Quentin Deplano

Philippe Tamarat

Brahim Lounis

Jean-Baptiste Trebbia
email: jean-baptiste.trebbia@u-bordeaux.fr

Sub nanosecond coherent optical manipulation of a single aromatic molecule at cryogenic temperature

Single molecules trapped in the solid state at liquid helium temperatures are promising quantum emitters for the development of quantum technologies, owing to their remarkable photostability and their lifetime-limited optical coherence time of the order of ten nanoseconds. The coherent preparation of their electronic state requires resonant excitation with a Rabi period much shorter than their optical coherence time. Sculpting the optical excitation with sharp edges and a high on-off intensity ratio (~ 3×10 5 ) from a single-frequency laser beam, we demonstrate subnanosecond drive of a single dibenzanthanthrene molecule embedded in a naphthalene matrix at 3.2 K, over more than 17 Rabi periods. With pulses tailored for a half-Rabi period, the electronic excited state is prepared with fidelity as high as 0.97. Using single-molecule Ramsey spectroscopy, we prove up to 5 K that the optical coherence lifetime remains at its fundamental upper limit set by twice the excited state lifetime, making single molecules suitable for quantum bit manipulations under standard cryogen-free cooling technologies.

INTRODUCTION

The coherent control of photon-matter interfaces is an interesting approach for the realization of quantum information networks. A large variety of optically addressable qubit platforms have been developed on the basis of ultra cold atoms [START_REF] Saffman | Quantum computing with atomic qubits and Rydberg interactions: progress and challenges[END_REF][START_REF] Henriet | Quantum computing with neutral atoms[END_REF] , Silicon-Vacancy defects in diamond [START_REF] Sipahigil | An integrated diamond nanophotonics platform for quantum-optical networks[END_REF][START_REF] Day | Coherent Interactions between Silicon-Vacancy Centers in Diamond[END_REF] or semi-conducting quantum dots [START_REF] Bayer | Coupling and Entangling of Quantum States in Quantum Dot Molecules[END_REF][START_REF] Tiranov | Collective super-and subradiant dynamics between distant optical quantum emitters[END_REF][START_REF] Press | Complete quantum control of a single quantum dot spin using ultrafast optical pulses[END_REF] . Among those systems, organic molecules trapped in condensed matter at liquid helium temperatures are particularly promising candidates [START_REF] Toninelli | Single organic molecules for photonic quantum technologies[END_REF] . They behave as two-level systems with excellent photostability [START_REF] Basché | Direct spectroscopic observation of quantum jumps of a single molecule[END_REF] and display zerophonon transition lines (ZPLs) with coherence times 𝑇 ! limited by the excited-state lifetime 𝑇 " (𝑇 ! = 2𝑇 " ) [START_REF] Tamarat | Ten Years of Single-Molecule Spectroscopy[END_REF] . These extended coherence times, up to tens of ns [START_REF] Basché | Photon Antihunching in the Fluorescence of a Single Dye Molecule Trapped in a Solid[END_REF] , make them suitable for quantum interference experiments [START_REF] Trebbia | Indistinguishable near-infrared single photons from an individual organic molecule[END_REF][START_REF] Lettow | Quantum Interference of Tunably Indistinguishable Photons from Remote Organic Molecules[END_REF][START_REF] Lombardi | Triggered emission of indistinguishable photons from an organic dye molecule[END_REF] . Moreover, they have proven their effectiveness for the generation of triggered single photons [START_REF] Brunel | Triggered Source of Single Photons based on Controlled Single Molecule Fluorescence[END_REF][START_REF] Trebbia | Efficient generation of near infra-red single photons from the zero-phonon line of a single molecule[END_REF] , with potential applications for quantum key distribution [START_REF] Bennett | Quantum cryptography: Public-key distribution and coin tossing[END_REF][START_REF] Ekert | Quantum cryptography based on Bell's theorem[END_REF] . The controlled interaction of single molecules with photonic structures [START_REF] Türschmann | Chip-Based All-Optical Control of Single Molecules Coherently Coupled to a Nanoguide[END_REF][START_REF] Lombardi | Photostable Molecules on Chip: Integrated Sources of Nonclassical Light[END_REF][START_REF] Ciancico | Narrow Line Width Quantum Emitters in an Electron-Beam-Shaped Polymer[END_REF][START_REF] Colautti | A 3D Polymeric Platform for Photonic Quantum Technologies[END_REF][START_REF] Boissier | Coherent characterisation of a single molecule in a photonic black box[END_REF] also leads to enhanced interaction efficiencies with light, offering interesting prospects for integrated quantum photonics. On the other hand, the possibility of fast, coherent optical manipulation of the molecular electronic states has been poorly investigated. To effectively invert the populations of the ground and excited states, or to create a superposition of these states with maximum coherence, we would need tailored resonant pulses associated with a quarter or half of a Rabi period, respectively. Near-unity fidelity in the state preparation will require that the damping of the transition dipole be negligible during the preparation time for the molecular state. It is thus crucial for the Rabi period to be much shorter than 𝑇 ! .

Up to now, optical nutation of single molecules have been monitored under resonant pulsed excitation [START_REF] Gerhardt | Coherent state preparation and observation of Rabi oscillations in a single molecule[END_REF][START_REF] Rezai | Detuning dependent Rabi oscillations of a single molecule[END_REF] , in a similar way to what was performed with atoms 26 or solid-state quantum emitters such as Silicon-Vacancy defect centers in diamond [START_REF] Becker | Ultrafast all-optical coherent control of single silicon vacancy colour centres in diamond[END_REF][START_REF] Zhou | Coherent control of a strongly driven silicon vacancy optical transition in diamond[END_REF] . Yet, their modest contrast would not allow to prepare the excited state with a fidelity better than ~ 0.7. In this letter, we report optical pulse shaping with sharp edges and a high on-off intensity ratio from a resonant single-frequency laser, and demonstrate sub-ns optical nutation in the fluorescence of an isolated single molecule. Negligible damping of the transition dipole over the first period shows the possibility to prepare the molecular excited state with a probability 0.97 after a ppulse excitation. Moreover, using single-molecule optical Ramsey spectroscopy [START_REF] Wang | Optical Ramsey spectroscopy on a single molecule[END_REF] , we directly measure the dephasing time 𝑇 ! and prove up to 5 K that it remains at its fundamental upper limit 2𝑇 " , making molecular electronic states suitable qubits for quantum logic gates operating under standard cryogen-free cooling technologies.

II. EXPERIMENTAL SETUP

The sample used in this study is prepared by heating a piece of naphthalene doped with dibenzanthanthrene (DBATT) (Fig. 1a) above the melting temperature and pressing it between two coverslips to obtain a thin layer with a thickness of a few μm. The sample is then cooled down to 3.2 K in a cryogen-free cryostat. A single-frequency tunable dye laser (linewidth ~1 MHz, wavelength ~618 nm) is used to excite the purely electronic transition of the molecules placed at the diffraction-limited focal spot (transverse FWHM ~ 500 nm) of a microscope objective with numerical aperture 0.95. The red-shifted fluorescence of the excited molecules is sent through a band-pass filter (bandwidth 100 nm centered on 700 nm) onto a single-photon avalanche photodiode with temporal resolution ~50 ps at half maximum. The response function of the avalanche photodiode to femtosecond pulses is displayed in Fig. S1.

The DBATT single molecule chosen for this study has its transition dipole moment aligned with the linear polarization of the laser beam, in order to keep the saturation intensity of the ZPL extremely low. This allows fast optical drive of its Bloch vector with modest excitation powers that will hardly heat the sample and generate thermal dephasing. Figure 1(b) shows its fluorescence excitation spectrum in the low saturation regime at 3.2 K. The optical resonance displays a Lorentzian profile with a FWHM ZPL width Δ𝜈 # = (22.3 ± 0.07) MHz, in agreement with the previous values reported in the literature for this host-guest system [START_REF] Jelezko | Pump-probe spectroscopy and photophysical properties of single di-benzanthanthrene molecules in a naphthalene crystal[END_REF] . From this linewidth, one can estimate the dephasing time of the optical transition 𝑇 ! = (𝜋Δ𝜈 # ) $" = (14.1 ± 0.05) ns. The evolution of the ZPL width with increasing excitation intensities 𝐼 (Fig. 1c) is consistent with the saturation law Δ𝜈(𝐼) = Δ𝜈 # 31 + 𝐼 𝐼 %&'

⁄ , which provides the saturation intensity 𝐼 %&' = 0.32 W cm -2 of this molecule. The absence of deviation from the saturation law up to the strongest excitation intensities ~ 10 ( 𝐼 %&' is a signature of negligible dephasing due to laser-induced local heating processes, such as non-radiative vibrational relaxation that accompanies fluorescence. After switching off the laser beam with an electro-optic modulator (EOM), the fluorescence signal of the molecule decays exponentially, as shown in the inset of Fig. 1c). This decay provides a measure of the electronic excited state lifetime 𝑇 " = (7.2 ± 0.3) ns, which is consistent with 𝑇 ! ≅ 2𝑇 " .

III. OPTICAL NUTATION

Coherent optical manipulation of a single molecule with a high fidelity requires strong excitation intensities with abrupt on and off switches, as well as a high on-off intensity ratio to avoid optical saturation of the molecule during the off periods. The experimental setup built to obtain fast, sub-ns optical nutation is depicted in Fig. 2a. As a first step, the resonant optical excitation is prepared by shaping the laser beam with just an EOM driven by a square-wave RF signal, leading to an on-off intensity ratio 𝜅 ~ 8×10 3 (See Fig. S2 for details on the EOM operation). Figure 2b shows the fluorescence time trace of the molecule subjected to resonant excitation for a duration of 30 ns with strong intensity 𝐼 ~ 6100 𝐼 %&' . This evolution is displayed after background subtraction (See Figs S3 andS4) and deconvolution from the response function of the avalanche photodiode, as described in Fig. S5. Rabi oscillations clearly show up with a period of 850 ps and are damped throughout the illumination slot as a result of optical coherence relaxation. At the end of the excitation, the molecule reaches its steady state characterized by a saturated population 0.5 of the excited state, which allows rescaling the fluorescence signal into excited-state population. Such normalization allows reproducing the experimental time trace with the computed evolution of the excited-state population (red curve), derived from the semi-classical Bloch equations of a two-level system under resonant excitation. Before the laser excitation is switched on (𝑡 < 0 in Fig. 2b), this population is kept at ~ 0.2 as a result of the residual laser intensity during the off-periods of the EOM. This artifact degrades the contrast of the Rabi oscillations and would limit to ~ 0.9 the probability to prepare the molecular excited state after pulsed excitation for half a Rabi period (p-pulse). The fidelity in such preparation is simulated in Fig. 2c as a function of the on-off intensity ratio 𝜅, for an intensity 𝐼 = 6100 𝐼 %&' . It shows that the fidelity is expected to approach unity (~ 0.97) for values of 𝜅 above 10 [START_REF] Bayer | Coupling and Entangling of Quantum States in Quantum Dot Molecules[END_REF] .

In order to properly initialize the molecule in the ground state before switching on the EOM, the on-off intensity ratio is enhanced by adding a fast acousto-optic modulator (AOM), which commutates the laser beam in its first diffraction order before injection into the EOM fiber (Fig. 2a). The electronic signals commanding these modulators are delivered by an arbitrary-wave generator and finely shaped to sculpt the excitation slots with flat-top profiles and with an on-off laser intensity ratio 𝜅 ~ 3×10 5 (See Fig. S6 for details on the pulse profile). With such pulse shaping, optical nutation of the molecule can be driven at high frequencies and with enhanced visibility, as demonstrated in Fig. 2d for 𝐼 𝐼 %&' ⁄ = 6100. At least 17 Rabi oscillations can be distinguished in Fig. 2d, which to our knowledge constitutes the largest number of optical Rabi oscillations observed to date with a single solid-state emitter. This time trace is well reproduced by the evolution of the excited-state population (red curve) under resonant excitation whose rising and falling edges match the characteristics of the experimental pulse (see Fig. S7). In particular, the slow rise in fluorescence before the front edge of the EOM is due to residual excitation of the molecule during the AOM rising edge. Nevertheless, our measurements demonstrate the ability of fast, sub-ns preparation of the molecular excited state with a probability as high as 0.97 if the duration of excitation is set to a half Rabi period. The red-shifted fluorescence is spatially and spectrally filtered, using respectively a pinhole and a bandpass filter centered on 700 nm with a width of 100 nm, then sent onto an avalanche photodiode (APD). b) Temporal response of the fluorescence signal under resonant laser pulses shaped with the EOM alone with an intensity 𝐼 = 6100 𝐼 !"# . For such excitation intensity, there is no thermally induced decoherence, as demonstrated in the saturation plot of Fig. 1. The time trace is monitored by time-correlated single photon counting, and its origin of time corresponds to the rising edge of the EOM-shaped pulse. The red curve is the simulated evolution of the excited-state population, taking a rectangular shaped illumination profile, 𝑇 % = 2𝑇 $ and Ω & 𝑇 $ = 55.2. c) The theoretical fidelity is derived from the time evolution of the master equation calculated for a flat-top pulse for a given on-off intensity ratio 𝜅 and corresponds to the maximum value reached by the population of the excited state. The black points are associated to the experimental configurations with the EOM alone (𝜅 ~ 8×10 3 ) and with both EOM and AOM (𝜅 ~ 3×10 5 ). With both EOM and AOM, we achieve the maximal fidelity obtainable with this excitation intensity. d) Optical nutation recorded under resonant excitation slots shaped with both the EOM and the AOM, at 𝐼 𝐼 !"# ⁄ = 6100. The red curve is the simulated excited-state population, using a realistic temporal profile for the excitation (see Fig. S7). The fluorescence time traces b, d are scaled to the excited-state population, after background subtraction (Fig. S3, S4) and deconvolution from the response function of the avalanche photodiode (Fig. S5).

IV. COHERENT MANIPULATION

Coherent manipulation of the molecular electronic state, such as population inversion or superposition of the ground and excited states with maximal coherence, requires controlled pulse areas 𝒜 = Ω ) 𝜏 * , where 𝜏 * is the duration of the flat-top excitation pulse and Ω ) is the Rabi frequency. With a view to fast coherent state preparation, we have submitted the molecule to a series of resonant pulses with fixed duration 𝜏 * = 1 ns and increasing excitation intensities, while collecting the fluorescence signal during the decay following each pulse, as sketched in Fig. 3a. Figures 3b-d ⁄ , 𝜋 and 2𝜋 respectively, while the laser frequency is locked at resonance on the ZPL within 1 MHz. In the case of pulsed excitation with 𝒜 = 𝜋 2 ⁄ (Fig. 3b), the population reaches ~0.5 after the pulse, which is consistent with the creation of a superposition between the molecular ground and excited states with maximal coherence. For pulsed excitation with 𝒜 = 𝜋 (Fig. 3c), the population of the molecule is inverted with a probability 0.97 at the end of the pulse. In the case of a 2𝜋 pulse (Fig. 3d), the molecular state is brought back to the ground state at the end of the pulse. The residual tail observed in the fluorescence temporal profile is explained by the finite slope of the AOM falling front (green profile in Fig. 3a), which causes residual excitation of the molecule at the end of the pulse. The evolution of the averaged fluorescence signal with increasing excitation intensities is shown in Fig. 3e (blue points). It is mainly characterized by oscillations, which reflect the tunability of the excited-state population after a controlled number of Rabi oscillations. Additionally, the heights of the minima increase with the pulse intensity due to the increased residual pumping of the molecule at the end of the pulse. Overall, the evolution of the population is reproduced with an exponentially decaying sinusoid with a linear offset (red curve in Fig. 3e), as in previous reports [START_REF] Gerhardt | Coherent state preparation and observation of Rabi oscillations in a single molecule[END_REF][START_REF] Allen | Optical Resonance and Two-Level Atoms[END_REF] but with a higher modulation depth. The fluorescence signal is scaled with that collected after a strong and long pulse producing a highly saturated population level (0.5) at its end. e) The normalized fluorescence signal, averaged over many pulse sequences, is displayed as a function of the pulse area. The data are fitted to an exponentially decaying sinusoid with a linear offset (red curve).

V. RAMSEY SPECTROSCOPY

In order to characterize the robustness of the molecular transition dipole against decoherence induced by electron-phonon interaction, we have investigated the temperature dependence of the dephasing rate using single-emitter Ramsey spectroscopy [START_REF] Press | Complete quantum control of a single quantum dot spin using ultrafast optical pulses[END_REF][START_REF] Zhou | Coherent control of a strongly driven silicon vacancy optical transition in diamond[END_REF][START_REF] Wang | Optical Ramsey spectroscopy on a single molecule[END_REF][START_REF] Stufler | Ramsey Fringes in an Electric-Field-Tunable Quantum Dot System[END_REF][START_REF] Becker | Long Exciton Dephasing Time and Coherent Phonon Coupling in CsPbBr 2 Cl Perovskite Nanocrystals[END_REF] . The evolution of the Bloch vector under the application of two pulses with duration 𝜏 * and interpulse delay 𝜏 is illustrated in Fig. 4a, where u and v respectively correspond to the in-phase and in-quadrature responses of the dipole to the driving laser field, while w is the population inversion between the excited and ground electronic states of the molecule. Between the two pulses, the Bloch vector undergoes a precession that is governed by the laser detuning 𝛿 ) with respect to the molecular resonance. As a consequence, the excited-state population obtained at the end of the second pulse oscillates with 𝛿 ) . This generates Ramsey fringes of the fluorescence signal as the laser frequency is swept across the molecular resonance, as exemplified in Fig. 4b for 𝜏 + = 1 ns and 𝜏 = 2.7 ns. In the case of infinitely short pulses, the fringe period is expected to evolve as 1 𝜏 ⁄ [START_REF] Wang | Optical Ramsey spectroscopy on a single molecule[END_REF] . Our calculations based on Bloch equations indicate that the Ramsey fringe periodicity should actually evolve as 1 ?𝜏 + 4𝜏 + 𝜋 ⁄ @ ⁄ , as demonstrated in Supplementary Note 1. This evolution is verified experimentally as the delay 𝜏 is increased (Figs. 4 b-d). Our numerical simulations of the excited-state population further support this analysis and reproduce well the experimental Ramsey fringes (red curves in Figs. 4 b-d).

Figure 4: a) The top frame exemplifies the evolution of the Bloch vector of a two-level system initially set in the ground state (Bloch vector pointing downwards) and submitted to two consecutive pulses. The blue line represents the evolution of the Bloch vector during the first pulse; the green curve traces its free evolution between the pulses, while the red curve depicts its trajectory during the second pulse. At the end of this pulse sequence, the excited state population is sensitive to the pulse area, the laser detuning and the delay. While the derivation of 𝑇 ! from the ZPL width may be distorted by spectral diffusion, single-molecule Ramsey spectroscopy provides a direct determination of 𝑇 ! . Indeed, the fluorescence signal measured after the second pulse decays with the delay 𝜏 between the two pulses, as a result of the decay of the coherence prepared by the first pulse, as illustrated in Fig. 5a. This decay time is exactly 𝑇 ! in the case of resonant 𝜋 2 ⁄ pulses [START_REF] Wang | Optical Ramsey spectroscopy on a single molecule[END_REF] . The upper inset of Fig. 5b shows the decay of the fluorescence signal as a function of 𝜏 at 3.2 K. The exponential fitting curve provides 𝑇 ! = 14.7 ± 0.7 ns, which reaches its upper limit 2𝑇 " and thus shows evidence for the absence of decoherence of the molecular transition dipole at this temperature. When raising the temperature to 9.8 K, 𝑇 ! drastically shortens to 1.6 ns, as evidenced in the lower inset of Fig. 5b. The temperature dependence of the dephasing rate (𝜋𝑇 ! ) $" measured with Ramsey spectroscopy increases exponentially with temperature, as shown in Fig. 5b (blue points). It is well reproduced with an Arrhenius law, with an activation energy 𝐸 , = 42.9 ± 0.8 cm -1 attributed to a quasi-local mode identified in the fluorescence spectrum [START_REF] Tamarat | Ten Years of Single-Molecule Spectroscopy[END_REF][START_REF] Jelezko | Pump-probe spectroscopy and photophysical properties of single di-benzanthanthrene molecules in a naphthalene crystal[END_REF] . This dephasing rate matches perfectly with the ZPL width (black points in Fig. 5b), which demonstrates the absence of spectral diffusion for this molecule.

VI. CONCLUSIONS

Interestingly, our Ramsey measurements of 𝑇 ! point to a vanishing dephasing rate at temperatures lower than 5 K, which are readily achievable with standard cryogen-free cooling technologies. This makes single molecules attractive electronic-state qubit systems that can be manipulated with near-unity fidelity and at rates much higher than the dephasing rate. Further studies will aim at exploring the coherence lifetime of entangled electronic states of dipoledipole coherently coupled molecules [START_REF] Hettich | Nanometer Resolution and Coherent Optical Dipole Coupling of Two Individual Molecules[END_REF][START_REF] Trebbia | Tailoring the superradiant and subradiant nature of two coherently coupled quantum emitters[END_REF] . In particular, the long-lived subradiant state that can be selectively excited [START_REF] Trebbia | Tailoring the superradiant and subradiant nature of two coherently coupled quantum emitters[END_REF] should open up extended coherence times for various quantum information schemes [START_REF] Zhang | Theory of Subradiant States of a One-Dimensional Two-Level Atom Chain[END_REF][START_REF] Guimond | Subradiant Bell States in Distant Atomic Arrays[END_REF] .

Figure 1 :

 1 Figure 1 : a) Illustration of the single-molecule microscopy setup. The naphthalene matrix doped with DBATT is crystallized on a glass coverslip. A microscope objective focuses the laser to excite the molecules and collect its red-shifted fluorescence. b) Fluorescence excitation spectrum of a single DBATT molecule at 3.2 K, recorded in the low saturation regime (𝐼 𝐼 !"# ⁄ ~ 0.2). Its signal-to-background ratio is greater than 200. The spectrum is averaged over 40 spectra (5 ms per bin) that have been centered to eliminate residual broadening by low-frequency spectral diffusion. The inset is a diagram of the ground and first excited electronic levels of the molecule, as well as the vibrational levels involved in the fluorescence transitions. c) Evolution of the homogeneous spectral broadening with increasing excitation intensities for the same molecule as in b). The red curve is a fit with the saturation law. The inset shows the fluorescence decay subsequent to a resonant excitation shaped with a single EOM (30 ns duration, start at time 0, 𝐼 𝐼 !"# ⁄ ~ 5), for the same molecule. Fitting with an exponential decay (red curve) is made several nanoseconds after the falling edge of the excitation to avoid deconvolution with the EOM falling edge. The exponential fit provides the excited-state lifetime 𝑇 $ = (7.2 ± 0.3) ns.

Figure 2 :

 2 Figure 2 : Fast, sub-ns optical nutation of a single molecule. a) Optical setup designed to sculpt optical pulses in a single-frequency excitation laser beam, whose frequency is controlled with a wave meter to within 1 MHz. A confocal microscope placed inside a cryostat is used to excite and collect the fluorescence from a single DBATT molecule.The red-shifted fluorescence is spatially and spectrally filtered, using respectively a pinhole and a bandpass filter centered on 700 nm with a width of 100 nm, then sent onto an avalanche photodiode (APD). b) Temporal response of the fluorescence signal under resonant laser pulses shaped with the EOM alone with an intensity 𝐼 = 6100 𝐼 !"# . For such excitation intensity, there is no thermally induced decoherence, as demonstrated in the saturation plot of Fig.1. The time trace is monitored by time-correlated single photon counting, and its origin of time corresponds to the rising edge of the EOM-shaped pulse. The red curve is the simulated evolution of the excited-state population, taking a rectangular shaped illumination profile, 𝑇 % = 2𝑇 $ and Ω & 𝑇 $ = 55.2. c) The theoretical fidelity is derived from the time evolution of the master equation calculated for a flat-top pulse for a given on-off intensity ratio 𝜅 and corresponds to the maximum value reached by the population of the excited state. The black points are associated to the experimental configurations with the EOM alone (𝜅 ~ 8×10 3 ) and with both EOM and AOM (𝜅 ~ 3×10 5 ). With both EOM and AOM, we achieve the maximal fidelity obtainable with this excitation intensity. d) Optical nutation recorded under resonant excitation slots shaped with both the EOM and the AOM, at 𝐼 𝐼 !"# ⁄ = 6100. The red curve is the simulated excited-state population, using a realistic temporal profile for the excitation (see Fig.S7). The fluorescence time traces b, d are scaled to the excited-state population, after background subtraction (Fig.S3, S4) and deconvolution from the response function of the avalanche photodiode (Fig.S5).

  exemplify fluorescence time traces of the molecule under pulsed excitation with 𝒜 = 𝜋 2

Figure 3 :

 3 Figure 3: a) Sketch of the pulse sequence applied to the molecule for coherent state preparation. An AOM pulse (green profile) is superimposed on each EOM pulse (square-shaped intensity profiles in blue) to improve the onoff extinction ratio and reduce the stationary population of the excited state to less than 0.02 in the off periods. The black curves illustrate the temporal evolution of the excited-state population. The shaded light grey areas correspond to the periods (duration 7 ns) where the fluorescence signal is integrated. The pulses are separated by a 70 ns delay to let the system relax to its ground state. b)-d) Normalized time traces of the fluorescence signal under pulsed excitation in the cases 𝒜 = 𝜋 2 ⁄ (b), 𝒜 = 𝜋 (c) and 𝒜 = 2𝜋 (d).The fluorescence signal is scaled with that collected after a strong and long pulse producing a highly saturated population level (0.5) at its end. e) The normalized fluorescence signal, averaged over many pulse sequences, is displayed as a function of the pulse area. The data are fitted to an exponentially decaying sinusoid with a linear offset (red curve).

  Figure4: a) The top frame exemplifies the evolution of the Bloch vector of a two-level system initially set in the ground state (Bloch vector pointing downwards) and submitted to two consecutive pulses. The blue line represents the evolution of the Bloch vector during the first pulse; the green curve traces its free evolution between the pulses, while the red curve depicts its trajectory during the second pulse. At the end of this pulse sequence, the excited state population is sensitive to the pulse area, the laser detuning and the delay. b)-d) Ramsey fringes are obtained from the fluorescence signal when the laser is scanned through the ZPL transition, while series of pulse pairs with fixed intensity and duration (𝜏 ' = 1 ns) are applied, using various delays 𝜏: 𝜏 = 2.7 ns in b), 𝜏 = 4 ns in c), 𝜏 = 10 ns in d). The corresponding periods of the Ramsey fringes are 250 MHz, 187 MHz and 86 MHz, respectively, in accordance with the expected period given by 1 2𝜏 + 4𝜏 ' 𝜋 ⁄ 4 ⁄ , considering a flat-top pulse.

Figure 5 :

 5 Figure 5: a) Scheme of the Ramsey pulse sequence (red bars), together with the evolution of the excited-state population (black curve) of a two-level system. The grey shaded areas after the second pulse indicate the periods over which the fluorescence signal is integrated to evaluate the excited state population. b) Evolution of the ZPL width (black points) and the dephasing rate (𝜋𝑇 % ) ($ measured from Ramsey spectroscopy (blue points), when temperature is increased. These evolutions match very well and are fitted with an Arrhenius law (𝜋𝑇 % ) ($ (𝑇) = (𝜋𝑇 % ) ($ (0) + 𝐴 𝑒𝑥𝑝(-𝐸 ) 𝑘 * 𝑇 ⁄ ), with 𝐴 = 92.2 GHz and 𝐸 ) = 1287 GHz, while (𝜋𝑇 % ) ($ (0) is set to Δ𝜈 + = 22.3 MHz (red curve). The upper and lower insets respectively display the decay of the fluorescence signal at 3.2 K and 9.8 K as a function of the inter-pulse delay 𝜏. The exponential fits (red curves) provide a measure of 𝑇 % : 14.7 ± 0.7 ns at 3.2 K and 1.6 ± 0.2 ns at 9.8 K.

Acknowledgments :

The authors acknowledge the financial support from the French National Agency for Research, Région Aquitaine (FAWN, 2018-1R50313), Institut Universitaire de France (B. Lounis), Idex Bordeaux (LAPHIA Program, Risky-MOmENTuM-AAP2020), the EUR Light S&T Graduate Program (PIA3 Program "Investment for the Future", ANR-17-EURE-0027), and GPR LIGHT (LIGHT-006-IMOON-2021). We thank Eric Cormier for the loan of the arbitrary wave generator (AWG7122C).

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Supplementary material :

See the supplementary material (Figures S1 to S7) for experimental details and the supplementary note 1 for the derivation of Ramsey fringe periodicity.

Author declaration:

Conflict of Interest

The authors have no conflicts to disclose.


Structural and acoustic response of sandwich plates by means of sublaminate models with variable kinematics

Lorenzo Dozio * , Riccardo Vescovini * , Michele D'Ottavio † * Department of Aerospace Science and Technology Politecnico di Milano via La Masa, 34, 20156 Milano, Italy [lorenzo.dozio, riccardo.vescovini]@polimi.it † LEME, UPL Université Paris Nanterre 50 rue de Sèvres, 92410 Ville d'Avray, France michele.d_ottavio@parisnanterre.fr

Summary

Introduction. Due to their high strength to weight ratio, sandwich panels are adopted as structural members in several engineering applications, such as space launchers and ship hulls. They can be considered a particular case of composite structures, featuring a fundamental pattern of two stiff and thin outer sheets, the skins, enclosing a relatively thick and compliant layer, the core. As opposed to the case of thin-walled panels, a complicating effect of sandwich panels is due to the strong mismatch of geometric and constitutive properties along the thickness, which induces sharp gradients across the thickness direction. For this reason, specific ad-hoc structural models are typically required to obtain accurate and reliable predictions of their mechanical behavior. The evaluation of the acoustic performance of sandwich panels can be also crucial in some applications. Indeed, interior noise quality and speech intelligibility has become increasingly important in modern light and fast transportation vehicles, where novel arrangements are designed through the adoption of sandwich configurations to achieve high structural efficiency. In particular, various solutions have been studying in the aerospace industry with the aim of improving the cabin acoustic comfort, especially for VIP aircrafts and helicopters. A common way to reduce the noise radiated by a vibrating fuselage panel is to add viscoelastic materials. The most efficient solution is the so-called constrained-layer damping treatment, where a viscoelastic layer is inserted between the vibrating structure and a constraining rigid layer, thus leading to a sandwich configuration. Another effective approach for noise reduction is to adopt inner fuselage panels, called trim panels, specifically designed for optimizing the acoustic absorption or the transmission loss (TL). Trim panels have typically a classical sandwich multilayer structure with stiff facesheets and a thick low-density core.

Motivation. In both previous approaches, the material and thickness of core and skins of the sandwich panel, the number of layers of external sheets, the topology of the core and the total and relative thickness are, among others, important design parameters affecting the acoustic performances of the structural component. The acoustic design can be further complicated by the adoption of less-conventional and innovative configurations involving multiple cores. Since an extensive experimental campaign is lengthy and costly, suitable numerical simulations can be carried out to guide the design process, i.e., to evaluate the effect of the main parameters and to study the nature and arrangement of layers. However, accurate 2-D modeling of multilayered sandwich panels for vibration and acoustic analysis is challenging, mostly due to their highly heterogeneous anisotropic constitution in the thickness direction and the wide frequency range of interest. In particular, it can be difficult to identify sufficiently simple yet reliable theoretical models without unnecessary complexity.

Mathematical formulation. In this work, a powerful and effective numerical tool for the acoustic analysis of composite panels of finite extent having classical and complex advanced sandwich configurations is presented [START_REF] D'ottavio | Bending analysis of composite laminated and sandwich structures using sublaminate variable-kinematic Ritz models[END_REF]. Main feature of the proposed formulation is the versatile kinematic representation of the displacement field along the thickness direction, consisting in the representation of the plate by means of sublaminates, i.e. arbitrary groups of adjacent plies composing the panel. Each sublaminate is associated with an independent and arbitrary kinematic description, either equivalent single-layer or layerwise, so that the use of refined high-order theories can be tailored to specific thickness subregions. Accordingly, each component of the 3-D displacement field in the generic ply p of the sublaminate k is postulated in a layerwise (LW) manner as follows:

u • (x, y, z p ,t) = F α u• (z p )u p,k •α u• (x, y,t) α u • = 0, . . . , N k u • (• = x, y, z) (1)
where z p is the local ply-specific thickness coordinate, F α u• (z p ) are thickness functions, u p,k •α u• is the kinematic coordinate of the adopted 2-D approximation, N k u • is the order of expansion taken as a free parameter, and the summation is implied for repeated theory's indexes α u • . It is noted that equivalent single-layer (ESL) sublaminate model can be recovered by setting z p equal to the sublaminate-specific thickness coordinate z k . The thickness functions are taken as a proper combination of Legendre polynomials so that the continuity between adjacent plies or sublaminates is easily imposed. The equilibrium equations are derived through the Principle of Virtual Displacements (PVD). Once a specific plate theory is postulated through Eq. ( 1), the corresponding displacement approximation is substituted into the PVD equation so that the original 3-D problem is transformed into a 2-D problem in the xy plane. The resulting variational form contains 2-D generalized kinematic coordinates, which are further expressed through a Ritz expansion as follows:

u p,k •α u• (x, y,t) = N u • i (x, y)u p,k •α u• i (t) i = 1, . . . , M (• = x, y, z) (2)
where N u • i (x, y) form a complete set of admissible functions, which are taken as the product of Legendre polynomials and proper boundary functions after defining the problem in the computational domain (ξ , η), with ξ , η ∈ [-1, 1]. After substituting Eq. ( 1) and Eq. ( 2) into the PVD, the discretized weak form of the dynamic equilibrium equations can be expressed in compact form by means of self-repeating building blocks, denoted as fundamental kernels of the formulation, which are invariant with respect to the number of sublaminates, the typology of the local kinematic description (ESL or LW) and the orders of expansion of each local displacement quantity. Accordingly, the proposed approach allows for the hierarchical generation of plate models with different 2-D kinematic descriptions from the same unified mathematical framework. The expansion and assembly procedure involves four main steps. The first step deals with the expansion of the kernels according to the summation related to the order of the kinematic description postulated in each sublaminate. The second step is the assembly of the ply-contributions in each sublaminate involving a cycling over the index p. All sublaminate contributions are subsequently stacked along the thickness coordinate to account for the continuity of the generic displacement variable at the interfaces between adjacent layers. The sublaminate contributions of different layers are always assembled in a LW manner. The assembly of the sublaminates contributions involves the cycling over the index k. The final step deals with expansion corresponding to the summation related to the Ritz series approximation of the kinematic quantities. Once the final set of equations governing the structural problem is obtained, the acoustic performance of the panel can be evaluated.

In particular TL simulations are performed by assuming the plate in an infinite rigid baffle with a negligible air loading and imposing a diffuse field on one side of the panel through a set of incident plane waves of same amplitude and different incidence angle (θ , φ ). The incident pressure field on the top side of the panel can be expressed as p = 2e -jk sin θ (x cos φ +y sin φ ) where k = ω/c 0 is the wavenumber. For each incident wave, the incidence transmission coefficient is computed as

τ(ω, θ , φ ) = 2ρ 0 c 0 P(ω, θ , φ ) S cos θ ( 3
)
where S is the panel area and the radiated sound power is evaluated in terms of elementary radiators [START_REF] Fahy | Sound and Structural Vibration[END_REF]. Finally, the diffuse transmission loss is computed as TL(ω) = -10 log(τ d (ω)), where τ d is the diffuse transmission coefficient derived by integrating the response of all incident plane waves over the incident angle and weighting them with the solid angle to account for the directional distribution. Illustrative example. Some vibration and TL results on different sandwich plates will be presented and discussed at the Symposium, including also comparison with experimental data, to show the effectiveness of the proposed approach. An illustrative example is here reported in Figure 1, which shows the stacking sequence of an acoustic trim panel of size 0.90 × 0.90 m 2 , thickness 21.7 mm, with clamped boundary conditions and composed of melamine foam placed between Nomex honeycombs and external fiberglass layers. The simulation of the TL and the experimental measurements are compared in Fig. 2. The numerical model is built by properly subdividing the plate into five sublaminates having first-order kinematics in the outer stiffer thickness regions and a higher-order normal and shear deformation theory in the internal soft layer. The high TL is assured thanks to a dilatation effect of the foam from medium frequencies and the static bending stiffness of the honeycombs. The peculiar behavior of a double wall resonance around 600-700 Hz is observed and well predicted by the simulation using a model suitably tuned to give an optimal balance between accuracy and computational cost.

Figure 1 :

 1 Figure 1: Stacking sequence of a sandwich acoustic trim panel.

Figure 2 :

 2 Figure 2: Diffuse transmission loss: comparison between numerical predictions and experimental data.


A Seguin

C Coulais

F Martinez

Y Bertho

P Gondret

Local rheological measurements in the granular flow around an intruder

Keywords: numbers: 45, 70, -n 83, 80, Fg

The rheological properties of granular matter within a two-dimensional flow around a moving disk is investigated experimentally. Using a combination of photoelastic and standard tessellation techniques, the strain and stress tensors are estimated at the grain scale in the time-averaged flow field around a large disk pulled at constant velocity in an assembly of smaller disks. On the one hand, one observes inhomogeneous shear rate and strongly localized shear stress and pressure fields. On the other hand, a significant dilation rate, which has the same magnitude as the shear strain rate, is reported. Significant deviations are observed with local rheology that justify the need of searching for a non-local rheology.

I. INTRODUCTION

The description of the flow properties of granular material remains a true practical and fundamental challenge. This challenge comes from the inherent non linear interactions betweens the grains and from the small separation of length scales. As a matter of fact, no (attractive) force exists between two grains when the contact is broken and the flow scale is often not much larger than the elementary grain scale. Many tools have been developed over the past decades to tackle this issue and extract the kinematic properties of grain flows, underlying the open problem of defining a single rheology for granular matter. In the past decade, a local rheology emerges based on a unique relation of the friction coefficient µ with the socalled inertial number I corresponding to the shear rate made dimensionless with confining pressure and density [1,2]. This rheology works quite well by rescaling various experimental and numerical data into a consistent picture for parallel flows -such as Couette flows or flows down inclines -or weakly non parallel flows in a wide range of flow rates [1,3,4]. Nevertheless, this local description falls short of describing non-parallel flows where the streamlines are far from parallel and also quasi-static regimes close to the "liquid-solid" transition [5]. Nonlocal effects, where the stress not only depends on the strain rate but also on its spatial variations, have been recently reported both experimentally and numerically [6][7][8]. These effects may arise from plastic rearrangements inducing long range correlations and cooperative behaviors. Non-local modeling for granular flows is thus now considered to extend the µ(I) rheology by the introduction of a diffusive term [8][START_REF] Henann | Proc. Natl. Acad. Sci[END_REF][START_REF] Henann | [END_REF][11]. This captures well the average flow fields reported in some numerical simulations and experiments where the flow is observed to be strongly inhomogeneous. Such developments were inspired by kinetic theory, where non-locality arises from the diffusion of a granular temperature into a granular flow [12,13].

In this paper, we present quantitative measurements in the two-dimensional flow of small photoelastic disks around a larger intruder disk, where the local strain rate and the local stresses are recorded simultaneously. This allows one to test the local rheology framework in the case of a strongly non-parallel flow. The four invariants of the local strain rate tensor and local stress tensor reveal that the flow is strongly localized around the intruder. A tentative rheological analysis sheds light on a significant amount of dilation and some spatial variations of the friction coefficient that bring out new challenges for modeling.

II. EXPERIMENTAL SETUP

The experiments consist in pulling an intruder disk at constant velocity V 0 into a two-dimensional assembly of about 8 × 10 3 smaller photoelastic disks in a set-up sketched in Fig. 1(a) which is adapted from [14]. This granular assembly is a bidisperse mixture of disks of diameters 4 and 5 mm to avoid any possible crystallization, and made of polyurethane (density ρ s = 1280 kg/m 3 ) which is photoelastic. These disks are placed in between two horizontal glass plates with a gap thickness of 4 mm, slightly larger than the disk height h = 3.2 mm, and closed by lateral walls that form a square area of L x ×L y = 400×400 mm 2 ≃ 90×90 d 2 g , where d g = 4.5 mm is the mean diameter of the granular assembly. By varying the number of disks, the global packing fraction φ 0 of the model granular medium has been varied in the range 0.76 ≤ φ 0 ≤ 0.80 that corresponds to a dense packing but remains always significantly smaller than the jamming point φ J ≃ 0.83 [14]. The larger intruder disk of diameter d = 16 mm corresponding to about 4d g is pulled in the x-direction with a steel wire attached to a linear stepper motor at a constant velocity V 0 in the range 10 -4 ≤ V 0 ≤ 3 mm s -1 . In order to prevent the intruder from tilting and to ensure minimal interaction between the wire and the grains, the steel wire is welded on the top face of intruder and we designed the intruder slightly taller than the disks. As a result, the overhead wire hardly sags and has minimal contact with the grain assembly. Since the wire is very stiff, the intruder does not tilt more than a few degrees. The steel wire is essentially inextensible and ensures that the pulling device does not deform when the intruder travels across the medium. A piezoelectric sensor records the time evolution of the drag force F (t) exerted by the granular medium onto the intruder disk during its displacement. Once the intruder placed in the granular medium at one side of the cell at its initial position (X i , Y i ) ≃ (-0.4L x , 0) and before its pulling motion to the other side of the cell towards its final position (X f , Y f ) ≃ (0.4L x , 0) close to the front wall, the whole packing is first homogenized by horizontal vibrations of the bottom plate in the transverse y-direction. All experiments presented here were performed at constant volume with a fixed cell size (constant φ 0 ). During the intruder displacement, snapshots are recorded using a high resolution camera (2048 × 2048 pixels) at a frame rate f such as V 0 /f < d g /10. To avoid any possible boundary effect in the results, images and subsequent data analysis are restricted to a central square region of interest of size 160 × 160 mm 2 ≃ L x /2.5 × L y /2.5, leading thus to a spatial resolution of about 0.1 mm ≃ d g /45. Each grain is thus composed by about 1.6 × 10 3 pixels in the images. Since the forces at stake are of the order of 0.1 N, the in-plane compression is smaller than 0.01%, so that we consider a constant diameter in the detection algorithm. The granular packing is backlit with a large uniform source of linear polarized light and a switchable circular analyzer is mounted on a stepper motor synchronized with the picture acquisition. Such a device gives one alternated cross-polarized and direct light pictures at the effective frame rate of f /2. Assuming little change in the packing between two successive pictures, one combines structural information extracted from the direct light pictures [Fig. 1(b)] and stresses measurements extracted from the cross-polarized pictures [Fig. 1(c)] to quantify stress and strain tensors at the grain scale. Considering the flow geometry, the results will be presented in polar coordinates (r, θ) centered on the intruder, with θ = 0 in the x-direction of the intruder motion.

III. IMAGE ANALYSIS

For each direct image, we extract the position of each grain including the intruder. By tracking each grain between two successive images, we compute its velocity from its displacement. To limit noise, the velocity gradient tensor ∇v at the grain location is not calculated by differentiation but by integration along a closed contour around the grain [15,16] :

∇v = 1 S v ⊗ n dl, (1)
where S is the surface delimited by the corresponding contour, dl is the elementary displacement on this contour for which the outward normal unit vector is n, and ⊗ denotes the tensor product. Here, as each grain is surrounded by a finite and discrete number of neighbors, we compute the integral of Eq. ( 1) along the edges of its Voronoï cell. The strain rate tensor G is then deduced for each grain by

G = 1 2 ∇v + ∇v T , (2)
where the exponent T refers to the transpose matrix. The strain rate tensor is finally interpolated along a cartesian grid of step dx = dy = 0.4 mm ≃ d g /10.

Another important part of the image processing is the computation of the stress tensor σ. From each direct image, the Delaunay triangulation and the Voronoï tessellation are constructed to extract the neighbor network and contact network of each grain at each time t. From the cross polarized image at the same given time t (more exactly that only differs by 1/f ), we determine then the contact forces between grains by calculating the spatial gradient of the intensity inside triangular sectors defined by one particle center and the corresponding side of the Voronoï cell [17]. The stress tensor for each grain is then computed as [15,18]

σ = 4 πd 2 i i =j f ij ⊗ r ij , (3)
where f ij is the contact force between the two grains i and j in contact and r ij is the vector of modulus d i /2 pointing from the center of grain i of diameter d i towards the center of grain j. The stress tensor σ is then interpolated on the same Cartesian grid as for the strain rate tensor G. Note that the present 2D stress is a force per unit length and not a force per unit area as the real stress σ/h would be. Finally both strain rate and stress tensors G and σ are decomposed into an isotropic part and a deviatoric part [START_REF] Landau | Theory of Elasticty[END_REF] :

G = εI + G d , (4)
σ = -pI + σ d , (5)
where I is the unit tensor, ε = 1 2 tr(G) = 1 2 div v is the dilation rate of the 2D grain flow, G d is the shear rate tensor, p is the pressure and σ d is the shear stress tensor. In the following, we shall present the results using only scalar quantities corresponding to the invariants of the strain rate tensor G and the stress tensor σ. The first invariants are the dilation rate ε and the pressure p and the second invariants corresponding to the deviatoric parts of the two tensors are the shear rate γ and the shear stress τ as named usually in rheology [START_REF] Andreotti | Granular media : between fluid and solid[END_REF] and defined as :

γ = 1 2 tr(G 2 d ) 1/2 , (6)
τ = 1 2 tr(σ 2 d ) 1/2 . ( 7
)
The drag force F on the intruder can be inferred from the image analysis described above by the integration of the local stress tensor around the intruder :

F = -σn • e x dl. (8)
For the image analysis, we leave out the images where the intruder is too close to the edges of the image and only focus on the images where the intruder is in a central zone (-60 X 60 mm, i.e. -L x /7 X L x /7). This allows one to have local information in the granular medium up to at least 22 mm ≃ 5d g away from the intruder center. This should be a large enough zone of analysis as the radial extension of the velocity field perturbation is expected to be here of about 2d g + d/4 ≃ 13 mm ≃ 3d g [START_REF] Seguin | [END_REF].

The time evolution of the drag force F (t) extracted from the image analysis [Eq. ( 8)] is reported in Fig. 2(a) together with the global measurements from the piezoelectric sensor. Both show strong fluctuations that are undoubtedly associated with the creation/breaking of force chains inside the granular material but no significant slow variations. The fluctuations of these two signals are rather well time-correlated even if their amplitudes are different which may be related to the very different measurement technics. As a matter of fact, the force sensor measures not only the drag on the intruder but also the friction of the intruder with the bottom and the possible friction of the wire attached to the intruder. By contrast, the photoelastic technique only measures intergrain contact forces. Anyway, the absence of slow time variations suggests that a stationary regime is reached where the force is constant on average, in agreement with previous observations for the motion of objects in a granular material [13,[START_REF] Seguin | [END_REF][22][23][24]. In the following, we only focus on time-averaged quantities along the intruder motion in the central part of the cell. Figure 2(b) shows that the mean drag forces F obtained from the force sensor and from the photoelastic signal are very close to each other whatever the pulling velocity V 0 , which gives confidence in our local image analysis. For V 0 ranging within about five decades, F (t) is roughly constant, in agreement with the experiments mentioned above.

In the next section, we will focus on the time-averaged local quantities that are extracted from image analysis of both direct and polarized images. These different quantities characterizing the granular flow will be normalized by the natural length scale d g and time scale d g /V 0 . As there is no external stress applied on the lateral boundaries and no "hydrostatic pressure" from gravity in our configuration, the typical stiffness k = (π/4)Eh of the contact between two cylinders [START_REF] Popov | Contact mechanics and friction : physical principles and applications[END_REF] will be taken as the stress scale, where E is the effective Young modulus of the disks of height h with here the value k ≃ 1 N mm -1 .

IV. LOCAL MEASUREMENTS

A typical velocity field of the granular medium around the moving intruder is displayed in Fig. 3(a). As expected from previous kinematic description [13,[START_REF] Seguin | [END_REF], the velocity perturbation induced by the intruder motion is found to be strongly localized around it. The corresponding local packing fraction φ is displayed in Fig. 3(b) : φ is observed to be quite homogeneous in space and close to the global value φ 0 except in a very narrow crown upstream of the intruder and in an elongated triangular wake downstream where φ is significantly smaller. This downstream wake has been reported in detail by [START_REF] Kolb | [END_REF] in similar 2D experiments and follows directly from the absence of confining pressure on the cell boundaries or gravity, as the present setup is horizontal. The radial and azimuthal profiles of φ are displayed in Figs. 4(a) and (b) along one azimuthal or radial position respectively. The data points have been obtained by digitizing the 2D interpolated grids onto annular sectors (r -dr, r + dr), (θ -dθ, θ + dθ), with dr = d g /8 and dθ = π/32 Figure 4(a) shows that φ is quasi-constant and very close to φ 0 = 0.76 upstream of the intruder except in a very narrow zone of size one grain diameter close to the intruder where φ decreases significantly and abruptly with r. The azimuthal profile close to the intruder [Fig. 4(b)] shows that there is no significant variation of φ around the intruder in a large azimuthal range (-π/2 θ π/2) except in the downstream wake observed in Fig. 3(b).

The maps of the four scalar invariants ε, γ, p and τ extracted from the strain rate tensor and stress tensor respectively are displayed in Fig. 5. All these quantities vary significantly upstream of the intruder with some differences. Concerning the strain rate field [Fig. 5(a)], one observes a zone of high positive ε extending about one intruder diameter upstream and in a quite large azimuthal extension (-π/2 θ π/2), with thus a "banana shape". The shear rate γ in Fig. 5 banana shape with a radial upstream extension of about half the intruder diameter and a similar azimuthal extension. This correlation between ε and γ is related to the fact that dense granular media are known to dilate upon transient shear [27,28]. Finally the stress fields [Figs. 5(cd)] differ from the strain fields. A zone of high pressure p elongated in the upstream direction appears in front of the intruder and a zone of high shear stress τ of similar shape is also observed. There is thus a clear link between p and τ . The relation between stress and strain appears however less clear from these four maps.

To quantify more precisely the spatial variations of the strain and stress tensors, the radial and azimuthal profiles of the normalized four scalar invariants are displayed in Fig. 6 for the same global packing fraction (φ 0 = 0.76) but three intruder velocities V 0 ranging over 4 decades. As previously, the data points have been obtained by digitizing the 2D interpolated grids onto annular sectors (r -dr, r + dr), (θ -dθ, θ + dθ), with dr = d g /8 and dθ = π/32. The rather good collapse observed in Figs. 6(a-d) of the three curves for the three velocities shows that ε and γ scales with V 0 as expected. By contrast, Figs. 6(e-g) confirm that p and τ do not depend significantly on V 0 which is consistent with the fact that the measured drag force F does not depend significantly on V 0 in Fig. 2(b). In the lin-log plots of Figs. 6(a,c,e,g), the strong monotonic decrease of ε, γ and p with increasing radial distance from the intruder is in agreement with the exponential radial decrease already reported for γ [13]. By contrast, τ displays a non- monotonic evolution with a first increasing and then decreasing evolution with radial increasing distance from the intruder surface. Figures 6(b,d,f,h) show that the four invariants have a maximal value in the direction of motion (θ = 0) but quite different variation with θ. The variations of ε and γ are close to a cosine shape. Such a cosine shape has been already observed for γ in the granular flow around a cylinder in vertical penetration motions [13,[START_REF] Seguin | [END_REF] which is thus a consistent result. The θ variations of p and τ are in contrast far from a cosine. The present observed strong localization of p and τ with both r and θ may explain the very small length scale reported previously for the force interaction of a moving object with both bottom [24] or lateral [29] walls. Note that for the lower speed experiment (V 0 = 10 -4 mm/s) the radial variations of ǫ and τ in Figs. 6(a) and (g) are quite different from the others, which will be discussed in the next Section.

V. RHEOLOGICAL CURVES

Let us now look at the relations between the four scalar invariants ε, γ, p, and τ to search for the existence of a possible local constitutive law for the granular medium in the present non-parallel flow around a moving intruder. Despite some dispersion, Fig. 7(a) shows that the dilation rate ε increases with the shear rate γ. This correlation between ε and γ is reminiscent of the so-called Reynolds dilatancy [30], where dilation usually increases quadratically with shear strain ε ∼ γ 2 [27]. With also some dispersion, Fig. 7(b) shows that the shear stress τ increases with the pressure p which justifies the introduction and use of a friction coefficient. Note that we ensure that the principal directions of the shear rate and shear stress tensors are nearly parallel in the upstream region where a local dilatancy coefficient D and a friction coefficient µ will be computed, in agreement with the requirements pointed out by [4].

In order to test the possible local rheology, we now look at both relations of the dilatancy coefficient D = ε/ γ and the friction coefficient µ = τ /p with the inertial number I = γd g ρ s φ/p which is the ratio of the inertial time scale d g / p/ρ s φ at the grain scale and the flow time scale 1/ γ , and which has been shown to be the relevant dimensionless number in simple parallel flows [1][2][3]. The data shown in Figs. 8 and In some directions [θ 45 • in Figs. 9(a) and (b)], one may even observe a significant decrease of µ with increasing I at very small values (I 10 -5 ). This is linked to the differences reported in Fig. 6 for the lowest speed experiments (lowest I). This behavior is expected to arise from theoretical models [31] but was not yet reported experimentally, and may explain the stick-slip like behavior observed for intruder dragged at very low velocities [23].

The key point is however the significant dependence observed for µ with the azimuthal direction θ. Indeed, a significant decrease by a factor about 3 is observed from the direction of motion where µ ≃ 0.15 [Fig. 9(a) for θ ≃ 10 • ] towards the equatorial direction where µ ≃ 0.05 [Fig. 9(d) for θ ≃ 80 • ]. This strong θ variation of µ(I) means that such a local rheology fails here to describe the entire granular flow around an intruder. This experi-mental result is consistent with recent theoretical findings [START_REF] Henann | [END_REF] where the friction coefficient is found larger in front and behind the intruder and smaller on the sides. Our present result confirms experimentally these theoretical predictions and strongly suggests that non-local rheology is thus needed to fully describe non-parallel flows.

VI. CONCLUSION

The present paper demonstrates the possibility of computation of the local stress tensor together with the local strain tensor in a two-dimensional granular packing of photoelastic disks from a detailed analysis of crosspolarized images. This possibility is here used to test a possible local rheology in a strongly non-parallel flow consisting in the grain flow around a larger intruder disk. The analysis of the spatial variations of the four invariants of the strain rate tensor and stress tensor reveals that the flow is strongly localized around the intruder and that the stress is very localized in the direction of motion. A positive dilation rate increasing roughly linearly with the shear rate is put in light and allows one for the determination of a dilatancy coefficient D for the present stationary non-parallel granular flow. This dilatancy coefficient is essentially independent of the azimuthal direction θ. By contrast, the friction coefficient µ that relates shear to normal stresses shows significant θ dependence. This spatial dependence demonstrates that local rheology may be not sufficient to describe strong non-parallel flows such as the present flow around a cylinder. While such spatial variations of µ have been recently predicted by non-local modeling in intruder geometry [START_REF] Henann | [END_REF], the finite level of dilatancy D of stationary flows has not been reported yet theoretically or numerically. The present finding may thus put further constraints on the formulation of continuum models by opening up novel challenge in modeling and understanding non-parallel compressible granular flows using compressible non-local rheology. Experimentally, checking for non-local rheology is challenging as the computation of Laplacian terms -at the core of these non-local models [8,[START_REF] Henann | [END_REF] -is hampered by finite resolution and large fluctuations.

Similar experiments with an imposed vibration of the packing could be developed in order to uncouple the effect of the inhomogeneous excitement created by the pulling of the intruder and the effect of homogeneous excitement [32][33][34][35]. These studies should help to a better understanding of the drag force with depth in the case of the vertical motion under gravity [36,37] and the direction of the motion in the penetration-extraction problem [38,39] in order to be extended for the understanding of the animal locomotion in sand [40].

Figure 1 .

 1 Figure 1. (Color online) (a) Sketch of the experimental setup. (b) Snapshot of the intruder moving from left to right through the granular packing. (c) Same image obtained between crossed polarizers and displaying transient chains of contact forces.

Figure 2 .

 2 Figure 2. (Color online) (a) Instantaneous drag force F (t) on the intruder obtained either (--) by the force sensor or (-) by the image analysis from the photoelastic images for a intruder of diameter d = 16 mm in motion at the velocity V0 = 10 -2 mm s -1 in a 2D granular medium of packing fraction φ0 = 0.76. (b) Time-averaged drag force F as a function of V0. Data from (▽) force sensor and ( ) photoelastic images together with (--) the mean value F = 0.22 N averaged over the explored velocity range 10 -5 < V0 < 10 mm/s.

Figure 3 .

 3 Figure 3. (Color online) (a) Velocity field around an intruder disk of diameter d = 16 mm moving from left to right at the velocity V0 = 2.7 mm s -1 into a granular medium of global packing fraction φ0 = 0.76. Vectors indicate the grain velocity direction and the colormap encodes the modulus of the dimensionless velocity v/V0. (b) Corresponding local packing fraction φ.

Figure 4 .

 4 Figure 4. (a) Local packing fraction φ as a function of the radial position r/dg from the intruder along the azimuthal direction θ = (10 ± 3) • . (b) Local packing fraction φ as a function of the azimuthal position θ for the radial position r = (11 ± 1) mm (i.e. r/dg ≃ 2.4 ± 0.2 from the intruder). Data are obtained for d = 16 mm, V0 = 10 -2 mm s -1 , and φ0 = 0.76.

Figure 5 .

 5 Figure 5. (Color online) Maps of the four dimensionless scalar invariants of the (a,b) strain rate tensor and (c,d) stress tensor around an intruder of diameter d = 16 mm moving from left to right at the velocity V0 = 2.7 mm s -1 through a granular packing of global packing fraction φ0 = 0.76 : (a) dilation rate εdg/V0, (b) shear rate γdg/V0, (c) pressure p/k, and (d) shear stress τ /k.

)Figure 6 .

 6 Figure 6. Variations with the normalized radial distance r/dg along the azimuthal direction θ = (10 ± 3) • , and with θ for r = (11 ± 1) mm of the four dimensionless scalar invariants : (a,b) dilation rate εdg/V0, (c,d) shear rate γdg/V0, (e,f) pressure p/k, and (g,h) shear stress τ /k, for φ0 = 0.76 and V0 = 10 -4 (×), 10 -2 (•), and 1 mm s -1 (+). (-) Cosine function as a guideline for the eye. The grey zone corresponds to the intruder extension up to r = d/2 = 8 mm.

Figure 7 .

 7 Figure 7. (a) Dimensionless dilation rate εdg/V0 as a function of shear rate γdg/V0 and (b) dimensionless shear stress τ /k as a function of pressure p/k for an intruder of diameter d = 16 mm. Same data points as in Figs. 6(a, c, e, g) together with (△) for φ0 = 0.80 and V0 = 10 -2 mm s -1 .

  9 are the D, µ and I values averaged over the radial range 2 r/d g 6 along different azimuthal directions, from the motion direction near θ = 0 [Figs. 8(a) and 9(a)] towards the equatorial direction near θ = π/2 [Figs. 8(d) and 9(d)]. One observes that D does not depend significantly on I with about the value D = 2 ± 1 that remains about the same whatever the azimuthal direction θ. Concerning the local friction coefficient, one does not observe any significant increase of µ with I, which is due to the very low I values explored here corresponding to a quasi-static regime (I < 10 -2 ).

2 IFigure 8 .Figure 9 .

 289 Figure 8. Local dilatancy coefficient D as a function of the inertial number I with an average over the radial extension r/dg ≤ 6 along four azimuthal directions : (a) θ = 10 ± 3 • , (b) θ = 45 ± 3 • , (c) θ = 60 ± 3 • , and (d) θ = 80 ± 3 • . Same data symbols as in Fig.6together with V0 = 10 -3 ( ), 10 -1 (⋄), and 2.7 mm s -1 (▽), for φ0 = 0.76.

ACKNOWLEDGMENTS

We are grateful to V. Padilla and C. Wiertel-Gasquet for the development of the experimental setup and we thank J. Crassous and A. Lemaitre for fruitful dis-cussions. This work is supported by a public grant of the French National Research Agency (ANR) No. 2010-BLAN-0927-01 (STABINGRAM project) and supported by Triangle de la physique contracts No. 2011-075T and No. 2012-063T (COMIGS2D and REMIGS2D projects).