Jean-Baptiste Camps
email: jbcamps@hotmail.com

Florian Cafiero
email: florian.cafiero@polytechnique.edu

Fabrice Rossi

Setting bounds in a homogeneous corpus: a methodological study applied to medieval literature *

The authors present here an exploratory and unspecific method that does not necessitate any a priori on the data -or any heavy transformation such as lemmatisation-that would have to be understood as a first step in the apprehension of a corpus. After a first phase of calibration, based on a control sample, the authors will introduce a method which heuristic value is to bring out, at different levels, internal divisions of different kinds (diachronic, diatopic, related to authorship or scribes,. . . ), that can then be analysed specifically. The authors illustrate this method by applying it to a corpus of Occitan medieval texts, the vidas. The corpus's authors and origins are in good part unknown.

Les auteurs présentent ici une méthode exploratoire, non spécifique et ne nécessitant pas d'a priori sur les données -ni de transformations lourdes comme la lemmatisation -, qui se définit comme une première étape dans l'appréhension d'un corpus. Après une phase de calibration, fondée sur l'utilisation d'un échantillon témoin, sera présentée une méthode dont la valeur heuristique est de pouvoir rendre apparentes, à différents niveaux, des divisions internes de différentes natures (diachroniques, diatopiques, autoriales, scribales,. . .), pouvant ensuite faire l'objet d'analyses spécifiques. Ils illustrent cette méthode par une application à un corpus à première vue assez homogène de courts textes occitans médiévaux, les vidas, dont les auteurs et les origines sont en bonne part inconnues. * The database used by the authors, along with the R scripts, and additional relevant material can be found at http://graal.hypotheses.org/617.

A peculiar corpus: the vidas

Elements of context

Amongst the relatively small corpus of Occitan prose literature from the XIII th century that has survived to this day, the vidas, along with the razos, constitute a corpus of roughly 225 texts known as the "biographies" of the troubadours since Raynouard's edition in 1820. Several debates surround these texts' origin: were they performed at first or were they created specifically to be integrated into manuscripts? Did they appear in Northern Italy or do they come from an older Languedocian tradition? Despite the lack of medieval unambiguous vocabulary, there is an apparent distinction between commentaries, explanation of a specific poem (a razo), and more general biographical narratives detached from a specific work (vidas as we call them).

Written in Northern Italy during the XIII th century (perhaps, for some of them, from some preexistent Languedocian material 1 ), vidas seem to answer a specific need for contextualisation, consistent with the needs of a public separated from the poems' authors both by time (up to over a century and a half), space, and native language. These vidas also participate in a larger movement linked with the transformation from an oral tradition based on performance to a written one: organisation of the poems, classification in various genres, attribution to the right authors, and even at times philological care for the text and its establishment. In this context, the vidas are used both to determine an adequate and unique explanation as well as an exegesis of the troubadour's poems based on the author's life. It also helps organising the troubadours in a hierarchical way, each author receiving a proper place, the esteem and rank he or she is entitled to by talent or birth. In this transformation, the oral living tradition of troubadour's lyric becomes a honourable literature, and the troubadours themselves are dignified and looked upon as respectable authors 2 .

The sample this paper focuses on is composed of 52 vidas, taken from the same Venetian manuscript of the final quarter of the XIII th century (Bibl. ap. vat., Reg. lat. 5232, known to philologists as the chansonnier A) 3 . Much like other luxurious Venetian manuscripts of the same period (in particular I and K ), A uses the vida, along with a miniature -often described as a "portrait" -to mark the beginning of the authors' sections and to introduce the poems 4 . The most recent datable vida in our corpus is from c. 1274, while the date of the earliest is uncertain. Their authorship, exact location and composition date are in question. Only a few vidas are signed, and none are in A. Still, in some manuscripts (but not in A, the vida of Bernart de Ventadorn is "signed" by a known author, Uc de Saint-Circ ( † c. 1257), one of the troubadours having lived in Italy. He is also considered as a possible author for several other vidas, although to a much debated extent 5 . The main purpose of this paper will be to address questions 1 See the brief summary of this question in Burgwinkle (1999, p. 252-523).

2 For more information about vidas, see, among numerous studies Wilson [START_REF] Poe | From Poetry to Prose in Old Provençal: The Emergence of the Vidas, the Razos and the Razos de Trobar[END_REF] and [START_REF] Meneghetti | Il Pubblico dei Trovatori, ricezione e riuso dei testi lirici cortesi fino al XIV secolo[END_REF].

3 We used the edition provided by Françoise Vielliard in [START_REF] Lemaître | Portraits de troubadours : initiales du chansonnier provençal A[END_REF]. 4 Both the vida and the miniature are part of a metatextual apparatus which Meneghetti calls "un doppio filtro metatestuale" (Meneghetti, 1992, p. 348).

5 [START_REF] Panvini | Le Biografie provenzali. Valore e attendibilità[END_REF] thinks that Uc can be considered author of 36 vidas , while [START_REF] Favati | Le Biografie trovadoriche : testi provenziali dei secc[END_REF] attributes him a majority of the vidas and nearly all razos. Meneghetti considers as certain the paternity of Uc on the vidas of Raimbaut d'Aurenga, Guillaume IX, Sordello (in the IK version) and Guilhem Figueira, and has even stated that it was "ragionevole, quantomeno a livello di ipotesi di lavoro, considerare come composte, o almeno rivedute, da Uc tutte le biografie contenenti allusioni ad avvenimenti anteriori al 1257" (Meneghetti, 1992, p. 243-245), while she recognised that historiography might have fallen, at some point, to a "furor attribuzionistico a senso unico", and has also proven the existence of other biographers of date, authorship and manuscript tradition, without having to develop an automated syntactic analysis, but rather by working from a graphical and lexicometrical point of view.

Lexicometric characteristics of our corpus

The vidas are brief narratives, telling us in a very standardised style the life and deeds of a troubadour, as an introduction to his poems. The fifty-two vidas of A are usually short (a few lines), but there are some variations in size (fig. 1,p. 4): the shortest one (the vida of Richartz de Tarascon) has only 22 words, and the longest one (Raimon Jordan, the viscount of Saint Antonin) has 676 words, the mean being 164 words (and the median 147). Only two especially long vidas have more than 400 words and they correspond to very specific stories -Guillem de Cabestaing and the story of the eaten heart 6 , Raimon Jordan and his impossible love. While the unusual length of these latter vidas might be an indication as to their somewhat later composition, the length of the vidas in itself is not necessarily always a clue as to its possible author or date. In many cases, the length of the text is determined from the available sources (mostly the poems attributed to the said troubadour), and also to the fame of the troubadour, the esteem he was given, and his rank in troubadour lyric. The fifty-two vidas amount for 8519 words and 1479 word-forms, of which 833 are hapax (respectively 9,78% and 56,32% of the total).

Vidas are globally composed according to the same patterns. The genre in itself is clearly recognisable, and they constitute, as Boutière and Schutz (1964, p. viii) stated, "un véritable genre littéraire, dont le moule, le squelette, la langue et la phraséologie ont été si bien respectés au cours du temps qu'il serait aisé d'en faire aujourd'hui des pastiches". In fact, they are built around the same structure and use a limited amount of words and formulae to describe the troubadours. Most of them start with the phrase "[Troubadour's name] si fo de [place] e fo [first designation of the troubadour]", and the rest of the text is, for the most part, constructed on this relatively standard and nearly always present social designation, along with a limited set of qualities, and narrative patterns attributed according to this designation 7 .

In terms of repeated segments, this leads to two antagonist factors: the lack of graphical standardisation -from modern standards -characteristic of medieval vernacular languages such as Old Occitan, causing the number of forms to increase; the limited amount of vocabulary and the stereotyped formulae, leading to a restriction in terms of lemmata. This is illustrated by the study of the Repeated Segments 8 . There is a total number of 1228 RS of length (in words) l ≥ 2 and frequency f req. ≥ 2 distributed according to Tab. 1.

The first observation to be made is both the elevated number of RS and the relatively low average frequency, which confirms the existence of these two antagonist fac- (Meneghetti, 2002, p. 148), contesting the stylistic comparative approach of [START_REF] Guida | Ricerche sull'attività biografica di Uc de Saint Circ[END_REF], potentially biased by the very standard and formulaic style of the vidas.

6 For more elements on this very famous story, see [START_REF] Gaunt | Le coeur a ses raisons": Guillem de Cabestanh et l'évolution du thème du coeur mangé'. In Scène, évolution, sort[END_REF], or, for a more general synthesis, in English, Gaunt (2006, part. p. 77-79).

7 In A, there is only one vida with no such designation, the vida of Marcabru, and it is justified by the fact that he was, as a child, "gitaz a la porta d'un ric home, ni anc non saup hom qui . l fo ni don" (was thrown at the door of a powerful man, and nobody ever knew who he was nor whence he came). In the other texts, this designation can be present either in its simpler form "he was a. . . " or under the filiation form, e.g. "he was the son of. . . ".

8 For a presentation of the concept of RS and their use, see [START_REF] Salem | L'inventaire des segments répétés d'un texte[END_REF]; the repeated segments were computed using the Lexico3 software, developed by the SYLED-CLA2T of University Paris 3 Sorbonne-Nouvelle. The shorter segments systematically comprised in a longer one (that is, preceded and followed always by the same two word-forms) were eliminated by the software. tors. If we take a closer look at the RS, we see that in many cases, if simple graphical or flexional changes are ignored, many different RS can be attributed to the same model. For instance, the formula meaning "and here are written of his songs" corresponds to different RS and to segments non identified as repeated due to these variations (see Appendix, p. 23, Tab. 5), for a total frequency of 10 and a form that could be summarised as " Et (aqui-aissi) son escriutas (granren-moutas-∅) de las soas (cansos-canssos-chanssos)". The variations can be inherent to the formula itself, as for the formula meaning "he was from . . . , from a castle called . . . ". It is very likely that lemmatisation and standardisation of the texts would reduce the number of different RS and greatly increase their frequency9 . Finally, it is to be noted that the RS amount to roughly 60% of the texts. If the vidas were to be stripped from all RS, the total amount of word-forms would go from 1479 to 1433, and more importantly the total amount of words from 8519 to 3448 (a decrease of approx. 59,526%). Had the texts been lemmatised, the RS count could have still been higher10 .

In order to analyse these 52 texts, we had to choose between standardising and/or lemmatising, or working directly from the word forms. Classifying texts depending on the vocabulary they use, and thus, working on lemmatised word forms, is a common approach, useful for content or theme-based clustering, but is not always the most relevant [START_REF] El-Bèze | Peut-on rendre automatiquement à César ce qui lui appartient ? Application au jeu du Chirand-Mitterrac[END_REF], and especially not here, given the lexical characteristics of the vidas. On the contrary, our goal was to identify all the elements that constitute clusters in the corpus: groupings due to branches of the tradition, scribes, diatopic or diachronic variations, or possible authors, and not to determine a priori which variations were relevant or not. This led us to work on both unlemmatised and unstandardised word forms, as any graphical variation could be an indication of the origin of a text and of its place in the manuscript tradition. For the same reasons, we chose not to ex ante rule out the most frequent words, including "stop words" or "function words", as variations through their usage could also be of interest regarding matters of style or morphosyntactic variations, and be helpful for author identification. In fact, studies tend to demonstrate that variation in the use of the "function words" is more relevant to authorship attribution than the study of the rarer, content-related, words [START_REF] Stamatatos | A survey of modern authorship attribution methods[END_REF][START_REF] Argamon | Measuring the usefulness of function words for authorship attribution[END_REF], notably because their usage is less conscious and less subject to variations due to change of theme or genre. This explains why, in a second time, we have chosen to restrict our database to the one hundred most frequent word-forms (see below, from 2.2 onwards). This is why the vidas have simply been transformed in a raw word frequency table 11 . Consequently, we had a total of 1480 different word forms across all 52 texts. Each text contained between 14 and 296 word forms, the mean being of 92 (fig. 1). The number of different word forms is obviously affected by the total number of words, and we can observe that they are linked by a non-linear relationship (fig. 2) 12 . This relationship seems to fit the intuition that, as though the number of words a text may contain is virtually infinite, the vocabulary a language possesses is finite. Consequently, the curvature of this relationship is a fair indication of the amount of vocabulary that an author had at his disposal. This is particularly true in the case of the vidas and reveals the standardisation of their vocabulary. The abnormal individual (the same abnormal individual as in the scatter plot, the vida of the viscount of Saint Antonin) possesses properties that substantially differ from the other texts 13 . Had we done the non-linear regression again while ignoring the abnormal individual, we would have had an even more efficient model (dashed line on fig. 2).

Term weighting and calibration

Two important characteristics of our corpus are the limited amount of vocabulary, a direct function of the total number of words, and the relatively important variation 11 Informations relative to sentences' length and use of punctuation were not relevant, since both of these are conditioned by the choices of the modern editor, not the author of the text.

12 The non-linear regression was done by minimising the non-linear least-squares using the Gauss-Newton algorithm. 13 The wider vocabulary used by this text seems to sustain its classification in a somewhat different group than the other vidas, being less of a short biographical narrative and more of a developed narrative, announcing Italian XIV th century novellas (such as those contained in Boccaccio's Decameron).

in sizes between texts, a known difficulty in term of text classification and authorship attribution methods [START_REF] Sanderson | Short text authorship attribution via sequence kernels, markov chains and author unmasking: an investigation[END_REF]. Thus, we had to seek a way to avoid the risk of seeing our clusters determined solely by the size of the texts. To solve this problem, we considered using some form of differential term weighting [START_REF] Dumais | Improving the retrieval of information from external sources[END_REF], but the most popular ones (including Dumais') are built to be useful in a theme-based approach different from ours. To compensate specifically for the different text lengths, we adopted a very basic weighting (let F ij be the frequency of term i in document j and T j the total of terms in document j, Fij Tj )14 and to compare its results with a simple centring and scaling of the variables. Results were roughly equivalent when used with Hierarchical Clustering (see fig. 5), but not with the same errors, but our weighting was superior when used with Kohonen's Self organising maps.

Another problem came from the difficulty to judge, ex ante as well as ex post, which method would be the most helpful to divide our corpus into relevant subgroups. This is why we constituted a control sample (see table 7 in appendix), composed of texts as close as possible to the vidas of A: texts from the same manuscript, likely copied by the same scribe in the same period and area. It is to be noted, though, that these texts are from a different literary genre, verse in lieu of prose, and have somewhat different properties as to their lexical richness. Notwithstanding, lacking a corpus of clearly attributed vidas, they were our least biased alternative15 . For this control sample to be effective we chose texts from four different troubadours. We picked them in order to have two strongly different groups (an earlier Northern Occitan group and a later Northern Italian group) so that the two troubadours in each group, while close one to the other, would still be separated from him in terms of chronology and geography. The first group is composed of the auvergnat Peire d'Alvernha (fl. c. 1150-1170) and the limousin Guiraut de Bornelh ( †1215); the second group is composed of Uc de Saint Circ ( †1257), a troubadour who was born in Quercy but composed most of his work in Northern Italy, and the Venetian Bertolome Zorzi ( †c. 1273).

A first approach: Factor Analysis

Homogeneity or dimensionality?

Our first approach was to use Factor Analysis. In addition to the more usual Principal Components Analysis, we computed a Correspondence Analysis. While partly similar in its principles to Pearson's PCA, it applies the same weighting to lines and columns (the χ 2 ) and is particularly relevant to the study of homogeneous data sets, contingency tables in general, and frequency tables in particular.

It seemed likely that we could experience here what has been known since Bellman as the "curse of multi-dimensionality". Our corpus is composed of a small number of individuals (i.e. objects described by a set of data). For each individual, we study several characteristics, since each word of the frequency table is a characteristic of its text. Geometrically, this results in a distribution of very few points in a high-dimensional space, very often giving results that prove unhelpful with respect to relevance or perceptible contrast (E. [START_REF] Chavez | Searching in metric spaces[END_REF]). Yet, it is possible a technique such as Factor Analysis would work. It has been shown that increasing dimensionality can even help as long as the added dimensions are relevant [START_REF] Houle | Can Shared-Neighbor distances defeat the curse of dimensionality[END_REF]. Are the number of occurrences of each word relevant dimensions? The only way to find out was to use them in the computation and check. In order to refine this analysis, we applied different forms of Differential Term Weighting on the frequency table [START_REF] Dumais | Improving the retrieval of information from external sources[END_REF]). Yet, this method failed to increase contrast and readability16 .

Reducing dimensionality

How can one explain this failure? First, this could come from our data: the individuals could simply be "too close to call" if they form distinct and coherent subgroups. This would be in favour of the hypothesis of one single author for most of the vidas. But it could also come from the dimensionality problem aforementioned. In order to avoid this problem, we implemented two methods.

First, we tried to recompute our two Factor Analyses, taking into account only the top one hundred most frequent words, as is often done in authorship attribution methods. Yet, this method failed to correctly sort our control sample. To avoid dimensionality problems, we then implemented a non-metric Multidimensional Scaling (fig. 4).

This procedure behaves quite well on our test corpus, and actually separates some groups from a core of texts in the vidas, which seems to show that there are indeed distinct parts in our corpus. Yet, the visualisation hardly helps us to determine what the more relevant groupings of these texts would be.

Determining an agenda

In order to obtain results on questions relative to authorship, scribes, and manuscript tradition, we needed to operate a clear-cut classification of the individuals, to establish proximities and distances between texts. This is why we sought a method that could allow us to represent these proximities and distances in an human-readable fashion, while still being able to render the hierarchy of the distances, and the abnormal individuals. We focused however on unsupervised learning, allowing us to gather the more similar individuals in relevant subsets or "clusters", without having to be given any characteristics of what these clusters should be. The method we have chosen to use is a classic type of non-supervised learning called hierarchical Clustering.

In the agglomerative version of this method, each individual is aggregated to another single individual considered the "closest", according to a criterion the user has to choose. Then, the clusters formed in this first step are once again aggregated to the closest cluster, the operation being repeated until the clusters are gathered in a unique group containing all the individuals. As we do not define a priori constraints on what the common features of individuals belonging to a cluster should be, similarity is here understood as a mere matter of distance between individuals: the closer individuals are on a geometrical viewpoint, the more similar they are.

We then used our test corpus to understand which precise technique of Hierarchical Clustering to use. Several decisions had to be taken: which distance measure? Which linkage criterion? Do we use our term weighting? Do we use all the word forms? Should we restrain to a hundred of them instead, as has often proved effective in authorship attribution methods [START_REF] Argamon | Measuring the usefulness of function words for authorship attribution[END_REF][START_REF] Stamatatos | A survey of modern authorship attribution methods[END_REF]? We ran many possible combinations 17 of these decisions to see which would best divide our test corpus, the best division being to gather texts this way: {{1, 2, 3}, {4, 5, 6}}, {{7, 8, 9}, {10, 11, 12}}. Our best fit was obtained when using only the one hundred most common forms, with Manhattan distance 18 , Ward's method 19 and weighted terms.

The results were then compared to those of a Self Organising Map [START_REF] Kohonen | Self-organized formation of topologically correct feature maps[END_REF] 20 and to a HC done on the prototypes corresponding to observations. We tested several grid-sizes (see.Tab. 2) and settings on our test corpus, using weighted frequencies and the one hundred most frequent word-forms. We were able to notice that the quantisation error decreased with the size of the map, ranging from 8.99e-46 for an 7 × 7 map, to close to 0 starting from a 16 × 16 map, while Kaski & Lagus' error [START_REF] Kaski | Comparing self-organizing maps[END_REF] seemed to attain a minimum to then stabilise. Nonetheless, the map whose results were the most relevant were obtained with a 8 × 8 (quantisation error 6.06e-51) and a 10 × 10 map (quantisation error 1.69e-102), corresponding to a maximisation of the HC agglomerative coefficient at 0.803519 and 0.8053272 (yet, the results obtained with a 9×9 map, agglomerative coefficient of 0.7982797, showed one slight misattribution). The 8 × 8 map was obtained using Principal Component based initialization, standard Best Matching Unit calculation approach, a radius of 3.968489, exponential-like annealing and a gaussian kernel (see fig. 6). The results obtained by using our weighting were better than with scaled frequencies. 17 We tested combinations of: single-linkage, complete-linkage, weighted-average linkage, UPGMA, McQuitty and Ward's method as linkage criterion; euclidean, maximum and Manhattan as distance; with all the words, the one hundred most common and the two hundred most common; with and without our weighting.

18 Manhattan distance is a concept coming from "taxicab geometry", where the distance between two points is the sum of the absolute differences of their coordinates:

n i=1 |x i -y i |
19 In Ward's method, an individual is not necessarily linked with the closest neighbour; pairs are formed so that they make the least increase to the error sum of squares at each step [START_REF] Ward | Hierarchical grouping to optimize an objective function[END_REF]. 20 To compute the Self Organising Map and obtain the distances between prototypes, we used the R Yasomi package (Rossi, 2012). We then applied the same method to the text of the vidas. Similarly to what happened with our test corpus, quantisation error tended to decrease with the size of the map, and Kaski & Lagus's error decreased to a minimum, to then stabilise (Tab. 3). Based on the experience obtained from the control corpus, we discarded the 18 × 18 map where errors were too high, as well as the 21 × 21 and 28 × 28 maps that showed increased errors when compared to their neighbouring maps. The analysis was founded on the 20 × 20 (Fig. 8), 23 × 23, 24 × 24, 27 × 27, 29 × 29 and 30 × 30 maps, which main structure was robust, while the 19 × 19, 22 × 22, 25 × 25 and 26 × 26 maps, that showed changes in structure were kept for comparison purposes, along with the HC done directly on the data (Fig. 7).

Results

The main division of the corpus in two large groups, the former more important than the latter, is common to all HC's, even if the exact individuals composing these groups may vary. Much like the control sample, this division of the corpus is considerably higher than the others (2.44 height against 1.68 for the second, see fig. 9). The possibility exists that here, as previously, this division is to be interpreted in terms of diachronic and/or diatopic difference. For these two main clusters, if we exclude the "moving" individuals (i.e. individuals that "move" between the different clusters in some of the maps listed above) 21 , and take only into account the stable ones (i.e. individuals that are always in the same cluster) 22 , and try to categorise the clusters according to all the word-forms of their texts (using a value test), the categorising variables with the highest probability values are, for the second cluster, the most frequent words of the core structure of the vidas (si, p = 0.0228, fo, p = 0.0083, de, p 0) and words related to the poetic activity which mostly occur as an introduction to the songs following the vidas and are also often part of the core structure of the vidas (sirventes, p = 0.0083, canssos, p = 0.0406, coblas, p = 0.0422). On the other hand, words characterising the first cluster denote a more complex structure (pronoun el, p = 1e -04, determiner lo, p = 3e -04, subordinating words que, p = 0.0029, don, p = 0.0282, preposition per, p = 0.0034) and a more developped narrative (verbs of movement venc, p = 0.0075, anet, p = 0.008, words related to time, lonc, p = 0.0396, temps, p = 0.0423). If we use a database constituted of informations relating to the dates, and the mentions of geographical origin of the troubadours, as well as some external informations concerning the tradition of the texts (notably the other manuscripts in which the texts are attested) and the supposed authors when they are know, none of these criteria are to be linked with a significant (> 0.05) probability value to one of these clusters. It is still interesting to acknowledge that 100% of the vidas relating to the older troubadours (first half of the XII th century) are in the first cluster, as well as 100% of the mention of Limousin as the geographical origin, and that 81.25% of the vidas of the first cluster are also to 21 The vidas of Aimeric de belenoi, Aimeric de Sarlat, Arnaut Daniel, Bernart de Ventedorn, Bertolome Zorzi, Bertran de Born, Giraut de Bornelh, la Comtessa de Dia, Peire Bermon lo Tort, Pons de Capduoill, Raimons de Miraval, Sordel, Uc Brunet.

22 For the first cluster, Aimeric de Peguillan, Albertet, Arnaut de Maruoill, Azemar lo Negre, Cadenet, Dalfin d'Alvernge, Folqet de Marseilla, Gaubert Amiel, Gaucelm Faidit, Gui d'Uissel, Guillem de Cabestain, Helias Cairel, Hugo de Pena, Jaufre Rudel, le Monges de Montaudo, le Monges Gaubertz, le Vescoms de Saint Antoni, Marcabrus, Peire d'Alvernge, Peire Raimon, Peire Rotgiers, Peire Vidals, Peirol, Perdigons, Raembautz de Vacheiras, Uc de Sain Circ; for the second, Berengier de Palasol, Bertran de Lamanon, Daurde de Pradas, Folqetz de Romans, Gauseran de Saint Ledier, Guillem Ademars, Guillem de Bergedan, Guillem de Saint Leidier, Na Castelloza, Peire de Bussinac, Peire de la Mula, Ricart de Berbesiu, Richart de Tarascon. Setting bounds in a homogeneous corpus be found in the XIV th century Languedocian manuscripts ER, which also points to more important texts, with a broader circulation. Finally, the more categorising variables are actually, in favour of the first cluster, the number of different word-forms (p = 0.0032), the length (p = 0.006) and number of hapax (p = 0.0032). Of this can be deduced that this main division opposes, in a first cluster, longer texts with a stronger authorial value, and in a second cluster, shorter texts (sometimes quite short, 22 words for the vida of Richart de Tarascon), that the method fails to attribute, texts lacking clear authorship, and probably some very short vidas that were composed by anonymous authors only to be integrated in the manuscripts. This main division seems to tell us little about the authors, and it will be necessary to consider the lower divisions of the tree. The second and third divisions are a bit lower (1.68 and 1.36, fig. 9) but still relatively high. They are found in the same fashion in all dendrograms 23 (except for one small change in the 24 × 24 map), resulting in four subgroups. The first subgroup is build around nine stable individuals 24 and is opposed to the second subgroup, build around twelve stable individuals 25 on several external criteria: for all the stable individuals, the second subgroup contains all the vidas making mentions of locations in Catalonia or Aragon, and the only vida in the stable subpopulation to be also present in the Catalonian manuscript Sg 26 and all the vidas for which there exists a strong and acknowledged suspicion for Uc de Saint Circ to be the author 27 . Moreover, this second subgroup 23 That is, the tree-like diagrams illustrating the groupings done by the method at each step. 24 Albertet, Azemar lo Negres, Dalfin d'Alvernge, Gaubert Amiel, Gaucelm Faiditz, Helias Cairel, Hugo de Pena, Peire d'Alvernge, Peire Vidal; Peirol is also almost always included in this subgroup, while Perdigon, Arnaut de Maruoill, lo Monges de Montaudo and Peire Raimon tend to move from this subgroup to the second, and Bertran de Born from this subgroup to the fourth.

25 Aimeric de Peguilhan, Cadenet, Folqet de Marseilla, Gui d'Uissel, Guillem de Cabestain, Jaufre Rudel, lo Monges Gaubert, lo Vescoms de Saint Antoni, Marcabrus, Peire Rotgier, Raembaut de Vacheiras and Uc de Sain Circ; Uc Brunet is also almost always included in this subgroup (once in the fourth), as is Sordel; many individuals tends to move between this subgroup and the fourth (Arnaut Daniel, Bernart de Ventadorn, Bertolome Zorzi, la comtessa de Dia, Sordel, Uc Brunet). 26 The vida of Raimbaut de Vaqueiras. It is to be noted that two out of the other four vidas of A also found in Sg are part of the moving individuals of this subgroup: nominally, Bernart de Ventedorn, Guiraut de Bornelh. Pons de Capdueil tends to move between the third and fourth subgroup as does Guilllem de Saint Leidier. 27 The vida of Uc himself, among the stable individuals [START_REF] Meneghetti | Uc e gli altri : sulla paternità delle biografie trobadoriche[END_REF], and the one of Bernart contains longer and more complex vidas with more vocabulary than the ones of the first subgroup. The third subgroup contains only three stable individuals 28 , all very short texts; the fourth subgroup also contains three 29 , in both cases vidas of a very limited diffusion, found only in the Venetian manuscripts 30 . As was previously stated, some individuals from the third subgroup tend to move towards the first, and individuals from the fourth towards the second. It is then likely we need to discard these third and fourth unclear subgroups in order to increase the ability to interpret the results. We then need to center our analysis on the first two groups, integrating some of the moving individuals from the third and fourth with a firm anchorage in one of the two first subgroups.

If we restrain our analysis on these first two subgroups, we can define them as containing stable (S) and moving (M) individuals and being themselves constituted, in most analyses, of two different subgroups ({{1a, 1b}, {2a, 2b}}); when the belonging of an individual to a subgroup was robust, it as been indicated, otherwise not, as shown in Tab. 4 (individuals moving between those two groups themselves, and individuals only rarely included in the subgroup have been excluded -they are shown in italics and are in the table of the group to which they belong more often and in both in case of equality). It is in 2b that we find the texts attributable to Uc with the most certainty (vidas of Uc, and of Bernart de Ventedorn, the last one signed in the manuscripts IKERSg, but not de Ventedorn among the moving ones. The vida of Rigaut de Barbezieux, sometimes attributed to Uc [START_REF] Guida | Ricerche sull'attività biografica di Uc de Saint Circ[END_REF] is not part of the second subgroup on the HC based on the SOM, but is on the HC directly based on the data, fig. 7.

28 Bertran de Lamanon, Folqet de Roman, Richart de Tarascon.

29 Daude de Pradas, Gauseran de Saint Ledier, Na Castelloza. 30 The vida of Bertran de Lamanon is attested only in A, those of Richart de Tarascon, Daude de Pradas, Gauseran de Saint Ledier in ABIK, the one of Folqet de Roman in AHIK, and for Na Castelloza in AIK.

in A, by Uc de Saint Circ), and part of the question is to know if this presumption of authorship can be extended to 2a and to undecided individuals. Several vidas of this cluster share undeniable similarities both in stylistic and factual terms: the vidas of Aimerics de Peguilhan, Bernart de Ventadorn, Cadenetz, Folquet De Marseilla, Peire Rogier, Raimbaut de Vacqueyras, Sordel, and Uc de Saint Circ. In stylistic terms, these vidas are distinguished by the use of the first person (singular), elsewhere seldomly present, along with the mention of direct testimony 31 ; also, the senhal of the troubadours and their ladies is quite often quoted 32 . On a more factual level, theses texts are characterised by a good level of information about the concerned troubadours, in terms of geography 33 or persons 34 . Moreover, these vidas seem to be mostly aware of the trips of the troubadours to Spain or Italy and to their participation in courts such as the court of Savaric de Mauleon, of the king Alphonse VIII of Castilla, or of the count Raimon V of Toulouse, even when this is not mentioned in the extant works of the troubadour 35 . These texts 31 Vidas of Folquet("q'ieu vos ditz"), Cadenetz ("E tot lo sieu faich eu saubi per vertat per auzir et per vezer"), Bernart de Ventadorn ("E tot so qu'ieu vos ai dich de lui si me comtet e . m dis lo vescoms n'Ebles de Ventedorn, que fo fills de la vescomtessa q'en Bernartz amet tant"). It is also present in the vida of Raimbaut d'Aurenga, for which a space was kept in ms. A, but was never copied and is kept only by N 2 , and in the vida of Gausbert Amiel, placed by the clustering in the first group, as will be discussed later. It is to be noted that the first person is used by the author, very likely Uc de Saint Circ, of the vida of Savaric de Mauleon, attested in mss IK. 32 Vidas of Cadenetz, "e fasia se apellar Bagas"; Raimbaut de Vaqueiras, "Et apellava la en sas chanssos mos bels cavalliers"; Sordel, "et apellava la en los sieus chantars que el fazia per lieis doussa enemia". It is also present in the vida of Raimon de Miraval, "si fon mout honratz e tengutz en car per lo bon comte Raimon de Tolosa, que . l clamava n'Audiartz et el lui" and Rigaut de Berbeziu, whose vida is sometimes attributed to Uc [START_REF] Guida | Ricerche sull'attività biografica di Uc de Saint Circ[END_REF], but was placed by some of our analyses in the fourth cluster, "et apellava la Mieils de dompna en totz sos chantars". One could also add the vida of Peire Rogier, that contains the mention "E la clamava Tort-n-avez" in other mss but not in A. 33 The vida of Folquet de Marseilla mentions a "rica abadia que es en Proenssa, que a nom lo Terondet", informations that are "précises et exactes" [START_REF] Stroński | Le troubadour Folquet de Marseille : édition critique, précédée d'une étude biographique et littéraire et suivie d'une traduction, d'un commentaire historique, de notes, et d'un glossaire[END_REF]; the vida of Uc de Saint Circ, a burg named "Tegra", the castle of "Sain Circ", "qu'es a pe de Sainta Maria de Rocamaor, qe fo destruiz e derrochatz per gerra"; the vida of Cadenet, a castle "que a nom Cadenet, qu . is en la riba de Durenssa (. . . ) destruitz e raubatz per la gen del comte de Tolosa" -"Les ruines d'un vieux château dominent aujourd'hui encore la localité de Cadenet (ch.-l. de cant. de l'arrond. d'Apt, Vaucluse), sise non pas "sur la rive", mais à quelque distance au N. de cette rivière" (Boutière and Schutz, 1964, p. 502) and the event in question can be identified as taking place between 1166 et 1176, during the fights between Raimon V de Toulouse and Alfonse II d'Aragon. The vida of Gausbert de Puycibot also mentions a monastery of "Sain Leonart", vraisemblablement "Saint-Léonard-des-Chaumes (près de La Rochelle) dont Savaric de Mauléon (. . . ) fut précisément un des bienfaiteurs"; precise informations can also be found here and there in this group, notably in the vida of Sordel.

34 The name of the father of many troubadour is given (vidas of Folquet de Marseilla, Uc de Saint Circ, Raimbaut de Vacheiras); the "adoptive father" of Cadenet is also mentioned as "un cavallier que avia nom Guillem del Lantar", likely Guillaume Hunaud de Lantar (. . . ) mort en 1222" (Boutière and Schutz, 1964, p. 502). The name of the lady and her husband is also mentioned here and there, as is in the vida of Raimbaut de Vacheiras "ma dompna Biatriz, qe fo moiller d'en Haenric del Carret". 35 Trips to "Peitieus", "Gascoingna", Catalonia, Aragon and Espaigna" are mentioned for Uc de Saint Circ, as well as a stay "en Proenssa" and a trip to "Lombardia et en la Marca" and "en Tervisana" (along with other precisions); for Raimbaut de Vacheiras, his trips to Montferrat and "Romania", as the trips of Americ de Peguilhan to Catalonia and Lombardia, or the trip of Gausbert de Puycibot to Spain and more modestly the trip of Cadenet to Provence; are also mentioned, not exhaustively, stays in the courts of Savaric de Mauleon (Uc de Saint Circ, Gausbert de Puycibot), Guillem de Berguedan (Aimeric de Peguilhan -also mentioned in the vida of Guilhem de Berguedan himself, in cluster 4 but perhaps to be included in this one), Alphonse VIII of Castilla (Folquet de Marseilla, Uc de Saint Circ, Peire Rogier, Aimeric de Peguilhan), Alphonse IX de Léon (Uc de Saint Circ), Raimon V of Toulouse (Folquet de Marseilla -not mentioned in his extant poems -, Peire Rogier), Richard I (Folquet de Marseilla). Other minor courts are mentioned here and there. For older courts, the court of Alienor, for instance, is mentioned for Bernart de Ventadorn and the court of the princes of Aurenga is mentioned for Peire Rogier or Raimbaut de Vaqueiras. Detailed informations of this nature are also present in the vida of Sordel.

can be dated according to internal criteria, in the state reflected by ms. A, from after 1238 (Cadenet entrance in the Order of the Hospital, which is mentioned) and perhaps before 1257 (death of Uc de Saint Circ, not mentioned), a period covering in good parts Uc's supposed period of literary activity in Italy (c. 1220Italy (c. -1253)). The attribution to Uc de Saint Circ of these vidas could be proposed with a reasonable doubt.

This attribution might be a bit less obvious for the vidas of Gui d'Uissel, Marcabru and Uc Brunet, even though they share some similarities with the previous ones 36 . A special consideration must be given to the vidas of Jaufre Rudel, Guilhem de Cabestanh, Raimon Jordan lo vescoms de Saint Antonin (and perhaps even Gausbert de Puycibot). These quite long and developed vidas seem to focus on a narration, whose historical accuracy is questionable, but which develops courteous patterns: amor de lonh of Jaufre and the countess of Tripoli, "martyrdom" of Guilhem de Cabestanh and his lover, love between Raimon Jordan and the countess of Pena, impeached by exterior factors. More romanced and longer, these vidas tend to draw closer to novellas than to biography, and could perhaps be a bit later in date, though it is not completely impossible to integrate them in the group build around Uc. Secondly, the presence, in some of the analyses, of the vida of Bertolome Zorzi, which cannot possibly be attributed to Uc, in this cluster, is worthy of some explanations. This seems to be caused by similarities between the vida of Bertolome and the one of Sordel, with which it is quite often coupled. Moreover, these two troubadours are close geographically and chronologically to Uc. Finally, a few words deserve to be said about some of the vidas included in cluster 4 -which individuals, as stated, tend to move towards cluster 2 -and the moving individuals excluded from the analysis in a first time, and for this a look directly at the SOM, as well as on the HC done directly on data might be useful. One might wonder, in fact, if some vidas of cluster 4 ought not to be integrated amongst those of this second cluster.

The groups 1a can be dated after 1222 (date of the death of Elias Cairel, which is mentioned here) and perhaps before 1235 (unmentioned death of Dalfin d'Alvernhe); the group 1b can be dated after 1221 (death of Albertet) and perhaps before the deaths of Gaubert Amiel or Uc de Pena. The texts of these groups also have some similarities in stylistic and historic terms. On what concerns the stylistic aspects, they are slightly shorter texts, with less details, and more notably, they do not integrate developed courteous narrative patterns or love stories (though weddings are mentioned from time to time) 37 . Instead, those texts focus on the aspects relevant to lyric composition, with a predilection for judgement on the quality of motz and sons and of the mesure 38 . Whether this aspects are to be attributed to a different focus of the author, to a chronological evolution, or simply to the fact that most troubadours of this groups are actually joglars 36 The vida of Marcabru mentions Aldric de Vilar as is adopted father, and gives the surname "Panperdut" -informations probably extracted from [BdT 16b,1] and [BdT 293,43]. The vida of Uc Brunenc mentions, though not nominally, the king of Aragon and the count of Toulouse, as well as Bernart d'Andusa and Dalfin d'Alvernha. The vida of Gui d'Ussel, even though it gives the names of Gui's brothers and of "Margarita d'Albusson, moiller del vescomte d'Albusson" and mentions the papal interdiction made by Pierre de Castelnau ("lo legatz del papa li fetz jurar que mais non fezes chanssos"), fails to mention his death c. 1225; moreover, on the HC done directly on the data, Gui d'Uissel is integrated in cluster 1 -none of this being, of course, absolutely definitive.

37 A narrative pattern common to two of these vidas is the fact to have lost all of its possessions by an excessive inclination towards games of chance: "E fetz se joglars per ochaion que el perdet a joc tot son aver" (Gaucelm Faidit), "mas el fo grans baratiers de jogar en taverna per q'el fo ades paubres e ses arnes" (Uc de Pena).

38 For instance "ben escrivia motz e sons" (Elias Cairel), "Albertetz si fetz assatz chanssos que agrons bons sons e motz de pauca valenssa" (Albertet), "e si fetz los sieus vers plus amesuratz que hom q'ieu anc trobes mais" (Gaubert Amiel), "fetz mout bos sons e bons motz" (Gaucelm Faidit), "aquel que fetz los meillors sons de vers que anc fosson faichs" (Peire d'Alvernha).

(jongleurs) can be argued. It is also to be noted that the first person singular seems to be used twice by the author in this group 39 . On a more historical point of view, an important element is the mention of Dalfin d'Alvernhe in several of the texts 40 . The case of the vidas of Peire Vidal and Peirol (and perhaps even of Peire d'Alvernha) seems worthy of some further estimation on their belonging to this group: even if their integration in it seems confirmed by the HC done on the SOM, they were both included in group 2 by the HC done directly on the data, and have in fact some similarities with the second group and some apparent dissimilarities with the texts of the first group.

Different hypotheses can be formulated regarding this cluster. Perhaps are we to look for a different author. Why not then Uc de Pena, whose vida is included in this cluster, and is perhaps an author of vidas himself. Formulated by Boutière and Schutz (1964, p. 258), the hypothesis that Uc de Pena could be an author of vidas rests on a sentence found in his own vida: "e saup granren de las autrui chanssos e sabia mout las generatios dels grans homes d'aqellas encontradas", which is to be compared with Uc de Saint Circ vida's "el apres tenssos e cansos e vers e sirventes e coblas, e . ls faitz e . ls ditz del valens homes que eron adoncs, ni que eron estat denan; et ab aqest saber el s'en joglari". Very few is otherwise know of this troubadour, and even his dates are subject to caution. The fact that his song [BdT 456,2] is imitated by the troubadour Palais gives us a terminus ante quem, except for the fact that the dates of Palais are also partially uncertain. If we follow [START_REF] Guida | Andrian de) Palais, trovatore lombardo? In Studi di filologia romanza offerti a Valeria Bertolucci Pizzorusso[END_REF], Palais, who is to be identified with Andrian de Palais, would have composed his songs in the end of the XII th and the first decades of the XIII th century (before 1235 for BdT 315,2), and was still alive in 1228. In this case, it would not be impossible to link the supposed date of the vidas of this group (after 1222) with the supposed literary activity of Uc de Pena. Moreover, some of the troubadours of this cluster are amongst the known inspirators of Uc de Pena (Gaucelm Faidit for instance, see Cura Curà (2007, p. 12 et passim)). On the other hand, nothing is known of a connection of Uc de Pena and the Dalfin d'Alvernha (contrarily to the established connection between Dalfin and Uc de Saint Circ), his vida seems to vague for an autobiography, and, finally, Uc de Pena has also been dated differently, by Cura Curà (2007, p. 10), who dates him c. 1248-1283, and notes concerning Boutière's hypothesis that "Si tratta di suggestione che, data la fragilità dell'ipotesi, non è stata ripresa in altri interventi sulla paternità delle biografie trobadoriche".

An other hypothesis could emphasise a chronological difference between those texts and the one found in the second cluster: could the opposition between this cluster and the second be caused by chronological and/or geographical differences more than authorial changes? This could explain the differences of datation, as well as some of the stylistical and historical differences. As for the second group, some individuals from the third cluster should perhaps be integrated to this group, as they are on the HC done directly on the data. 39 The vida of Gausbert Amiel has, in A, "fetz los sieus vers plus amesuratz que hom qu'ieu anc trobes mais" (there was in his verses more mesure than in any man I could ever find since), but IK have there instead (a reading prefered by Boutière) "e fetz los sieus vers plus (a)mesuratz de hom que anc mais trobes" (there was in his verses more mesure than in any man that has since composed poetry), and the double meaning of the verb trobar ("find" and also "compose poetry") as well as the absence of difference between the first and third person of the subjonctive imperfect of the verbs in -ar does not allow in itself for a choice between the two; the vida of Peire d'Alvernha has a first person along with a mention of testimony, "segon qu . m dis lo Dalfins d'Alvernge en cui terra el nasquet". 40 To the vida of Dalfin himself should be added the mention of the testimony of Dalfin in the vida of Peire d'Alvernha, and, in the vida of Peirol (undecided between 1a and 1b, and of which the belonging to this cluster will be discussed), the stay of Peirol at Dalfin's court and his love for Dalfin's sister, Sail de Claustra.

Though cutting the dendrogram at an uniform arbitrary height is customary, the tests on the control corpus tend to demonstrate that it is not always the preferable choice -especially when precise distinctions between clusters are needed -and that the observation of the shape of the dendrogram itself, whenever allowed by a limited amount of individuals, may be a more relevant solution. In any case, one should not be satisfied by the analysis of the clusters generated by a single cut of the dendrogram. As different divisions may contain different informations (perhaps even translate different dimensions), one should probably try to analyse both the main divisions of the population up above and the more consistent subgroups in the bottom. As such, in the case of homogeneous corpus with few individuals and possibly a lot of variables, the heuristic value of Hierarchical Clustering can be put forward in comparison to other, more widespread, methods. In particular, in the study of the "genetics" of texts, the way the divisions inside the corpus are showed seems easier to read and better adapted to the nature of the data than most other clustering methods.

Figure 1 :

 1 Figure 1: Bar plots and Functions of the Empirical Discrete Distribution for the Number of words, of different word-forms, and of hapax, for the vidas of A.

Figure 2 :

 2 Figure 2: Linear and nonlinear regression of the number of different word forms by the total number of words

Figure 3 :

 3 Figure 3: CA and PCA on weighted word frequencies of the control sample

Figure 4 :

 4 Figure 4: Non-Metric multidimensional scaling with word weighting on our control sample and on the vidas

Figure 5 :

 5 Figure 5: HC of control corpus, computed for the one hundred most frequent words, with Manhattan distances and Ward's method; left with weighted frequencies, right with scaled and centred frequencies.

Figure 6 :

 6 Figure 6: Kohonen's SOM based on the weighted frequencies and the one hundred most frequent word-forms and HC done with Ward's method on the prototypes corresponding to observations, labelled according to the number of the text to which they corresponded.

Figure 7 :

 7 Figure 7: HC done directly on the data (one hundred most frequent word-forms), with Manhattan distances, Ward's method and weighted terms.

Figure 8 :

 8 Figure 8: SOM (20 × 20 grid) based on weighted frequencies for the one hundred most frequent word-forms (left) and HC done on the prototypes, with Ward's method (right).

Figure 9 :

 9 Figure 9: Heights for the CAH done on the prototypes of the 20 × 20 map.

  length

Table 1 :

 1 Length, number of different RS, maximum, average (rounded) and total frequency, for the vidas of A.

	l	RS Max freq. Average freq. Total freq.
	9	1	2	2,00	2
	8	7	3	2,29	16
	7	10	6	2,40	24
	6	23	4	2,39	55
	5	52	7	2,40	125
	4 131	10	2,66	348
	3 313	23	3,17	992
	2 691	50	4,59	3174

Table 2 :

 2 Variation in sizes of the SOM's; maps whose HC delivered the expected results on the control corpus are highlighted.

Table 3 :

 3 Variation in sizes of the SOM's for the vidas. Maps that were not retained for analysis are in italics.

Table 4 :

 4 The two first subgroups.

Table 6 :

 6 The 10 most common RS, for each length.

		form.	freq.	length	form.	freq.
	9	hom fo de letras e de sen natural E	2	5	de l evescat de Peiregos	3
	8	Et aqui son escriutas de las soas canssos	3	5	fetz se joglars et anet	3
	8	Et aqui son escriutas de las soas chanssos	3	5	Et enamoret se d una	3
	8	de l abadia d Orllac E l abas	2	5	un chastel que a nom	3
	8	d Alvernge d un chastel que a nom	2	4	de l evescat de	10
	8	d amor e de dompnei e de totz	2	4	de las soas chanssos	9
	8	hom de letras e de sen natural e	2	4	Et aqui son escriutas	9
	8	pois el se rendet a l orden de	2	4	d en Guillem de	6
	7	Et aqui son escriutas de las soas	6	4	fetz se joglars e	6
	7	de grans bens e de grans mals	2	4	d un castel que	5
	7	de l evescat de Peiregos d un	2	4	el s en anet	5
	7	en trobar et en chantar et en	2	4	si fo de l	5
	7	si fo de Tolosa fills d un	2	4	e fetz se joglars	4
	7	fetz se joglars e trobet bonas chanssos	2	3	si fo de	23
	7	son escriutas granren de las soas chanssos	2	3	de las soas	14
	7	amor e de dompnei e de totz	2	3	que a nom	13
	7	totz temps volia qu il agesson gerra	2	3	las soas chanssos	12
	7	Alvernge d un chastel que a nom	2	3	de l evescat	11
	6	de letras e de sen natural	4	3	que avia nom	11
	6	d un castel que a nom	4	3	fills d un	10
	6	si fo de l evescat de	4	3	e lai el	9
	6	d un chastel que a nom	3	3	el s en	9
	6	fo avinens hom de la persona	3	3	fetz se joglars	9
	6	se rendet a l orden de	3	2	si fo	50
	6	e fo fills d un borzes	2	2	e de	47
	6	e lai el definet e moric	2	2	de la	43
	6	en Espaigna ab lo bon rei	2	2	d un	41
	6	d en Guillem de Saint Leidier	2	2	q el	36
	5	Et aqui son escriutas de	7	2	e l	32
	5	fo de l evescat de	5	2	s en	32
	5	se rendet a l orden	4	2	e fetz	31
	5	un castel que a nom	4	2	fo de	30
	5	avinens hom de la persona	4	2	d en	26
	5	e fetz se joglars e	3
	Number	Incipit		BdT	Attribution in ms.
	1	Bella mes la flors daiguilen		323.5	Peire d'Alvernha
	2	En estiu qan cridal iais		323.17	Peire d'Alvernha
	3	Abans qeil blanc puoi sion uert.		323.1	Peire d'Alvernha
	4	Er auziretz encabalitz chantars		242.17	Guiraut de Bornelh
	5	Ben mera bels chantars		242.20	Guiraut de Bornelh
	6	A ben chantar couen amars		242.1	Guiraut de Bornelh
	7	Nuills hom no sap damic tro la perdut.	457.26	Uc de Saint-Circ
	8	Anc enemics qieu agues.		457.3	Uc de Saint-Circ
	9	Seruit aurai longamen.		457.34	Uc de Saint-Circ
	10	Pron si deu mais penssar almieu semblan.	74.14	Bertolome Zorzi
	11	Entre totz mos cossiriers.		74.5	Bertolome Zorzi
	12	Mout fai sobreira foillia.		74.9	Bertolome Zorzi

Table 7 :

 7 The control sample

For the most common RS, see Appendix, p. 23, Tab. 6; and for the complete set, see the online reference given in the first footnote. For a more detailed analysis of the structure of the vidas, the use of formulae and narrative patterns, see[START_REF] Camps | Peirols si fo uns paubres cavalliers : lire les troubadours dans les chansonniers A, I et K[END_REF].

It is worthy to be noted that the use of these formulae itself for authorship attribution of the vidas, attempted, manually, by[START_REF] Guida | Ricerche sull'attività biografica di Uc de Saint Circ[END_REF], has been severely contradicted byMeneghetti (2002, part. p. 149).

This was allowed by the relationship between the number of word forms and total number of words: length of the texts would still be rendered by their lexical richness, while not being the most important element of all subsequent analysis.

Problems due to differences in terms of subject matter are easier to handle with authorship attribution methods, for instance by working with the most frequent word-forms ("function words").

A common solution, bypassing the difficulty rather than confronting it, would be to run the analysis anew ignoring these individuals. But such a method would inevitably result in a loss of information and would deprive us from a complete apprehension of the corpus.

Appendix

length form. freq. Total freq. 9

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Et aissi son escriutas granren de las soas chanssos 1 9

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Table 5: Various forms of the same formula.


T Bienaimé

M Petruzzo

D Bigerni

N Piovella

R Kaiser
email: robin.kaiser@inln.cnrs.fr

Atom and photon measurement in cooperative scattering by cold atoms

Keywords: Cold atoms, Dicke superradiance, cooperative scattering

In this paper, we study cooperative scattering of low intensity light by a cloud of N two level systems. We include the incident laser field driving these two level systems and compute the radiation pressure force on the center of mass of the cloud. This signature is of particular interest for experiments with laser cooled atoms. Including the complex coupling between dipoles in a scalar model for dilute clouds of two level systems, we obtain expression for cooperative scattering forces taking into account the collective Lamb shift. We also derive the expression of the radiation pressure force on a large cloud of two level systems from an heuristic approach and show that at lowest driving intensities this force is identical for a product and an entangled state.

Introduction

Cooperative scattering by an assembly of resonant systems has been studied in detail for many years and is based on the seminal work by R. Dicke in 1954 (1). Related superradiance effects and collective level shifts have been studied in the context of atomic physics in the 70s (2)(3)(4). In the last decade, this topic has seen renewed interest (5)(6)(7)(8)(9)(10)(11)(12)(13)(14) with novel experiments in nuclear physics (15) and in laser cooled clouds of atoms (16)(17)(18)(19)(20), applications in quantum information (21) and quantum phase transitions (22,23). As we are mainly concerned with applications on laser cooled atomic samples, we focus in this paper on specific parameters and observables which are of interest in such experiments. We therefore derive expressions of the radiation pressure force acting on the center of mass of the atomic cloud, as well as the scattered electric field. We go beyond past approximations including the complex kernel for the coupling terms between N atoms (3,8), described by two level systems in a scalar approach. Neglecting the complete vectorial nature of the dipole dipole coupling seems a priori more justified in a dilute sample of atoms, where near field corrections are small (19). Furthermore, we obtain the force and the radiation field as quantum operators, which may be useful for studying fluctuations and diffusion effects in radiation forces and scattered emis- sion. Also, the imaginary part of the complex kernel, describing the collective Lamb shift, is evaluated for a gaussian density profile.

This paper is organized as follows : in section 2, we specify the Hamiltonian used and discuss our approximations. In section 3, we introduce the observables relevant for experiments with cold atoms, namely the radiation pressure forces on the center of mass of the atomic cloud and the scattered light intensity. The evaluation of these observables is done for specific atomic states in section 4. We derive the result for this cooperative radiation pressure force from a more heuristic approach in section 5. In section 6 we discuss the relevance of the Timed Dicke State compared to a product state for this cooperative pressure force in the low intensity limit before concluding in section 7.

Hamiltonian and operator equations

Our system consists of a gas of N two-level atoms (with random positions r j , lower and upper states |g j and |e j with j = 1, . . . , N , transition frequency ω a with linewidth Γ = d 2 ω 3 a /2π 0 c 3 , where d is the electric dipole matrix element), driven by a uniform resonant radiation beam with wave vector k 0 = k 0 êz , frequency ω 0 = ω a + ∆ 0 and electric field E 0 (see fig. 1).

The atom-field interaction Hamiltonian in the rotating-wave approximation (RWA) is

Ĥ = Ĥ0 + Ĥ1 (1)
where

Ĥ0 = N j=1 - ∆ 0 2 σ3j + Ω 0 2 σj e -ik0•rj + σ † j e ik0•rj Ĥ1 = N j=1 k g k â † k σj e i(ωk-ω0)t-ik•rj + σ † j âk e -i(ωk-ω0)t+ik•rj . (2)
Here Ω 0 = dE 0 / is the pump Rabi frequency, âk is the photon annihilation operator with wavenumber k and frequency

ω k = ck, g k = d ω k /(2 0 V ph ), V ph
the photon volume, σj = exp(i∆ 0 t)|g j e j | and σ3j = |e j e j | -|g j g j |. Instead of solving the Schrödinger equation introducing some ansatz for the system state |Ψ(t) (18), we write the motion equations of the atomic and field operators,

dσ j dt = 1 i [σ j , Ĥ] = i∆ 0 σj + iΩ 0 2 σ3j e ik0•rj + i k g k σ3j âk e -i(ωk-ω0)t+ik•rj (3) dσ 3j dt = 1 i [σ 3j , Ĥ] = iΩ 0 σj e -ik0•rj + 2i k g k â † k σ j e i(ωk-ω0)t-ik•rj + h.c. ( 4
)
dâ k dt = 1 i [â k , Ĥ] = -ig k e i(ωk-ω0)t N m=1 σm e -ik•rm . (5)
We consider the atoms initially in their ground state and we assume weak excitation (Ω 0 Γ), so that we approximate σ3j (t) ≈ -Îj , where Îj is the identity operator for the jth atom. This approximation amounts to neglect saturation and multiexcitation, i.e. all the processes generating more than one photon at the same time (linear regime). Integrating Eq.( 5) and substituting it into Eq.( 3), neglecting a k (0) (since the initial field state is vacuum) we obtain

dσ j dt = i∆ 0 σj - iΩ 0 2 Îj e ik0•rj - k g 2 k N m=1 e ik•(rj-rm) t 0 dt σm (t -t ) e -i(ωk-ω0)t . (6)
The last term in Eq.( 6) describes the effect of the spontaneously emitted photons on the atoms, and it is well known in the quantum electrodynamic literature [START_REF] Scully | Quantum Optics[END_REF][START_REF] Agarwal | Quantum Statistical Theories of Spontaneous Emission and their Realation to other Approaches[END_REF]. In the Markov approximation (i.e. when the photon transit time through the atomic sample is much shorter than the excitation decay time (26)), we assume under the integral σm (tt ) ≈ σm (t). The time integral then yields a real part (with a term δ(kk 0 ) ) and an imaginary part (corresponding to the principal part of the integral). Taking into account these two terms is at the origin of the exponential kernel whereas the real part alone would lead to a sin kernel in Eq.( 9) below. We then transform the sum over the modes k into an integral, k → (V ph /8π 3 ) dk. The real and imaginary parts of the double integral over t and k yield the cooperative decay and frequency shift (collective Lamb shift), respectively. The proper expression of the cooperative frequency shift has been obtained adding to the Hamiltonian (2) the not-RWA contributions associated to virtual photons exchanged between different atoms. It results the following relation (12):

k g 2 k e ik•R ∞ 0 dt e -ic(k-k0)t -→ Γ 2ik 0 |R| e ik0|R| (7)
where Γ = V ph g 2 k0 k 2 0 /(πc). Using Eq.( 7) in Eq.( 6) we obtain (12),

dσ j (t) dt = i∆ 0 σj (t) - iΩ 0 2 Îj e ik0•rj - Γ 2 N m=1 γ jm σm (t). ( 8
)
where

γ jm = -i cos(k 0 r jm ) + sin(k 0 r jm ) k 0 r jm = e ik0rjm ik 0 r jm . (9)
and r jm = |r jr m |. Eqs. (8) describe the time evolution of the atomic operators for N weakly excited atoms scattering radiation. The real part of γ jm describes the spontaneous emission decay and the imaginary part of γ jm describes the energy shift due to resonant dipole-dipole interactions. A slightly different approach can be used to derive this result as shown in appendix A. Note that even though this result will yield a density dependent collective shift of the resonance, we use a scalar model for the field, neglecting thus any polarization and near field dependence (10,13).

Detailed calculations for small and large samples of various geometries however show that near field and far field contributions as well as resonant and antiresonant terms need to be taken properly into account for quantitative predictions (3,13,14), and the present model thus needs to be considered with care illustrating only a part of the dipole-dipole coupling for real systems.

Eq.( 8) can also cast in the form

dσ j dt = 1 i [σ j , Ĥ 0 + Ĥeff ], (10)
where

Ĥ 0 = N j=1 -∆ 0 σ † j σj + Ω 0 2 σj e -ik0•rj + σ † j e ik0•rj Ĥeff = Γ 2 j,m e ik0rjm ik 0 r jm σ † j σm . ( 11
)
and the commutation rules in the linear regime are [σ j , σ † m ] = δ jm .

Observables

Among the different observables of the system, scattered light and radiation pressure force contain important signatures of cooperative scattering. Concerning scattered radiation, the positive-frequency part of the electric field is defined as

Ê(r, t) = i k E k âk (t)e -iωkt+ik•r (12)
where E k = ω k /2 0 V ph is the single-photon electric field. By integrating Eq.( 5) and inserting it in Eq.( 12) we obtain

Ê(r, t) = k E k g k N m=1 e ik•(r-rm)-iω0t t 0 dt e -i(ωk-ω0)t σm (t -t ) (13)
Using Eq.( 7), the Markov approximation leads to

Ê(r, t) ≈ -i dk 2 0 4π 0 N j=1 e -iω0(t-|r-rj|/c) |r -r j | σj (t) (14)
which has a transparent interpretation as the sum of wavelets scattered by N dipoles of position r j and detected at distance r and time t. In the far field limit,

|r -r j | ≈ r -(r • r j )/r and Ê(r, t) ≈ -i dk 2 0 4π 0 r e -iω0(t-r/c) N j=1 e -iks•rj σj (t) (15)
where k s = k 0 (r/r).

The radiation pressure force acting on the jth-atom has been calculated from Eq.( 1) as Fj = -∇ rj Ĥ = Faj + Fej where ( 18)

Faj = i k 0 Ω 0 2 e -ik0•rj σj -h.c. ( 16
)
Fej = i k kg k â † k σj e i(ωk-ω0)t-ik•rj -σ † j âk e -i(ωk-ω0)t+ik•rj (17)
where Faj and Fej result from the recoil received upon absorption of a photon from the pump and from the emission of a photon into any direction k, respectively. Eliminating the field using Eq.( 5), Eq.( 17) becomes

Fej (t) = - k kg 2 k N m=1 e -ik•(rj-rm) t 0 dt e i(ωk-ω0)t σ † m (t -t ) σj (t) + σ † j (t) N m=1 e ik•(rj-rm) t 0 dt e -i(ωk-ω0)t σm (t -t ) .( 18
)
Assuming the Markov approximation, σm (tt ) ≈ σm (t), then Eq.( 18) becomes

Fej (t) = - N m=1 k kg 2 k σ † m (t)σ j (t)e -ik•rjm t 0 dt e i(ωk-ω0)t + σ † j (t)σ m (t)e ik•rjm t 0 dt e -i(ωk-ω0)t ( 19
)
where r jm = r jr m . The force (19) acting on the jth atom has a single-atom contribution F(self) ej (term m = j in the sum) accounting for its own photon emission recoil, and a contribution F(int) ej (terms m = j) accounting for coupling between the jth atom and all the other atoms. Note that this dipole-dipole interaction can occur via a coupling to common vacuum modes of radiation. The interference terms in the total scattered field can leave a fingerprint on the forces acting on the atoms inside the cloud. The first contribution yields

F(self) ej ≈ -Γ |k|=k0 k σ † j σj , (20)
where the sum is over all the randomly oriented modes k = k 0 k and we have omitted the self-energy shift (Lamb shift) coming from the principal part term of the time integral in Eq (19). Noting that for m = j we have ik exp(ik • r jm ) = ∇ rj exp(ik • r jm ), the second contribution to Eq.( 19) can be written as

F(int) ej (t) = -i ∇ rj m =j k g 2 k σj (t)σ † m (t)e -ik•rjm t 0 dt e i(ωk-ω0)t -h.c. .( 21
)
Using Eq.( 7) in Eq.( 21), Eq.( 19) becomes

Fej (t) = F(self) ej (t) -∇ rj m =j Vjm (t). ( 22
)
where

Vjm (t) = - Γ 2 σ † j (t)σ m (t)e -ik0rjm + σj (t)σ † m (t)e ik0rjm k 0 r jm ( 23
)
is the effective interaction energy between jth and mth atoms. Since ∇ r [exp(ik 0 r)/r] = r(ik 0 r -1) exp(ik 0 r)/r 3 , Eq.( 22) becomes

Fej = F(self) ej - k 0 Γ 2 N m=1 njm (k 0 r jm ) 2 σ † j σm (1 + ik 0 r jm )e -ik0rjm + h.c. , (24)
where njm = r jm /r jm . The emission force acting on the jth atom has two contributions: a) a self-force due to its own photon emission; b) a force due to the dipole-dipole interactions with all the other atoms. This second force has a term decreasing as 1/r jm and one decreasing as 1/r jm 2 .

Atomic state

The linear approximation assumed in the equations of the atomic operators σj , Eq.( 8), suggests that we may restrict the Hilbert space of the N atoms to the subspace spanned by the ground state |g = |g 1 , . . . , g N and the single-excitedatom states |j = |g 1 , . . . , e j , . . . , g N with j = 1, . . . , N . Hence, we set

|Ψ(t) = α(t)|g + e -i∆0t N j=1 β j (t)|j ( 25
)
where we will approximate α ≈ 1 after having evaluated the different expectation values, e.g. σj ≈ β j and σ † j σm ≈ β * j β m . So, Eq.( 8) yields

dβ j (t) dt = i∆ 0 - Γ 2 β j (t) - iΩ 0 2 e ik0•rj - Γ 2 j =m γ jm β m (t), (26)
with initial conditions β j (0) = 0. The self-interaction term, Γγ jj = Γ -i∆Ω LS yields the single-atom spontaneous decay Γ and the single-atom Lamb shift ∆Ω LS , which can be reabsorbed in the definition of the atomic frequency ω a , and will be neglected in the present analysis.

Considering the force applied to the center-of mass of the atomic ensemble, F = (1/N ) j Fj , from Eqs.( 16) and [START_REF] Scully | Quantum Optics[END_REF] the components along the z axis are

Faz = k 0 Ω 0 N N j=1 Im e ik0•rj β * j (27) Fez = - k 0 Γ 2N j =m ẑjm j 1 (k 0 r jm )i(β * j β m -c.c.), (28)
where j 1 (z) = sin(z)/z 2cos(z)/z is the first order spherical Bessel function and ẑjm = (z jz m )/r jm . Note also that the self-force (20) has zero average since k k = 0 (although in general its fluctuations are different from zero). Also, from Eq.( 15) it is possible to obtain the average intensity of the scattered radiation as a function of the atomic wave function,

I(r, t) = 0 c Ê † (r, t) Ê(r, t) = d 2 ω 4 0 16π 2 0 c 3 r 2 N j=1 e -iks•rj β j (t) 2 . ( 29
)
The state (25) may be conveniently expressed in the timed Dicke (TD) basis, introduced originally by Dicke (1) and successively considered by R. Friedberg and coworkers (3) for their study on cooperative Lamb shift and by M.O. Scully and coworkers (6,9) to describe cooperative decay of N atoms prepared in a symmetric phased state. The completely symmetric TD state is |+ k0 = (1/ √ N ) j exp(ik 0 • r j )|j and Eq.( 25) can be written as

|Ψ(t) = α(t)|g + e -i∆0t β T D (t)|+ k0 + e -i∆0t N -1 s=1 γ s (t)|s k0 , (30)
where |s k0 groups all the states orthogonal to |+ k0 (6).

A numerical analysis of eq.( 26) shows that, for a constant driving field Ω 0 and for atomic cloud sizes much larger than the optical wavelength, the occupation probability of the states |s k0 is only a small fraction of the atomic state (20) and it is in general negligible, so that Eq.( 26) becomes

dβ T D dt = - i 2 √ N Ω 0 + i (∆ 0 -∆ N ) β T D - 1 2 ΓN s N β T D , (31)
where

s N = 1 N 2 N j,m=1 sin(k 0 |r j -r m |) k 0 |r j -r m | e -ik0•(rj-rm) = 1 4π 2π 0 dφ π 0 dθ sin θ |S N (k 0 , θ, φ)| 2 (32)
∆ N = - Γ 2N N j =m cos(k 0 |r j -r m |) k 0 |r j -r m | e -ik0•(rj-rm) = - ΓN 8π 2 P ∞ 0 dκκ 3 κ -1 2π 0 dφ π 0 dθ sin θ |S N (k 0 κ, θ, φ)| 2 (33)
where

κ = k/k 0 , S N (k) ≡ 1 N N j=1 e -i(k-k0)•rj (34)
is the factor form and the integral over κ in Eq.( 33) is evaluated as a principal part. The term ∆ N is the collective Lamb frequency shift (3,10). At steady state we find

β T D = Ω 0 √ N 2(∆ 0 -∆ N ) + iN Γs N , (35)
and

Fz = Faz + Fez = k 0 Γ Ω 2 0 4(∆ 0 -∆ N ) 2 + N 2 Γ 2 s 2 N N [s N -f N ] (36)
where

f N = 1 N 2 j =m ẑjm j 1 (k 0 r jm ) sin(k 0 z jm ). ( 37
)
The cooperative radiation force can be obtained from the standard single-atom radiation pressure force F 1 = k 0 ΓΩ 2 0 /(4∆ 2 0 +Γ 2 ) substituting the natural linewidth by the collective linewidth, Γ N = ΓN s N , and multiplying it by 1f N /s N , where f N /s N is the probability to observe the photon emitted in the forward direction. Isolating the term j = m,

|S N (k)| 2 = 1 N + j =m e i(k0-k)•(rj-rm) ≈ 1 N + |S ∞ (k)| 2 (38)
where the factor form S ∞ (k) is evaluated for a continuous approximation with density distribution n(r),

S ∞ (k) = 1 N V drn(r)e i(k0-k)•r . (39)
Then,

s N ≈ (1/N ) + s ∞ and f N ≈ f ∞ where, s ∞ = 1 4π dΩ k |S ∞ (k)| 2 , f ∞ = 1 4π dΩ k cos θ|S ∞ (k)| 2 (40)
and Eq.(36) becomes

F z = k 0 ΓΩ 2 0 4(∆ -∆ N ) 2 + Γ 2 N 1 + N 4π dΩ k (1 -cos θ)|S ∞ (k)| 2 . ( 41
)
The factor form S ∞ (k) and the integrated factors s ∞ and f ∞ have been calculated in ref. (18) for a Gaussian density distribution with ellipsoidal profile, n(r

) 0 exp[-(x 2 +y 2 )/2σ 2 r -z 2 /2σ 2 z ], yielding S ∞ (k 0 , θ) = exp{-σ 2 [sin 2 θ+η 2 (cos θ- 1) 2 ]/2}
, where σ = k 0 σ r and η = σ z /σ r is the aspect ratio. For spherical and large clouds (η = 1 and σ 1),

s ∞ ≈ 1/(4σ 2 ), f ∞ ≈ s ∞ -1/(8σ 4
) and the collective Lamb shift is ∆ N ≈ ∆ ∞ where (see (14) and Appendix B)

∆ ∞ = - ΓN 4 √ πσ 3 , (42)
which is a redshift, proportional to the number of atoms in a cubic wavelength (3), i.e. atomic density and not optical thickness b 0 = 3N/σ 2 . These values for s ∞ , f ∞ and ∆ ∞ can be compared to numerical evaluation of the s N , f N and ∆ N for a finite number of atoms and a specific configuration. In Fig. 2 we show the distribution of these values for different sample size.

In our numerical simulations shown in Fig. 3 we observe strong configuration dependent fluctuations for the value of the collective Lamb shift. A precise comparison with our analytical expression, valid for large clouds, is thus cumbersome and did not allow us to validate precise predictions of the numerical factor in Eq. (42).

Normalizing the radiation pressure force with respect to the single atom value, we obtain for large atomic samples, Fz

F 1 = 4∆ 2 0 + Γ 2 4(∆ 0 -∆ N ) 2 + Γ 2 (1 + b 0 /12) 2 1 + b 0 24σ 2 (43)
Finally, from Eq.( 29) we obtain the scattered intensity

I(r) = I 0 16π 2 k 2 0 r 2 Γ 2 4(∆ 0 -∆ N ) 2 + Γ 2 (1 + b 0 /12) 2 N + N 2 |S ∞ (k s )| 2 . (44)
This expression of the scattered intensity illustrates the role of the shape of the atomic cloud for the modified emission diagram. The emission diagram of the TD state is shown in Fig. 4. It illustrates the strong forward emission by the cloud when its size exceeds a few optical wavelengths, reminiscent of Mie scattering, or more precisely of Rayleigh-Debye-Gans [START_REF] Van De Hulst | Light Scattering by Small Particles[END_REF]. As we will discuss in the following section, a modified emission diagram yields a modified radiation pressure force, as the recoil of the scattered photon (partially) compensate the recoil effect at absorption.

Heuristic approach

The result (36) can be interpreted heuristically considering the momentum balance in a given time interval δt (29). During δt, N two-level atoms with positions r j (j = 1, . . . , N ) do δN florescence cycles, each time absorbing a photon with momentum k 0 from the laser and emitting a photon with momentum k i (i = 1, . . . , δN ) in a random direction Ω i , with probability P i,j = P (Ω i , r j ). The momentum variation for the jth atom after δN cycles is

δp j = ( k 0 )δN - δN i=1 ( k i )P i,j . (45)
For a single isolated atom the emission is isotropic and P i,j = 1, but for N atoms the emission can be not isotropic depending on the atomic distribution. Also, the excitation could be not uniform if the phase front of the driving beam is getting distorted by the refractive index changes in the atomic cloud. Assuming for simplicity that the excitation is uniform over the entire atomic ensemble and neglecting phase distortion effects [START_REF] Gordon | [END_REF]31), δN will be the same for all the atoms and |k i | = k 0 .

Considering the momentum variation along the direction of the incident photon (z axis), after averaging over the atoms

δp z = 1 N N j=1 δp j,z = ( k 0 )δN -( k 0 ) δN i=1 P i cos θ i (46)
where P i = (1/N ) j P i,j = P (cos θ i ) is the emission probability along the angle θ i . Considering cos θ i and δN as independent random variables, the statistical average of Eq.( 46) is

δp z = ( k 0 ) δN -( k 0 ) δN • cos θ (47)
where we assumed i cos θ i ≈ δN • cos θ. Hence, the pressure force is

F z = δp z δt = ( k 0 ) δN δt 1 -cos θ (48)
Comparing with Eq.(36) we found the following correspondence

δN δt = Ω 2 0 Γ N 4(∆ -∆ N ) 2 + Γ 2 N , cos θ = f N s N (49)
where Γ N = ΓN s N . So, the scattering rate (δN /δt) is equal to the excitation probability,

ρ ee = Ω 2 0 /[4(∆ -∆ N ) 2 + Γ 2 N ]
, times the collective decay rate, Γ N . The radiation pressure force (36) is equal to the momentum photon, k 0 , multiplied by the scattering rate and by a geometrical factor 1cos θ taking into account the directionality of the scattered light. Cooperativity modifies both the scattering rate, enhancing the decay rate and shifting the resonance frequency, and the scattering direction. Small samples tend to radiate isotropically whereas extended samples radiate superradiantly in forward direction (11,18). These cooperative effects can be revealed measuring radiation pressure force by monitoring center-of-mass motion of large atomic clouds released by magneto-optical traps (16,17), and then identifying fast decay, shifts and modified emission diagrams described by Eqs.( 36) and (44).

Product state

It has been noted that the same results obtained for a symmetric TD state could be obtained assuming a product state for N atoms (5, 13) (named also 'Bloch state' by some authors (7)):

|Ψ(t) c = N j=1 α c (t)|g j + β c (t)e ik0•rj-i∆0t |e j , (50)
where α c (t) and β c (t) are the same for every atom, with |α c (t

)| 2 + |β c (t)| 2 = 1.
The ansatz of Eq. (50) assumes each jth atom driven into the excited state with equal probability |β c (t)| 2 and phase φ j = k 0 • r j -∆ 0 t. As it happens for the symmetric TD state [START_REF] Gordon | [END_REF], the driving field imposes a coherence in the photons emitted spontaneously by each atom, so that superradiance arises because the state is symmetric under exchange of particles [START_REF] Sargent | Laser Physics[END_REF]. However, it is expected that the quantum statistic of the symmetric TD state will be quite different from that of the 'quasi-classical' product state. Notice that for |β c | 1 the product state (50) can be written in the following form (7,14) , so for α c ≈ 1 they coincide with those obtained from the symmetric TD state. Differences between the product and the symmetric TD states should appear when higher-order expectation values are observed, as for instance σj σm , which is zero for the TD state and α * 2 β j β m ≈ β j β m for the product state. Notice that operator ordering produces different results in high-order expectation values if scattered pho-tons or atomic forces are measured. These features and non classical effects studies in cooperative scattering by cold atoms will be the object of a future investigation.

|Ψ c = α N c |g + α N -1 c β c j e ik0•rj-i∆0t |j + α N -2 c β 2 c j =m e ik0•(rj+rm)-2i∆0t |j, m + . . . (51

Conclusion

In this paper, we have included a more precise kernel to evaluate the cooperative radiation pressure force on a cloud of two level systems. The collective Lamb shift leads to a shift ∆ N of the resonance, which is proportional to the spatial density. As we have used a scalar model in this paper, near field and polarization effects are neglected. One thus needs to consider this shift with some scepticism as the numerical factor for this shift in a real system will be strongly modified by the vectorial nature of the light (3). For dilute clouds, we recover previous results (16), where these density effects are negligible. We also presented a simple model to estimate the radiation pressure force from the modified emission diagram and assuming coupling to the single photon superradiant (Timed Dicke) state (6). This approach can be useful to estimate not only average forces but also fluctuations and dissipation. Finally, we noted that in the low intensity limit, the average result we derived for the cooperative radiation pressure force can be obtained either by assuming a driven Timed Dicke state or a product state (5,7,13), with no entanglement required. Looking for non classical features in cooperative scattering of light by a cloud of two level system thus requires studies of higher orders either by using higher intensities or looking at correlations or fluctuations of the force. from 0 to -∞, since the relevant values of k are around k 0 . Using the expression above, we write I(r jm ) = Γ 2ik 0 r jm t 0 dt σm (tt )e ick0t δ(tr jm /c)δ(t + r jm /c) = Γ 2 e ik0rjm ik 0 r jm σm (tr jm /c). (A4)

where Γ = V ph k 2 0 g 2 k0 /(πc). We observe that this approach does not require to assume the Markov approximation before solving the time integral, as in the standard approach (12). On the contrary, this approach allows to obtain the retarded (or not local) kernel, which, when the 'rapid transit approximation' is assumed, i.e. σm (tr jm /c) ≈ σm (t), reduces to the exponential kernel of Eq.( 8). For σ 1 it is approximated by

∆ ∞ ≈ - ΓN 8πσ 2 P ∞ -∞ dx 2 + 1 x e -σ 2 x 2 ≈ - ΓN 4 √ πσ 3 , (B3)
in agreement with the result of Friedberg and Manassah (14).

Figure 1 .

 1 Figure 1. (color online) Experimental configuration considered : a cloud of two level atoms is driven by an incident laser detuned by ∆ 0 from the atomic resonance ωa, with wavevector k 0 .

Figure 2 .Figure 3 .

 23 Figure 2. (color online) Distributions for values of s N , f N and ∆ N for N = 50 atoms, plotted for 10000 configurations for a size corresponding to σ = 10 (blue curves) and σ = 2 (green curves).

Figure 4 .

 4 Figure 4. (color online) Emission diagram computed according to Eq. (29) for the Timed Dicke state |+ k 0 with N = 40 atoms : σ = 0.4 (blue), σ = 1 (red) , σ = 8 (green).

  ) where |j, m = |g 1 , . . . , e j , . . . , e m , . . . , g N . Hence, the product state can be expanded in the symmetric TD states with 1 to N excited atoms. Only in the limits α c ≈ 1 and |β c | 1 the product state reduces to the symmetric single-excited atom state |ψ ≈ |g + β c √ N |+ k0 if only the first two terms of Eq. (51) are retained. The expectation values for the state (50) are σj = α * c β c and σ † m σj = |α c | 2 |β c | 2

dθ 1 e 1 e-σ 2 x 2 -

 112 Appendix B. Collective Lamb shift for a Gaussian distributionLet consider Eq.(33) for a continuous distribution:∆ ∞ =sin θ |S ∞ (κ, θ, φ)| 2 . (B1) A spherical Gaussian distribution, n(r) 0 exp(-r 2 /2σ 2 R ), yields S ∞ (κ, θ, φ) = exp[-σ 2 (κ 2 + 1 -2κ cos θ)/2], where σ = k 0 σ R . Inserting it in eq.(B1) we obtain ∆ ∞ = --σ 2 (κ-1) 2e -σ 2 (κ 2 +1) -σ 2 (κ-1)2e -σ 2 (κ+1) e -σ 2 (2+x) 2 .(B2)

Acknowledgements

We acknowledge fruitful discussions with E. Akkermans, P. Courteille, M. Havey, I. Sokolov and stimulating presentations on this topic at the PQE 2011 conference.

Appendix A. Evaluation of the integral kernel in Eq. (6) Let's consider the last term in Eq.( 6) and pass to the continuous frequency approximation:

We exchange the integration order and introduce spherical coordinates, dk = dkk 2 dφ dθ sin θ. After integration of the angular part, we obtain

where r jm = |r jm |. We approximate the k integral as

dk e -ick(t -rjm/c)e -ick(t +rjm/c) , (A3)

where we made the following approximations: a) we assumed the spectrum centered around k ≈ k 0 , so that kg 2 k ≈ k 0 g 2 k0 ; b) we extended the lower integration value