Veronique Conedera
email: conedera@laas.fr

Fabien Mesnilgrente
email: fabien.mesnilgrente@laas.fr

Magali Brunet
email: mbrunet@laas.fr

Mathias Borella
email: mathias.borella@altatech-sc.com

STUDY OF A HIGHLY LOCALIZED ACTIVATED CARBON DEPOSITION PROCESS USING INKJET PRINTING TECHNOLOGY

Keywords:

An activated carbon deposition technique based on inkjet printing is proposed for the fabrication of supercapacitors electrodes. By mastering the activated carbon suspension in ethylene glycol through the use of a surfactant and by functionalizing the substrate with OTS, 10µm wide structures were obtained with 50 µm ink jet nozzles.

I. INTRODUCTION

Supercapacitors come usually in roll-type or button cell type package [1] compatible with large-size systems. For low profile applications, integration of supercapacitors on silicon substrate can be of great interest. Microtechnology approaches have been proposed in the literature [2,3]. One of the difficulties lies in the deposition of activated carbon. As a result, a deposition process based on inkjet printing was developed with the objective of limiting the number of steps needed in terms of technology, low-temperature elaboration and collective treatment. Another challenging task is the integration of the highest number of devices onto the smallest possible surface. Unfortunately, a major constraint imposed by the state of the art inkjet technology is the limited diameter of the nozzle head orifice to 15 µm. Thus, the use the activated carbon powders with particle diameters as high as 10µm can hardly be considered and jetting becomes almost impossible due to inevitable clogging caused by the accumulation of particles in the nozzle of the print head. To circumvent these obstacles, a regular 50µm diameter nozzle was selected and work focused on maintaining the particles in suspension in the emulsion by adding surfactant Triton X 100 and on locating the emulsion during the impact on the gold electrode by means of a localized OTS (otadecyltrichlorosilane) hydrophobic treatment on silicon oxide.

II.IMPLEMENTATION TECHNOLOGY

A..Ink preparation

Ink is an ethylene glycol emulsion of activated carbon and 5% DuPont PTFE aqueous solution which provides the ligant for carbon particles [1]. Surfactant concentration was optimized by measuring the contact angle on the OTS-treated silicon surface and on a gold contact. Figure 1 shows how this angle changes with Triton concentrations from 0.0001% to 5% in weight. Likewise, we have monitored the behaviour of the solution in test tubes. Following storage for 48 hours, three behavioural types were found:

-from 0.0001 % to 0,1%, three phases were observed (ie, an upper transparent phase, an intermediate homogeneous phase and a lower deposition phase), -from 0.1 % to 1 % , two phases were noted (an upper homogeneous phase and a lower deposition phase);

-for a concentration in excess of 1% , only one homogeneous phase remained. It can be pointed out that above 3% Triton concentration, the contact angle stabilizes at 60° on silicon oxide and becomes almost zero on gold. As a result, a 3% Triton concentration was selected, with an activated carbon emulsion of 3% (beyond this value, jetting conditions tend to be more random).

B.Surface preparation

A silicon substrate was used with a 1000 Å thermal oxide layer; the metal electrodes were obtained by means of vacuum deposition of a 150Å Cr and 2000Å Au. These electrodes were patterned by photolithography ; the patterns obtained consist of interdigitated lines with the width varying from 10µm to 50µm. Surface functionalization is the result of quenching of a 2% OTS solution in trichorethylen .

C. Results

a. Deposition at room temperature Figure 2 shows the behaviour of grain distribution after heating to evaporate glycol. The so-called 'coffee cup effect' can be noted with a specific distribution of particles around pattern edges [3] b.Deposition at 140°C Distribution is much more homogeneous, as shown in Fig. 3. 10µm wide structures were obtained.

III. CONCLUSION

A method was validated for the deposition of large size particles in suspension. Thanks to a tune surface treatment and by adding a surfactant, pattern dimensions as low as 10 µm were obtained with the ink jet printing technology with nozzles that are much larger than the final structure. This method could be applied to other types of materials while potentially paving the way for a higher resolution.

IV. ACKNOWLEDGEMENT.

The authors would like to thank Patrice Simon and Pierre-Louis Taberna from CIRIMAT laboratory in Toulouse, for providing the activated carbon solutions and for valuable advices.

V REFERENCES

Fig. 1 :

 1 Fig. 1: Variation of the angle of contact of Au and Si as a function of percent concentration in Triton X 100 weight.


Véronique Conédéra
email: conedera@laas.fr

Fabien Mesnilgrente
email: fabien.mesnilgrente@laas.fr

Magali Brunet
email: mbrunet@laas.fr

Norbert Fabre
email: nfabre@laas.fr

Fabrication of Activated Carbon Electrodes by Inkjet Deposition

The inkjet deposition technology turns out to be interesting in various applications when one attempts to reduce manufacturing costs, simplify technological steps, and implement materials that could not be deposited in any other way. One of the limitations however, is the issue raised by resolution which is directly linked to the ejection head manufacturing technology. In this paper, a method of activated carbon deposition based on inkjet is presented. Activated carbon is extensively used in chemistry for its impurity absorption capabilities and in other applications like the fabrication of supercapacitors. It is shown in particular that by using a localized surface treatment with OTS and an activated carbon suspension stabilized by a surfactant, one can get 10 µm structures, which are, therefore, much smaller than the ejection head nozzle diameter. To implement and optimize the process, the contact angle measurement technique is utilized.

Introduction

With its high porosity, activated carbon is used as the electrode material for supercapacitors [START_REF] Pandolfo | Carbon properties and their role in supercapacitors[END_REF]. Standard assembly of supercapacitors is based on stacking up solid active material layers (activated carbon) with a separator between two collectors ("coin cell" package). The resulting coin cell device features a diameter of a few centimetres. The fabrication of miniaturised supercapacitors integrated on silicon substrate is of great interest in diverse microsystems applications that require energy storage. The integrated supercapacitor design that is proposed is then composed of interdigited fingers (collectors) on which the active material has to be deposited. However, the deposition of activated carbon on the gold electrodes is a real challenge. In this work, the deposition of activated carbon was investigated through use of the inkjet technique developed for polymers, oxides, metals and nanoparticles [START_REF] Whel | Ink-jet printing: the present state of the art[END_REF][START_REF] Adams | Ink jet Printing Technologies[END_REF][START_REF] Morrin | The fabrication and characterization of ink-jet printed polyaniline nanoparticle films[END_REF][START_REF] Kayadanova | Direct-write inkjet printing of b a barium strontium titanate-based tunable circuits[END_REF][START_REF] Yang | RFID Tag and RF Structures on a Paper Substrate Using Inkjet-Printing Technology[END_REF]. So far, no study of the deposition of activated carbon based on inkjet has been reported for grain sizes between 1µm and 10µm. To be able to use this method, a suspension that remains stable over time is required. This suspension will consist of an ink obtained by mixing activated carbon (AC) in ethylene glycol solution (carrying fluid), polytetrafluoroethylene (PTFE) which provides a mechanical link between the carbon particles in the substrate [START_REF] Adams | Ink jet Printing Technologies[END_REF] and a surfactant, Triton X100 ( p- (1,3,3,tetramethylbutyl) phenoxy-poly(ethylene glycol)) which contributes to the wettability and stability of the emulsion. The aim is first to obtain a stable AC suspension as a function of the surfactant concentration, and then to project this ink onto a gold electrode which is more hydrophilic than the surrounding hydrophobic silica surface. Hydrophobic silica was obtained by means of an octadecyltrichlorosilane (OTS) treatment aiming to concentrate the ink onto hydrophilic zones with a high surface energy that is, onto the gold electrodes. Through evaporation of ethylene glycol, one gets an activated carbon deposition located exclusively on the metal electrode.

Materials and means

To form the ink, a ethylene glycol solution from Sigma Aldrich was used to mix an alcoholic PTFE nanoparticle suspension from DuPont, with a Triton X100 surfactant from Sigma Aldrich and activated carbon from Kuraray Chemical Co. The OTS was provided by Sigma Aldrich. To measure the contact angle, a Digidrop machine was used while ink deposition was achieved with an Altadrop machine from Altatech Semiconductor. This is an automatic machine cassette to cassette equipped with a single nozzle head. Substrate temperature can reach 150°C. The temperature of the ejection heads can be adjusted up to 120°C. Thanks to a special tromboscopic system on the machine, the droplet formation and production rate could be controlled. Also, a movie camera was used to display the deposition and to align the drop on the pattern.

Ink and surface preparation

Ink preparation

Suspensions were prepared by mixing first in ethylene glycol the Triton X100 with PTFE. The activated carbon was added afterwards [START_REF] Bonnefoi | Electrode compositions for carbon power supercapacitors[END_REF]. The percentage of PTFE was 5 wt% of the activated carbon Weigth. This suspension was agitated for 12 hrs and placed in a test tube. The suspension behaviour was observed over time. Solutions featuring Triton compositions from 0.0001% to 5% were used. The AC composition was set to 3% weight relative to ethylene glycol. Higher concentrations yielded inks that were extremely difficult to jet.

Surface preparation

Silicon dioxide (1500Å) was first grown on silicon wafer by thermal oxidation followed by the evaporation of 150 Å of chromium and 3000Å of gold. The electrodes patterns on which activated carbon had to be fixed were formed by photolithography and etching of the chromium/gold layer. Pattern widths varied from 50µm to 10µm. On the silicon wafer, zones were defined to make contact angle measurements on the metal or on the oxide. The silicon dioxide or glass hydrophobic treatment has been extensively reported in the literature [START_REF] Cras | Comparison of chemical cleaning methods of glass in preparation for silanization[END_REF]. Following cleaning in a sulfochromic solution (2min), samples were soaked into an activation solution with 50% methanol and hydrochloric acid. After drying under nitrogen, samples were dipped for 30 min into an OTS solution diluted 2% in trichloroethylene. To get rid of the remaining solvent and OTS, acetone and deionised water were used.

Results and discussion

Ethylene glycol is a fluid that can easily be projected by inkjet. Its viscosity is 19.86mP.s at 20°C and can equally be reduced by rising temperature. With a boiling temperature at 200°C, depositions can be achieved at ambient temperature and the behaviour of the droplets deposited on the surface can be observed. At 140°C, rapid evaporation was achieved and AC deposition was almost instantaneous.

Variation of the contact angle and of the ink stability as a function of Triton X100 concentration

The curves in figure 1 show that the contact angle of a suspension on SiO 2 or on Au starts to decrease with a 0.5% Triton concentration and stabilizes from 2% onwards. With respect to sedimentation, three situations may develop according to the Triton concentration after storage 48 hrs.

-from 0.0001 % to 0.01% Triton, three phases can be observed: an upper phase (transparent) ; a lower phase (opaque) and a carbon precipitation at the bottom of the test tube; -from 0.01 to 1% Triton, two phases can be seen: an upper phase (opaque) with numerous particles in suspension and deposition at the bottom of the test tube; -above 1%, there is only one phase in suspension. It follows from these findings that only with a percentage in excess of 1% can a suspension be used. At lower concentrations, the precipitation of the ink storage tank can only cause clogging of the ejection head. The curve of figure 2 shows that the difference in contact angles for gold and silica is highest between 1 and 3% Triton, thereby facilitating a selective deposition of the ink droplet on the gold electrode. The surface energy between gold and the ink tends to attract the ink droplet. If the contact angles on Au and on SiO 2 are too small and/or if the difference of contact angles between these two surfaces is too small, the ink droplet will remain in its original position after impact with the surface. As seen in figure 3 with 30° angles on Au and SiO 2 , a denser activated carbon particle distribution appears on the electrode edges. This phenomenon is also referred to as 'the cup of coffee effect' [START_REF] Antje | Geometric control of inkjet printed features using a gelating polymer[END_REF].

Figure 3 Image with 30°contact angles on Au and SiO 2.

If the ink droplet is projected outside the pattern onto a surface with a high contact angle, one gets, after ethylene glycol evaporation, a mount as shown in figure 4.

Importance of substrate temperature

Given the high boiling point of ethylene glycol, one can observe the deposition behaviour at room temperature and then have the ethylene glycol evaporated by a hot plate bake. The results can then be compared to a deposition where the substrate is heated at 140°C so as to cause an almost instantaneous evaporation of the droplets during impact on the surface. These operations were achieved with a ink containing 3% of activated carbon, 3% Triton, and 5% PTFE.

Deposition at room temperature

Figure 5 shows six ink droplets deposited at room temperature on a 50µm diameter nozzle on a 40µm wide and 650µm long electrode. Clusters of droplets bigger than the 40µm gold line can be observed. This could be a drawback for the case of interdigited electrodes (risks of short-circuits). As ethylene glycol evaporation takes place, one can see that the droplets were joining up figure 5

Deposition at 140°C

A homogeneous deposition on the gold electrodes was obtained at 140°C by varying the density of the droplets. This was achieved by increasing the number of passes or increasing the number of droplets. Patterns similar to those of figure 6 were obtained on a 30µm wide electrode. This approach was selected for 10 µm wide electrodes as shown in figure 7. The main particles originate from the heterogeneous AC distribution with grain size of 10µm. with a spacing of 40µm between each finger. These structures are then wire bonded for electrical characterisation as supercapacitors [START_REF] Brunet | Ink-jet printed carbon-based micro-supercapacitor for mobile self-powered modules[END_REF]. It is also a way for checking that the interdigited electrodes are not short-circuited by the activated carbon.

Conclusion

The ink jet deposition technology features unique capabilities in terms of materials deposition that cannot be achieved in any other way; the activated carbon example given in this paper demonstrates that good results can only be obtained if the ink jet deposition and the substrate surface condition are fully controlled. This method allows depositions smaller than the nozzle diameter to be achieved, resulting in a higher resolution for a technology which is limited by the inkjet head fabrication constraints or which requires parameter control conditions for the jets [START_REF] Rensch | Creation of Small Microdrops[END_REF] which are difficult to master and impossible to implement on certain machines. The developed process consisting in surface functionalizing and suspension improvement through surfactant can be extended to other types of materials in suspension such as nanoparticles. This technique could be adapted for batch processing use of multihead could assure a high throughput.

Figure 1 Figure 2

 12 Figure 1 Contact angle variation on Au and SiO 2 versus % Triton X100 weight.

Figure 4

 4 Figure 4 CA deposition onto a high contact angle surface of SiO 2 .

Figure 5 a

 5 Figure5shows six ink droplets deposited at room temperature on a 50µm diameter nozzle on a 40µm wide and 650µm long electrode. Clusters of droplets bigger than the 40µm gold line can be observed. This could be a drawback for the case of interdigited electrodes (risks of short-circuits). As ethylene glycol evaporation takes place, one can see that the droplets were joining up figure 5 (b) leading to a relatively inhomogeneous AC particle distribution.

Figure 6

 6 Figure 6 SEM of 30µm wide electrodes.

Figure. 7

 7 Figure. 7 SEM of 10µm wide electrode.

Figure 8

 8 Figure 8 depicts an interdigited electrodes structure

Acknowledgement

The authors would like to thank Patrice Simon and Pierre-Louis Taberna from CIRIMAT laboratory in Toulouse, for providing the activated carbon and PTFE solutions and for valuable advices.


Mateus Habermann
email: mateus.habermann@hds.utc.fr

Vincent Frémont

Elcio Hideiti Shiguemori

Clustering-based Unsupervised Hyperspectral Band Selection using Single-Layer Neural Networks

Keywords: Sélection de bandes, non-supervisé, clustering Band selection, unsupervised, clustering

Les images hyperspectrales fournissent des détails de la scène observée en exploitant les bandes spectrales continues. Cependant, le traitement de ces images est long à cause de leurs grandes dimensions. Donc, la sélection des bandes est une pratique commune qui est adopteé avant qu'un traitement soit fait. Ainsi, dans ce travail une nouvelle approche non-supervisé pour la sélection des bandes, basée sur le clustering et les réseaux de neurones, est proposée. Une comparaison avec quatre autres aproches de sélection de bandes montre la validité de l'algorithme proposé.

Introduction

Hyperspectral images (HSI) are composed of many continuous bands of the electromagnetic spectrum [START_REF] Chang | Hyperspectral Imaging : Techniques for Spectral Detection and Classification[END_REF]. Thus, a broad range this spectrum can be covered, which provides lots of information about the scene under analysis. Consequently, those images are useful to Remote Sensing (RS), enabling tasks such as image classification and target detection.

Nevertheless, a large amount of bands can bring problems for data processing, storage and transmission. Moreover, the high dimensionality of HSI can cause the curse of dimensionality [START_REF] Hughes | On the mean accuracy of statistical pattern recognizers[END_REF]. Those facts may cause difficulties in many RS applications, thus, dimensionality reduction is normally adopted. When it comes to dimensionality reduction, feature extraction (FE) is a commonly used method. According to the FE methodology, new features are created by combining the original ones. The resulting features lie in lower dimension, and still they keep much of the original information. One very popular FS technique is Principal Component Analysis (PCA). Since FS changes the original image information, it should not be used in cases that demand the physical meaning of individual bands [START_REF] Khalid | A survey of feature selection and feature extraction techniques in machine learning[END_REF]. Band selection (BS) is another approach for dimensionality reduction. The dimension of the feature space is decreased, whereas the original band information is kept [START_REF] Li | Challenges of feature selection for big data analytics[END_REF]. BS methods are classified in two types : supervised and unsupervised. Normally, supervised BS approaches yield better results compared to unsupervised ones [START_REF] Yang | An efficient method for supervised hyperspectral band selection[END_REF]. But, supervised methods need labeled training samples, which are expensive to be collected. Therefore, unsupervised methods are a wise alternative for band selection. Since unsupervised BS methods don't have class labels at their disposal, the band selection is made taking into account the data structure [START_REF] Chandra | Exploring autoencoders for unsupervised feature selection[END_REF]. K-Means algorithm is an unsupervised technique that clusters data samples according to the data set structure in the feature space [START_REF] Theodoridis | Pattern Recognition[END_REF]. This paper proposes an easily implementable unsupervised BS framework using K-Means clustering and single-layer neural networks (SLN). Initially, the whole data set is clustered by K-Means into two groups. Then, a SLN is used to find a hyperplane that separates those two clusters, and the bands linked to the biggest and smallest coefficients of the hyperplane equation are selected. This procedure is considered as one iteration of the method, and two bands are selected at each iteration. If more bands are needed, another iteration is started using either of the clusters just generated, instead of the whole data set. The contributions of this work are summarized as follows : i) It is a novel band selection framework based on K-Means clustering and single-layer neural networks ; ii) We compare the proposed method with other unsupervised band selection algorithms. The rest of the paper is organized as follows : In Section 2, a literature review is presented. Section 3 describes the proposed method. In Section 4, one can find the dataset, classifiers used, the competitors and the results. Finally, Section 5 provides a conclusion to his paper.

State-of-the-art approaches

The proposed band selection framework is unsupervised, so only recent state-of-the-art unsupervised approaches are cited in this Section. In the literature, one finds lots of works addressing BS following many different perspectives and methodologies. Due to the high HSI dimensionality, the different classes present in the image may lie in manifolds embedded in subspaces of the original feature space. Furthermore, it is also possible to explore the sparsity of the data set in order to find a more meaningful data representation. For example, in [START_REF] Wang | Salient band selection for hyperspectral image classification via manifold ranking[END_REF], the authors propose a new method in which they look for salient bands. The number σ of selected bands is user-defined. Then, the band selection algorithm has two steps. Firstly, β bands are selected by means clone selection algorithm, which seeks to minimize the Euclidean distance amongst elements of the same class, whereas maximizing the distance of elements from different classes. After that, if β < σ, those β bands already chosen will serve as seeds to a Manifold Ranking (MR) algorithm. MR sorts the remaining bands, and the most dissimilar band is added to the β group. This step is repeated until β = σ. In [START_REF] Wang | Unsupervised band selection using block-diagonal sparsity for hyperspectral image classification[END_REF], the authors propose a BS framework based on sparsity. Initially, the most representative bands are obtained according to the correlation matrix, whereas the block-diagonal structure is measured to segment bands into subspaces. Then, a method for band selection based on trace LASSO and spectral clustering is used. In [START_REF] Sun | Band selection using improved sparse subspace clustering for hyperspectral imagery classification[END_REF], the authors propose a method that initially represents data instances as sparse coefficient vectors by solving a L2-norm optimization using the least squares regression (LSR) algorithm. Then, a correct segmentation of band vectors is made using the resulting LSR matrix with sparse and blockdiagonal structure. After that, a similarity matrix is constructed by angular similarity measurement, and then the size of the band subset is calculated by the distribution compactness plot algorithm. In [START_REF] Gan | Dissimilarityweighted sparse representation for hyperspectral image classification[END_REF], the authors state that all HSI bands can be represented by a band subset. Thus, they propose a sparse representation of bands with row-sparsity constraint. Besides, a dissimilarity-weighted regularization term is integrated with the self-representation model, to avoid contiguous bands. The problem is solved by the alternating direction method of multipliers, and the representative bands can be chosen. In [START_REF] Sun | Fast and robust self-representation method for hyperspectral band selection[END_REF], a fast and robust self-representation framework to select a band subset is proposed. It is assumed the separability structure of the spectral bands, thus the problem may be seen as non-negative matrix factorization. After that, an optimizing convex problem is addressed and augmented Lagrangian multipliers are used to select the band subset. In [START_REF] Zhu | Hyperspectral band selection via rank minimization[END_REF], the authors propose a BS framework that can capture the inter-band redundancy through low-rank modeling. Then, by using an affinity matrix and concepts of data quality, the most representative bands are selected. In [START_REF] Wang | Unsupervised hyperspectral image band selection via column subset selection[END_REF], a BS method based on column subset selection is proposed. By means of column subset selection problem, it is possible to select some bands maximizing the volume of the selected subset of columns. The high dimensionality decreases the contrast amongst bands, thus Manhattan distance is used to get a higher quality in the BS process. In [START_REF] Cao | Automatic band selection using spatial-structure information and classifier-based clustering[END_REF], the authors propose a framework that removes low-discriminating bands that normally need to be discarded manually. Based on the spatial structure of the data set, it is possible to determine which bands have low-discriminating power. Then, a new clustering algorithm is proposed in order to define the optimum number of bands to be selected.

Another criterion that can be used in BS strategies is the HSI data information analysis. For example, in [START_REF] Sui | Unsupervised band selection by integrating the overall accuracy and redundancy[END_REF], the authors propose a framework that integrates both the overall accuracy and redundancy. Thus, an optimization problem using adaptive balance parameter is devised to handle the trade-off between the overall accuracy and redundancy. Furthermore, an unsupervised overall accuracy prediction method was adopted. In [START_REF] Sun | A new band selection method for hyperspectral image based on data quality[END_REF], the authors propose a framework that merges the concept of noiseadjusted principal components with maximum determinant of covariance matrix. A new index to measure the HSI quality is also proposed, taking into account signal-to-noise ratios (SNR) and correlation of bands. Based on the new index, the authors devise an unsupervised band selection method, which considers the quality of the data set as selection criterion. It selects bands with both high SNR and low correlation. In [START_REF] Santos | Efficient unsupervised band selection through spectral rhythms[END_REF], the authors propose a BS method based on the dissimilarity amongst neighboring bands. They use an intermediary representation named spectral rhythm, which can take advantage of a pixel sampling strategy, what ends up improving its efficiency without reducing the selected bands quality. In [START_REF] Luo | Information-assisted density peak index for hyperspectral band selection[END_REF], the authors propose a method based on information-assisted density peak index. It takes into account the intraband information entropy into the local density and intercluster distance to ensure cluster centers with a high quality. Besides, the channel proximity and band distance are integrated to control the local density compactness. The bands with top-ranked scores may get clear global distinction, good local density and also high informative quality. In [START_REF] Chang | Channel capacity approach to hyperspectral band subset selection[END_REF], the authors formulate the BS as a channel capacity problem. After constructing a band channel with the original bands. Then, some bands are selected by Blahut's algorithm, which iteratively finds a feature space that provides the best channel capacity. Thus, neither band prioritization nor interband decorrelation are required. Two iterative methods are devised to find the best band subset, which avoid an exhaustive search.

Using graph theory, in [START_REF] Yuan | Discovering diverse subset for unsupervised hyperspectral band selection[END_REF] the authors propose a multigraph determinantal point process (MDPP). The aim is to capture the structure amongst bands and find the optimal band subset. For this, multiple graphs are designed to capture the intrinsic relationship amongst bands. Besides, the proposed MDPP is used to model the multiple dependencies in graphs, providing an efficient search strategy for the BS process.

Evolutionary computation with optimization have been largely used BS methods. For example, in [START_REF] Xu | A new unsupervised hyperspectral band selection method based on multiobjective optimization[END_REF], the authors propose an incorporated rank-based multiobjective band selection framework, to avoid conflicting objective functions, such as Jeffreys-Matusita (JF) and Bhattacharyya distances. During the processing, the spectral bands are transformed into binary vectors, whose elements are subjected to flipping with a certain probability. In [START_REF] Gong | Unsupervised band selection based on evolutionary multiobjective optimization for hyperspectral images[END_REF], the authors propose a framework that handles two conflicting objective functions. One function is designed to represent the information contained in the selected bands, by means of entropy. The other function is set as the number of selected bands. Both objective functions are optimized simultaneously by a multiobjective evolutionary algorithm. In [START_REF] Su | Hyperspectral band selection using firefly algorithm[END_REF], the authors propose a framework for band selection which employs two objective functions using JF. During the search process, the spectral bands are treated as firefly variables. In [START_REF] Zhang | Unsupervised hyperspectral band selection by fuzzy clustering with particle swarm optimization[END_REF], a framework for band selection based on fuzzy clustering and swarm optimization in proposed. The authors devise a modified fuzzy clustering method for band selection, whose drawbacks are alleviated by swarm optimization. In [START_REF] Zhang | Memetic algorithm based feature selection for hyperspectral images classification[END_REF], the authors proposed a BS framework based on memetic algorithms (MA). Firstly, MA is used to select a subset of spectral bands. Also, a objective function is designed to select bands considering both bands information and redundancy deduction. The authors claim that this method is not only computationally faster than exhaustive search approaches, but also has comparable performances.

Finally, clustering techniques can also be used in band selection methods. For instance, in [START_REF] Datta | Clustering based band selection for hyperspectral images[END_REF] the authors propose a framework that removes redundancy amongst bands by means of clustering. Then, from each cluster one representative band is selected. After that, the bands are ranked according to their classification capabilities. In [START_REF] Yuan | Dual-clusteringbased hyperspectral band selection by contextual analysis[END_REF], the authors propose a framework based on dual clustering that takes into account the contextual information. For this, a novel descriptor that reveals the image context is devised, in order to select the representatives of each cluster, taking into consideration the mutual effects of each cluster. This paper uses K-Means to generate clusters according to intrinsic structure of the data set in the feature space, taking into account only the spectral information.

Proposed Framework

Definitions

Let C (0) be the whole data set corresponding to a hyperspectral image, whose elements are vectors x i ∈ R 1×d that contain spectral signatures, where d is the number of bands. Let S be the set containing the selected bands, and G the set with bands highly correlated to those in S. Let A be the set that has the original spectral bands a k , with k = 1, 2, ..., d. And let σ be a previously determined quantity of bands to be selected. Let f : F -→ L be a single-layer neural network, and F is the feature space comprised by A \ (S ∪ G), and L = {0, 1}. The input to f is a vector x and its output is a scalar defined by

L = f (z) = 1 1 + e -z , (1)
with z = W T x + b, where W and b are the weights and bias of the neural network, respectively. According to (1), L ∈ [0, 1], and to give it a binary value, these criteria are used :

-if z < 0 =⇒ f < 0.5 =⇒ L ← 0, -if z ≥ 0 =⇒ f ≥ 0.5 =⇒ L ← 1.
Therefore, the signal of z indicates whether an input sample will be assigned to class 0 or 1. Since the input data is normalized into [0, 1], the weights w l ∈ W , with l = 1, ..., p, in the equation

z = x 1 i w 1 + x 2 i w 2 + ... + x p i w p + b (2)
cause a strong impact in determining the signal of z, and, consequently, the estimate Li for x i . In (2), p represents the cardinality of A \ (S ∪ G).

The cost function of f is cross-entropy. The training is conducted by using stochastic gradient descent and the back-propagation algorithm.

Lastly, let C

(l) g be a partition of C (0) , where l is the partition level, and

C (l) 1 ∪ C (l) 2 ∪ • • • ∪ C (l) g = C (0) , and C (l) p ∩ C (l)
q = ∅, ∀p = q. There may be several levels, that is, l = 1, 2, 3, . . . , and for each level the number of partitions g is given by g = 2 l .

Description

General view. The proposed framework begins with an empty subset of selected bands, that is, S = ∅, to which the bands selected from A will be added. At the first iteration, C (0) is split by the K-Means algorithm into two partitions, C

(1)

1 and C (1)
2 . Then, we use a single-layer neural network to find a hyperplane that separates those two partitions. After that, two bands are selected, and consequently discarded from the data set. If more bands are needed, we keep repeating this procedure in deeper levels. As the proposed method is based on both clustering and single-layer neural networks, we shall call it CSLN. Its characteristics are described below. is to be solved by the function f . Since two bands are selected at each iteration, one needs to repeat the process until the desired number of bands σ is attained. When σ is even, one needs σ/2 iterations. If σ is an odd number, (σ + 1)/2 iterations are necessary, and the first σ selected bands are kept.

Iterations. CSLN is an iterative band selection method. At each iteration, a binary classification problem between K-Means-generated partitions C

Selection of bands.

After the training of the neural network, it is possible to give degrees of importance to all a k ∈ A\(S ∪G). As every element x l ∈ x is directly linked to w l , for l = 1, ..., p, the magnitude of w l is a indicator for the band a l . In (2), the largest and the smallest weights constitute the most important contributions to the signal of z. Thus, the bands linked to those weights are also considered the most important, and, consequently they are added to the set S. The feature space F can be updated by A \ (S ∪ G).

Avoiding highly correlated bands. The bands of a hyperspectral image are contiguous, which causes a high correlation among neighboring bands. Bearing this in mind, we adopt a method that avoids the selection of highly correlated bands. For each band a k ∈ F we construct a vector v k , whose elements are the bands indices in a descending order in relation to the correlation to the band a k . Thus, the following procedure is adopted :

-At a certain iteration, a band a k is selected, so S ← a k ; -G ← a v k [START_REF] Chang | Hyperspectral Imaging : Techniques for Spectral Detection and Classification[END_REF] , and G is, at the beginning, an empty set ; -After this iteration, the feature space is updated by A \ (S ∪ G).

6:

Update the feature space F by A \ (S ∪ G) 7: Return : S Fig. 1 depicts the proposed framework. Initially, the whole data set is split into two clusters by K-Means. Then, a single-layer neural network f is used to find a hyperplane that separates the clusters. This process is repeated until the desired number of bands is selected.

Results

The results of the proposed method are exhibited in this Section. Moreover, they are compared with other band selection approaches by considering the accuracy of two supervised classifiers, namely, Classification and Regression Trees (CART) and K-Nearest Neighbors (KNN). The input data are the selected bands. The image used is the Indian Pines. It consists of 145 × 145 pixels and 224 spectral reflectance bands in the 0.4 -2.5 µm wavelength range. As for the ground truth, we count on 16 classes, and they are used only for classification comparison purposes.

Competitors

The performance of the proposed method is compared with four other BS approaches. The first method is also clustering-based [START_REF] Martãŋnez-Usãşmartinez-Uso | Clusteringbased hyperspectral band selection using information measures[END_REF], and it will be referred to as WaLuDi. The second approach uses both ranking and clustering for band selection [START_REF] Datta | Combination of clustering and ranking techniques for unsupervised band selection of hyperspectral images[END_REF], and we will call it CR. The third competitor resorts to band elimination with partitioned image correlation [START_REF]Band elimination of hyperspectral imagery using partitioned band image correlation and capacitory discrimination[END_REF], and it will be referred to as EM. Lastly, the fourth competitor relies on information divergence, and this method will be called ID [START_REF] Chang | Constrained band selection for hyperspectral imagery[END_REF]. As already stated in Section 3.2, our proposed method will be referred to as CSLN.

Selected bands

The bands selected by the proposed framework are displayed in Table 1. We have only the first 18 bestranked bands of our competitors, thus the analyses of results are restricted to this number of bands. The bands in Table 1 are sorted according to the order they were selected. For example, at the first iteration, the bands 2 and 42 were selected. The results comparisons are made with different quantities σ of selected bands, that is, σ s = s × 3, with s = 1, 2, 3, 4, 5, 6. Thus, for σ 2 = 6, for example, the first six bands of Table 1 are used.

Results comparison

The classification results exhibited throughout this paper are the mean values over ten runs. The standarddeviation values are also displayed. In Table 2, the results of the KNN classifier are shown. The proposed method CSLN has the best results using 3, 6 and 9 bands. It is illustrated in Fig. 2 (a). In Table 3, the overall results achieved by the CART classifier are exhibited. The proposed framework achieves the best results with 3 and 9 bands. Fig. 2 (b) provides a visual perspective of the results.

Remarks about the results

KNN results are, in general, superior than that of CART : 73.26% and 63.13%, respectively. This may be attributed to the fact that CART splits the feature space into regions that correspond to the classes. Therefore, if x i is found in a region corresponding to a class α, for example, it will be classified as α, even if it belongs to class β. Whereas, in this same situation, KNN would analyze the K nearest neighbors of x i before assigning it a label. Consequently, KNN outperforms CART when the class boundaries are highly non-linear. Moreover, according to Tables 2 and3, one can notice that, in general, the accuracies get better as more bands are used. Both facts are depicted in Fig. 3.

As for the band selection methods, using the KNN classifier, the proposed BS framework achieves the best results in three situations. Furthermore, by analyzing the standard-deviation values of Table 2, one can notice that the CSLN method have statistically better results. With regard to the CART classifier, our method has the best accuracy with 3 and 9 bands. With 6, 12, 15 and 18 bands, CSLN has similar performance in relation to its best competitors.

Conclusion

The fine spectral details provided by HSI allow for a good characterization of objects in the scene. But, the large amount of spectral information can also bring inconveniences in terms of processing and storage. Therefore, this paper proposed a band selection framework to decrease the image dimensionality. The proposed unsupervised BS method is based on K-Means clustering and single-layer neural networks. It started by clustering the whole data set into two groups. Then, a single-layer neural network was used to find a separating hyperplane between the clusters. The bands linked to the biggest and smallest coefficients of the hyperplane equation were selected. Then, we kept repeating this procedure using the generated clusters to select the desired number of bands. By analyzing the results, one can see that the KNN classifier outperforms CART for the Indian Pines image. Moreover, the more bands, the better the classifier accuracy. We emphasize that 3 up to 18 selected spectral bands were investigated. When it comes to the proposed method, the best results were achieved in three situations out of six using KNN. With CART, the proposed method got the best results in two situations out of six. With regard to the future works, we will investigate other clustering algorithms and binary classifiers and use them in our framework.

Figure 1 -Algorithm 1 1 : 3 : 4 :

 11134 Figure 1 -Flowchart of the proposed BS framework. At each binary clustering, a single-layer neural net f is used to select the bands.

Table 1 -

 1 The selected bands according to the order of selection by the proposed method.

	Selected bands 2, 42, 6, 39, 22, 58, 25, 62,
	71, 101, 94, 151, 111, 203,
	156, 183, 171, 215.

Acknowledgements

This work was carried out in the framework of the Labex MS2T and DIVINA challenge team, which were funded by the French Government, through the program Investments for the Future managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02). We are also thankful for the support provided by Brazilian Air Force and Institute for Advanced Studies.


Frédéric Leroux

Kanhaiya Pandey

Riadh Rebhi

Frederic Chevy

Christian Miniatura

Benoît Grémaud

David Wilkowski

R Rehbi

Non-Abelian adiabatic geometric transformations in a cold Strontium gas

Introduction

In 1984, M. V. Berry published the remarkable discovery that cyclic parallel transport of quantum states causes the appearance of geometrical phase factors [START_REF] Michael | Quantal phase factors accompanying adiabatic changes[END_REF]. His discovery, along with precursor works [START_REF] Pancharatnam | Generalized theory of interference and its applications[END_REF][START_REF] Aharonov | Significance of electromagnetic potentials in the quantum theory[END_REF], unified seemingly different phenomena within the framework of gauge theories [START_REF] Simon | Holonomy, the quantum adiabatic theorem, and berry's phase[END_REF][START_REF]Geometric Phases in Physics[END_REF]. This seminal work was rapidly generalized to non-adiabatic and noncyclic evolutions [START_REF]Geometric Phases in Physics[END_REF] and, most saliently for our concern here, to degenerate states by F. Wilczek and A. Zee [6]. In this case, the underlying symmetry of the degenerate subspace leads to a non-Abelian gauge field structure. These early works on topology in quantum physics have opened up tremendous interest in condensed matter [START_REF] Mario | Giant magnetoresistance of (001) fe/(001) cr magnetic superlattices[END_REF][START_REF] Yuichiro | Observation of the spin hall effect in semiconductors[END_REF][START_REF] König | Quantum spin hall insulator state in hgte quantum wells[END_REF][START_REF] Hsieh | A topological dirac insulator in a quantum spin hall phase[END_REF][START_REF] Chang | Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator[END_REF] and more recently in ultracold gases [START_REF] Hadzibabic | Berezinskiikosterlitz-thouless crossover in a trapped atomic gas[END_REF][START_REF] Lin | A spin-orbit coupled bose-einstein condensate[END_REF][START_REF] Aidelsburger | Measuring the chern number of hofstadter bands with ultracold bosonic atoms[END_REF][START_REF] Jotzu | Experimental realization of the topological haldane model with ultracold fermions[END_REF][START_REF] Mancini | Observation of chiral edge states with neutral fermions in synthetic hall ribbons[END_REF][START_REF] Nguyen | Controlling and probing non-abelian emergent gauge potentials in spinor bose-fermi mixtures[END_REF][START_REF] Wu | Realization of twodimensional spin-orbit coupling for bose-einstein condensates[END_REF][START_REF] Song | Spin-orbit-coupled two-electron fermi gases of ytterbium atoms[END_REF][START_REF] Li | A stripe phase with supersolid properties in spin-orbit-coupled bose-einstein condensates[END_REF] and photonic devices [START_REF] Wang | Observation of unidirectional backscattering-immune topological electromagnetic states[END_REF][START_REF] Kuhl | Dirac point and edge states in a microwave realization of tight-binding graphene-like structures[END_REF][START_REF] Schine | Synthetic landau levels for photons[END_REF].

Moreover, it has been noted that geometrical qubits are resilient to certain noises, making them potential candidates for fault-tolerant quantum computing [START_REF] Zanardi | Holonomic quantum computation[END_REF][START_REF] Jones | Geometric quantum computation using nuclear magnetic resonance[END_REF][START_REF] Duan | Geometric manipulation of trapped ions for quantum computation[END_REF][START_REF] Solinas | On the stability of quantum holonomic gates[END_REF]. So far, beside some recent proposals [START_REF] Mark A Kowarsky | Non-abelian geometric phase in the diamond nitrogen-vacancy center[END_REF][START_REF] Sjöqvist | Nonadiabatic holonomic single-qubit gates in off-resonant λ systems[END_REF], experimental implementations have been performed for a 2-qubit gate on NV-centers in diamond [START_REF] Zu | Experimental realization of universal geometric quantum gates with solid-state spins[END_REF] and for a non-Abelian single qubit gate in superconducting circuits [START_REF] Jr | Experimental realization of non-abelian non-adiabatic geometric gates[END_REF]. These experiments were performed following a nonadiabatic protocol allowing for high-speed manipulation [START_REF] Sjöqvist | Nonadiabatic holonomic single-qubit gates in off-resonant λ systems[END_REF][START_REF] Zhu | Implementation of universal quantum gates based on nonadiabatic geometric phases[END_REF][START_REF] Sjöqvist | Non-adiabatic holonomic quantum computation[END_REF]. Recently, coherent control of ultracold spin-1 atoms confined in optical dipole traps was used to study the geometric phases associated with singular loops in a quantum system [START_REF] Hebbe | Singular loops and their non-abelian geometric phases in spin-1 ultracold atoms[END_REF]. If non-adiabatic manipulations are promising methods for quantum computing, they prevent the study of external dynamic of quantum system in a non-Abelian gauge field, where non-trivial coupling occurs between the internal qubit state dynamics and the center-of-mass motion of the particle.

Here, we report on non-Abelian adiabatic geometric transformations implemented on a non interacting cold fermionic gas of Strontium-87 atoms by using a 4-level resonant tripod scheme set on the 1 S 0 , F g = 9/2 → 3 P 1 , F e = 9/2 intercombination line at λ = 689 nm (linewidth: Γ = 2π × 7.5 kHz). About 10 5 atoms are loaded in a crossed optical dipole trap, optically pumped in the stretched Zeeman state |F g = 9/2, m g = 9/2 and Doppler-cooled down to temperatures T ∼ 0.5 µK [START_REF] Chalony | Doppler cooling to the quantum limit[END_REF][START_REF] Yang | A high flux source of cold strontium atoms[END_REF], see Methods. A magnetic bias field isolates a particular tripod scheme in the excited and ground Zeeman substate manifolds. Our laser configuration consists of two co-propagating beams (with opposite circular polarizations) and a third linearly-polarized beam orthogonal to the previous ones. These three coplanar coupling laser beams are set on resonance with their common excited state |e = |F e = 9/2, m e = 7/2 .

Results

Dark states basis. For any value of the amplitude and phase of the laser beams, the effective Hilbert space defined by the four coupled bare levels contains two bright states and two degenerate dark states |D 1 and |D 2 . These dark states do not couple to the excited state |e and are thus protected from spontaneous emission decay by quantum interference. For equal Rabi transition frequencies, we conveniently choose

|D 1 = e -iΦ13(r) |1 -e -iΦ23(r) |2 √ 2 |D 2 = e -iΦ13(r) |1 + e -iΦ23(r) |2 -2|3 √ 6 , (1)
where |i ≡ |m g = i + 3/2 (i = 1, 2, 3). Φ ij = Φ i -Φ j , where the space-dependent laser phases read Φ i (r) = k i • r + ϑ i . k i is the wavevector of the beam coupling state |i to |e and ϑ i its phase at origin, see Fig. 1. To implement non-Abelian transformations on the system, the two independent offset phases tuned by the Electro-Optic Modulators (EOM), shown in Fig. 1a, are

φ i = ϑ i -ϑ 3 (i = 1, 2).
In a first set of experiments, we probe and quantify the thermal decoherence of the dark states induced by the finite temperature of our atomic sample. In a second set of experiments, we analyze the non-Abelian character of geometric transformations within the dark-state manifold. To do so, we consider a certain phase loop in the parameter space defined by the two relative phases φ 1 and φ 2 of the tripod lasers, and we compare the final populations of the internal atomic states when the cyclic sequence is performed, starting from two different initial points on the loop. In all experiments, we monitor the subsequent manipulation and evolution of the atomic system in the dark-state manifold by measuring the bare ground-state populations with a nuclear spin-sensitive shadow imaging technique, see Methods.

Thermal decoherence. Starting from state |3 , we prepare the atoms in dark state |D 2 after a suitable adiabatic laser ignition sequence, see Methods. We assume that the atoms do not move significantly during the time duration of this sequence, see Fig. 2. Following [START_REF] Ruseckas | Non-abelian gauge potentials for ultracold atoms with degenerate dark states[END_REF][START_REF] Juzeliūnas | Double and negative reflection of cold atoms in non-abelian gauge potentials[END_REF], the subsequent evolution of the atoms is described by the Hamiltonian

H = (p1 1 -A) 2 2M + W (2)
where p = -i ∇ is the momentum operator, 1 1 is the identity operator in the internal dark-state manifold, M the atom mass, A the geometrical vector potential with The error bars correspond to a 95% confidence interval. Solid lines: theoretical predictions given by equations [START_REF]Geometric Phases in Physics[END_REF]. The temperature T , the initial and the final populations of each spin state are the fit parameters. The dashed lines, at early times, extrapolate the fits into the time window t0. We get a temperature T = 0.5(1) µK meaning that the atoms do not move significantly during the dark state preparation sequence since vt0/λ ≃ 0.08. matrix entries A jk = i D j |∇D k and W the geometrical scalar potential with matrix entries

W jk = 2 ∇D j |∇D k -(A 2 ) jk 2M . (3)
With our laser geometry, A, A 2 , and W have the same matrix form, and are uniform and time-independent, see Methods. Thus, we can look for states in the form |ψ ⊗ |p where p = M v is the initial momentum of the atoms and |ψ some combination of dark states. Denoting by P 0 (v) the initial atomic velocity distribution, we find that the population of state |2 remains constant while the two others display an out-of-phase oscillatory behavior at a velocity-dependent frequency

ω v = 2 3 [k(v x -v y ) + 2ω R ]: P 1 (v, t) = 5P 0 (v) 12 1 - 3 5 cos ω v t , P 2 (v, t) = P 0 (v) 6 , P 3 (v, t) = 5P 0 (v) 12 1 + 3 5 cos ω v t , (4)
where ω R = k 2 /(2M ) is the recoil frequency and k = 2π/λ is the laser wavenumber. The frequency component proportional to k(v x -v y ) comes from the momentum-dependent coupling term A • p/M in equation (2) (Doppler effect) whereas the other frequency component, proportional to ω R , comes from the scalar term A 2 /(2M ) + W . With our laser configuration, lightassisted mechanical forces can only come from photon absorption and emission cycles between a pair of orthogonal laser beams. Such photon exchanges would induce a population change of state |2 . Since no force is acting here on the centre-of-mass of the atoms (the Abelian gauge field is uniform and can be gauged away), the population P 2 (v, t) must stay constant, as predicted by equation ( 4). Since photon absorption and emission cycles between the pair of co-propagating laser beams do not impart any net momentum transfer to the atoms, population transfer between states |1 and |3 is possible and P 1 (v, t) and P 3 (v, t) change in time, their sum being constant due to probability conservation. Averaging over the Maxwellian velocity distribution of the atoms, the bare state populations of the thermal gas read

P1 (t) = 5 12 - 1 4 cos 4 3 ω R t exp - 4 9 (kvt) 2 , P2 (t) = 1 6 , P3 (t) = 5 12 + 1 4 cos 4 3 ω R t exp - 4 9 (kvt) 2 , (5)
where v = k B T /M is the thermal velocity of the gas at temperature T . We see that P1 and P3 converge to the same value at long times. This means that the thermal average breaks the tripod scheme into a Λ-scheme coupled to the two circularly-polarized beams and a single leg coupled to the linearly-polarized beam. As a consequence, quantum coherence partially survives the thermal average. Our experimental results confirm this behavior even if P1 and P3 do not merge perfectly, see Fig. 2. This discrepancy can be lifted by introducing a 10% imbalance between the Rabi transition frequencies in our calculation. The population difference P3 -P1 measures in fact the Fourier transform of the velocity distribution along the diagonal direction xŷ. It decays with a Gaussian envelope characterized by the time constant τ = 3/(2kv), as predicted by equations ( 5). This interferometric thermometry is similar to some spectroscopic ones such as recoil-induced resonance [START_REF] Courtois | Recoil-induced resonances in cesium: An atomic analog to the free-electron laser[END_REF][START_REF] D R Meacher | Method for velocimetry of cold atoms[END_REF] or stimulated two-photons transition [START_REF] Kasevich | Atomic velocity selection using stimulated raman transitions[END_REF][START_REF] Peters | Thermometry of ultracold atoms by electromagnetically induced transparency[END_REF]. From our measurements, we get T = 0.5(1) µK, τ ≃ 24 µs and v ≃ 6.9 mm/s.

Non-Abelian transformations. We now investigate the geometric non-Abelian unitary operator U acting on the dark-state manifold when the relative phases of the tripod beams are adiabatically swept along some closed loop C in parameter space. For a pinned atom (M → ∞), U is given by the loop integral along C of the 2 where P is the path-ordering operator [6].

× 2 Mead- Berry 1-form ω ≡ [ω jk ] ≡ [i D j |dD k ] U = P exp i C ω , (6)
As before, the system is initially prepared in dark state |D 2 . Then, starting from the origin, the phase loop is cycled counterclockwise, see Fig. 3a. Each segment is linearly swept in ∆t = 4 µs and the phase excursion is φ 0 . The total duration 3∆t of the loop is thus less than the thermal decoherence time τ discussed above. In Fig. 3b, we plot the bare state populations measured right after the phase loop as a function of φ and their com- parison to theoretical predictions for pinned atoms and for atoms at finite temperature under the adiabatic assumption. This clearly shows that thermal effects are an important ingredient to reproduce the experimental results and that the adiabatic approximation is well justified, see Methods. Note that the mismatch with pinned atoms decreases with increasing φ 0 . This is because the thermal decoherence is quenched by the increasing geometrical coupling among the dark states when the sweep rate γ = φ 0 /∆t > kv, ω R 1 . As a further approximation, we now disregard thermal decoherence and consider that the system after the phase loop is described by a pure quantum state |ψ out = j=1,2 d j |D j 2 . As shown in Figs. 3b and3c, one can easily extract the dark state populations and the absolute value of the azimuthal angle ϕ = Arg(d 2 ) -Arg(d 1 ) from the measured bare state populations, see Methods. When φ 0 0.7π, the values for ϕ match well with the prediction for a pinned atom confirming the quenching of thermal decoherence. At φ 0 = π, the two dark state populations are almost equal. In the language of the Bloch sphere representation, this corresponds to a rotation of the initial south pole state |D 2 to the equatorial plane.

We now reconstruct the full geometric unitary operator U for φ 0 = π. Up to an unobservable global phase, we 1 The thermal decoherence quenching can be quantified by the bare population distance ∆P = 3 i=1 (P i -P 0i ) 2 , where P i and P 0i are the experimental and pinned-atom populations. At φ 0 = π, we get ∆P = 0.04 [START_REF]Geometric Phases in Physics[END_REF]. This value increases when φ 0 decreases, reaching ∆P = 0.19 [START_REF]Geometric Phases in Physics[END_REF] at φ 0 = 0 2 A pure state is denoted by a density matrix ρ fulfilling Tr{ρ 2 } = 1. For a finite-temperature gas, we find Tr{ρ 2 } = 0.95 at φ 0 = π. This value decreases when φ 0 decreases, reaching Tr{ρ 2 } = 0.8 at φ 0 = 0 write:

U = α β -β * α * (7

Discussion

Using a tripod scheme on Strontium-87 atoms, we have implemented adiabatic geometric transformations acting on two degenerate dark states. This system realizes a universal geometric single-qubit gate. We have studied SU(2) transformations associated to laser beams phase loop sequences and shown their non-Abelian character. In contrast to recent works done in optical lattices [START_REF] Aidelsburger | Measuring the chern number of hofstadter bands with ultracold bosonic atoms[END_REF][START_REF] Jotzu | Experimental realization of the topological haldane model with ultracold fermions[END_REF][START_REF] Mancini | Observation of chiral edge states with neutral fermions in synthetic hall ribbons[END_REF][START_REF] Nguyen | Controlling and probing non-abelian emergent gauge potentials in spinor bose-fermi mixtures[END_REF][START_REF] Wu | Realization of twodimensional spin-orbit coupling for bose-einstein condensates[END_REF][START_REF] Song | Spin-orbit-coupled two-electron fermi gases of ytterbium atoms[END_REF][START_REF] Li | A stripe phase with supersolid properties in spin-orbit-coupled bose-einstein condensates[END_REF], our system realizes an artificial gauge field in continuous space. Depending on the laser field configuration, different manifestations of artificial gauge fields can be engineered such as spin-orbit coupling [START_REF] Juzeliūnas | Double and negative reflection of cold atoms in non-abelian gauge potentials[END_REF][START_REF] Jacob | Cold atom dynamics in non-abelian gauge fields[END_REF], Zitterbewegung [START_REF] Juzeliūnas | Double and negative reflection of cold atoms in non-abelian gauge potentials[END_REF], magnetic monopole [START_REF] Ruseckas | Non-abelian gauge potentials for ultracold atoms with degenerate dark states[END_REF] or non-Abelian Aharomov-Bohm effect [START_REF] Jacob | Cold atom dynamics in non-abelian gauge fields[END_REF] (see [START_REF] Dalibard | Colloquium: Artificial gauge potentials for neutral atoms[END_REF][START_REF] Goldman | Light-induced gauge fields for ultracold atoms[END_REF] for reviews). A generalization to the SU(3) symmetry is also discussed in [START_REF] Hu | U (3) artificial gauge fields for cold atoms[END_REF]. Some of these schemes might be difficult to implement in optical lattices. Gauge fields generated by optical fields come from a redistribution of photons among the different plane waves modes and involve momenta transfer comparable to the photon recoil. Observing mechanical effects of non-uniform or non-Abelian gauge fields would thus require atomic gases colder than the recoil temperature and thus cooling techniques beyond the mere Doppler cooling done here [START_REF] Desalvo | Degenerate fermi gas of sr 87[END_REF][START_REF] Meng Khoon Tey | Double-degenerate bose-fermi mixture of strontium[END_REF]. However, the gauge field is still driving the internal state dynamics regardless of the temperature of the gas provided the adiabatic condition is fulfilled. Noticeably, this internal state dynamics is still present when the gauge field is Abelian and uniform. It led us to an interferometric thermometry based on the Fourier transform of the velocity distribution of the gas.

Methods

Cold sample preparation and implementation of the tripod scheme. The cold gas is obtained by laser cooling on the 1 S 0 → 3 P 1 intercombination line at 689 nm (linewidth Γ = 2π × 7.5 kHz). Atoms are first laser cooled in a magneto-optical trap and then transferred into an ellipsoidal crossed optical dipole trap at 795 nm (trapping frequencies 150, 70, and 350 Hz) where they are held against gravity. Atoms are then optically pumped in the stretched m g = F g = 9/2 magnetic substate and subsequently Doppler cooled in the optical trap using the close m g = F g = 9/2 → m ′ e = F ′ e = 11/2 transition, see Fig. 5. The atomic cloud contains about 10 5 atoms at a temperature T = 0.5 µK (recoil temperature T R = ω R /k B ≈ 0.23 µK, where k B is the Boltzmann constant). A magnetic field bias of B = 67 G is applied to lift the degeneracy of the Zeeman excited states. Because the Zeeman shift between levels in the excited manifold F e = 9/2 is large, one can isolate a tripod scheme between three ground-state levels and a single excited state, namely |e = |F e = 9/2, m e = 7/2 , as indicated in Fig. 5. The Zeeman shift of the ground-state levels (Landé factor g = -1.3 × 10 -4 ) is weak (12 kHz) and is compensated by changing accordingly the frequencies of the three tripod laser beams. The lasers are finally tuned at resonance and their polarizations are chosen according to the electrical dipole transition selection rules. In practice, the two laser beams with right and left circular polarizations, respectively addressing the m g = 5/2 → m e = 7/2 and m g = 9/2 → m e = 7/2 transitions, are copropagating. The laser beam with linear polarization, aligned with the magnetic bias field, addressing the m g = 7/2 → m e = 7/2, is orthogonal to the circularly polarized beams, see Fig. 1b. The plane of the lasers is chosen orthogonal to the direction of gravity. The two independent laser offset phases φ 1 and φ 2 (see main text) can be tuned by using two electro-optic modulators.

Adiabatic approximation.

The two independent laser offset phases φ 1 and φ 2 are ramped from 0 to φ 0 ≤ 1.2π at a constant rate γ during the sweep time ∆t = 4 µs. The AC-Stark shifts of the bright states is given by √ 3Ω = 2π × 780 kHz. Since γ = φ 0 /∆t ≤ 2π × 150 kHz, we have √ 3Ω/γ ≥ 5.2 and the adiabatic approximation is well justified. Initial dark state preparation. Starting with atoms in the |m g = 9/2 stretched state, the tripod beams are turned on following two different sequences. The first sequence prepares dark state |D 2 , see equation [START_REF] Michael | Quantal phase factors accompanying adiabatic changes[END_REF]. More precisely, we first turn on the two laser beams connecting the empty bare states |m g = 5/2 and |m g = 7/2 to the excited state |m e = 7/2 and then adiabatically ramp on the last laser beam. This projects state |m g = 9/2 onto |D 2 with a fidelity of 95%. Since the bare state |m g = 9/2 is only present in |D 2 , our choice of basis in the dark-state manifold is well adapted to understand the dark state preparation. A different ignition sequence is used to prepare a combination of dark states |D 1 and |D 2 . By turning on sequentially abruptly the left-circular beam, and adiabatically the right-circular beam, we create a coherent (dark) superposition of the state |m g = 5/2 and |m g = 9/2 . Finally, we turn on abruptly the linearly-polarized beam and we expect to produce the linear combination (|D 1 + √ 3|D 2 )/2. In practice, a systematic phase rotation occurs once the last beam is turned on which adds an extra mixing among the dark states. Performing the bare state population analysis, we find that this initial state corresponds in fact to 0.6|D 1 + 0.8e i0.15π |D 2 . Spin sensitive imaging system. The bare state populations in the ground-state are obtained with a nuclear spin-sensitive shadow imaging technique on the F g = 9/2 → F ′ e = 11/2 line, see Fig. 5. First we measure the population of state |m g = 9/2 with a shadow laser tuned on the closed m g = 9/2 → m ′ e = 11/2 transition. Then, using the same atomic ensemble, we measure the population of state |m g = 7/2 by tuning the shadow laser on the m g = 7/2 → m ′ e = 9/2 transition. This transition is open but its large enough Clebsch-Gordan coefficient ( 9/11 ∼ 0.9) ensures a good coupling with the shadow laser. The population of state |m g = 5/2 is measured in the same way (m g = 5/2 → m ′ e = 7/2 open transition, its Clebsch-Gordan coefficient 36/55 ∼ 0.8 being still large enough). The shadow laser beam shines the atoms during 40 µs with an on-resonance saturation parameter I/I s = 0.5 (saturation intensity I s = 3 µW/cm

2 ). With such values, the average number of ballistic photons scattered per atom is less than one and optical pumping can be safely ignored, ensuring an accurate measurement of the ground-state populations. To achieve a good statistics, the same experiment was repeated 100 times and the corresponding data averaged. The error bars on the bare state populations correspond to a 95% confidence interval.

Dark states and unitary matrix reconstruction.

A state in the dark-state manifold takes the form |ψ = Do note that the normalization of |ψ restricts the possible values of the Pi summing up to 1. The sign of ϕ is determined using the prediction of equation (6) for a pinned atom (M → ∞).

To reconstruct the unitary matrix U , as expressed in equation ( 7), we perform the phase loop sequence on two different initial dark states (their representative points on the Bloch sphere should not be opposite) and perform the dark state reconstruction for each of them. The two phase terms in U are reconstructed up to a sign. As for the dark state reconstruction, we rely on the prediction for a pinned atom to lift this sign ambiguity.

Gauge fields and adiabatic Schrödinger equation. The time-dependent interaction operator for the resonant tripod scheme, in the rotating-wave approximation, has the following expression:

H(t) = Ω(r, t) 2 3 i=1 |e i| + H.c. (8)
We assume here that the laser Rabi frequencies coupling the ground-states |i = |m g = i + 3/2 to the excited state |e = |m e = 7/2 have all the same amplitude denoted by Ω. The time dependency comes from the cyclic ramping sequence of the two offset laser phases φ j (j = 1, 2). Neglecting transitions outside the dark-state manifold (adiabatic approximation), the system is described by a quantum state |ψ(r, t) = Σ j=1,2 Ψ j (r, t)|D j (r, t) , where Ψ j is the wave function of the centre-of-mass of the atom in an internal state |D j . In this basis, the adiabatic Schrödinger equation for the column vector Ψ = (Ψ 1 , Ψ 2 ) T reads:

i Ψ = (p1 1 -A) 2 2M + W -ω t Ψ, (9)
where the dot denotes time derivative. The first two terms on the right-hand side describe the dynamics of an atom subjected to the synthetic gauge field. The last term ω t ≡ [ω jk ] ≡ [i D j | Ḋk ] is due to the cyclic ramping sequence of the laser phases. Only this term remains for a pinned atom (M → +∞), in which case one recovers equation ( 6). The general expressions of A and W are given in the main text. With equal and constant Rabi frequencies amplitude, and for the orientation of our laser beams, one finds:

A = 2 (k 2 -k 1 ) 3 M, A 2 2M = 8E R 9 M, W = - 4E R 9 M,
where E R = ω R = 2 k 2 /(2M ) is the recoil energy and k j is the wavevector of laser beam j (see main text). As one can see, all these operators have the same matrix form. The matrix M reads:

M = 1 1 + s • σ 2 = 3/4 - √ 3/4 - √ 3/4 1/4 ( 10
)
and satisfies M 2 = M, its unit Bloch vector being s = (-√ 3/2, 0, 1/2). As a consequence, all these operators can be diagonalized at once by the same transformation and amenable to the simple projector matrix form:

M → M D = 1 0 0 0 . ( 11
)
Because of our laser beams geometry, the vector potential A is in fact Abelian since its only non-zero matrix component is along k 2k 1 .

In contrast, the operator ω t has a different matrix form. Indeed, the two offset phases φ j of the lasers (see main text) can be addressed at will. Following [START_REF] Dalibard | Colloquium: Artificial gauge potentials for neutral atoms[END_REF], we get:

ω t = 2 φ1 + φ2 ( φ1 -φ2 )/ √ 3 ( φ1 -φ2 )/ √ 3 ( φ1 + φ2 )/3 . (12)
In particular, we note that ω t leads to non-Abelian transformations. An immediate consequence is that, for a given phase loop in parameter space, the geometric unitary operator associated with a cycle of phase ramps depends on the starting point of the cycle on the loop. Different starting points lead to different, though unitarily related, non-commuting geometric unitary operators.

Figure 1 .

 1 Figure 1. Tripod scheme. (a) Propagation directions of the laser beams and their polarizations along a magnetic bias field B. Two Electro-Optics Modulators (EOM) are used to sweep the two independent relative phases of the laser beams. (b) Bare energy level structure of the tripod scheme, implemented on the intercombination line of Strontium-87. The magnetic bias field shifts consecutive excited levels by ∆B ≃ 760Γ = 5.7 MHz. It allows each tripod polarized laser beam to selectively address one of the magnetic transitions marked by the black arrows.

Figure 2 .

 2 Figure 2. Ballistic expansion. Time evolution of the bare state populations after the tripod ignition sequence (duration t0 ≃ 8 µs) is completed and laser beams have reached equal Rabi frequencies Ω = 2π×250 kHz. The blue circles, the green squares, and the red stars correspond to the populations P1, P2, and P3 with |i ≡ |mg = i+3/2 (i = 1, 2, 3), respectively.The error bars correspond to a 95% confidence interval. Solid lines: theoretical predictions given by equations[START_REF]Geometric Phases in Physics[END_REF]. The temperature T , the initial and the final populations of each spin state are the fit parameters. The dashed lines, at early times, extrapolate the fits into the time window t0. We get a temperature T = 0.5(1) µK meaning that the atoms do not move significantly during the dark state preparation sequence since vt0/λ ≃ 0.08.

Figure 3 .

 3 Figure 3. Geometric gate operation. (a) Phase loop in parameter space (φ2, φ1). φi (i = 1, 2) are the two independent offset phases tuned by the EOMs shown in Fig. 1a. We have performed two counterclockwise cycles: the first one is a → b → c and starts from the origin, the second one is c → a → b and starts from the upper corner. The loop is completed in 12 µs and its excursion is φ0. (b) Measured bare state populations P1 (blue circles), P2 (green squares) and P3 (red stars) as a function of φ0 for the first cycle. The Rabi frequencies are Ω = 2π × 450 kHz and T = 0.5 µK. Dark state reconstruction as a function of φ0. (c) Population of |D1 (blue circles) and |D2 (red squares). (d) Azimuthal phase ϕ. The solid and dashed curves in panels (b), (c), and (d) are the theoretical predictions for a pinned atom and for a gas at temperature T respectively. The error bars correspond to a 95% confidence interval.

Figure 4 .

 4 Figure 4. Unitary operators reconstruction. (a) Elements of operator U given by equation (7) for the phase loop a → b → c at φ0 = π, see Fig. 3a. The blue, red, and green bars correspond to a pinned atom, a gas at temperature T = 0.5 µK, and the experimental data respectively. (b) Same as (a) but for the phase loop c → a → b. The error bars correspond to a 95% confidence interval.

) with |α| 2 +

 2 |β| 2 = 1. The previous dark state reconstruction, done after the phase loop applied on |D 2 , gives access to |α|, |β| and Arg(α) -Arg(β), see Methods. To obtain Arg(α) and Arg(β) and fully determine U , we start from a linear combination of dark states |D 1 and |D 2 , perform the phase loop and process the new data. The results are shown in Fig. 4a and compared to the theoretical predictions for a pinned atom and a gas at finite temperature. The good agreement with our data validates the expected small impact of temperature for φ 0 = π. Probing non-Abelianity. With the previous phase loop protocol, we have U = U c U b U a where a, b, and c label the edges of the loop, see Fig. 3a. To illustrate the non-commutative nature of the transformation group, we will cycle the phase loop counterclockwise starting from the upper corner. We then reconstruct the corresponding unitary operator U ′ = U b U a U c like done for U . The results are depicted in Fig. 4b and show that U and U ′ are indeed different, though unitarily related, confirming the sensitivity of these geometric transformations to path ordering. The Frobenius distance between the two unitaries is D = 2 -|Tr(U † U ′ )| = 1.27(25) and is in agreement with the theoretical result for a pinned atom (D = 1.09) and for a finite-temperature gas (D = 1.14). These values have to be contrasted with the maximum possible Frobenius distance D = 2.

Figure 5 .

 5 Figure 5. Energy levels and experimentally relevant transitions. A magnetic bias field B = 67 G lifts the degeneracy of the different Zeeman manifolds and allows to address each transition individually. The Landé factors g are indicated for each hyperfine level. The black arrows correspond to the tripod beams (see main text for more details). The dashed red arrows indicate the transitions used for the shadow spin-sensitive imaging system. The dash-dotted purple arrow is the red-detuned cooling transition used in the far-off resonant dipole trap.

j=1, 2

 2 d j |D j with |d 1 | 2 and |d 2 | 2 = 1-|d 1 | 2 the populations of states |D 1 and |D 2 and ϕ = Arg(d 2 ) -Arg(d 1 ) the azimuthal angle. Using equation (1), we immediately find |d 2 | = 3 P3 /2, cos ϕ = ( P1 -P2 )/ P3 (2 -3 P3 ).

Acknowledgements. F. C. thanks CQT and UMI MajuLab for their hospitality. The Authors thank Mehedi Hasan for his critical reading of the manuscript. C. M. is a Fellow of the Institute of Advanced Studies at Nanyang Technological University (Singapore). This work was supported by the CQT/MoE funding Grant No. R-710-002-016-271.

Data availability. The authors declare that the main data supporting the findings of this study are available within the article. Extra data are available from the corresponding author upon request.

Competing interests: The authors declare nonfinancial interests.