measured by observing the longitudinal motional side-bands of the trapped atoms [START_REF] Blatt | Rabi spectroscopy and excitation inhomogeneity in a one-dimensional optical lattice clock[END_REF]: their spacing and shape give a direct measurement of the trap depth (at the trap centre) and of longitudinal and transverse temperatures of the atoms [START_REF] Le | Optical lattice clock with strontium atoms : a second generation of cold atom clocks[END_REF].

From these three parameters, we can deduce the average trap depth experienced by the atoms.

Repeating the experiment for various values of θ enables to deduce the differential coefficient, as reported in [START_REF] Westergaard | Lattice-induced frequency shifts in Sr optical lattice clocks at the 10 -17 level[END_REF]:

|∆κ v | = 0.22 ± 0.05 Hz/E R , (29)
in agreement with the theoretical estimate reported in equation ( 23). The uncertainty is limited by the knowledge of the polarization state ε in the lattice cavity. In the usual clock operation, B s is orthogonal to the wave vector k (i.e. cos θ 0). This configuration minimizes the first order term of equation ( 28), but it also maximizes its quadratic term. This latter term can therefore easily be observed, even for moderate trapping depths below 200 E R .

Figure 4 shows such a quadratic dependence for an elliptical polarization. In order to fit the data points of this figure, the two Landé factors of 3 P 0 and 1 S 0 have to be known. We performed a precise measurement of the ratio between these coefficients by locking the clock on σ ± and π transitions, as shown on figure 5:

g( 3 P 0 ) g( 1 S 0 ) = 1.58794(7). ( 30
)
Given the Landé factor of the fundamental state g( 1 S 0 ) = 184.4(1) [START_REF] Boyd | Nuclear spin effects in optical lattice clocks[END_REF][START_REF] Olschewski | Messung der magnetischen Kerndipolmomente an freien 43 Ca-, 87 Sr-, 135 Ba-, 137 Ba-, 171 Yb-und 173 Yb-Atomen mit optischem Pumpen[END_REF], this gives: g( 3 P 0 ) = 2.928(2)MHz/T, and ∆g = 1.0842(7)MHz/T. (31) in agreement with previously reported values [START_REF] Boyd | Nuclear spin effects in optical lattice clocks[END_REF]. The measurement of the tensor shift was conducted with a purely linear polarization for the trapping light, and therefore with a quantization axis along the bias field B s . The tensor component is extracted by measuring the average clock frequency shift for various values of β. The results are reported in table I

: ∆ν = ν( 3 P 0 , m) -ν( 1 S 0 , m) + ν( 3 P 0 , -m) -ν( 1 S 0 , -m) 2 = (∆κ s + β∆κ t )U 0 . (32)
where: I: Geometrical configurations chosen to measure the tensor shift. A strictly linear polarization is selected by the enhancement cavity of the optical lattice by using an intra-cavity birefringent element. For this polarization, the β coefficient reduces to β = (3 sin 2 θ cos 2 φ-1)(3m 2 -F (F +1)) = (3 cos 2 -1)α(3m 2 -F (F +1)) where α is the angle between the polarization and the bias field Bs. We switch between cos φ = 1 and cos φ = 0 by changing the polarization eigen-mode of the lattice cavity. The angles are derived from the relative values of the Zeeman shift when the magnetic field is changed.

∆κ s = - ∆α s α s = - α s ( 3 P 0 ) -α s ( 1 S 0 ) α s (33)
is the remaining differential scalar light shift due to a possible detuning of the trapping light from the magic wavelength. The resulting value for the tensor shift coefficient is then, as reported in [START_REF] Westergaard | Lattice-induced frequency shifts in Sr optical lattice clocks at the 10 -17 level[END_REF]:

∆κ t exp = ∂∆ν ∂β = (-57.7 ± 2.3) µHz/E r , (34)
in agreement with the theoretical value shown in equation (23) well within the experimental error bar. We now consider the behaviour of the tensor light shift when the light polarization ε is not perfectly linear. In this case, the quantization axis e Z for the excited state 3 P 0 is aligned with the effective magnetic field b eff and thus varies with the trap depth U 0 . Consequently, the β coefficient depends on U 0 through:

| ε• e Z | 2 = ε • B s || b eff || 2 | ε• e Bs | 2 1 -2 κ v ξ cos θ U 0 gB s + o (U 0 ) . ( 35
) where e Bs is the unit vector along B s . Because of this coupling effect between the vector and tensor light shifts, the average clock frequency contains a frequency shift ∆ν vt quadratic with the trapping potential reading:

∆ν vt = -γ vt | ε • e Bs | 2 ξ cos θ U 2 0 , (36)
with

γ vt = 3m 2 -F (F + 1) 6κ t ( 3 P 0 )κ v ( 3 P 0 ) g( 3 P 0 )B s , (37)
assuming that 1 S 0 exhibits no vector shift. For 87 Sr we have γ vt = 11 µHz/E 2 R for B s = 0.1 mT. This quadratic frequency shift on the average clock frequency is three orders of magnitude smaller than the quadratic term of the Zeeman shift written in equation (28). But this effect is quite significant when compared to the hyperpolarizability coefficient γ = 0.4 µHz/E 2 R [START_REF] Westergaard | Lattice-induced frequency shifts in Sr optical lattice clocks at the 10 -17 level[END_REF][START_REF] Targat | Comparison of Sr optical lattice clocks at the 10 -16 level[END_REF]. However, it is dramatically reduced when the polarization is linear, B s ⊥ k and B s ⊥ ε. Its experimental demonstration is for now challenging, as it is predominant in a configuration where the vector light shift is large and blurs the atomic resonances.

Finally, repeated measurements of the total light shift over two years with our two strontium clocks [START_REF] Targat | Experimental realization of an optical second with strontium lattice clocks[END_REF] confirmed the experimental value for the magic wavelength published in [START_REF] Westergaard | Lattice-induced frequency shifts in Sr optical lattice clocks at the 10 -17 level[END_REF], for which the differential scalar light shift is cancelled (i.e. ∆κ s = 0):

ν magic = 368 554 725 ± 3 MHz. ( 38
)
The uncertainty on this value is limited by our knowledge of β used to subtract the tensor light shift from the total light shift from equation (32). In addition, we measured the sensitivity of the scalar light shift with the trapping light frequency ν latt :

∂∆κ s ∂ν latt = -15.5 ± 1.1 µHz/E R /MHz. ( 39
)
This value is in agreement with the estimation given in [START_REF] Middelmann | High accuracy correction of blackbody radiation shift in an optical lattice clock[END_REF] from Monte-Carlo simulation using transition strengths and scalar polarizabilities of the clock states at various frequencies.

Conclusion

In this paper, we have conducted a theoretical estimate of the vector and tensor polarizabilities of the 87 Sr clock states in agreement with experimental measurements. We have shown that the differential vector polarizability is largely due to state mixing of the excited clock state 3 P 0 with 3 P 1 . The tensor polarizability is mainly due to state mixing of 3 P 0 with 3 P 2 , with a small contribution from the hyperfine structure of 1 P 1 on 1 S 0 .

FIG. 3 :

 3 FIG.3: Left: parametrization of the light polarization ε and the static bias magnetic field Bs. ex (resp. ey) is a unit vector along the major (resp. minor) axis of the trapping light polarization ε, such that the polarization ellipsis (red) is included in the ( ex, ey) plane. Right: projection in the plane defined by the trapping light wave vector k and Bs. The fictitious magnetic field representing the vector light shift adds to Bs to form an effective magnetic field b eff . If the vector component of the Hamiltonian dominates its tensor component, the Hamiltonian is diagonal for a quantization axis eZ defined along b eff . When the polarization ε is elliptical, the quantization axis therefore depends on the trap depth U0, introducing non-linear effects in the Zeeman shift and average clock frequency as a function of U0.

FIG. 4 :FIG. 5 :

 45 FIG.4: Differential Zeeman shift ∆Zs of the 87 Sr clock resonance as a function of the trap depth U0 for an elliptical polarization |ξ| 0.9 and |m = 9/2|. The experimental data points are fit with equation (27), assuming κ v ( 1 S0) = 0. These four data points are sufficient to determine the three free parameters of this equation that describe the geometry of the experimental setup, here Bs = 355.0 ± 0.2 µT, ξκ v ( 3 P0) = 212 ± 2 mHz/ER, and θ = 63.9 ± 0.5 mrad.

  1 S 0 and 3 P 0 on m

	5s5p ¹P₁ 5s5p ¹P₁ ...	5s6s ³S₁ 5s10s ³S₁ ...	5p² ³P₀	5s4d ³D₂ 5s4d ³D₂ 5p² ³P₂ ... 5s9d ³D₂ ... 5s9d ³D₁ 5p² ³P₁ 5s9d ³D₂ ...
				5s4d ³D₁
	MOT λ = 461 nm	5s5p ³P₂ 5s5p ³P₁ 5s5p ³P₀ 2nd stage Clock λ = 689 nm λ = 698 nm cooling
	5s² ¹S₀
	FIG. 1: Energy levels for Sr. The two clocks states are rep-
	resented in red. The F = 9/2 hyperfine component of the
	three green states are involved in the vector and tensor po-
	larizabilities of the excited clock state 5s5p 3 P0, as explained
	in section II. The other states are considered to calculate the
	polarizability coefficients of the clock transition.

We also described a non-linear behaviour of these light shifts that are largely cancelled in normal clock operation, but that have to be considered as the accuracy of optical lattice clocks continues to improve.

SYRTE is UMR CNRS 8630 between Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, and Observatoire de Paris. The Laboratoire National de Métrologie et d'Essais is the French National Metrology Institute. This work is supported by CNES, IFRAF and Nano-K-Conseil Régional Île-de-France and ESA under the SOC project.