C Sarraf

H Nouri

F Ravelet

F Bakir

Experimental study of blade thickness effects on the overall and local performances of a Controlled Vortex Designed axial-flow fan

Keywords: Axial-flow fan, turbomachinery, blades thickness, overall performance, pressure fluctuations, phase-averaged anemometry

The purpose of this work is to study the effects of blade thickness on the performances of an axial-flow fan. Two fans that differ only in the thickness of their blades were studied. The first fan was designed to be part of the cooling system of an automotive vehicle power unit and has very thin blades. The second fan has much thicker blades compatible with the rotomoulding conception process. The overall performances of the fans were measured in a test bench designed according to the ISO-5801 standard. The curve of aerodynamics characteristics (pressure head versus flow-rate) is slightly steeper for the fan with thick blades, and the nominal point is shifted towards lower flow-rates. The efficiency of the thick blades fan is lower than the efficiency of the fan with thin blades but remains high on a wider flow-rate range. The mean velocity fields downstream of the rotors are very similar at nominal points with less centrifugation for the thick blades fan. Moreover, the thick blades fan maintains an axial exit-flow on a wider range of flow-rates. The main differences concern local properties of the flow: Phase-averaged velocities and wall pressure fluctuations strongly differ at the nominal flow-rates. The total level of fluctuations is lower for the thick blades fan that for the thin blades fan and the spectral decomposition of the wall fluctuations and velocity signals reveal more harmonics for the thick blades fan, with less correlation between the different signals. For this kind of turbomachinery, the use of thick blades could lead to a good compromise between aerodynamic and acoustic performances, on a wider operating range.

Introduction

The low-speed axial-flow fans, used for instance in the cooling system of automotive vehicles power units, have very complex three-dimensional geometries that can affect both their overall and acoustical performances. An example of one parameter that dramatically modifies the performances is for instance the blade sweep [START_REF] Fukano | Noise generated by low pressure axial flow fans, iii: Effects of rotational frequency, blade thickness and outer blade profile[END_REF][START_REF] Kergourlay | Experimental investigation of the 3d unsteady flow field downstream of axial fans[END_REF][START_REF] Vad | Aerodynamic effects of forward blade skew in axial flow rotors of controlled vortex design[END_REF][START_REF] Hurault | Experimental and numerical analysis of the sweep effect on three-dimensional flow downstream of axial flow fans[END_REF]. A second parameter that should affect the performances is the blade thickness. Low-speed axial-flow fans usually have conventional thin blades. However, following motivations such as active control of the blade shape or the use of the rotomoulding process [START_REF] Rao | Principles of rotational molding[END_REF][START_REF] Tcharkhtchi | Structure-processibility relationships during rotational moulding of plastics[END_REF], the designers would have now to face profiles with thicker blades. The control of blade shape may help to reduce the radiated noise or to extend the operating range and the need for actuators then implies thicker blades. The rotomoulding processpreviously tested for wind turbine blades-leads to thick hollow blades and may be more economic because of material gain. The blades can have more stiffness with greater inertia moment. Complex shapes with blade recovering are moreover easier to manufacture than with traditional plastic injection methods, and the blades can be filled for instance with noise-absorbing foam.

In aeronautics and in the automotive industry, changing the blade thickness has been used for many years as an efficient way to modify the lift and drag characteristics and the boundary layers detachment process [START_REF] Thwaites | Incompressible Aerodynamics, An Account of the Theory and Observation of the Steady Flow of incompressible Fluid past Aerofoils, Wings, and Other Bobies[END_REF][START_REF] Hoerner | Fluid-Dynamic Lift[END_REF]. Applied to low-speed axial-flow fans, one could a priori think that the extra thickness may increase the dissipation of energy in the von Kármán street behind the blades so that the performances of the fan decrease and the pressure fluctuations in its wake increase. The effects of blade thickness on the overall performances are not so well known [START_REF] Roelke | The effect of rotor blade thickness and surface finish on the performance of a small axial flow turbine[END_REF][START_REF] Estevadeordal | Flow field in a low-speed axial fan: a dpiv investigation[END_REF][START_REF] Sarraf | Thickness effect of naca foils on hydrodynamic global parameters, bound-ary layer states and stall establishment[END_REF][START_REF] Lee | Design optimization of low-speed axial flow fan blade with three-dimensional RANS analysis[END_REF] and to the best of our knowledge the influence on the dynamics remains an open question.

Therefore this study intends to compare the aerodynamical performances of two axial-flow fans that only differ in the thickness of their blades. The geometry of the fans and the experimental set-up are presented in § 2. The results on the mean features are presented and discussed in § 3: the overall characteristics of the two fans are presented in § 3.1; the mean velocity fields downstream of the fans are compared in § 3.2 and these results are briefly discussed in § 3.3. Some results on local fluctuating quantities are presented and discussed in § 4: measurements of the wall pressure fluctuations that are an image of the acoustical performances are presented in § 4.1 and finally, phase-averaged velocity signals that are measured on one point are then compared in § 4.2. Concluding remarks are then given in § 5. The fans used in this study are two prototypes developed for automotive engine cooling system applications following the methods detailed in [START_REF] Noguera | Design and analysis of axial pumps[END_REF]. The fans are made out of aluminium and have the same geometrical characteristics except the blade thickness. The thin blades axialflow fan is the reference and is labeled F A . The thick blades fan is labeled F B . Pictures of the thick blades fan are displayed in Fig. 1 with a drawing of a section of the F A and F B blades at hub, mid-span and tip. The main characteristics of the blade cascade are given in Tab. 1.

Experimental setup

Geometry of the two fans

Each of these fans has six blades. The rotor is built up from blades of circular-arc camber lines, with NACA-65 profiles of mean chord c = 74 mm clipped at 0.95 x/c. The hub-to-tip radius ratio is R int /R max = 0.365 with tip radius R max = 179 mm. The rotors are moreover shrouded with a zigzag circular collar in order to close the radial gap in the free-flow experimental facility (see § 2.2 below). The spanwise load of the blades is prescribed following a controlled vortex design method [START_REF] Vad | Aerodynamic effects of forward blade skew in axial flow rotors of controlled vortex design[END_REF][START_REF] Noguera | Design and analysis of axial pumps[END_REF]: in the present case the absolute tangential velocity downstream of the rotor V θ increases linearly with the radius such that the Euler work per unit mass ωrV θ increases quadratically from the hub to the tip of the blades. The blade loading that is the ratio of the Euler work to the square of the peripheral velocity is moreover constant along the radial direction. With this design the hub losses are reduced and the blades are less twisted [START_REF] Vad | Aerodynamic effects of forward blade skew in axial flow rotors of controlled vortex design[END_REF]. The blades are stacked on the trailing edge. The blades are finally swept forward in order to benefit from lower tonal noise as suggested by several studies [START_REF] Fukano | Noise generated by low pressure axial flow fans, iii: Effects of rotational frequency, blade thickness and outer blade profile[END_REF][START_REF] Kergourlay | Experimental investigation of the 3d unsteady flow field downstream of axial fans[END_REF][START_REF] Vad | Aerodynamic effects of forward blade skew in axial flow rotors of controlled vortex design[END_REF][START_REF] Hurault | Experimental and numerical analysis of the sweep effect on three-dimensional flow downstream of axial flow fans[END_REF]. The forward sweep may also increase or decrease the overall performance, depending on the blade loading distribution, as reported in Refs. [START_REF] Vad | Aerodynamic effects of forward blade skew in axial flow rotors of controlled vortex design[END_REF][START_REF] Hurault | Experimental and numerical analysis of the sweep effect on three-dimensional flow downstream of axial flow fans[END_REF].

The maximum blade thickness of the thin blade axialflow fan F A is 4 mm, i.e. 5.6% of the chord length at mid-span (conventional axial-flow fans usually have a relative maximum thickness varying from roughly 10% at hub to 7% at tip [START_REF] Noguera | Design and analysis of axial pumps[END_REF]). The blades of the second fan F B are thickened to reach a maximum thickness of 10 mm, i.e. 13.5% of the chord length at mid-span. The thickness distribution e of fan F B is: e B = 2 e A + 2mm. The Fan F A was designed to meet the specifications point ∆p des = 270 Pa, Q N , des = 0.736 m 3 .s -1 for a nominal angular velocity ω N = 260 rad.s -1 . The corresponding specific speed

Ω = ω √ Q (∆p/ρ) 3/4 is Ω = 3.89.

Normalized test benches and measurement techniques

Free-flow experimental facility

The Figure 2a Pressure rise ∆p is measured with an absolute precision of ±0.1 Pa. The torque C on the fan shaft is measured with a HBM strain gauge transducer. The uncertainty of torque measurements corresponds to 0.1% of the maximum measured torque. The angular velocity ω is measured with a tachymeter of relative precision ±0.2%. The power adsorbed on the fan shaft is then estimated through the product Cω. The accuracy of the whole system leads to a determination of the efficiency within about ±0.5%.

The mean velocity field in the wake of the fans has been measured with Two-Component Laser Doppler Anemometry (LDA), using a DANTEC "FlowExplorer" system. The method to get the three components of the mean velocity field downstream of the fan relies on the axisymmetry of the mean flow, that is a posteriori checked. The definition of the laboratory Cartesian coordinate system {X ; Y ; Z} and of the two measuring planes is pictured in Fig. 3 Y axis and on the Z axis. The measurement is first performed in the {Y ; Z} plane at a certain radial distance r from the axis of rotation, and at an (axial) distance z from the trailing edges of the fan in the Y direction (point 1 in Fig. 3). The measured velocities, in polar coordinates, are thus the radial velocity V r (r, z) and the axial velocity V z (r, z). The measuring volume is then precisely traversed to the same radial and axial distances r and z, but in the {X ; Y } plane (point 2 in Fig. 3). Special care has been given to the alignment of the system. The velocities that are measured at this second point, in polar coordinates, are thus the opposite of the tangential velocity -V θ (r, z) and the axial velocity V z (r, z) that is measured two times.

We then check that the two measurements give the same mean axial velocities: the difference between V z (r, z) in the two planes is less than 1% everywhere. Constant Temperature Anemometry (CTA) with a 2D Dantec 55R51 hot fiber film probe has also been used to study more specifically the velocity fluctuations and the phase-averaged velocity signals. The method to get the three velocity components is the same as described in Refs. [START_REF] Kergourlay | Experimental investigation of the 3d unsteady flow field downstream of axial fans[END_REF][START_REF] Hurault | Experimental and numerical analysis of the sweep effect on three-dimensional flow downstream of axial flow fans[END_REF] and is based on two measurements at different angular positions of the probe.

Ducted-flow experimental facility

In order to compare the behavior of the two fans in a ducted-flow configuration and to measure the wall pressure fluctuations, a second test bench was built according to the ISO-5801 standard (see Fig. 2b).

It consists of a cylindrical pipe of inner diameter D = 380 mm. A bell mouth is flush-mounted at the inlet of the duct. The upstream face of the fan is at a distance D from the pipe inlet. A DC-motor is hidden in a casing of diameter 0.3D and length 0.7D, with a warheadshape end. The binding to the tube is ensured by five rods of diameter 8 mm, i.e. 0.02D, in order to minimize their influence, regardless the flow-rate. The distance between the upstream face of the fan and the binding rods is 0.26D. An anti-gyration device made of eight metal sheets of thickness 1.5 mm and length 2D is placed 2D downstream of the rotor-stator set. It prevents the outgoing flow from having any rotating component. The static pressure of the axial fan is measured 1D downstream of the anti-gyration device, with an average over four flushmounted pressure taps. The flow-rate is measured with a normalized diaphragm, located 10D downstream of the anti-gyration device and 5D upstream of the pipe outlet. The diaphragm has a diameter of 0.73D. Other diaphragms of various sizes are placed at the exit of the pipe to vary the test-bench hydraulic impedance and thereby to vary the operating point of the studied axial-flow fan.

Wall pressure fluctuations are measured simultaneously by eight microphones that are evenly distributed on the circumference of the duct. They are mounted downstream of the fan, halfway between the fan and the five rods that ensure the binding to the tube (see Fig. 2b). The microphones are G.R.A.S 40BP 1/4" polarized pressure microphones of sensitivity 1.65 mv.Pa -1 , with G.R.A.S 26AC preamplifiers and a G.R.A.S 12AG power supply module. The signals are amplified with a gain of +30 dB and highpass filtered with a three-pole Butterworth filter with cutoff frequency of 20 Hz. The signals are then digitalized using a NI Data Acquisition Card (M 6211, 16 bits) at a sample rate of 12 kHz.

Comparison of the mean features of the two fans

Overall characteristics

The characteristics of the two fans are shown in Fig. 4a together with the design operation point of the reference fan. The flow-rate coefficient (Φ) and pressure rise coefficient (Ψ) are defined respectively by equations 1 and 2:

Φ = Q π ω R 3 max (1) Ψ = ∆p 1 2 ρ ω 2 R 2 max ( 2
)
The design point thus corresponds to Φ des = 0.156 and Ψ des = 0.200 and is indicated by a dot in Fig. 4a.

The curves are the results of five measurements at the nominal rotation rate ω N = 260 rad.s -1 . The error bars in Fig. 4a stand for the standard deviation of the five measurements of Φ and Ψ, magnified by a factor three for better visualization. The pressure coefficient distributions are relatively similar to each other. The thin blades fan F A matches the specifications point plotted in Fig. 4. At the design flow-rate, the thick blades fan F B produces a pressure rise that is 8% below the specified pressure rise. The curve of F B is moreover steeper than the curve of F A and there is a cross-over between the two fans: at partial flow-rates below Φ = 0.11, F B produces more pressure rise than F A ; at overflow-rates, approximately beyond Φ = 0.2, F A produces this time more pressure rise than F B and the difference of pressure rise between the two fans remains almost constant.

(b) (a)
The efficiency coefficient (η) is defined by equation 3:

V θ η = ∆p Q C ω (3)
The efficiencies of both fans are shown in Fig. 4b with the flow-rate coefficient of the design point highlighted by the vertical dash-dotted line. The efficiencies distributions have a classical shape, with a slow evolution for partial flow-rates and a strong drop after the nominal flow-rate. The nominal flow-rate for F A is Φ N , A = 0.18 and the corresponding pressure coefficient is Ψ N , A = 0.17. The efficiency is then η max , A = 0.58 ± 0.005 which is pretty fair for this type of turbomachinery [START_REF] Noguera | Design and analysis of axial pumps[END_REF]. The maximum of η is shifted towards overflow with respect to the design point. This point is discussed below in § 3.3.

The maximum efficiency for the thick blades fan F B is η max , B = 0.55 ± 0.005. The thickening of the blades has led to a decrease of 0.03 of η max . The nominal flowrate of F B is shifted toward a partial flow-rate of about 17% with respect to the nominal flow-rate of F A and the thick blades fan thus gives its best efficiency for a flow-rate Φ N , B = 0.15 very close to the design point Φ des = 0.156. The corresponding pressure coefficient is Ψ N , B = 0.19. The distribution of the efficiency of F B is moreover flatter than that of F A : quantitatively, the range of flow-rates corresponding to a decrease of 3% of η from η max is ±0.3 Φ/Φ N for F B , while it is only ±0.2 Φ/Φ N for F A .

As a partial conclusion please also note that although the thick blades fan F B does not meet the specification point at the nominal angular velocity ω N = 260 rad.s -1 , the desired point in a dimensional point of view (∆p = 270 Pa, Q N = 0.736 m 3 .s -1 ) can be reached with this fan rotating at ω = 270 rad.s -1 . Since the efficiency of the two fans is the same at such flow-rates (η ≃ 0.55), the use of F B at ω = 270 rad.s -1 in the automotive cooling application would lead to the very same power consumption as the use of F A at ω N = 260 rad.s -1 .

Comparison of the mean exit-flow fields

The mean exit-flow downstream of the rotors may give further information that help to understand the differences that have been observed on the overall characteristics. To compare the two fans F A and F B , two sets of measurement of the flow field have been performed. Each set consists in choosing one dimensional operating point (fixed diaphragm at the inlet of the free-flow experimental facility, flow-rate Q and pressure rise ∆p) and in adjusting the rotation rates of the two fans to make them operating at this very same point.

The first set was made very close to the nominal flowrate of F B (Φ N , B = 0.15) and is presented in Fig. 5. The second set was made very close to the nominal flow-rate of F A (Φ N , A = 0.18) and is presented in Fig. 6. The flow fields have been measured on a Cartesian grid that extends typically from z = 20 mm to z = 150 mm with a spacing of 10 mm (14 points in the axial direction) and from r = 40 mm to r = 220 mm or from r = 65 mm to r = 255 mm with a spacing of 5 mm (38 points in the radial direction). The mean data rate is 2000 Hz and the signal has been acquired for 15 s, i.e. roughly 500 revolutions of the rotor, at each position. The mean value and the standard deviation are thus well converged.

Discharge flow components

The flow fields in meridional plane, i.e. the discharging component V z (r, z) and the centrifugal component V r (r, z), are displayed in Figs. 5a-b and6a-b.

One common feature of the two fans, regardless of the flow-rate, is the presence of a large recirculation, with small upstream velocities close to the hub: the discharged flow exits only for r > 100 mm. Measurements of the flow field close to the hub with CTA that do not give access to the velocity direction should be considered with caution. The main difference lies in the centrifugation of the flow by the two fans.

The meridional flow for F A at a partial flow-rate of Φ/Φ N , A = 83% is plotted in Fig. 5a. The flow downstream of the fan already has a very important radial component as noticed by Kergoulay et al. [START_REF] Kergourlay | Experimental investigation of the 3d unsteady flow field downstream of axial fans[END_REF]. The discharging flows for the two fans at their respective nominal flowrate (Fig. 6a for F A and Fig. 5b for F B ) are very similar with smaller centrifugation for F B . The meridional flow for F B at an overflow-rate of Φ/Φ N , B = 120% is plotted in Fig. 6b and presents barely no centrifugation. There thus seems to be a tendency for the thick blades fan F B to maintain a mainly axial exit-flow on a wider range of operating points.

Mean tangential velocity

Contours of the tangential velocity field V θ (r, z) are plotted in Figs. 5c-d and6c-d. As a common feature for both fans, one can notice that the tangential velocity downstream of the rotors has significant values, of the order of the discharge velocity Q/(πR 2 max ). At the design flow-rate, an extra static pressure rise of roughly 1/2ρV 2 θ = 30 Pa could be obtained by the use of a stator. That corresponds to 20% of the pressure rise and would lead to a significant increase of the efficiency. The tangential velocity fields are moreover strongly correlated to the meridional velocity field for both fans, regardless of the flow-rate. It is almost zero outside the discharge jet. The 3D structure of the exit-flow thus resembles an annular rotating jet. Close to the fans, there are two local maxima for nominal (Figs. 5d for F B and 6c for F A ) and overflow-rates (Fig. 6(d) for F B ). The radial positions of these maxima are adjacent to the edges of the discharge jets.

The differences between the two fans lie in the magnitude and the axial attenuation of the tangential velocities. The velocities are stronger for the thick blades fan F Bwhich is consistent with its overall lower efficiency-and the axial velocity decay is also slower. These two points are discussed below in § 3.3.

Standard deviation of the tangential velocity

Contours of the standard deviation (or the root mean square rms) of the tangential velocity field V θ rms are plotted in Figs. 5e-f and 6e-f. The maxima of fluctuations are located at the edge of the discharge jet where the radial gradient in tangential velocity is steep, and the core of the jet corresponds to a local minimum of fluctuations. Finally, one can notice that the global level of fluctuations in the exit-flow of F A is stronger than that in the exit-flow of F B .

Remarks and discussion on the modifications of overall performances and of exit-flow topology

The nominal flow-rates of the two fans are different and furthermore do not match the design point (see Fig. 4b). The latter could be an effect of the forward sweep as noticed in Refs. [START_REF] Vad | Aerodynamic effects of forward blade skew in axial flow rotors of controlled vortex design[END_REF][START_REF] Hurault | Experimental and numerical analysis of the sweep effect on three-dimensional flow downstream of axial flow fans[END_REF] : for comparable Controlled Vortex Design fans -of relative thickness 10% and pressure rise coefficient Ψ N ≃ 0.3 for Vad [START_REF] Vad | Aerodynamic effects of forward blade skew in axial flow rotors of controlled vortex design[END_REF] and of relative thickness 9% and pressure rise coefficient Ψ N ≃ 0.3 for Hurault et al. [START_REF] Hurault | Experimental and numerical analysis of the sweep effect on three-dimensional flow downstream of axial flow fans[END_REF]-both studies reported a decrease of the flow-rate and efficiency with respect to the design point when the fan blades are swept forward. However, the trend is not systematic and first depends on the blade loading that is directly linked to half of the pressure rise coefficient Ψ N . Indeed, for a more loaded fan with more load near the tip, Hurault et al. [START_REF] Hurault | Experimental and numerical investigation of downstream turbulence in axial flow fan[END_REF] reported an increase of the nominal flow-rate with forward sweep. The fact that the nominal flow-rates of F A is greater than the design pointcomputed with a method that had been validated for conventional thickness of roughly 7% [START_REF] Noguera | Design and analysis of axial pumps[END_REF]-while it is lower for F B in the present study suggests that the shift of nominal flow-rate with respect to the design point could not be ascribed to the blade sweep alone but that blade thickness also plays a major role because it somehow modifies the blade loading and the balance between lift and losses. Some results concerning the effects of thickness on the aerodynamic behavior of isolated profiles may help to explain the relative position, i.e. the cross-over between the two curves (see Fig. 4a). In general the thickening of isolated profiles leads to a reduction of the aerodynamic efficiency (lift to drag ratio) [START_REF] Thwaites | Incompressible Aerodynamics, An Account of the Theory and Observation of the Steady Flow of incompressible Fluid past Aerofoils, Wings, and Other Bobies[END_REF][START_REF] Hoerner | Fluid-Dynamic Lift[END_REF]. Even at the most efficient angle of attack, significant boundary layer separations are observed on thick profiles -of relative thickness above 10%-while not on thin profiles [START_REF] Sarraf | Thickness effect of naca foils on hydrodynamic global parameters, bound-ary layer states and stall establishment[END_REF] and the lift is reduced. The viscous dissipation is moreover greater for thick profiles, especially when they have rounded trailing edge as it is the case here [START_REF] Estevadeordal | Flow field in a low-speed axial fan: a dpiv investigation[END_REF]. Despite these drawbacks, the thick profiles have the advantage of having a more progressive stall than thin profiles, at higher angles of attack. The first two points are most likely responsible for the lower pressure rise and the lower efficiency of F B at nominal and overflow-rates where the angle of attack of the blade cascade is low: at high flow-rates the lift is increased for the thin blades fan F A that means more pressure rise and the viscous losses are reduced and this could explain the shift of the nominal point towards higher flow-rates. The last feature is almost probably at the origin of the gain of pressure rise and of efficiency at partial flow-rates as observed for F B in the present experiment: the losses that are dominated by the separation on the suction side are reduced and the lift is increased.

The effects of the thickness on the overall performances close to the nominal flow-rates could also be informed by the modifications of the exit-flow topology (see Figs. 5 and6). The tangential velocities downstream of the fan F B (Fig. 5d and Fig. 6d) are stronger than for the fan F A (Fig. 5c and Fig. 6c). Since the velocity fields for the two fans have been measured at the same flow-rate and pressure rise, F B rotating a little bit faster than F A for that, the Euler work for F B is thus greater than the Euler work for F A and its static efficiency is lower. However, if one would have to use a stator, the fan F B would lead to a greater recovery of static pressure which would increase the static efficiency of the stage. The regions of high velocity also extend further away axially and are moreover larger in radius for F B : a stator could be positioned in a wider range of axial distance from the fans, which should be beneficial regarding interaction noise. And at last, the fact that the thick blades fan F B maintains a mainly axial flow on a wider range of flow-rates (less centrifugation) would also increase the efficiency of a stator downstream of it.

Study of local fluctuations

The increase of blade thickness thus leads to strong differences in the mean-flow features of those two fans. The purpose of the present section is to study the influence of the blade thickness on local dynamics such as pressure and velocity fluctuations.

Wall pressure fluctuations

The results of this paragraph concern the two axial-fans operating in a ducted-flow configuration (see Fig. 2b). In that case there is a radial gap of 1.5 mm between the pipe and the collar surrounding the fan. It should be also noted that for technical reasons, the ducted experiments have been performed for an angular velocity ω = 209.5 rad.s -1 (2000 rpm). The presence of the gap substantially reduces the efficiency of the fans without modifying the nominal flow-rate: F A reaches a maximum efficiency η = 0.49 for a flow coefficient Φ = Φ N , A = 0.18. The pressure coefficient is Ψ = 0.15, slightly lower than in the free-flow experiment which presents no radial gap. The F B fan reaches a maximum efficiency η = 0.48 at a flow coefficient Φ = Φ N , B = 0.15 and a pressure coefficient Ψ = 0.18. The characteristic curves show similar behavior as in the free-flow case for nominal and overflow-rates. The behaviors at partial flow-rates are completely different owing to the presence of the tube that prevents the flow from being centrifuged. The results reported here are thus limited to flow-rates above Φ = 0.15.

The wall pressure fluctuations are compared for two different flow-rates, corresponding to the nominal point of the thin blades fan Φ N , A = 0.18 and to the nominal point of the thick blades fan Φ N , B = 0.15. In every case, for a fixed diaphragm at the pipe exit, the angular velocity of one of the two fans has been set to ω = 209.5 rad.s -1 and the angular velocity of the second fan has been adjusted in order to give the same dimensional pressure rise and flow-rate for the two fans.

The squared amplitude spectra of wall pressure at flowrate Φ N , A are plotted in Figure 7. The spectra S(f ) are expressed in dB using the formula:

S(f ) = 20 log 10 (p ′ (f )/p 0 )
p ′ (f ) being the modified Welch-average Fourier Transform of the fluctuating pressure. The sampling time is 10 s, corresponding to approximately 330 fan rotations, and the spectra are an average of eight 50% overlapping parts, windowed with a Hamming window. The spectra are displayed between 150 and 850 Hz. The reference pressure is p 0 = 1 Pa.

Both fans exhibit relatively high discrete peaks, corresponding to the blade passing frequency f bpf (200 Hz for F A and 202 Hz for F B ) and its harmonics. The corresponding values are reported in Tab. 2. They are high above the broadband noise. The two fans differ however strongly by the repartition of energy between the different harmonics. For F A , only the fundamental frequency f bpf and the first harmonic 2f bpf are visible. The level at f bpf is 21 dB, and the level at 2f bpf is -1.5 dB.

For F B fan, the first three harmonics clearly appear. At the blade passing frequency f bpf the level is 12 dB. At 2f bpf it is 2 dB, at 3f bpf it is -4 dB and at 4f bpf it is -11 dB. For this flow-rate, the increase of blades thickness thus seems to lead to a decrease of the first peak level, and to an increase of the harmonics. The global level of fluctuations, i.e. 10 log 10 (σ 2 p ′ /p 2 0 ), is moreover lower for the thick blades fan (F B : 19 dB) than for the thin blades fan (F A : 23 dB).

The previously described features regarding both the spectral composition and the relative fluctuations levels could be due to the fact that the F A fan is working at its nominal flow-rate while the F B fan is working at overflowrate. To answer this question we have performed the same experiment at the nominal flow-rate of the F B fan.

The squared amplitude spectra of wall pressure at the nominal flow-rate of the thick blades fan Φ tck = 0.15 are plotted in Figure 8 and the levels of the discrete peaks are reported in Tab. 3. The blade passing frequency is now f bpf = 200 Hz for F B and 193 Hz for F A . Please note that the angular velocity ratio needed to reach the same dimensional static pressure rise and flow-rate ω B /ω A = 1.037 is the same as the ratio discussed in § 3.1 dedicated to the fan overall characteristics.

The results are very similar to the previously described flow-rate: for the F A fan, the fundamental frequency f bpf and the first harmonic 2f bpf are still the only visible peaks in the spectrum. The level at f bpf is 21 dB, and the level at 2f bpf is -2 dB. For the F B fan, the first three harmonics clearly appear. At the blade passing frequency f bpf the level is 17 dB. At 2f bpf it is 5 dB, at 3f bpf it is -2.5 dB and at 4f bpf it is -9.5 dB. The decrease of the first peak level and the increase of the level of the harmonics is also present at this flow-rate. The global level of fluctuations is 20 dB for the thick blades fan and 22.5 dB for the thin blades fan.

f bpf 2f bpf 3f bpf 4f bpf Total F A 21 -1.5 - - 23 F B 12 2 -4 -11 19
Table 2: Level of wall-pressure fluctuations at the blade passing frequency and its harmonics (dB) for both fans at flow-rate Φ N , A . 2b). Red curves: cross-correlation function of pressure fluctuations between position 1 and position 5 (see Fig. 2b). Top graph corresponds to F A at Φ N , A = 0.18, bottom graph corresponds to F B at Φ N , B = 0.15. The time is made non-dimensional with reference to one period of rotation:

t ′ = t ω 2π
The black curves correspond to the autocorrelation function:

A(τ ) = p ′ 1 (t) p ′ 1 (t -τ ) σ 2 p ′ 1
and the red curves correspond to the cross-correlation function between microphones 1 and 5:

C 1-5 (τ ) = p ′ 1 (t) p ′ 5 (t -τ ) σ p ′ 1 σ p ′ 5
One can notice that the functions present a periodicity that corresponds to six events in one rotation that is the blade passing frequency. The time lag for the main peak in the cross-correlation function does match the angular distance between the microphones which is one half of a turn here. Another common feature between the two fans is that secondary peaks height for the autocorrelation function are close to the peaks height for the cross-correlation function. Some strongly coherent flow structures might be involved in this zone between the rotor and the binding static part. The strong difference between the F A fan and the F B fan is the value of the cross-correlation coefficient. It is significantly lower for the thick blades fan. Local velocity measurements in the wake of the fans may help to better understand this difference and the links between exit-flow dynamics and wall pressure fluctuations. to give the same flow-rate and pressure rise, that corresponds then to a flow coefficient Φ B = 0.170 Time-resolved signals of the tangential velocity downstream of the fans have been measured by Constant Temperature Anemometry (CTA) at an axial distance z = 24 mm downstream of the fan and at a radius r = 120 mm that corresponds to mid-span. The rotation rates of the fans have been adjusted to give the same operating point (fixed diaphragm at the inlet of the free-flow experimental facility, flow-rate Q and pressure rise ∆p). This point is close to the nominal flow-rate of F A (Φ N , A = 0.18) and the fans rotate at an angular velocity close to ω = 209 rad.s -1 (2000 rpm). An amount of 2 19 samples have been recorded at an acquisition rate of 120 kHz. That corresponds to roughly 150 revolutions of the fans and to approximately 3600 samples per revolution. The rotation rates have been measured with an encoder, synchronized to the CTA data acquisition, in order to perform analysis of phase-averaged signals [START_REF] Kergourlay | Experimental investigation of the 3d unsteady flow field downstream of axial fans[END_REF]. This point has been chosen as close as possible to the fans blades, and such that the three components of the velocity are close for both fans, as can be seen in Fig. 6. The phase-averaged signals of the tangential velocity V θ (φ) are plotted in Fig. 10, φ being the phase on one revolution, in degree. Portions of signal of length corresponding to one revolution have been averaged. The signals for F A (Fig. 10a) and for F B (Fig. 10b) are periodic with a periodicity of 60 o that corresponds to the blade passages. The shape of the signal for each blade is very similar owing to the very good symmetry and alignment of the system. The mean velocity is respectively 7.4 m.s -1 for F A and 7.7 m.s -1 for F B . For F A , the signal has a more regular shape than for the thick blades fan F B , and the peakto-peak amplitude is bigger. These peak-to-peak phaseaveraged fluctuations give the biggest contribution to the total standard deviation of the velocity. They should also give the biggest contribution to the peaks at blade passing frequency and its harmonics in velocity spectra as reported by Kergoulay et al. [START_REF] Kergourlay | Experimental investigation of the 3d unsteady flow field downstream of axial fans[END_REF] and as can be seen from the phaseaveraged rms of the tangential velocity V θ rms (φ) plotted in red in Fig. 10.

Local velocity measurements in the wake of the fans

The local tangential velocity spectra are plotted in Fig. 11. The peaks at the blade passing frequency and its harmonics are clearly visible and are reported in Tab. 4. The same features as for the wall pressure fluctuations studied in § 4.1 are present: F A has a stronger peak at f bpf but smaller and fewer secondary peaks. This is consistent with the shape of the phase-averaged velocity signals (Fig. 10) and the fact that the main contribution to the peaks arises from this phase-averaged component of the velocity signal.

f bpf 2f bpf 3f bpf 4f bpf 5f bpf F A -1.

Conclusions

Two fans that differ only in the thickness of their blades were studied in order to highlight the effects of thickness on overall performances and the pressure and velocity fluctuations that are generated. The first fan, which has very thin blades, was designed to be part of the cooling system of an automotive vehicle power unit. The second compatible with the rotomoulding conception process, that generates only hollow parts with large edge radius, has much thicker blades.

The performances of the fans were measured and the results indicate that the overall performances were substantially equivalent, with a drop of 8% of pressure rise at conception flow-rate for the thick blades fan and a maximum efficiency that is 3% lower than the efficiency of the thin blades fan and that is shifted towards lower flow-rates.

Contrary to what one could a priori think the thick blades fan does present a certain number of interesting features:

1. The overall performances are only slightly decreased.

The conception point can be reached by a relative increase of rotational speed of 4% for the same power consumption. The fact that the characteristic curve is steeper could be positive: the flow-rate is more stable with respect to small variations of the circuit hydraulic impedance. Finally, at very partial flowrates, the thick blades fan produces more pressure rise than the thin blades fan, which could be beneficial concerning stall instabilities. 2. The mean exit-flow fields present significant diffferences. The tangential velocity component is stronger and remains high further away downstream of the thick blades fan F B , which may increase the efficiency of a potential stator. The exit-flow moreover remains axial on a wider operating range. The thin blades fan F A leads to a more centrifuged discharging flow, which may be a configuration of interest in some applications, for instance for automotive cooling with the fan sucking air through the heat exchanger.

3. The wall pressure fluctuations just downstream of the fan are lower for the thick blades and their spectral decomposition is completely different. The same features are observed on velocity fluctuations. The spectral decomposition that shows stronger peaks at the blade passing frequency and less harmonics for the thin blades fan than for the thick blades fan, together with the phase-averaged analysis of velocity signals suggest that the pressure fluctuations are due to strong coherent wakes that are more dissipated with the use of thick blades. These measurements will be used as a benchmark for high-order finite volume CFD codes developed for aeroacoustics in the laboratory [START_REF] Ravelet | Study of the aerodynamics/aeroacoustics of an axial-flow fan: Experimental validation of a les/lpce/brinkman penalization method[END_REF], and data are also available on request.

Further numerical and experimental investigations are planed to better understand the origins of the observed features that are reported in the present Article. The open questions that remain concern the influence of the blade thickness and of its distribution from leading edge to trailing edge on the blade loading and how this affects the efficiency and shift of the nominal flow-rate, the centrifugation of the exit-flow and the modifications of the pressure and velocity spectra.

Figure 1 :

 1 Figure 1: Views of the two fans. Front view, sections of thick and thin profiles at various span locations and thickness distribution law along the meridional coordinate.

  shows the experimental facility used to determine the overall performances of axial-flow fans in open-flow configuration. The air suction test bench was designed and built at the Dynfluid laboratory of Arts et Métiers ParisTech according to the ISO-5801 standard. It consists of a cuboid of size 1.3×1.3×1.8 m. A pierced plate inside it prevents the flow from having prerotation. The fans are placed at one end, the zigzag circular collar being tightly fitted to the bench. The air flow-rate is set and measured according to the ISO-5167 standard by setting the hydraulic impedance of the bench through diaphragms of various sizes that are placed at the other end.

Figure 2 :

 2 Figure 2: ISO-5801 test benches. (a): open-flow facility. The dimensions are 1.3 × 1.3 × 1.8 m. The fans suck the flow through the test bench. (b): ducted-flow facility. This configuration is dedicated to the local measurements of wall pressure fluctuations. The fans blow the flow into the pipe.

Figure 3 :

 3 Figure 3: Measurement of the mean exit-flow field with a Twocomponent LDA system. The measurements are taken in the two planes that are displayed with white lines. The origin of the coordinate system is centered on the rotation axis, and the axial position of the origin corresponds to the plane that contains the fans blades trailing edges. The LDA system measures the velocity in the Y and Z directions. At point 1 this corresponds to the axial Vz and radial velocities Vr. At point 2 this corresponds to the axial Vz and to the opposite of the tangential velocities -V θ

Figure 4 :

 4 Figure 4: Fans characteristics: (a) Pressure rise coefficient Ψ vs. flow-rate coefficient Φ and (b) Fans efficiency η vs. flow-rate coefficient Φ. (⋄): thin blades fan F A , ( ): thick blades fan F B . (•): design specification point. The reproducibility is evaluated from 5 independent experiments and displayed are errorbars magnified by a factor 3 for (a) and a factor 6 for (b)

Figure 5 :Figure 6 :

 56 Figure 5: Mean exit-flow measured by LDA close to the nominal point of F B . (a-c-e): F A rotating at ω = 209.5 rad.s -1 for a flow-rate Q = 0.557 m 3 .s -1 and a pressure rise ∆p = 164 Pa corresponding to a flow coefficient Φ A = 0.148. The discharge velocity Q/(πR 2 max ) is 5.6 m.s -1 . (b-d-f): F B rotating at ω = 214.2 rad.s -1 to give the same flow-rate and pressure rise, that corresponds then to a flow coefficient Φ B = 0.144. (a-b): Mean meridional flow {Vr(r, z) ; Vz(r, z)}. (c-d): mean tangential velocity V θ . (e-f): Standard deviation of the tangential velocity V θ rms . The blue patch in each figure stands for the position of the blades

Figure 7 :

 7 Figure 7: Square amplitude spectra of wall pressure fluctuations for thin blades axial-flow fan F A (a) and thick blades axial-flow fan F B (b) at a flow-rate Φ = Φ N , A = 0.18. F A rotates at ω = 209.5 rad.s -1 and F B rotates at ω = 211.5 rad.s -1 in order to give the same flowrates and pressure rises.

Figure 8 :

 8 Figure 8: Square amplitude spectra of wall pressure fluctuations for thin blades axial-flow fan F A (a) and thick blades axial-flow fan F B (b) at a flow-rate Φ = Φ N , B = 0.15. F B rotates at ω = 209.5 rad.s -1 and F A rotates at ω = 202.7 rad.s -1 .

Figure 9

 9 Figure 9 presents autocorrelation and cross-correlation functions of wall pressure fluctuations for the two fans.

fTable 3 :

 3 bpf 2f bpf 3f bpf 4f bpf Total F Level of wall-pressure fluctuations at the blade passing frequency and its harmonics (dB) for both fans at flow-rate Φ N , B .

Figure 9 :

 9 Figure 9: Black curves: autocorrelation function of pressure fluctuations at position 1 (see Fig.2b). Red curves: cross-correlation function of pressure fluctuations between position 1 and position 5 (see Fig.2b). Top graph corresponds to F A at Φ N , A = 0.18, bottom graph corresponds to F B at Φ N , B = 0.15. The time is made non-dimensional with reference to one period of rotation: t ′ = t ω

Figure 10 :

 10 Figure 10: Phase-averaged tangential velocity V θ (φ) (blue) and phase-averaged rms of the tangential velocity V θ rms (φ) (red) at an axial distance z = 24 mm downstream of the fan and at midspan r = 120 mm. The operating point is close to the nominal point of F A . (a): F A rotating at ω = 205.5 rad.s -1 for a flow-rate Q = 0.620 m 3 .s -1 and a pressure rise ∆p = 125 Pa corresponding to a flow coefficient Φ A = 0.178. (b): F B rotating at ω = 209 rad.s -1 to give the same flow-rate and pressure rise, that corresponds then to a flow coefficient Φ B = 0.170

Figure 11 :

 11 Figure 11: Square amplitude spectra of V θ (t) under the conditions of Fig. 10. (a): F A fan. (b): F B fan

4 Table 4 :

 44 4 -9.7 -17.1 -26.5 -F B -4.8 -7.8 -12.3 -19.7 -25.Level of tangential velocity fluctuations at the blade passing frequency and its harmonics (dB) for both fans close to flow-rate Φ N , A .

Table 1 :

 1 . The LDA system measures the velocities on the Blade cascade parameters for the reference fan F A . Radius R (mm). Chord length c (mm). Cascade solidity σ.

		R	c	σ	γ	profile	D
	Hub	65.4 66.6 0.97 53 NACA 65(07)06 0.372
	Mid-span 122.2 74.0 0.58 66 NACA 65(10)05
	Tip	179.0 81.3 0.43 70 NACA 65(11)05 0.650
							Stagger angle γ
	( o ). Profile designation according to the nomenclature given in Ref. [13]: NACA65(xx)yy with (xx) representing the relative camber and yy
	standing for the relative thickness. Lieblein's diffusion factor D.


Henri Debrat
email: henri.debrat@loria.fr

Bernadette Charron-Bost

Stephan Merz
email: stephan.merz@loria.fr

Formal Verification of Consensus Algorithms in a Proof Assistant

I. INTRODUCTION

Distributed algorithms are often quite subtle, both in the way they operate and in the assumptions under which they work correctly. Indeed, several algorithms have been found to be erroneous, and numerous misunderstandings have arisen due to different interpretations of the precise objectives of the algorithms and of the underlying hypotheses. Formal verification is therefore crucial in distributed computing.

In a distributed system, individual processing nodes communicate among each other by exchanging messages along point-to-point channels or links. Standard interleaving models represent executions of distributed algorithms as linear sequences of events local to the nodes: local computation, message sending and reception. It is well known that this fine-grained semantics leads to state explosion in model checking approaches to verification, making it impractical to verify more than small finite instances. It also complicates deductive verification, because all possible intermediate system states must be accounted for in the invariant that is at the base of correctness proofs, yielding large and hard to understand invariants.

We are therefore interested in models that allow us to verify algorithms over a coarser-grained representation, taking advantage of the causal order [START_REF] Lamport | Time, clocks, and the ordering of events in a distributed system[END_REF] between events of a distributed system instead of explicitly considering all possible linearizations. An important insight comes from the work of Elrad and Francez [START_REF] Elrad | Decomposition of distributed programs into communication-closed layers[END_REF] who proposed to structure distributed algorithms into rounds that constitute communication-closed layers: messages are delivered only in the round in which they were sent. Indeed, many distributed algorithms are based on this principle, which helps algorithm designers to understand the possible global configurations a system can be in.

Consensus is regarded as the fundamental problem that must be solved to implement a fault-tolerant distributed system. It requires nodes to eventually agree on a common value among the initial values held by each of them. Agreement should be reached despite the failures of some components (process or link). Depending on the timing and failure models, numerous Consensus algorithms have been proposed in the literature. We are studying techniques for proving the correctness of these algorithms using the interactive proof assistant Isabelle [START_REF] Nipkow | A Proof Assistant for Higher-Order Logic[END_REF], aiming for a high level of automation.

II. BENIGN ERRORS: THE HO MODEL

Charron-Bost and Schiper [START_REF] Charron-Bost | The Heard-Of model: computing in distributed systems with benign faults[END_REF] proposed the Heard-Of (HO) model of distributed algorithms that tolerate failures. In marked contrast to standard models, the HO model does not record the failures that occur during an execution, but focuses on what "goes well". The authors show that in this way the model can uniformly represent all benign errors, which include process crashes and message losses.

More precisely, an algorithm in the HO model is represented as an infinite sequence of communication-closed rounds. In each round, every node sends messages to all other nodes, receives messages from other nodes sent in the current round, and processes them in a local computation step. The set of nodes from which node n receives messages in round r is called the heard-of set HO (n, r). Assumptions about the failures tolerated by the algorithm are captured by a communication predicate, which is expressed in terms of the sets HO(n, r) that occur during an execution, and correctness is formally asserted relative to specific communication predicates.

The HO model thus combines the communication-closed round structure, which underlies most (if not all) known Consensus algorithms, with a uniform and elegant way of representing failure models in terms of correctness hypotheses and properties. Important properties of algorithms, including Consensus, can be verified in terms of a coarse execution model that models entire rounds as being performed atomically [START_REF] Chaouch-Saad | A reduction theorem for the verification of round-based distributed algorithms[END_REF]. This allows us to construct finite-state representations of executions for finite algorithm instances, despite the presence of unbounded round numbers, and to validate algorithms using model checking.

We have represented [START_REF] Charron-Bost | Formal verification of a Consensus algorithm in the Heard-Of model[END_REF] the HO model as a locale in the interactive proof assistant Isabelle/HOL, and have since instantiated this model for several Consensus algorithms that appear in [START_REF] Charron-Bost | The Heard-Of model: computing in distributed systems with benign faults[END_REF]. Making use of the coarse-grained representation of HO algorithms, and of generic lemmas proved once and for all in the Isabelle locale, the verification of a new algorithm now takes about one person-week, and our proofs are shorter by roughly an order of magnitude compared to proofs of similar algorithms in an interleaving model [START_REF] Jaskelioff | Proving the correctness of Disk Paxos[END_REF]. Most importantly, we find the proofs much more readable, and indeed comparable to a careful pencil-and-paper proof.

III. TOWARDS MALICIOUS ERRORS: THE SHO MODEL

The HO model does not account for so-called value faults (sometimes called malicious or Byzantine faults) that result from values being corrupted during local computation or message transmission. In this case, a node n receiving a message m from some node n ′ has no guarantee that m corresponds to what n ′ should have sent according to the algorithm, be it that n ′ is deliberately cheating, that a fault occurred during its computation, or that the message was corrupted during transmission.

Biely et al. [START_REF] Biely | Tolerating corrupted communication[END_REF] proposed an extension of the HO model that takes into account these more general kinds of faults. The basic idea is again to not identify a "culprit" responsible for the fault, and to limit failures to transmission faults. Indeed, nodes are supposed to perform state transitions correctly, i.e., following their program code. In order to express the discrepancies between the values that should have been sent and those that have been received, the model introduces sets SHO(n, r) of all nodes n ′ from which n receives the expected message for round r. Assumptions on the occurrence of benign and value faults are formally stated in terms of the sets HO (n, r) and SHO(n, r). Because a node n cannot determine if a value fault has occurred, it cannot refer to the sets SHO(n, r) in its code.

We are currently validating this model by encoding in it some standard Consensus algorithms that tolerate value faults. We will then extend the existing Isabelle locale to take into account the SHO model. In future work, we are mainly interested in two aspects: (1) a more detailed study of the properties that are preserved by the reduction to coarsegrained representations of runs, and (2) a higher degree of automation in verifying such algorithms, for example by representing elementary data structures such as HO sets in theories that are amenable to automatic reasoning using SMT solvers or saturation-based theorem provers.


T Vanderbruggen

S Bernon

A Bertoldi

A Landragin

P Bouyer

Spin-squeezing and Dicke state preparation by heterodyne measurement

Keywords: numbers: 42.50.Lc, 42.50.Dv, 42.50.St, 03.67.Bg, 03.65.Ta, 05.10.Gg B. Analytical

We investigate the quantum non-demolition (QND) measurement of an atomic population based on a heterodyne detection and show that the induced back-action allows to prepare both spinsqueezed and Dicke states. We use a wavevector formalism to describe the stochastic process of the measurement and the associated atomic evolution. Analytical formulas of the atomic distribution momenta are derived in the weak coupling regime both for short and long time behavior, and they are in good agreement with those obtained by a Monte-Carlo simulation. The experimental implementation of the proposed heterodyne detection scheme is discussed. The role played in the squeezing process by the spontaneous emission is considered.

I. INTRODUCTION

Generation of entangled collective spin states such as spin-squeezed states (SSS) [1,2] and Dicke states [3][4][5] is of first importance for quantum information [6] and quantum metrology [7][8][9][10]. A non-linear evolution of the system is required to create collective entanglement: common approaches exploit interactions between the particles [11,12], probing with squeezed light [13] or cavity mediated interaction [9]. The strong non-linearity of a measurement determined by the state renormalization can also lead to such states [14][15][16][17].

Measurement induced squeezing of atomic ensembles has been performed using either a Mach-Zehnder interferometer and simultaneous interaction with several optical frequencies [18], or cavity transmission modulation [19,20]. Nevertheless, the experimental realization of highly engineered atomic states is still challenging and a deep understanding of the underlying stochastic process of the measurement is needed. The dynamic of the atomic state collapse can be described as a succession of partial measurements. Several methods have been used to model this collapse process, like stochastic Schrödinger equation [4,21], master equation [22], or wavevector formalism [14].

In this article, we propose to use the frequency modulation (FM) spectroscopy technique [23][24][25][26] to generate both spin-squeezed and Dicke states. A laser beam is phase modulated to produce frequency sidebands; one sideband is placed close to an atomic transition and experiences a phase-shift passing through the atomic sample. The detection of the beatnote at the modulation frequency allows then to estimate the atomic population of the probed state.

A theorical analysis of the measurement process shows that a heterodyne detection performed on an atomic sample allows to prepare new quantum states. The study is based on the wavevector formalism previously introduced for homodyne detection [14] and shows similarities with the method used in [27,28] to reconstruct the photonic state from the results of a QND measurement sequence. We study the quantum trajectories of the atomic state caused by the repeated interaction with single-photons sent in the QND apparatus. In the weak coupling limit an analytical expression is derived for the evolution of the state variance during the measurement, it completely describes the atomic wavefunction collapse. The result is in good agreement with a Monte-Carlo simulation of the measurement process.

II. THE MEASUREMENT PROCESS

The principle of the QND measurement is presented in Fig. 1. An atomic system interacts coherently with an optical probe. As a result, the orientation component of the atomic state (such as the atom number in a well defined hyperfine level), becomes weakly entangled with the probe photons, and the photodetection gradually provides the observer with information about the spin state. Continuous observation conditionally reduces the orientation uncertainty with respect to the QND measurement outcome thanks to the measurement quantum back-action. In any individual measurement trajectory, the initial wavefunction will thus collapse in a single squeezed state whereas the ensemble averaged uncertainty will not be reduced, because the measurement outcomes over many QND trajectories are distributed within the variance of the initial state. Our model will allow to study the dynamics of this collapse by considering single photon detection events.

Back-action

Measurement result

Atoms

Single photons FIG. 1: Basic scheme of the measurement process: the phaseshift induced by the atomic sample is estimated using the repeated interaction with single-photons. The back-action on the atomic state increases with the precision of the phase measurement.

A. Heterodyne QND detection

We consider the QND measurement of the population difference between two atomic levels using an heterodyne detection. The measurement is realised by estimating the phase-shift between the two frequency components of the detection beam. This configuration is equivalent to a frequency space interferometer in which each path corresponds to a single frequency mode. In our model, we consider photons individually sent into this interferometer and reconstruct the output beatnote photon after photon by comparing their arrival time with a time reference. The model for the QND apparatus is shown in Fig. 2. The input optical state is composed of two spectral modes. The probe mode (0) at a frequency ω 0 /2π is close to the frequency of the atomic transition, and will therefore be strongly phase shifted when traveling through the atoms. The reference mode (Ω) is at a far-off resonance frequency (ω 0 + Ω) /2π and will be unaffected. A singlephoton state |1 in the mode (0) is associated to the photon annihilation operator a 0 , and the vacuum state |0 in the mode (Ω) is associated to a Ω . A single sideband op-tical phase modulator B is used as a spectral beamsplitter to generate spectrally mode-entangled single-photons (see App. A). A local oscillator (LO) drives B with frequency Ω/2π. We consider the driving signal as a noisefree classical field. After passing through the atoms, the two-channel-optical-state is sent to a single-photon detector which is assumed to be ideal. The detection of a photon in the mode (0) is associated with the operator u 0 , whereas the detection of a photon in the mode (Ω) is associated with u Ω . If the coherence length of the optical source is much larger than the LO wavelength, the two modes (0) and (Ω) are in a coherent superposition, and the spatial overlap of these two modes generates a beating at frequency Ω/2π. A pulse generator, in phase with the LO, delivers regularly spaced pulses (at time interval τ = 2π/Ω) used as a reference to measure the arrival time of the photons. The histogram representation of the photon arrival time allows to reconstruct the beating signal.

Model of the apparatus

Atomic state

The atomic ensemble is a collection of a fixed number N at of two-levels atoms {|a i , |b i }, described as fictitious 1/2-spins {j

(i) x , j (i) y , j (i)
z } where:

j (i) x = 1 2 (|b i a i | + |a i b i |) , (1)
j (i) y = i 2 (|a i b i | -|b i a i |) , (2)
j (i) z = 1 2 (|b i b i | -|a i a i |) . (3)
The atomic ensemble is characterized by the collective spin operators

J k = i j (i) k , k = x, y, z.
A collective state |n is an eigenstate of both J z and J 2 = J 2

x +J 2 y +J 2 z , and is called a Dicke state. It verifies:

J 2 |n = N at 2 N at 2 + 1 |n (4)
J z |n = n |n (5)
where -N at /2 ≤ n ≤ N at /2. After N p photons detected the atomic state is |ψ at (N p ) = n c n (N p ) |n . The initial atomic state is a coherent spin state (CSS) polarized along J x , which means an average population difference J z = 0, and its explicit expression is [START_REF] Zare | Angular momemtum[END_REF]:

c n (0) = 1 2 Nat/2 N at ! Nat 2 + n ! Nat 2 -n ! . ( 6
)
From the Moivre-Laplace theorem, if N at ≫ 1 the coefficients c n (0) are well approximated by a gaussian distribution:

c n (0) ∝ exp - n 2 N at . (7)
The spectral beamsplitter B creates a coherent superposition of the two input spectral modes (0) and (Ω), see App. A. Its action is described by the following matrix:

B ∝ √ T - √ R √ Re iΩt √ T e iΩt , (8)
where, in analogy with a spatial beamsplitter, R is the probability for a photon in the probe mode (0) to be in the reference mode (Ω) at the output of B, and T is the probability for a photon to stay in the carrier mode.

We neglect any absorption and dispersion in B so that

R + T = 1.
When ω 0 + Ω is far from any atomic transition, the reference beam experiences no phase-shift and the QND interaction can be written as

U QND (J z ) = e iφJz 0 0 1 . (9)
φ is the optical phase-shift induced by a number diference of two (n = 1), and depends on the coupling strength of the transition at ω 0 and on the optical density of the atomic cloud. Spontaneous emission induced by the probe beam is neglected in this analysis; the reduction it causes on the coherence of the atomic state is considered in Sec. IV. The scattering matrix of this system is S (J z ) = U QND (J z ) B and it furnishes the output modes of the interferometer once the input ones are given: (u 0 (J z ) , u Ω (J z )) = S (J z ) (a 0 , a Ω ).

Measurement back-action

A click at the detector corresponds to the annihilition of a photon either in mode (0) or (Ω). After the measurement the state of the system satisfies the relation:

|0 0 , 0 Ω ⊗ |ψ at (N p + 1) ∝ [u 0 (J z ) + u Ω (J z )] |1 0 , 0 Ω ⊗ |ψ at (N p ) . ( 10
)
The stochastic recurrence relation which defines the atomic state evolution after one measurement follows from the expression of S (J z ):

|ψ at (N p + 1) ∝ √ T + √ R cos φJ z 2 - Ω t 2 +i √ T - √ R sin φJ z 2 - Ω t 2 |ψ at (N p ) , ( 11
)
where t is the time at which the photon has been detected.

This expression contains a beatnote signal at frequency Ω/2π with a contrast C = 2 √ RT . The reconstruction of the beatnote photon after photon is achieved by comparing the arrival time t k of the k-th photon with the last time reference pulse pτ , where p is the number of pulses counted between t = 0 and t = t k . We estimate the phase-shift ϕ k ≡ Ω δt k from the time delay δt k = t k -pτ between the last pulse of the time reference and the photon detection.

The atomic state after the detection of N p photons is obtained applying N p times the relation of Eq. ( 11) on the initial atomic state |ψ at (N p = 0) . Expressed as a superposition of collective states, it results |ψ at (N p ) = n F Np (n) c n (0) |n , where

F Np (n) 2 ∝ Np k=1 [1 + C cos (φn -ϕ k )] (12)
is the back-action function [40]. Eq. ( 12) leads straightforwardly to the probability P (ϕ) to measure a phase-shift ϕ when the (N p + 1)-th photon is detected:

P (ϕ) = 1 2π n |c n (N p )| 2 (1 + C cos (φn -ϕ)) . ( 13
)
Using this expression and the recurrence relation of Eq. ( 11), we simulated the quantum trajectories followed by the atomic state (see App. C). Typical results are presented in Fig. 3.

In the weak coupling limit (φN at ≪ 1), it is possible to derive an analytical expression for the F Np (n) 2 function (see App. B):

F Np (n) 2 ∝ exp -2M 2 N p n 2 + 2δ ϕn/φ , (14)
where

δ ϕ = lim N t Np →0 N t N p Np/Nt j=1 δ ϕ j (15)
is the average mean position over the followed trajectory, and

M 2 = φ 2 4 1 -1 -C 2 , ( 16
)
is the measurement strength. This expression of the back-action function contains the complete description of the atomic state evolution in the weak coupling regime and allows to quantitatively study the atomic wavefunction collapse, that is the squeezing process.

III. DYNAMICS OF THE WAVEFUNCTION COLLAPSE

The atomic state evolution is studied by comparing the initial atomic state distribution c n (0) of Eq. ( 7) with The solid line is the average over 1000 trajectories (only 20 are plotted) while the dashed line is the result of the analytical calculation of the variance at short time, given by Eq. (18). At short time scale, the evolution is deterministic as predicted by the analytical study. At long time scale, the variances become stochastic but remain bounded. In addition, the average variance over the trajectories is below the short time behavior. (c) The histogram of the mean positions at the end of the trajectories is compared to the initial distribution (solid line). The measurement process satisfies the Born probability rule

Pn = | n|ψat (Np = 0) | 2 .
F Np (n). Depending on their relative width, two regimes appear during the atomic state evolution: the short time limit -where the width of the atomic distribution ∆J 2 z is large compare to one and the atomic state is in a superposition of many atomic levels -and the long time limit -where the distribution is very narrow and the state is split on a few levels of amplitude F Np (n). The boundary between these two regimes occurs for ∆J 2 z ∼ 1, that is for N p ∼ M -2 .

A. Short time limit

In the short time limit, defined by the condition

N p ≪ M -2 , F Np (n)
2 is broad and |c n (N p )| 2 is a gaussian distribution characterized by the following mean position and variance:

J z = -C 2 ξ 2 κ 2 δ ϕ/φ, ( 17
) ∆J 2 z = ξ 2 N at /4, (18)
where κ 2 = M 2 N at N p is the signal-to-noise ratio and ξ 2 = 1/ 1 + κ 2 is the squeezing factor. For N p ≥ 1, the squeezing factor drops below unity and the initial CSS collapses into a SSS as a consequence of the measurement process. The remarkable result is that ∆J 2 z is independent of the stochastic parameter δ φ at first order and is thus deterministic, as shown with the numerical simulation in Fig. 3 (b).

B. Long time limit

In the long time limit (N p ≫ M -2 ) the atomic distribution is very narrow (∆J 2 z ≪ 1) and the atomic state c n (N p ) ∼ F Np (n) is spread over a few eigenstates. The evolution strongly depends on the distance between the mean value J z and the closest eigenvalue of J z . Two extreme cases can be considered (Fig. 4): either J z is in the middle of two eigenvalues, which turns out to be an unstable equilibrium state of the system, or it is an eigenvalue of J z , which corresponds to the stable equilibrium state (the stability of the attractors in treated in [4,[START_REF] Adler | [END_REF]).

In the first case the distribution is centered between two eigenstates of J z , namely |n 0 and |n 0 + 1 . The atomic state is well approximated by a superposition of these two states |ψ at (N p ) ≈ (|n 0 + |n 0 + 1 ) / √ 2, since the amplitude of the other states decreases exponentially. In this case J z = n 0 + 1/2 and the variance is ∆J

2 z = ψ at (N p )| J 2 z |ψ at (N p ) -J z 2 = 1/4
, which results to be the upper bound value in the long time limit. This limit clearly appears in the numerical calculations, as shown in Fig. 3 (b).

The second case represented in Fig. 4 (b) corresponds to an atomic distribution centered on an eigenstate |n 0 of J z . The atomic state is then a superposition of three eigenstates

|ψ at (N p ) ≈ k=-1,0,1 F Np (n 0 + k) |n 0 + k . ( 19
)
From the normalization condition, the symmetry of the state around n 0 , and Eq. ( 14), we have:

k=-1,0,1 F Np (n 0 + k) 2 = 1 ( 20
)
F Np (n 0 ± 1) 2 = F Np (n 0 ) 2 e -2M 2 Np . ( 21
)
Since J z = n 0 , it follows ∆J 2 z = 2 exp -2M 2 N p , which is the lower bound for the variance evolution as plotted in Fig. 3 (b).

IV. EXPERIMENTAL CONSIDERATIONS

The main experimental challenge in a QND detection is to limit the spontaneous emission and so the optical power while reaching the shot-noise limit with a sufficient bandwidth. In this paragraph, we show how a heterodyne detection allows to reach the shot-noise of a weak probe beam and how the existence of two sidebands rejects all the frequency independent common dephasing sources. We finally discuss the effect of the spontaneous emission on the squeezing performances.

In a realistic setup, different configurations for the measurement can be implemented depending on the relative position of the carrier and the sidebands with respect to the atomic transitions. Two examples of possible schemes are presented in Fig. 5, where the case of alkaline atoms is considered.

A. Shot-noise limited detection

The intrinsic advantage of a heterodyne detection is that it makes possible to reach the shot-noise for a weak probe beam by beating it with a much stronger reference beam. In FM spectroscopy, the reference beam is the strong carrier and the probe beam is the weak sideband. If N c is the number of photons in the carrier, N s that one in the sideband, and N e the photon equivalent noise due to the detection electronics, the signal-to-noise ratio of the optical detection is:

SNR ∝ N c N s N c + N s + N e . ( 22
)
If N c ≫ N s , N e , then SNR ∼ √ N s : by using a strong reference, the detection can thus be limited to the shotnoise of the weak probe. Nevertheless, the reference must be far from the atomic transition to strongly reduce its absorption. A high modulation frequency Ω is thus required, which results to be the main technical limitation of the method. Nowadays, photodiodes designed for high-speed optical communications provided with integrated transimpedance amplifiers with gains of a few kilo-ohms, set an indicative upper limit of 10 GHz for the achievable bandwidth.

B. Common mode noise rejection

All the systems where the phase induced by the atoms on a probe beam is measured by comparison with the phase of a reference beam require a precise control of all the sources of relative phase noise between the two beams. The geometrical splitting of the two components, like in a Mach-Zehnder interferometer, requires to precisely stabilize the relative length between the two paths. On the other hand, if the splitting is in frequency and the two components are spatially overlapped, all the noise sources acting on the optical path length (like acoustic noise, optical index fluctuations and mechanical displacements) are rejected, since they are common mode. In the specific case of a two sidebands modulator, where only one sideband is close to an atomic resonance, the signal detected on the photodiode is the difference between the beatnotes of each sideband with the carrier. The differential detection leaves only the atomic contribution to the optical index, an active stabilisation of the interferometer is not required.

In [18], an amplitude modulation method in a Mach-Zehnder interferometer is proposed which similarly rejects the common mode noise using two probe beams. Nevertheless, it requires a demanding good spatial over-lap, a well balanced power and an active phase control of the two probes; all these features are intrinsic in the method we propose.

The proposed scheme could be further improved by injecting the probe in an optical cavity. The signal-to-noise ratio for the same destructivity is increased by a factor square-root of the finesse [25]. The injection in another cavity resonance can be used to perform a differential measurement when studying amplitude fluctuations of a DC signal as shown in [19,20]. In our scheme, a differential measurement that rejects the phase noise due to cavity length fluctuations is realized by placing the carrier in the middle of two cavity resonances and injecting the two sidebands in two different resonances.

C. Spontaneous emission

The performance of a QND detection is reduced by the spontaneous emission induced by the probe. The atomic decay causes a decoherence of the atomic state since the detection of the emitted photon allows to determine in which state the atom has been projected. The problem of quantifying this effect has been treated in [14,31,32].

We now analyse the performance of scheme (a) in Fig. 5 when the atomic decay is taken into account. The dephasing induced by the atomic cloud on the optical probe is given by [33]:

∆Φ = λ 2 4πA (N 1 S 1 + N 2 S 2 ) , (23)
where N F is the population of the |F state, λ the optical wavelength, A the probe beam area, and S F the probe coupling the |F state:

S F = F ′ γ∆ F F ′ ∆ 2 F F ′ + γ 2 S F F ′ . (24)
In the last equation γ is the atomic linewidth and S F F ′ is the dipole transition strength of |F → |F ′ given by:

S F F ′ = (2F ′ + 1) (2J + 1) J J ′ 1 F ′ F I 2 , (25)
where I and J are the nuclear and total angular momentum, respectively. The detunings ∆ 1F ′ and ∆ 2F ′ are choosen so that S 1 = -S 2 ≡ S. Introducing ǫ so that N 1 = N at (1 + ǫ) /2 and N 2 = N at (1 -ǫ) /2, the optical dephasing can be written as ∆Φ = ρ 0 Sǫ, where ρ 0 = λ 2 N at /4πA is the resonant optical density. The phase-shift φ induced by a population difference of one atom between the two atomic states is obtained for ǫ = 2/N at :

φ = λ 2 S 2πA . ( 26
)
The integrated probability for an atom to scatter a photon when probed by N p photons is, in the limit C ≪ 1:

η = ρ 0 N at C 2 N p 2 L, ( 27
)
where L is the atomic lineshape:

L = F,F ′ γ 2 ∆ 2 F F ′ + γ 2 S F F ′ . (28)
The signal-to-noise ratio can be rewritten as a function of the decoherence: κ 2 = µρ 0 η where µ = S 2 /L. In [31], the squeezing factor is estimated to be:

ξ 2 = (1 -η) 2 1 + µρ 0 η + 1 -(1 -η) 2 . ( 29
)
Considering 10 7 atoms of 87 Rb optically trapped with a beam waist of 20 µm and probed on the D 2 (5 2 S 1/2 → 5 2 P 3/2 ) line at λ = 780 nm with a detuning ∆ 13 = 3.2 GHz then µ ∼ 1 and φ = 4.1 10 -7 rad. The resulting resonant optical density is ρ 0 ∼ 2400 from which a squeezing factor ξ 2 = 0.06 (about 12 dB) can be achieved for an optimum decoherence η ∼ 0.01. It corresponds to a measurement of N p ∼ 10 7 photons in the total beam for a modulation depth of 1%.

The optimum squeezing factor strongly depends on the optical density of the atomic sample. Using a Bose-Einstein condensate (BEC), a much higher squeezing can be achieved but much less photons have to be used during the measurement to reach the optimum.

With regard to the possibility of exploring the long time regime behavior to prepare Dicke states, the decoherence due to spontaneous emission can be seen as a constraint that would make this regime hard to reach. Nevertheless the discussion above holds mainly for a large atom number, if the initial sample contains only a few atoms (for example a BEC of 100 atoms) the variance of the coherent state is of the order of a few units so that entering the Dicke state regime in the presence of spontaneous emission is realistic.

V. CONCLUSION

In this article we analyse a QND heterodyne detection of an atomic sample to prepare spin-squeezed states. A theorical model of the measurement process is proposed that consists on the sequential detection of single photons that causes a back-action on the atomic wavevector. One should note that in Eq. ( 12) no asumption has been done on the atom-light coupling strength. An analytical solution is obtained in the weak coupling regime and completely describes the atomic state evolution. Remarkably, two regimes occur during the wavefunction collapse: at short time spin-squeezed states are prepared whereas it converges to Dicke states at long time.

Finally, experimental considerations to implement the detection scheme have been presented, pointing out the main advantages and limitations of the method as well as the role played by the atomic decay and its influence on the squeezing factor. The requirement of an active stabilisation of the separated geometrical paths used in standard interferometric methods, such as the Mach-Zehnder configuration, is replaced by the need for a stable RF field. Moreover, the heterodyne detection allows to reach the shot-noise of a weak beam by mixing it with a stronger one.

This detection method could be used to generate other kinds of non-classical states such as Schrödinger cats or NOON states. Such states could be employed in quantum enhanced metrology to approach the Heisenberg limit sensitivity [34]. Moreover, sequential measurements allow to prepare almost deterministic states by implementing a quantum feedback loop [4,[35][36][37], and can be exploited for quantum error correction [38].

(φ 2 N at N t ≪ 1). In the weak coupling limit the number of photons in each sequence can be large since φ 2 N at ≪ 1.

The detection probability at the end of the j-th sequence of N t measurements is P j (ϕ) = P 0 (ϕ) + δP [δ ϕ j ] (ϕ). If we choose a phase measurement resolution π/m (m ∈ N and m ≫ 1) we find from Eq. ( 12 The number of photons has to be large in the intervals [-π, -π + π/m] and [π -π/m, π], where P (ϕ) is close to zero. This is verified for N t ≫ m 3 , since the following condition must be satisfied when C ∼ 1:

N t π π-π m P 0 (ϕ) dϕ ∼ N t 1 -C 2m + π 2 C 12m 3 ≫ 1. (B3)
In that case N j,l ≫ 1 and the second order term for the Taylor expansion of f (j,l)

Np (n) in φn can be identified with that of a gaussian distribution: By converting the discrete sum over l into an integral and using Eq. (B1) and Eq. (B4), we obtain Eq. ( 14).

f (j,l) Np (n) ∝ exp -2M 2 l N j,l (n -n l )

FIG. 2 :

 2 FIG.2: Model for the QND based heterodyne detection. Single-photons are sent through a spectral beamsplitter B and interact non-destructively with the atomic ensemble (UQND). The detection time of each photon is measured by comparison with a time reference signal.

FIG. 3 :

 3 FIG. 3: Trajectories of the atomic distribution momenta with a Monte-Carlo simulation, once set = 200, φ = 10 -3 rad, and R = T = 1/2. (a) Mean position of the atomic population Jz as a function of the number of measured photons. For a long enough measurement the state converges to a Dicke state. (b) Variance evolution during the measurement sequence.The solid line is the average over 1000 trajectories (only 20 are plotted) while the dashed line is the result of the analytical calculation of the variance at short time, given by Eq.(18). At short time scale, the evolution is deterministic as predicted by the analytical study. At long time scale, the variances become stochastic but remain bounded. In addition, the average variance over the trajectories is below the short time behavior. (c) The histogram of the mean positions at the end of the trajectories is compared to the initial distribution (solid line). The measurement process satisfies the Born probability rule Pn = | n|ψat (Np = 0) | 2 .

FIG. 4 :

 4 FIG. 4: Extreme cases for long time evolution. The mean position of the atomic distribution is in the middle of two eigenvalues of Jz, (a) or centered on an eigenvalue of Jz (b).

FIG. 5 :

 5 FIG. 5: Two possible measurement schemes. (a) One sideband is in the middle of the two probed states whereas the carrier and the other sideband are far from the transition (∆ F F ′ ≪ ∆c). (b) Each sideband is close to an atomic transition, the carrier is in the middle of the two probed states.

  Np (n) = [1 + C cos (φn -πl/m)] N j,l .(B2)N j,l = π m P j πl m N t is the number of measurements giving the result ϕ = πl/m during the j-th sequence.

  2 ,

	where
	M 2 l = n l =	Cφ 2 4 1 φ 1 + C cos (πl/m) C + cos (πl/m) (1 + C cos (πl/m)) 2 , C + cos (πl/m) sin (πl/m) .	(B5) (B6)

Acknowledgements

We acknowledge Isabelle Bouchoule, Carlos Garrido Alzar and Luca Pezzé for useful discussions. This work was supported by IFRAF, DGA, the European Union (with STREP program FINAQS), and ESF (EURO-QUASAR program). A. B. acknowledges support from EU under an IEF Grant.

Appendix A: The phase modulator as a single-photon spectral beamsplitter

The electro-optic modulation relies on the Pockels effect, a second order non-linear interaction between an optical and a quasi-static, radio-frequency (RF) field. The Hamiltonian associated with this process is:

where a m , a 0 and a 1 are the annihilition operators for the RF field, the input field and the generated field, respectively and g is a coupling constant (depending on the non-linear crystal, the field frequencies, . . . ). This Hamiltonian is the result of two processes: a down conversion from an optical to a quasi-static electric field, that is an optical rectification, and a frequency sum with a quasi-static electric field, which is the Pockels effect. Let t be the propagation time of the optical wave in the non-linear crystal. The evolution of an incident state |ψ in through the crystal in the weak coupling limit (gt ≪ 1) gives the following output state:

Considering a coherent RF field and a single-photon optical state in input |ψ in = |α m ⊗ |1 0 , 0 1 , the output field is:

The phase modulator thus splits the input photon wavefunction in two spectral components, whose superposition determines a beating of the photon wavefunction with itself at the modulation frequency of the RF field, as shown in Fig 6 .      RF FIG. 6: Quantum beats of the single-photon wavefunction generated by a phase modulator.

Appendix B: Derivation of an analytical expression of FN p (n) 2 in the weak coupling limit

The general stochastic measurement process described by Eq. ( 12) is non-Markovian. In the weak coupling limit (φN at ≪ 1) the process becomes Markovian [START_REF] Gardiner | Quantum Noise[END_REF] but is not stationary. Using again the weak coupling approximation we show that the overall process can be split into a sequence of stationary sub-processes, which brings to an analytical expression of F Np (n)

2 .

Evaluation of the probability P (ϕ)

In the weak coupling regime the phase shift determined by the whole atomic sample is small, then φn ≪ 1 for -N at /2 ≤ n ≤ N at /2. We can expand Eq. ( 13) to the first order in φn around a centered probability distribution P 0 (ϕ) = (1 + C cos ϕ) /2π. We find P (ϕ) = P 0 (ϕ) + δP [δ ϕ] (ϕ), where δP [δ ϕ] (ϕ) = -C sin ϕ δ ϕ/2π and δ ϕ = -φ J z is a stochastic parameter depending on the followed trajectory. The chain of stochastic variables { ϕ k } 1≤k≤Np is reduced to a single parameter δ ϕ Np which proves that the process is Markovian. P 0 (ϕ) is the deterministic contribution for a zero phase detection, whereas δP [δ ϕ] (ϕ) describes the atomic phase-shift.

Decomposition into stationary sub-processes

To study the evolution of the atomic state, we split the measurement into sequences of N t photon detections, each inducing a negligible evolution of the atomic wavefunction (the sub-processes are stationary). A small signal-to-noise ratio is required for each sequence The numerical simulation of the quantum trajectories shown in Fig. 3 adopts the state of Eq. ( 7) as initial one. For each photon, the probability P Np+1 (ϕ) to measure a phase ϕ for the next detected photon is given by Eq. ( 13). More explicitly, we use the cumulative distribution function associated to this density of probability, which is:

By generating a random number with an uniform distribution over [0, 1] and numerically inverting F Np+1 we get the phase detected for the (N p + 1)-th photon. The recurrence relation given by Eq. ( 11) yields the new atomic distribution. The quantum trajectory is obtained iterating the sequence.


Julien Le Duigou
email: julien.leduigou@irccyn.ec-nantes.fr

Alain Bernard
email: alain.bernard@irccyn.ec-nantes.fr

Nicolas Perry
email: nicolas.perry@iut.u-bordeaux1.fr

Jean-Charles Delplace
email: jean-charles.delplace@cetim.fr

Generic model for the implementation of PLM systems in mechanical SMEs

Keywords: Product Lifecycle Management (PLM), Modelling Framework, Generic Model, Information System

To improve the performance of extended enterprises, SMEs must be integrated. This integration must be done on several levels, including the PLM level. PLM is underdeveloped in SMEs mainly because of their difficulties in implementing information systems. This work aims to provide a modelling framework to facilitate the implementation of PLM systems in mechanical SMEs. Our approach offers a generic model for the creation of processes and data models. These models are explained, and then the scope and the improvement they provide are considered.

modelling skills are not sufficiently present in SMEs.

• The lack of interoperability between PLM and CAD, ERP and PLM of partner companies. To solve these problems, section 2 explore existing product models for PLM. Section 3 explains the proposed modeling framework. The first part introduces the global research approach used and the generalization method. The second part presents the generic model obtained before explaining how to use it.

PLM models

The product models structure information of the company. The product models support the business functionalities required by SMEs, facilitate the creation of specific models and improve the interoperability of the PLM system with other information systems.

The product modelling has been introduced in Europe in the early 90s by Dupinet [START_REF] Krause | Product Modeling[END_REF] and Krause [START_REF] Bernard | Contribution à la modélisation des produits et à l'intégration des technologies productrices dans un environnement multi-acteurs[END_REF], then by Tichkiewitch and Bernard to manage product information in multi-views and multi-actors context [START_REF] Labrousse | FBS-PPRE, an Enterprise Knowledge Lifecycle Model, Methods and Tools for Effective Knowledge Life-Cycle Management[END_REF][START_REF] Noël | A dynamic multi-view product model to share product behaviors among designers: how process model adds semantic to the behavior paradigm[END_REF].

All these models use information, some aimed at structuring information (such as FBS-PPRE [START_REF] Gzara | Les patterns pour l'ingénierie des systèmes d'information produit[END_REF] and IPPOP [13]), others focus on reuse (patterns, e.g. Gzara patterns [14]), or others focus on exchange (like AP214 [START_REF] El Khalkhali | An information infrastructure to share product models using STEP standard, 9th IPSE international conference on concurrent engineering: research and applications[END_REF] or PLCS [START_REF] Euler Chelpin | Information modelling for the manufacturing system life cycle[END_REF] both from STEP). In the context of PLM systems, these generic models should be adapted. The product models can be used, but they are both too complex and too generic for SMEs.

The use of these models in industrial context requires modifications, adaptations and specialisations. Some works deal with the specialisation and consolidation of generic models, especially STEP [START_REF] Zachman | A framework for information systems architecture[END_REF][START_REF] Kosanke | CIMOSA modelling processes[END_REF], but use a top down approach to complete them, and do not fully meet the industrial needs. Others, using a bottom up approach, use frameworks like Zachman [START_REF] Bacha | De la gestion des données techniques pour l'ingénierie de production[END_REF] or Cimosa [START_REF] Abdmouleh | Composants pour la modélisation des processus métier en productique, basé sur Cimosa[END_REF] to create their own models [21,[START_REF] Duigou | Global approach for product data management, application to ship equipment part families[END_REF] but they lose the benefit of interoperability of the generic model.

The next section will propose a framework adapted from GERAM [23] containing a bottom up method to get a generic model and a second method to specialize the generic model for a specific domain or company.

Modelling framework

Research approach and generalisation method

To propose a framework that facilitates business processes modelling and business objects modelling by improving the functionalities of PLM and system interoperability, the proposed approach reverses the traditional V cycle (Figure 1). The approach begins with a rising and inductive step formalizing the generic model. Three month of immersions in three representative companies give results for the needs of SMEs about PLM, processes to implement in companies to address these needs and data structure that can automate these processes [24]. These results are then aggregated to create a generic model.

Figure 1 Research approach

To generalise from three specific models to one generic model, the following method is applied:

1. Select the objects that deal with PLM. Indeed the specific models contain specific objects, on the company's business. 2. Group objects by theme and identify generic objects representing each group.

This phase is performed by a cognitive analysis of the modeller based on a semantic analysis. 3. Construct generic objects. This step is performed by an extraction of the intersection of attributes and methods. 4. Link the different generic objects. This is done by identifying the various links between specific objects of different groups. This method is applied on the three specific models to obtain a generic model that could be used in the three representative pilot companies. The next section will explain the model obtained.

Presentation of the generic model and specialisation method

The different identified objects are activity, product, material, function, resource, trigger, organization and document. The activity specializes in task and operation, the organization in customer and supplier and the resource in material, human and software. They are represented in a UML class diagram (Figure 3).

Figure 3 Simplified model

The product model (Figure 4), the activity model (Figure 5) and the resource model (Figure 6) will be explained.

The product is composed of other products, which may be optional. A component can have alternative components in an assembly context. The product has different versions. These versions have states (creation, in validation, validated, obsolete…) and attached documents. The product is linked to the activity through input and output. A product may be the output of an activity and the input of another. The structure of the product by product (assembly) -activity -product (component) decomposition gives different structures depending on the decomposition activity (assembling, maintaining, decommissioning…). The product is also related to the function, as a technical function or a service function via the selected link. A technical function for a component in an assembly becomes a service function for this component when it is alone. This design allows a functional view of the product structure by a decomposition product (assembly) -technical function = service function-product (component).

Figure 4 Product model

As the product model, activity has versions that have states and documents. The activity can be composed of other activities which may be optional. Activity can have an alternative activity in a parent activity context (i.e. a process). In addition, the activities are ordered through the link "next activity". A trigger activates this link to go from one activity to the other. The resource has also versions, with states and documents. The resource can be composed of other resources which may be optional. Moreover, the resource may have an alternative resource for a given activity.

Figure 6 Resource model

Figure 8 shows the framework adapted from GERAM with an abscissa representing the needs, processes and business objects. The ordinate represents the instantiation of components: generic, partial or particular levels. And the depth represents the different views of the system: function, information, resource and organization.

To reach the specific needs of the companies and obtain specific models, the instantiation view specialises the objects of the generic model for each domain and for each company. Three levels are used:

• To create the company-specific models, the proposed method is as follow (Figure 9). It begins with needs analysis then continues with the formalisation of the processes necessary to address these needs. The third step is the extraction of business objects involved in these processes.

1. Identifying needs: Needs identification is made through interviews, observations and immersions in the company. Mind maps are used to formalise the requirements with the experts of the company. This identification is facilitated by the possibility of selecting the generic needs proposed in the generic map. 2. Process definition: Process modelling is done with the experts. IDEF0 diagrams are used to formalise the processes. These processes are deduced both the needs identified and the corresponding generic processes. 3. Definition of business objects: The identification of objects is done by extracting the inputs and outputs of processes defined in step 2. Based on the generic model, the specialisation of the generic objects leads to specific objects of the company. The UML class diagrams are used to formalise the objects of the company. All these steps are based on generic models, the difficulty of formalizing the processes and creation of data model for a PLM system is reduced.

Discussion

The contribution of this work is on two levels: on one hand a method of generalisation of models for a generic model and specialisation of it to particular models and on the other hand a generic model for mechanical SMEs.

The generalization method creates a generic model of various specific models. Different departments or companies may have a common view on their product-processresource information. The exchange and sharing of information is then facilitated. The difficulty of the method is the grouping of objects that inherit from a single generic object. This is done by a semantic analysis of objects names and attributes, but in the present study, it was facilitated by the fact that the same modeler did the three specific models.

The specialization of the generic model gives a specific model adapted to the business of the company, but keeping the interoperability with other companies around the generic level. The choice of languages for the specialization is done to improve communication with company experts. The elected languages are the mind map formalism to extract requirements, IDEF0 for processes formalization and UML class diagram for the business objects representation. These languages were also used during the modeling phase in the three pilot companies and gives validated results. The use of one language such as SysML with requirement diagrams, sequential diagrams and class diagrams could be done but we prefer languages already known by the experts.

The second point is generic model for mechanical SMEs that meet needs identified in the three pilot companies. It can be specialized in a particular model for each new company or to each department who need a specific model. The link with the rest of the extended enterprise is provided with the link to the generic model. The main weakness of this model is that it is based only on three cases, even if they are representative.

A PLM demonstrator integrating the approach and the data model has been developed. This demonstrator will be the base of a new product developed by a new society created at the end of the study.

Conclusion

The mechanical SMEs need to integrate more and more extended enterprises PLM systems. For interest in PLM system, a framework to facilitate the processes and data modelling and to improve interoperability between their information systems is proposed. This work aims to give them a framework to integrate their functionalities through the specialization of their data model according to their field of activity. It also provides a method of specialization to create their own particular data model ensuring greater interoperability through the alignment of high semantic models (for the PLM) with specialised models for other applications (ERP, CAD and KBS).

The proposed Product Process Resource model is the result of the concatenation of models of three companies representing the needs of mechanical SMEs in terms of PLM. It may be specialized in specific models for each individual company but keeping the same structure for a high level of interoperability.

The next contribution will be using this framework and the generic model to solve the needs identified in mechanical SMEs. It will carry the processes to meet these needs with demonstrator software integrating the proposed framework and the generic model.

Figure 5

 5 Figure 5 Activity model

  The generic level: The generic model has been defined in this paper. The objects of this model are generic and can be used in any kind of company. • The partial level: This model is specialized from the generic model and expresses all the objects of one particular domain, e.g. machining, forging, stamping… • The particular level: It is the company model. It is defined from the partial model (or directly from the generic model, if the partial model does not exist). The generic model provides a common point of view of function, information and resource. It also incorporates the organizational aspect which coordinates the other points of views. The representation of the framework could then have only two dimensions, removing the view dimension. But this axis represents the different views that are modelled, even if they are all integrate in the generic model.

Figure 8

 8 Figure 8 Modelling framework

Figure 9

 9 Figure 9 Specialisation of the modelling framework

Acknowledgement

We would like to thank companies PSL CONCEPT, CAPRICORN and SMP for allowing us to carry out our study in their companies and for their technical support.