Hugo Pfister

Laurent Madec

Pierre Le Cann

Nathalie Costet

Martine Chouvet

Stéphane Jouneau

Laurent Vernhet
email: laurent.vernhet@univ-rennes1.fr

Factors determining the exposure of dairy farmers to thoracic organic dust

Keywords: Mutualité Sociale Agricole, GM: geometric mean, TWA : time-weighted average, PPI: parallel particle impactor, CFU: colony-forming unit, LOQ: limit of quantification respiratory disease, organic dust, dairy farms, thoracic particles, determinants

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Several studies have shown that dairy farmers are more likely to suffer from respiratory disorders like chronic bronchitis, non-atopic asthma and chronic obstructive pulmonary disease (COPD) than is the general population [START_REF] Dalphin | Prevalence of asthma and respiratory symptoms in dairy farmers in the French province of the Doubs[END_REF][START_REF] Gainet | Twelve-year longitudinal study of respiratory status in dairy farmers[END_REF][START_REF] Omland | New-onset asthma and the effect of environment and occupation among farming and nonfarming rural subjects[END_REF][START_REF] Reynolds | Systematic Review of Respiratory Health Among Dairy Workers[END_REF][START_REF] Guillien | Prevalence and risk factors for COPD in farmers: a cross-sectional controlled study[END_REF]. This increased risk of pulmonary disease may result from repeatedly inhaling organic dust [START_REF] Eduard | Chronic bronchitis, COPD, and lung function in farmers: the role of biological agents[END_REF][START_REF] Thaon | Influence of hay and animal feed exposure on respiratory status: a longitudinal study[END_REF][START_REF] Jouneau | On-site screening of farming-induced chronic obstructive pulmonary disease with the use of an electronic mini-spirometer: results of a pilot study in Brittany, France[END_REF][START_REF] Marescaux | Impact of farm modernity on the prevalence of chronic obstructive pulmonary disease in dairy farmers[END_REF].

Organic dust contains particles of plant, animal and/or microbial origin [START_REF] Douwes | Bioaerosol health effects and exposure assessment: progress and prospects[END_REF]. The most widely investigated microbial agent in organic dust from dairy farms, is endotoxin, a major component of the outer membrane of Gram-negative bacteria.

Lipopolysaccharides, that contain endotoxins and lipoglycans, induce severe inflammation in murine models and can trigger lower respiratory tract symptoms in farmers [START_REF] Donham | Respiratory dysfunction in swine production facility workers: Dose-response relationships of environmental exposures and pulmonary function[END_REF][START_REF] Vogelzang | Endotoxin Exposure as a Major Determinant of Lung Function Decline in Pig Farmers[END_REF][START_REF] May | Respiratory Health Effects of Large Animal Farming Environments[END_REF]. However, components of Gram-positive bacteria (peptidoglycans) and fungi (glucans) are also present in organic dust and probably help trigger chronic inflammation [START_REF] Larsson | Gram positive bacteria induce IL-6 and IL-8 production in human alveolar macrophages and epithelial cells[END_REF][START_REF] May | Respiratory Health Effects of Large Animal Farming Environments[END_REF][START_REF] Poole | Immunological and Inflammatory Responses to Organic Dust in Agriculture[END_REF].

It is essential to know how much of each type of particle reaches those bronchial areas where COPD, asthma and chronic bronchitis are thought to develop in order to investigate the impacts of exposure on disease development. Particle size is the main factor determining how organic dust interacts with the respiratory tract and only a fraction of inhaled particles is thought to reach the bronchial tract. The European EN481 standard (CEN, 1993) has defined three aerosol fractions linking particle size to the distance particles can penetrate into the respiratory tract. One is the inhalable fraction: particles that can penetrate throughout the respiratory tract, including extra-thoracic particles that are deposited up to the larynx. The second is the thoracic fraction: particles that reach the bronchial region and below. And the third is the respirable fraction: the fraction that can enter the alveolae. The thoracic fraction thus appears to be the most suitable for describing the exposure of dairy farmers to organic dust.

Most studies on the factors determining the exposure of dairy farmers to bioaerosols have collected inhalable fractions of organic dust while the farmer has been working [START_REF] Spaan | Exposure to inhalable dust and endotoxins in agricultural industries[END_REF][START_REF] Samadi | The influence of bedding materials on bio-aerosol exposure in dairy barns[END_REF][START_REF] Garcia | Occupational exposure to particulate matter and endotoxin for California dairy workers[END_REF][START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF]. They have clearly demonstrated that several operations expose dairy farmers to inhalable particles but did not identify the specific tasks that produced the thoracic dust. We have recently shown that 3-10 µm diameter thoracic dust is the main product of mechanical straw spreading [START_REF] Pfister | Granulometry, microbial composition and biological activity of dusts collected in French dairy farms[END_REF], a task that greatly increases exposure to inhalable dust [START_REF] Garcia | Occupational exposure to particulate matter and endotoxin for California dairy workers[END_REF][START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF]. This suggests that workers on dairy farms are indeed exposed to thoracic dust although the factors determining such exposure remain to be identified.

We therefore measure the exposure of dairy farm workers to thoracic particles, determined the concentrations of endotoxin, culturable bacteria and fungi in these particles, and identified the farm features and tasks that influence exposure to them. We repeatedly measured the quantities and components of thoracic dust around workers on 29 French dairy farms as they carried out specific tasks, the features of these farms and analysed our data using linear and logistic mixed-effect statistical models.

Materials and methods

Sampling

We studied 42 dairy farmers on 29 dairy farms randomly selected from the Breton "Mutualité sociale agricole" (MSA), the French agricultural social security system. All the farms were in the department of Ille-et-Vilaine region of Brittany. Each participating farm was visited three times during the year to cover the various activities and climate changes. Two or three farmers from farms that were not run by a single farmer took part in the study.

Measurements were done as defined by the annual calendar of Brittany dairy activities published by the MSA. Winter (November 1 to February 28), during which cows are kept in cowsheds on most farms and involving many activities in barns. Spring (March 15 to May 31), when cows are let out to pasture and farmers prepare for arable crops (seeding, ploughing, fertilising, etc). Summer (July 1 to October 31) when the cows are mostly outdoors and less work is done in barns. Measurements at each period were planned independently of their location or activity and were limited only by the farmer. The study was approved by the local ethics committee (registration number 14.72).

Exposure monitoring

We performed 3 to 7 measurements on each farm, always in the morning as the activities during the morning and afternoon were generally similar. The samples covered all tasks performed by dairy farmers in the cowshed and outdoors. A total of 37-38 measurements were recorded at each season for a total of 110-112 measurements of thoracic dust, endotoxins, cultivable Gram-positive bacteria and fungi. Each sampling session took 227 min (SD = 71 min) in winter, 216 min (SD = 68 min) in spring and 247 min in summer (SD = 94 min).

Thoracic dust, endotoxins, cultivatable bacteria and fungi were collected on 37-mm glass fibre filters (pore size: 0.8 µm; Millipore, Billerica, MA, USA) using a thoracic parallel particle impactor (PPI)-T (Tecora, Paris, France) connected to a sampling pump (Aircheck 2000, Tecora, Paris, France) operating at 2 L/min [START_REF] Görner | Sampling Efficiency and Performance of Selected Thoracic Aerosol Samplers[END_REF]. Airflow was checked before and after sampling. The air inlet of the PPI was attached at shoulder level in the personal breathing zone. The sample-bearing filters were transferred from the PPI to a 37 mm diameter cassette and kept at 4°C for transport to the laboratory. One field negative control and one laboratory negative control were processed with each week's samples.

Gravimetric analysis and extraction of thoracic dust

The glass fibre filters were weighed before and after sampling. Filters were placed in a controlled environment (35-50% humidity, 18-22°C) overnight prior to each weighing, They were then weighed on an electronic micro-balance (model Precisa 2000, Mettler Toledo, Colombus, Ohio, USA) just before and after dust sampling. The limit of quantification (LOQ) of the method was 0.08 mg per filter. Weights below the LOQ (n=15) were assigned an imputed value following the EN 689 standard (CEN 1996). Microorganisms were extracted from the filters by placing them in 10 mL pyrogen-free water (Lonza, Walkersville, USA) containing 5% Tween-20 (Sigma-Aldrich, Saint-Quentin Fallavier, France) immediately after weighing. The filters and extractants were shaken (2000/min) for 1 hr at room temperature and centrifuged at 2000 g for 10 min at 4°C. The resulting pellets were suspended for microbial analyses.

Endotoxin measurements

Endotoxins were quantified using the kinetic Limulus Amebocyte Lysate test (Lonza, Walkersville, USA). Suitably diluted samples (from 1:1 to 1:1000) were tested in duplicate.

According to manufacturer, the LOQ of the method was 0.005 EU/ml. As each filter was extract with 10 ml pyrogen-free water, the final LOQ was 0.05 EU per filter.

Cultivable bacteria and moulds

Cultivatable bacteria were grown on Trypton soy agar (Biokar, Beauvais, France). Suspensions (100 µl) were tested in duplicate by plating out on agar petri dishes at 1:10, 1:100, and 1:1000 dilutions and incubating them for 48 hr at 36°C. Gram-positive were discriminated from Gram-negative bacteria by staining with gentian violet, Lugol, 96% alcohol (V/V) and carbol fuchsin reagents (Millipore, Radnor, Pennsylvania, USA). Cultivable fungi and spores were grown on dichloran-glycerol agar (Biokar, Beauvais, France). Duplicate (100 µl) suspensions were plated out on agar (1:10, 1:100, 1:1000 dilutions) and incubated for 3-7 days at 24°C. Colonies were counted on days 3, 5 and 7 after inoculation. As each filter was extracted with 10 ml pyrogen-free water, the LOQ was 100 CFUs per filter for both bacteria and fungi. Values below the LOQ (n = 29 for bacteria and n = 47 for moulds) were assigned an imputed value following the EN 689 standard (CEN, 1996).

Farms

The selected dairy farms were typical family-run Breton farms; nine were run by a single farmer and the others by 2 to 5 farmers. Our measurements were done on a maximum of 3 farmers per farm. A typical farm had free stalls, a main cowshed for the dairy cows and one to five other compartments for calves or feed storage. The mean number of dairy cows was 78 (SD = 31). The main cowsheds were generally ventilated by opening in the walls. We used two indicators of ventilation: the area of wall opening and the area of wall opening normalised to cowshed floor area.

Collection of data on determinants

All data were recorded on formatted sheets, prepared according to the specific Breton dairy practises and to the results of previous studies describing the exposure of farmers [START_REF] Jouneau | On-site screening of farming-induced chronic obstructive pulmonary disease with the use of an electronic mini-spirometer: results of a pilot study in Brittany, France[END_REF][START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF][START_REF] Samadi | The influence of bedding materials on bio-aerosol exposure in dairy barns[END_REF][START_REF] Garcia | Occupational exposure to particulate matter and endotoxin for California dairy workers[END_REF]. The time each farmer took to perform each task during sampling was recorded by the technician (Institut Technique des Gaz et de l'Air, Saint-Grégoire, France) conducting the measurements.

Particular attention was paid to the tasks that could be done manually or mechanically using a tractor. The characteristics of feeding systems, cowsheds, slurry and manure management, and milking methods were also recorded.

Data analysis

The time taken to complete the morning shift was 210 min, but it varied greatly from one farm to another and between farmers. We therefore expressed the exposure of farmers using the 210-min time-weighted average. Natural logarithms of the time-weighted average (TWA) values were also calculated to normalize the distribution and perform statistical analyses. The geometric mean (GM) and geometric standard deviation (GSD) were used to describe exposure.

Effects of tasks and farm characteristics on exposure to contaminants

All tasks and farm features with less than 5 observations were excluded from the analyses. The effect of tasks and farm features on exposure to thoracic dust, endotoxins and cultivable bacteria were investigated with mixed-effect linear models. As the number of potential exposure determinants was large, we used a 2-step procedure: (1) For each contaminant, a univariate analysis was performed with each determinant. Farms and their workers were considered to be nested random effects. We used a likelihood ratio test to compare a naïve mixed-effect model (including only random effects) with another model including each determinant as a fixed effect. Determinants with a P-value below 0.25 were selected as candidates for multivariate analysis. (2) multivariate linear mixed-effect models were then implemented. The candidate variables selected from the univariate analysis were introduced iteratively following a forward process based on the minimization of the Akaike Information Criterion (AIC). The tasks were treated as quantitative variables, using their duration. Most farms characteristics were treated as dichotomous variables (presence = 1, absence = 0). However, humidity, temperature, number of cows, and area of the wall opening were all treated as quantitative variables using their respective units. The wall opening area, normalized to cowshed area was used as a quantitative variable or a dichotomous variable (using the 75 th percentile value of 0.36m²/m² as cut-off). The variability produced by the multivariate models was calculated to evaluate their quality.

A logistic mixed-effect model was used instead of a linear mixed-effect model because many of the fungus measurements were below the LOQ (n = 47, 41%). Exposure to fungi was recoded as a dichotomous variable using 1000 CFU/m 3 value as cut-off to define exposed (= 1) and unexposed (= 0) farmers. This cut-off was derived from the Swiss guide value for occupational exposure to moulds [START_REF] Suva | Valeurs limites d'exposition aux postes de travail (Occupational limit values at workstations)[END_REF]. The previous procedure was then applied (univariate selection followed by the iterative forward procedure) to investigate the potential impact of determinants on exposure to fungi. The area under the ROC curve was used to assess the discriminating ability of the multivariate logistic model. All statistical analyses were performed using the R package lme4 [START_REF] Bates | Fitting Linear Mixed-Effects Models using lme4[END_REF] and ROCR [START_REF] Sing | ROCR: visualizing classifier performance in R. Bioinforma[END_REF].

Possible correlations between tasks and farm features were determined prior to modelling by calculating the Spearman correlation coefficients.

Results

Farm features and tasks performed by workers

The quantitative features of the 29 dairy farms that took part in the study are summarized in Table 1a and the qualitative farm features and tasks, together with their relative frequencies are listed in Tables 1b and2. The corn (maize) silage or the corn silage/haylage mix used as fodder on most dairy farms was distributed in the cowshed. Two farms used a fodder system composed of only raw grassland or haylage. Fodder was usually distributed mechanically using a tractor. Grain or dry compounds, mixed manually or mechanically with silage in a feed mixer, were also frequently added to feed. Most farmers kept their cows feeding out in meadows as long as the weather was favourable, but cows were permanently housed in cowsheds on farms equipped with an automatic milking system. Cows were milked manually in herringbone parlours on 25 farms. The other farms had automatic milking systems. The main bedding was straw on 26 farms and sawdust or shredded straw on the other three.

Exposure to thoracic dust, endotoxins and microbial agents.

Table 3 shows the mean amounts of thoracic dust, endotoxins, bacteria and fungi collected on the 29 farms. Quantifiable amounts of Gram-negative bacteria were found in only two samples and are thus not shown. The exposure to the three microbial contaminants varied according to the season. The exposure to thoracic dust in winter and spring were very similar but exposure was higher in summer. Most endotoxins were inhaled in winter, with less in spring and the least in summer; it thus tended to decrease as the temperature increased. In contrast, exposure to Gram-positive bacteria and fungi was highest in summer and lowest in spring. The numbers of values below the LOQ were also greatest in spring.

Univariate statistical analysis

The results of the univariate statistical analysis assessing the effects of tasks and farm features on the exposure of farmers are summarized in Tables S1 andS2. Only results with pvalues below 0.25 were used for multivariate analysis as explained in Materials and Methods.

Determinants of exposure to thoracic dust, endotoxins and Gram-positive bacteria

The results of the multivariate linear mixed-effect models for exposure to thoracic dust, endotoxins and cultivable Gram-positive bacteria are shown in Table 4. The hierarchical sampling design produced three components of variance: "between-farms", "between-workers" and "within-worker" (day-to-day). The "within-worker" variance was the dominant variance of exposure to thoracic dust, endotoxins and Gram-positive bacteria, accounting for 69%, 60% and 76% of their respective variations. The "between-farms" and "between-workers" components accounted for 5%, 15% and 24%, and 26%, 25% and 0% of the variations in exposure to thoracic dust, endotoxins and bacteria, respectively.

The multivariate models used five tasks, of which grain or feed handling and mechanical bedding spreading were significant exposing tasks common to the three models. The 10-min mechanical bedding spreading increased exposure to thoracic dust by 60%, to endotoxins by 120%, and to Gram-positive bacteria by 46%. Ploughing and hay handling increased exposure to dust and Gram-positive bacteria by over 90%. More generally, handling grain, silage, bedding and slurry were all major contributors to exposure to airborne contaminant. The models also contained three to four farm features depending on the contaminant. The only characteristic that reduced exposure to all three contaminants was the systematic use of grass for feed. Exposure to Gram-positive bacteria was 70% lower in cowsheds with cubicles than in those with deep litter pens. Main cowsheds with large wall openings (> 0.36 m 2 /m 2 ) reduced exposure to dust, but the reduction was not statistically significant. In contrast, automated milking systems increased exposure to endotoxins and gram positive bacteria with over 150%.

In addition, storing manure in the cowshed and using an automatic scrapper in alleyways increased exposure to endotoxin by 126% and to Gram-positive bacteria by 149%. Increasing the number of cows housed in cowsheds also significantly enhanced exposure to Gram-positive bacteria. Finally, environmental parameters influenced exposure to thoracic dust and endotoxins. A 15% increase in relative humidity (corresponding to the interquartile range value) reduced exposure to thoracic dust (by 21%), as did a 10°C increase in temperature (corresponding to the interquartile range value) for exposure to endotoxin (by 45%).

Their respective multivariate models explain almost half the variations in exposure to thoracic dust (46%), endotoxin (48%) and Gram-positive bacteria (40%) (Table 4). The "between farms" variability of the three airborne contaminants was totally explained by the multivariate models, as was the "between workers" variance for exposure to dust. These models accounted for 22% (dust), 45% (endotoxin), and 22% (bacteria) of "within farmer" variations in exposure.

Determinants of exposure to fungi

The determinants affecting exposure to fungi (> 1000 CFU/m 3 ) were assessed using a multivariate logistic mixed-effect model because so many of these values were below the LOQ.

Many determinants increased the probability of exposure to fungi on dairy farms (Table 5).

Handling grain, silage, bedding and slurry/manure probably promoted exposure, as for the other contaminants. The 10-min mechanical spreading of bedding had the greatest effect; it should increase chance of exposure to fungi by an OR of 3.16. Mechanical manure scraping and manual spreading of bedding also increased exposure to fungal contaminants but only marginally.

Workers who mainly fed cows with corn silage were more likely to be at risk of exposure to fungi/mould (OR = 3.81) than their counterparts who used a fodder system (grass or mix grass + corn silage). In contrast, workers on farms with cowsheds with cubicles were less likely to be exposed to moulds (OR = 0.25) than those on farms with other housing systems. Lastly, a 100 m² increase in wall opening was linked to much lower exposure to fungi (OR= 0.24).

Finally, the area under the ROC curve, used to determine the accuracy of the multivariate logistic mixed-effect model, was 0.86, indicating that the model was highly discriminatory.

Discussion

Our Breton dairy farmers were exposed to thoracic dust containing endotoxins, cultivable Gram-positive bacteria and fungi. We measured the types and durations of occupational activities to describe precisely the factors determining exposure and analysed the findings with mixed-effect linear and logistic regression models to show that exposure to thoracic organic dust was mainly controlled by specific occupational tasks and farm features.

These Breton dairy farmers were exposed to concentrations of thoracic dust (GM = 240 µg/m 3 ) three to four times lower than the dust (GM = 812 to 1000 µg/m 3 ) inhaled by American or other European dairy farmers [START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF][START_REF] Garcia | Occupational exposure to particulate matter and endotoxin for California dairy workers[END_REF][START_REF] Samadi | The influence of bedding materials on bio-aerosol exposure in dairy barns[END_REF].

Thus, thoracic dust might not account for most of the particles inhaled by Breton dairy farmers during their work. However, their levels are substantially higher than respiratory dust (PM2.5) levels (GM = 35.3 to 48 µg/m 3 ) found in Californian dairies [START_REF] Garcia | Occupational exposure to particulate matter and endotoxin for California dairy workers[END_REF]. The particles in thoracic dusts may thus be small enough for them to be deposited in the bronchial region but not in the alveolae. This is supported by the results of our study showing that spreading straw bedding in Breton dairy farms mainly released organic dust particles with diameters of 3 to 10 µm, which are unlikely to enter the alveolae [START_REF] Pfister | Granulometry, microbial composition and biological activity of dusts collected in French dairy farms[END_REF]. The endotoxin concentrations (GM = 128 EU/m 3 ) in the thoracic dust collected on Breton dairy farmers were also lower than those in inhalable dusts in Californian, Dutch or Danish barns (GM = 329-453 EU/m 3 ) [START_REF] Basinas | Exposure to inhalable dust and endotoxin among Danish livestock farmers: results from the SUS cohort study[END_REF][START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF][START_REF] Garcia | Occupational exposure to particulate matter and endotoxin for California dairy workers[END_REF][START_REF] Samadi | The influence of bedding materials on bio-aerosol exposure in dairy barns[END_REF]. Nevertheless, thoracic dust collected on Breton dairies contained more endotoxin than that calculated for inhalable dusts from other dairy farms when the endotoxin concentrations are normalized to mg/dust. We found 512 EU/mg endotoxins in our thoracic dust, while it was 360 EU/mg inhalable dust from Danish dairy farms [START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF]. These results agree with those of [START_REF] Madsen | Airborne endotoxin associated with particles of different sizes and affected by water content in handled straw[END_REF] who reported that thoracic dust from straw storage halls at power plants contained more endotoxins than did inhalable dusts [START_REF] Madsen | Airborne endotoxin associated with particles of different sizes and affected by water content in handled straw[END_REF]. In addition, the concentrations of cultivable Gram-positive bacteria (9.6E+02 CFU/m 3 ) and fungi (6.9E+02 CFU/m 3 ) in thoracic dust were about one order of magnitude lower than those in inhalable dust from Dutch dairy barns (1.11E+02 and 2.37E+03 CFU/m 3 ) [START_REF] Samadi | The influence of bedding materials on bio-aerosol exposure in dairy barns[END_REF]. These differences may be linked to the type of bedding and/or feed used by Breton and other workers [START_REF] Samadi | The influence of bedding materials on bio-aerosol exposure in dairy barns[END_REF]. The concentrations of both cultivable Gram-positive bacteria and fungi also varied widely, perhaps due to the poor survival of some microorganisms. Gram-negative bacteria were detected in only two of our 110 dust samples. This is not surprising since cultivable Gram-negative bacteria collected in cow and calf sheds have short survival periods [START_REF] Clark | Airborne bacteria, endotoxin and fungi in dust in poultry and swine confinement buildings[END_REF][START_REF] Zucker | Airborne Gram-Negative Bacterial Flora in Animal Houses[END_REF]. However, we recently sequenced bacterial ribosomal 16S-RNA and found that two Gram-negative phyla Proteobacteria and Bacteroides accounted for the majority of bacteria in thoracic dust released from straw bedding in four of five Breton dairy farms [START_REF] Pfister | Granulometry, microbial composition and biological activity of dusts collected in French dairy farms[END_REF]. We also measured significant amounts of endotoxins in most samples. Thus, Gram-negative bacteria were widespread in thoracic dust collected on Breton dairy farms, although they probably could not be cultivated.

Our results showed that the variability in exposure to thoracic dust was mainly due to the "within-worker" variance component (Table 4), as observed by [START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF]. However, the variance components accounting for the variations in total dust (69%) and endotoxin (60%) in our models are higher than those measured by [START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF]. These differences may arise because the activities of Breton farmers frequently changed from day to day, while Danish dairy farmers perform similar tasks every day. The variability in exposure to Grampositive bacteria is also mainly due to "within-worker" variance. In contrast, the model on dust exposure explained all (100 %) the "between workers" variability. This variability was however small and accounts for only 26 % of the total variance in the naïve model for dust. This indicates that there was little difference in the exposure of workers to dust on our 29 dairy farms. Our mixed-effect linear model describes 40% to 48% of the overall variability in exposure of the investigated airborne contaminants. The accuracy of our predictive model is probably due to the methodology used. The nature and duration of the tasks performed by the farmers were recorded directly during dust sampling, unlike the protocols of previous studies in which task information was recorded retrospectively during a post-sampling interview. Our direct on-farm observations may have provided more accurate measurements of task times.

Nevertheless, a substantial part of the overall variability remained unexplained due to unrecorded parameters like specific feeding practices or stable management, physicochemical characteristics of grain or straw, and weather conditions. Another factor may be the mathematical model misspecification inherent to the use of log TWA-exposure in the present statistical analysis [START_REF] Burstyn | Measurement Error and Model Specification in Determining How Duration of Tasks Affects Level of Occupational Exposure[END_REF]. Indeed, this model implied that the duration of each task had an exponential effect on the untransformed TWA-exposure. Since it is admitted that the mean exposure level during a task is constant, the impact of the duration of a task on the TWAexposure is thus linear and not exponential [START_REF] Burstyn | Measurement Error and Model Specification in Determining How Duration of Tasks Affects Level of Occupational Exposure[END_REF]. Consequently, this model may not correctly adjust to some measurement values.

The mixed-effect linear and logistic model results suggest that specific tasks and farm features markedly increased exposure to dust and microorganisms. They confirm the effects of grain handling and bedding-related tasks on exposure to thoracic dust and endotoxins. These tasks were previously identified as the greatest contributors to inhalable dust and endotoxins in dairy and pig farms [START_REF] Garcia | Occupational exposure to particulate matter and endotoxin for California dairy workers[END_REF][START_REF] Basinas | Exposure to inhalable dust and endotoxin among Danish pig farmers affected by work tasks and stable characteristics[END_REF][START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF]. Our data also show that they contributed to exposure to the Gram-positive bacteria and fungi in the thoracic fractions of inhaled dust. Silage handling was particularly associated with increased probability of exposure to fungi, which suggests that silage is an important source of fungal contaminants.

We found that a higher outdoor temperature was associated with a lower exposure to thoracic endotoxins, in agreement with the findings for inhalable endotoxins on dairy and pig farms [START_REF] Preller | Modeling long-term average exposure in occupational exposure-response analysis[END_REF][START_REF] Basinas | Exposure to inhalable dust and endotoxin among Danish pig farmers affected by work tasks and stable characteristics[END_REF][START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF][START_REF] Basinas | Exposure to Inhalable Dust, Endotoxin, and Total Volatile Organic Carbons on Dairy Farms Using Manual and Automated Feeding Systems[END_REF]. This is probably because thoracic dust contains more endotoxins in cold weather (winter) than when it is warm (summer). The endotoxin content in thoracic dust were 714 EU/mg in winter, 547 EU/mg in spring and 393 EU/mg, in summer (data not shown). And exposure to dust varied inversely with the outdoor humidity, perhaps because a high humidity seems to limit the aerosolisation of particles released during work, so reducing exposure to thoracic dust. A similar association was found on Estonian dairy farms [START_REF] Kaasik | Concentrations of airborne particulate matter, ammonia and carbon dioxide in large scale uninsulated loose housing cowsheds in Estonia[END_REF]. Greater areas of open walls, which favour stable ventilation, was associated with a much lower probability of exposure to fungi (OR = 0.24) and better ventilation is also likely to reduce indoor humidity, producing less favourable conditions for mould growth.

Our data also show a positive, very significant association between the use of an automatic milking system and exposure to endotoxins (change factor = 3.50, p < 0.007), in agreement with the report for inhalable endotoxins on Danish farms [START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF].

These authors suggest that working with milking robots makes it possible to have more cows per farm, so increasing the time spent by the farmer doing more exposing work. This is supported by our finding of a significant positive correlation between the presence of an automatic milking system and the time spent manually spreading straw (r = 0.63, p <0.001, data not shown). The presence of milking robots also required cows to be kept in the main cowshed, which probably increased the volumes of manure or slurry and the production of particles heavily contaminated with endotoxins when the cows moved.

The mechanical scraping of slurry and manure also had a major influence on exposure to both endotoxins and Gram-positive bacteria, while having a manure storage pit in the cowshed significantly enhanced exposure to endotoxins. As most of the endotoxins in manure are thought to come from anaerobic Gram-negative bacteria [START_REF] Zucker | Airborne Gram-Negative Bacterial Flora in Animal Houses[END_REF], the solid structure of manure and the unstirred nature of the manure in storage pits could foster the growth of anaerobic Gram-negative bacteria, so increasing the endotoxin concentration.

Having an automatic scrapper in alleyways also seems to enhance exposure to Gram-positive bacteria, while the repeated scraping of alleyways could help produce small particles of fresh slurry contaminated with Gram-positive bacteria. In contrast, cowsheds divided into stalls had much lower (70-75%) Gram-positive bacteria and mould concentrations. Cows housed in cubicles were usually cleaner and their movements should result in fewer suspended particles contaminated with bacteria and mould than those of cows in deep litter pens.

But the factors determining increased thoracic dust and exposure to endotoxins in our study were not always the same as those identified in previous similar studies. [START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF] showed that milking was positively correlated with exposure to inhalable endotoxins, whereas our multivariate linear mixed-effect model did not identify milking. Forcing the duration of milking in our thoracic endotoxins model resulted in a very low, non-significant correlation ( = 0.0002, p = 0.91), without substantially modifying the impact of other determinants (data not shown). [START_REF] Basinas | Exposure-affecting factors of dairy farmers' exposure to inhalable dust and endotoxin[END_REF] also reported that using an automatic scrapper in alleyways reduced exposure to inhalable dust by 40%, while we find that this device had no effect on exposure to thoracic dust on Breton dairy farms and increased exposure to Gram-positive bacteria. These differences are probably due to the different particle fractions (thoracic/ inhalable) sampled. Inhalable dust may contain mostly extra-thoracic particles that cannot be collected by thoracic samplers, as reported recently in a study showing that most of the particles produced during milking are extra-thoracic particles [START_REF] Schaeffer | Size, Composition, and Source Profiles of Inhalable Bioaerosols from Colorado Dairies[END_REF]. Farm tasks or features that selectively increase the production of extra-thoracic dust are not selected by multivariate models investigating the determinants of exposure to thoracic dust. Moreover, extra-thoracic particles are unlikely to enter the bronchio-tracheal regions where COPD and asthma are believed to develop. Thus, the results of studies investigating the risk factors for lower airway diseases like COPD and asthma may be misinterpreted if the proportions of extra-thoracic and thoracic fractions in inhalable dust are not determined after sampling, especially when the inhalable dust is mainly made up of extra-thoracic dust. In contrast, studying the thoracic fraction may strengthen the specificity and relevance of determinants correlated with exposure to organic dust. Therefore, it may be essential to compare directly the factors determining exposure to inhalable and thoracic dusts collected on a particular dairy farm. The main limiting factor of this work may be that we did not do these comparisons.

Conclusions

Our study of the exposure of dairy farmers to the thoracic fraction of inhalable organic dust and their quantitative and qualitative determinants shows that the mechanical handling of bedding, grain and dry feed markedly increased the exposure of farmers to thoracic dust, endotoxins, Gram-positive bacteria and fungal material. However, certain farm features, such as the presence of cubicles in the main cowshed and large open wall areas can result in much lower exposure to these harmful agents. Further studies are required to confirm the roles of these determinants and to identify strategies that protect dairy farmers from exposure to organic dusts. Lastly, it is essential to determine whether exposure to thoracic dust is correlated with a dairy farmer's respiratory status.

Funding

This work was funded by the Institut Technique des Gaz et de l'Air (ITGA). Hugo Pfister holds a fellowship from Association Nationale Recherche Technologie (ANRT).

Acknowledgements

We thank Dr Valérie Lecureur and Professor Olivier Fardel for helpful comments and critically reading the manuscript and Dr Owen Parkes for editing the English text.

Declarations of interest: none

Ventilation

  .36 m²/m² (75 th percentile value) b 31Slurry/manureAutomatic scrapper in alleyway 54Solid manure pit in stable

Table 3 : Arithmetic Mean (AM), Geometric Mean (GM) and Geometric Standard Deviation (GSD) of personal exposure levels to the thoracic fraction of dusts, endotoxins, Gram-positive bacteria and mould. Limit of quantification (LOQ)

 3

	Period	Dust (mg/m 3 )

Table 4 : Multivariate mixed-effect linear model analysing the effects of tasks and farm characteristics on the log-transformed personal exposure levels to the thoracic fraction of dusts, endotoxins and Gram-positive bacteria. Blank spaces indicate that the determinant was not selected in model. Variance components of naive models e

 4

	Dusts	Endotoxins	Gram-positive bacteria

a change factor in exposure was calculated as exp(β) if not specified otherwise b change factor in exposure was calculated as exp (β x median time of the task duration). c change factor in exposure for an increment of the interquartile value of the determinants was calculated as exp(β x interquartile value). Interquartile values are 15%, 10°C and 45 cows for relative hygrometry, temperature and cow numbers, respectively d SWON = Surface of wall opening normalized on stable surface e models without fixed effects (bf : between-farm, bw: between-worker, ww: within-worker) f explained variability was calculated as 1-[variance component (complete model)] / [variance component (naive model)]

Table 4 : continued Table 5: Multivariate mixed-effect logistic model analysing the effects of performed tasks and farm characteristics on the probability of mould exposure > 1000 CFU/m 3 with n = 112 and 32 measurements > 1000 CFU/m 3 (28.5%)

 4 Odd Ratio was calculated as exp(β) if not specified otherwise b Odd Ratio was calculated as exp (β x median time of task duration). c Odd Ratio for an increment of 100 m² was calculated as exp(β x 100).

	Determinants	β (se)	OR (CI 95%) a	P-value
	Background level	-1.840 (0.717)		0.010
	Tasks (min)
	Manual loading/unloading grain of dry feed	0.131 (0.053)	1.93 (1.31 -3.24) b	0.014
	Manual loading/unloading of silage	0.059 (0.029)	1.34 (1.02 -3.18) b	0.040
	Manuel spreading of bedding materials	0.054 (0.029)	1.72 (0.97 -3.03) b	0.065
	Mechanical spreading of bedding materials	0.115 (0.038)	3.16 (1.50 -6.65) b	0.003
	Mechanical scrapping of slurry/manure	0.054 (0.03)	2.13 (0.94 -4.85) b	0.073
	Farm characteristics (presence of absence)
	Fodder = mainly corn silage	1.337 (0.651)	3.81 (1.84 -13.64)	0.040
	Floor = concrete + cubicle	-1.387 (0.667)	0.25 (0.07 -0.92)	0.038
	Farm characteristics (quantitative)
	Surface of wall opening (m²)	-0.006 (0.003)	0.24 (0.3 -0.99) c	0.036
	Model evaluation
	Area under ROC curve	0,89

a

Table S1 : Univariate effects of task duration on log-transformed personal exposure levels to the thoracic fraction of dusts, endotoxins and Gram- positive bacteria.

 S1 effect assessed in mixed-effect linear model. P-values are associated to likelihood ratio test (naïve model vs bivariate model) b effect assessed in mixed effect logistic model. P-values are associated to likelihood ratio test (naïve model vs bivariate model) c only P-values < 0,25 are shown

	Farm characteristics	Dust a	Endotoxins a	Gram-positive Bacteria a	Mould b
		β	P-value a	β	P-value a	β	P-value a	β	P-value a
	Qualitative
	Floor = concrete + cubicle					-0.678	0.167	-0.852	0.085
	Floor = concrete + mattresses					-0.737	0.234	-1.07	0.085
	Automatic milking	0.80	0.032	0.998	0.059	2.34	< 0.001	2.215	0.001
	Milking parlor directly connected to stable	0.34	0.143
	Usage of high pressure washing							-0.671	0.134
	Automatic scrapper in alleyways					0.71	0.132
	Manure storage in stable			0.829	0.011
	Cows pasturing at least a part of the day	-0.297	0.114	-0.516	0.045			-0.665	0.04
	Fodder = only haylage or grass	-0.814	0.069	-1.962	0.002	-1.671	0.035
	Fodder = mainly corn silage							-0.931	0.014
	SWON > 0.36	-0.410	0.114
	Quantitative
	Cow number					0.012	0.129
	Surface of wall opening (per 10m 2 )							-0.044	0.055
	SWON > (m²/m²) d	-0.555	0.241
	Environmental parameters
	Hygrometry (%)	-0.007	0.243					0.195	0.006
	Temperature (°C)			-0.044	0.014

a

Table S2 : Univariate effects of farm characteristics on log-transformed personal exposure levels to the thoracic fraction of dusts, endotoxins, Gram-positive bacteria and mould.

 S2

a effect assessed in linear mixed effect model. P-value associated to likelihood ratio test (naïve model vs bivariate model) b effect assessed in logistic mixed effect model. P-value associated to likelihood ratio test (naïve model vs bivariate model) c only P-values < 0.25 are shown d SWON = Surface of wall opening normalized on stable surface

Stable characteristics

Mean

Median (min-max)


Maxence Lepers

Véronique Zehnlé

Jean Claude Garreau

Kicked rotor quantum resonances in position space

We present an approach of the kicked rotor quantum resonances in position-space, based on its analogy with the optical Talbot effect. This approach leads to a very simple picture of the physical mechanism underlying the dynamics and to analytical expressions for relevant physical quantities, such as mean momentum or kinetic energy. The ballistic behavior, which is closely associated to quantum resonances, is analyzed and shown to emerge from a coherent adding of successive kicks applied to the rotor thanks to a periodic reconstruction of the spatial wavepacket.

I. INTRODUCTION

The kicked rotor is a simple system that plays a central role in studies of classical and "quantum chaos". The latter is defined as the quantum behavior of a system whose classical counterpart is chaotic. In its simpler form, a kicked rotor (KR) is formed by a particle orbiting a fixed circular orbit to which an instantaneous force (a kick ) is applied periodically. The corresponding classical dynamics is found to be regular (periodic) for weak kick intensities, for intermediate kick intensities chaotic regions develop in limited zones of the phase space, and for strong enough forcing an ergodic diffusion appears [1]. In 1995, Raizen and co-workers [2] established this system as a privileged ground for studies of quantum chaos by realizing experimentally a kicked rotor in the quantum regime with laser-cooled atoms placed in a kicked (i.e. rapidly turned on and off) laser standing wave.

Despite its apparent simplicity, the quantum kicked rotor (QKR) has very remarkable dynamical properties. One of the most studied is the so-called "dynamical localization": in contrast to the classical case, the ergodic diffusion does not last forever in the quantum case. After a characteristic "localization time", the diffusion is stopped by destructive quantum interferences [2,[START_REF] Casati | Stochastic behavior of a quantum pendulum under a periodic perturbation[END_REF]4]. Another feature of the QKR that has been the object of a recent burst of theoretical and experimental activity is the existence of quantum resonances (QRs), whose most dramatic manifestation is the appearance, for specific values of the parameters, of a ballistic motion, instead of a diffusive or a localized behavior. Quantum resonances are the main subject of the present paper.

The experimental realization of the QKR has triggered in the last decade an impressive number of studies involving dynamical localization [5,6,7,8,9,10,11], quantum transport [12,13,14,15], ratchets [16,17,18], chaosassisted tunneling [19,20], classical and quantum resonances [21,22,23,24]. Quantum resonances have been used in studies of fundamental aspects of quantum chaos such as quantum stabilization [25] or measurements of the gravitation [26]. "High-order" quantum resonances were also observed recently [27,28,29] both with a Bose-Einstein condensate and laser-cooled atoms.

In the present work, we show that the kicked rotor quantum resonances have a very simple and intuitive interpretation in position space, as opposed to the more common momentum space representation (see [30] and references therein). QRs in position space have been previously considered by Izrailev and Shepelyansky [4,31].

Here, we extend their approach and build a simple physical picture that enlightens the underlying physics and allows the calculation of quantities of experimental relevance, such as average momentum or the kinetic energy.

II. PHYSICAL ORIGIN OF QUANTUM RESONANCES

In this section we describe the fundamental properties of the kicked rotor dynamics and discuss the physical origin of quantum resonances. We aim at a simple presentation that puts into evidence the physical mechanisms even for the reader unfamiliar with quantum resonances. Some aspects of the following discussion may be considered as trivial, but the discussion is nevertheless necessary to introduce ideas and notation non-ambiguously.

The atom-optics realization of a (quantum) kicked rotor consists in placing laser-cooled atoms in a far-detuned laser standing wave. In such conditions, the atoms feel the light intensity as a mechanical potential affecting their center-of mass degree of freedom [START_REF] Cohen-Tannoudji | Fundamental Systems in Quantum Optics[END_REF][START_REF] Metcalf | Laser Cooling and Trapping[END_REF][START_REF] Meystre | Atom Optics[END_REF]. The standing wave is pulsed periodically, being on for a time interval so short that the motion of the atoms can be neglected; in such conditions, the pulses can be considered as delta functions (kicks). The standing wave sinusoidal modulation of intensity generates a spatial potential in sin (2k L x), where k L = 2π/λ L is the wavenumber of the radiation forming the standing wave. As the de Broglie wavelength of laser-cooled atoms is about λ L /3, it is comparable to the periodicity of the potential, λ L /2, and the system is in the quantum regime, provided that decoherence is negligible during the experiment. In practice, this implies using a far-detuned radiation to reduce spontaneous emission to acceptable levels.

We shall use a normalized spatial coordinate X = 2k L x = 2x/λ L which plays in fact the role of a cyclic variable: the spatial periodicity of the potential implies that the physics is the same if translations by a multiple of λ L /2 are performed. We can thus use either a "linear" ("unfolded") representation of the KR, using the linear variable X or a "cyclic" or "folded" representation using the angular variable θ = X[mod 2π] (x[mod λ L /2] in usual units) [35].

Choosing, moreover, units such that the mass of the particle is 1 and the time-period of the forcing is T = 1, the Hamiltonian of the KR reads

H = P 2 2 + K cos X N -1 n=0 δ(t -n) ( 1
)
where P is the momentum (scaled by a factor M/2k L T ) X the position in the periodic potential, K (usually called "stochasticity parameter") the intensity of the kicks, and n a discrete time corresponding to the n th kick (n is an integer in normalized units). Labeling X t , P t the position and momentum immediately after the t th kick, and integrating the classical Hamilton equations of motion corresponding to Eq. ( 1) produces the so-called Chirikov's "standard map" [1]:

X t+1 = X t + P t P t+1 = P t + K sin X t+1 . (2)
The classical dynamics is found to be periodic below a critical value of the kick intensity K c ≈ 0.9716. Chaotic regions appear in phase-space for K > K c , and progressively grow as K increases. For K 5 the islands of stable dynamics are barely visible, and the classical dynamics becomes an ergodic diffusion in phase space. The average kinetic energy then increases linearly with time (or kick number t): P 2 /2 = D c t, where D c ≈ K 2 /2 is a diffusion coefficient that can be explicitly calculated [START_REF] Rechester | [END_REF].

The standard map presents well-known classical resonances, also called accelerator modes. For example, set K = 2π, and consider a particle with initial conditions X 0 = π/2 and P 0 = 0. Iteration of Eqs. (2) shows that P t = 2πt and X t = 2π(t 2 -t) + π/2. The momentum increases linearly with time and the kinetic energy P 2 /2 increases quadratically with time, which is a signature of a ballistic dynamics, in contrast with the linear increase in the diffusive case. The origin of the ballisticity is easily seen: for this particular choice of K and of the initial conditions, the particle is always kicked at the same position (modulus 2π), that is sin X t = sin 2π(t 2 -t) + π/2 = 1, and thus receives the same amount of linear momentum per kick. For arbitrary values of K or of the initial conditions, the particle is kicked in different positions, and the effect of some kicks compensate the effect of other kicks, leading to a slower increase of the energy. One can easily convince oneself that, in general, there is a classical resonance for K = 2πp with integer p and adequate initial conditions. The classical resonance was experimentally observed in the atom-optics realization of the KR [21]. There are also classical antiresonances: e.g. for the above initial condition and K = 3π/2, successive kicks have opposite directions, and the momentum jumps endlessly between two values, P t = 0 and P t+1 = 3π/2.

In order to study the quantum dynamics of the KR, one considers the one-period evolution operator, also called Floquet operator:

U = e -iH/k = exp -i K k cos X exp -i P 2 2k (3)
where k = 4 k 2 L T /M is the normalized Planck's constant resulting from the definition of normalized variables satisfying the commutation relation [X, P ] = ik; it thus describes the "quanticity" of the system (k→ 0 is the classical limit). The above operator relates the quantum state after the t th kick to the quantum state after the (t -1) th kick [37]. It is a remarkable fact that this operator factorizes into the product of two exponentials, a free evolution followed by the kick effect, despite the fact that X and P do not commute. This is a consequence of the δ-function time dependence: during the instantaneous kick, the evolution related to the kinetic energy term is negligible. If one starts e.g. in the P -representation, the free evolution (from t + to (t + 1) -) corresponds to simply adding a phase. One can then convert to the Xrepresentation where the kick operator is diagonal and also simply adds a phase; one then goes back to the Prepresentation. The calculation of the quantum evolution over a period thus "costs" only two Fourier transforms and two multiplications. It is this formal and numerical simplicity that made the QKR so popular. Despite this simplicity, the cross-action of these two operators makes the QKR dynamics very rich: the free evolution operator mixes space components and the kick operator adds new momentum components, generating a complex quantuminterference pattern.

An important property of the kick operator is that it is periodic in space. This means that its eigenstates have a Bloch-wave (BW) structure. This is formally seen by developing it in terms of Bessel functions (noting that e inX is the momentum translation operator

e inX |P = |P + nk ) exp (-iκ cos X) = +∞ m=-∞ (-i) m J m (κ)e imX
where we introduced the quantity

κ ≡ K k . (4)
Therefore, if one starts from an initial state of welldefined momentum |P 0 , only states of the form |P 0 + mk (m integer ) will appear in the dynamics. We can write any arbitrary momentum P 0 in the form

P 0 = (m + β)k
with m integer and β ∈ [-1/2, 1/2) (i.e. β is in the "first Brillouin zone"). This introduces the quasimomentum kβ which, as just demonstrated, is a constant of motion. The particle wavefunction ψ(X) = X| ψ can be written as

ψ(X) = 1 √ 2π ∞ -∞ e ikx ψ(k)dk,
where k = P/k, and can thus be decomposed in:

ψ(X) = 1 √ 2π +1/2 -1/2 dβ +∞ m=-∞ ψ β (m)e i(m+β)X = +1/2 -1/2 dβψ β (X), (5)
where we introduced the quasimomentum component ψ β (X):

ψ β (X) = 1 √ 2π ∞ m=-∞ ψ β (n)e i(m+β)X (6)
with

ψ β (X) = e iβX u β (X). (7)
The function

u β (X) = 1 √ 2π ∞ m=-∞ ψ β (m)e imX (8)
is of a 2π-periodic function. Eq. ( 6) is a direct manifestation of the Bloch theorem: the particle is described by a Bloch wave, ψ β (X), which is the product of a planewave of well-defined quasimomentum β and a periodic function u β (X).

In most experimental realizations of the kicked rotor with laser-coled atoms the initial velocity distribution is larger than the Brillouin zone of the system and all quasimomenta are present; it may thus be necessary to average observable quantities over the quasimomentum. It is shown in the App. A that these averages values can be simply expressed in terms of the BW decomposition:

P (t) = 1/2 -1/2 dβ P β (9a) E (t) = 1/2 -1/2 dβ E β (9b)
where the subscript β indicates an average over ψ β .

In order to unveil the physical origin of quantum resonances, let us take, for simplicity, an initial state of quasimomentum β = 0 and apply to it the free evolution exp(-ikP 2 /2):

ψ 0 (X, t = 1 -) = 1 √ 2π m ψ 0 (m, t = 0) exp -ik m 2 2 exp (imX) . (10)
If we set in the above expression k 2 = 2πℓ, with ℓ arbitrary positive integer, we see that the argument of the exponential phase factor in Eq. ( 10) is always an integer multiple of 2π. This means that the free evolution over a period leaves the wavepacket invariant. Fig. 1 shows an initial BW that is spatially localized [38] in the first Brillouin zone and which evolves freely with time. The BW first becomes completely delocalized but, as seen above, it "focalizes" back to its initial shape just before the kick! Our using of an optics language is more than fortuitous: a monochromatic beam diffracted by a grating also reforms after a certain propagation length, an effect known in optics as the "Talbot effect". The quantum resonance is the atom-optics analog of the optical Talbot effect [22]. The only possible evolution comes from the kicks, whose effect is to add new momentum components ("sidebands") separated by a multiple of k. As the effect of the kick is diagonal in position representation; it adds a position-dependent phase to the wavefunction:

ψ 0 (X, t = 1) = exp (-iκ cos X) ψ 0 (X, t = 0).
As the wavepacket has always the same shape when the kick is applied, it acquires the same phase from each kick, and the kick effects add coherently. The quantum resonance condition is thus a constructive interference effect (in contrast with dynamical localization for instance, which results from destructive interferences that "freezes" the wavefunction evolution).

The above discussion puts into evidence an analogy (despite its very different nature) between quantum and classical resonances: in both cases, for particular values of the parameters, the dynamics is such that the particle (the wavepacket in the quantum case) is always kicked at the same position and the effect of the kicks adds (constructively interferes) to produce a linear increase of the momentum. An important difference, however, is that in the classical case the effect is related to the intensity of the kicks -the resonance condition thus depends on the parameter K -whereas in the quantum case it is related to the constructive accumulation of quantum phases and the resonance condition thus depends on the value of k.

In the following sections we shall develop an approach to calculate and interpret the behaviors of the system for different types of QRs. In section III, the so-called "sim-ple" resonances" corresponding to k = 2πℓ (ℓ positive integer) are analyzed, in which the initial wavepacket"refocuses" in a single replica of the initial wavepacket. In section IV, we focus on the so-called "high-order resonances" (k = 4πr/s, r, s integers), where the free evolution of an initial wavepacket generates various spatially-separated replicas of the initial wavepacket.

III. ANALYSIS OF "SIMPLE" QUANTUM RESONANCES

Quantum resonances obeying the condition k = 2πℓ (with ℓ a positive integer) are named "simple" quantum resonances (SQR). With this form for k, it is shown that the shape of BW remains invariant in the free propagation step between two kicks and that the effects of the kicks may lead to a ballistic (linear) growth of the momentum; the kicking period is a multiple of the half-Talbot time defined in classical optics, which is the condition for the optical (integer) Talbot effect [39,40].

A. Time-evolution

Let us consider, at some (integer) time (t -1) a state ψ β (X, t-1) corresponding to the general form of Eq. ( 6), and apply to it the free-evolution operator with k = 2πℓ :

exp -i P 2 2k ψ β (X, t -1) = 1 √ 2π m ψ β (m, t -1) exp -iπℓ (m + β) 2 e i(m+β)X = 1 √ 2π exp -iπℓβ 2 m ψ β (m, t -1) exp -iπℓm 2 exp (-i2πℓmβ) e i(m+β)X
As ℓm 2 and ℓm have the same parity, the first exponential factor under the sum is equal to exp (-iπℓm) ; it can thus be combined with the second term, yielding exp (-imk (β + 1/2)). We obtain

ψ β (X, t -) = 1 √ 2π exp -i kβ 2 2 m ψ β (m, t -1) exp (-imkβ ′ ) e i(m+β)X
with β ′ ≡ β + 1/2. This last expression can be rewritten, using Eq. ( 6), as

ψ β X, t -= exp i kβ(β + 1) 2 ψ β (X -kβ ′ , t -1) .
(11) Applying now the kick operator (which is diagonal in the X representation), we obtain a recurrence relation linking the wavepackets at times t and (t -1):

ψ β (X, t) = e -iκ cos X exp i kβ(β + 1) 2 ψ β (X -kβ ′ , t -1) . (12)
The above result shows that in the conditions of a simple resonance |ψ(X,

t)| 2 = |ψ(X -v, t -1)| 2 = |ψ(X -vt, 0)| 2 :
the square-modulus of the wavefunction remains invariant immediately before the kick, except for

v = k β + 1 2 . (13)
Moreover, the wavefunction acquires a positiondependent phase due to the kick. By iterating Eq. ( 12) down to t = 0, we can express ψ β (x, t) in terms of the initial wavefunction

ψ β (X, t) = exp (-iκΦ(X, t)) ψ β (X -vt, 0) , (14)
where the accumulated phase is given by [START_REF]The constant phase factor in Eq. (12)[END_REF] Φ(X, t)

= t-1 s=0 cos (X -vs) . (15)
A remarkable property is obtained if

v = 2π p q (16)
with p,q integer, e.g. for rational values of the quasimomentum. Then, the phase defined in Eq. ( 15) takes the values Φ(X, t = q) = 0 if q = 1 and Φ(X, t = q) = cos X if q = 1. Let us first consider the case q = 1. After a recurrence time t r = q the BW is given by:

ψ β (X, t r ) = ψ β (X -2πp, 0) = e i2πpβ ψ β (X, 0),
and the particle comes back to its initial state after t r kicks, leading to a periodic evolution. To show the physical origin of this periodicity, let us take the simple case v = π, corresponding to t r = 2. From Eqs. ( 14) and ( 15), the BW evolution over two successive kicks is

ψ β (X, t = 2) = e -iκ cos X ψ β (X -π, t = 1)
and

ψ β (X -π, t = 1) = e iκ cos X ψ β (X -2π, t = 0),
which means that the phase added by the kicks simply cancels after two kicks. An analogous cancellation happens for any other value of the velocity v obeying Eq. ( 16) with q = 1: the force acting on the particle averages to zero after t r kicks, and the motion of the wavepacket is a simple oscillation of period t r . This behavior, that cannot lead to a ballistic behavior, is called "anti-resonance". If v = 2πp [or q = 1 in Eq. ( 16)], corresponding to

β = p ℓ - 1 2
the evolution of the BW is given by

ψ β (X, t r = 1) = e -iκ cos X ψ β (X -2πp, 0) = e -i2πpβ e -iκ cos X ψ β (X, 0).
The wavepacket exactly recovers its initial shape after each kick and the particle is subjected to an identical potential at each kick. In contrast with to the former case, the kicks will add in a coherent way, which, as we shall demonstrate in Sec. III B, causes a linear increase of the average momentum, or ballistic behavior. This is the quantum resonance. If Eq. ( 16) is not fulfilled, our picture still allows to guess the general shape of the asymptotic evolution for large numbers of kicks (t ≫ 1). If v/2π is not a rational number, the phase Φ(X, t) will tend to 0 as t → +∞. Once again, the contribution of the successive kicks will average to zero and the wavepacket just "drifts" (for stroboscopic times) while keeping its initial shape (see Fig. 2).

B. Averages

Let us now focus on the time evolution of average values. From Eq. ( 12) we can easily obtain a recurrence relation for the BW average position

X (t) = X (t -1) + v . (17)
A recurrence relation for the average momentum is obtained using Eq. ( 12):

P β (t) = P β (t-1)+K π -π dX sin X |ψ β (X -v, t -1)| 2 = P β (t -1) + K π -π dX sin(X + vt) |ψ β (X, t = 0)| 2 .
(18) This expression can be iterated down to t = 0, leading to:

P β (t) = P β (0) + K t n=1 π -π dX sin(X + nv) |ψ β (X, 0)| 2 = P β (0) + K sin(tv/2) sin(v/2) Im e i(t+1)v/2 π -π dXe iX |ψ β (X, 0)| 2 . ( 19
)
The role of interferences is clearly seen in this last result:

indeed, if v = k(β + 1 2 ) = 2π [mod 2π
], the momentum increases linearly with the (stroboscopic) time t, i.e

P β (t) = P β (0) + Dt ( 20
)
where the slope

D = K π -π dx sin x |ψ β (x, 0)| 2 (21)
clearly appears as a force averaged over the initial spatial distribution. The observed momentum transport shows that well-controlled diffusion can be obtained by a suitable choice of initial conditions [i.e of ψ β (x, 0)].

For v = k(β + 1 2 ) = π [mod 2π], destructive interference occurs. This can be seen from Eq. ( 19): P β (t) = P β (0) (t even) or P β (t) = P β (0) -D (t odd). In the general case (v = 2π[mod 2π]), the momentum is frozen around its initial value.

Quantum resonance effects can also be analyzed through the temporal evolutions of the average kinetic energy. Using Eq. ( 14), one obtains:

E β (t) = E β (t = 0) + K 2 2 π -π dX t n=1 sin(X + nv) 2 |ψ β (X, t = 0)| 2 + K π -π dX t n=1 sin(X + nv) J(X, t = 0) (22)
where we introduced the current

J(X, t) = i k 2 ψ β (X, t)∂ X ψ * β (X, t) -c.c. .
The constructive interference case v = 2π [mod 2π] then leads to an average kinetic energy increasing quadratically with time:

E β (t) = E β (t = 0) + 1 2 K 2 t 2 π -π dX sin 2 X |ψ β (X, t = 0)| 2 + Kt π -π dX sin XJ(Xx, t = 0) (23)
which is the quantum-mechanical analog of the ballistic motion observed -in different conditions -for a classical resonance. The ballistic growth is seen to be proportional to the quantum average of the square of the force K 2 sin 2 X.

C. Map

Inspection of Eqs. (17) and Eq. ( 18) suggests that the dynamics of position and momentum averages of a Bloch-wave can be described by a map. In the following, we assume that the initial BW is sharply localized around its mean position X β (t = 0) = X 0 . Noting P t = P β (t) and X t = X β (t) we have from Eq. ( 18), P t = P t-1 + K sin(X 0 + vt) (with the normalization con-dition π -π dX |ψ β (X, t = 0)| 2 = 1). We then obtain

P t = P t-1 + K sin X t (24a) X t = X t-1 + v, (24b)
which evokes the classical map (see Eq. 2), with the important difference that the position in Eq. (24b) does not depend on P t but solely on the drift velocity v. This produces after t kicks

X t = X 0 + vt P t = P t-1 + K sin(X 0 + vt).
Iterating in turn this momentum equation produces

P t = P 0 + K t n=1 sin(X 0 + nv).
Ballisticity is found if v = 2π (mod 2π):

P t = P 0 + tK sin X 0 = P 0 + Dt with D = K sin X 0 . (26)
For a localized packet, ballisticity emerges, as in the classical case, if the particle is always kicked at the same position. In the classical case, this is possible if K = 2πp (p integer) and X 0 = ±π/2. In the quantum case, the resonance condition depends on the quantum parameters k and β (through v), but the kick intensity K and the initial position X 0 determine only the growth rate of the average momentum. Recalling that t is an integer (stroboscopic) variable counting the kicks, one can find also a whole class of periodic behaviors. An example is k = 2π and β = 0 (v = π), with a recursion time t r = 2, which has already been analyzed in sect. III A. More generally, for rational quasimomentum values β = (p, q integer), after a number of kicks t r , the average momentum and kinetic energy come back to their initial values. In such cases the dynamics is periodic, due to the effect of kick compensation discussed above.

IV. THE k = π HIGH-ORDER RESONANCE

"High-order"QRs are the quantum-mechanical analogs of the fractional optical Talbot effect [39]. In contrast with the situation found in simple resonances, after a free propagation the initial packet does not reconstruct in an identical packet, but forms two or more replicas of the original one. The action of the kick in these subpackets generates different quantum phases and produces quantum interference effects during the subsequent free propagation. This makes high-order quantum resonances fundamentally different from, and more complex than, simple ones.

High-order quantum resonances (HQRs) correspond to a dimensionless Planck's constant of the form k = 4πr/s, with r and s > 2 integers. A calculation similar to that leading to Eq. (12) shows that after the free propagation the initial packet refocalizes into s uniformly spaced subpackets if s is odd, and s/2 subpackets if s is even. We shall consider here only the the simplest case k = π. The method presented below can in principle be generalized to more complicated cases, but the algebra involved quickly gets very cumbersome. As in the preceding section, our approach gives a simple picture of the physical mechanism of HQRs.

Using the general expression for BW at time (t -1), Eq. ( 6), and applying the free-evolution operator produces in the case k = π:

ψ β (X, t -) = 1 √ 2π n ψ β (n, t -1) exp -i π 2 (n + β) 2 e i(n+β)X = 1 √ 2π exp - iπβ 2 2 n ψ β (n, t -1) exp - iπn 2 2 exp (-iπnβ) exp (i(n + β)X) . ( 27
)
We show in the Appendix B that the above expression can be written as

ψ β (X, t) = e iπβ 2 /2 e -iκ cos X √ 2 e -i π 4 ψ β (X -kβ, t -1) + e i π 4 e iβπ ψ β (X -kβ -π, t -1) (28)
or

ψ β (X + wt, t) = e -iκφ(X,t) √ 2 e -iπ/4 ψ β (X + w(t -1), t -1) + e iπ/4 e iβπ ψ β (X + w(t -1) -π, t -1) ( 29
)
where w is the packet (stroboscopic) drift velocity defined by w = kβ (30) and we introduced the "local" phase [43] φ(x, t) = κ cos(X + wt)

Eq. ( 29) shows that at any time t, ψ β (X + wt, t) is the superposition of two subpackets having the same shape as the initial BW, centered at X = 0 and X = π. Each of these two subpackets is multiplied by a phase factor which is the sum of the accumulated phases, producing a complex interference pattern. The principle of the following calculation is to keep track of the coefficient of each subpacket, as we know that the shape of the subpackets is fixed. We then write:

ψ β (X + wt, t) = c 1 (X, t)ψ β (X, 0) + c 2 (X, t)ψ β (X -π, 0), (32)
where c 1 (X, t) and c 2 (X, t) are 2π-periodic complex amplitudes [START_REF]The periodicity of c1,2(x, t) follows from Eq. (32) and the condition ψ β (x + 2π[END_REF]. The above expression corresponds to a coupled two-level model where the "particle", initially in "level 1" (i.e c 1 = 1 at t = 0) is progressively "transferred" to level 2" (c 2 = 0) and then back again to "level 1", performing a kind of Rabi oscillation.

Let us define the state vector

c t = c 1 (X, t) c 2 (X -π, t)
It is shown in App. B 2 that these amplitudes obey a matrix recurrence relation :

c t = M t c t-1
where M t is a matrix depending on time and space having the form

M t = e -iπ/4 M t where M t = 1 √ 2 e -iφ ie -iφ e -iβπ ie iφ e iβπ e iφ , ( 33
)
with φ(X + wt) given by Eq. (31). The matrix M t can be recast as

M t = 1 √ 2
e -iφ 0 0 e iφ 1 ie -iβπ ie iβπ 1 .

The rightmost matrix in the above product stands for the free propagation that induces a coupling between the two subpackets; it is thus responsible of the interference effects. The leftmost one represents the effect of the kick and is obviously diagonal in x-representation. Analytical results can be obtained in the case β = 0. As in this case w = 0, the matrix M t is time-independent. It is then easy to write c t as a function of the initial condition c 0 = 1 0 :

c t = [M t ] t c 0 = e -itπ/4 M t t c 0
The eigenvalues of M t are: λ = exp(±iΘ) where the phase Θ depends on X and is given by

cos Θ = cos φ √ 2 = cos(κ cos X) √ 2
(note that π/4 ≤ Θ ≤ 3π/4). If P is the diagonalizing matrix, one can write (see Sec. B 3 in App. B)

c t = e -it π 4 P e itΘ 0 0 e -itΘ P -1 c 0 . ( 34
)
After a straightforward calculation, the amplitudes are found to be

c 1 (X, t) = e -it π 4 cos(tΘ) - i √ 2 2 sin φ sin Θ sin(tΘ) (35)
and

c 2 (X -π, t) = i √ 2 2 e iφ sin Θ e -it π 4 sin(tΘ). (36)
From the above result we can calculate the average momentum, but the algebra involved is cumbersome (see App. C):

P β (t) = P β (t -1) + K π -π dX sin X |ψ β (X, t)| 2 .
In the general case, it is difficult to analyze the behavior of the amplitudes c 1 (X, t) and c 2 (X, t) since they depend implicitly on X in a quite complicated way. Analytic results can however be obtained in the limit which the initial BW is formed of spatially narrow wavepackets.Let us assume that ψ β (X, t = 0) is well localized at some position X 0 in the interval [-π, π] (i.e its width ∆X ≪ 2π) and express the average momentum in terms of the coefficients c 1 and c 2

P β (t) = P β (t -1) + K π -π dx sin(X + wt) |c 1 (X, t)| 2 -|c 2 (X -π, t)| 2 |ψ β (X, 0)| 2 (37)
This expression shows that if the two subpackets have the same weight, the momentum shift per kick is zero: the subpackets are localized around positions X 0 and X 0 + π, and subjected to opposite forces +K sin(X 0 )/2 and -K sin(X 0 )/2, an effect that is characteristic of the high-order resonances and obviously does not exist for simple resonances. This expression is valid if the two subpackets are well separated so that they do not interfere significantly. For a wavepacket that is strongly localized at X 0 , the amplitudes in the decomposition Eq. ( 37) can be evaluated at X 0 and depend only on time, while the phases φ and Θ take constant values (φ = κ cos X 0 ). Hence, the average momentum evolution is

P β (t) = P β (t -1) + K sin (X 0 + wt) 1 -2 |c 2 (X 0 -π, t)| 2 , (38)
which, in our simple case β = 0, takes the explicit form

P β=0 (t) = P β=0 (t -1) + K sin X 0 1 - sin 2 (tΘ) sin 2 Θ .
Note that the expression inside parenthesis in the above expression is characteristic of the diffraction on a grating, which puts into evidence the "wavelike" nature of the dynamics. Iterating down to t = 0 produces

P β=0 (t) = P β=0 (t = 0) + K sin X 0 sin 2 φ 1 + sin 2 φ t + 1 1 + sin 2 φ sin [(2t + 1)Θ] 2 sin Θ - 1 2 .
-0.4 (39) ] as a function of X0 for 3 values of κ: 0.5 (full line), 1.0 (dots) and 2.0 (dashed).

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 π π/2 0 -π/2 -π D / K x 0 κ = 0.5 κ = 1.0 κ = 2.0
This expression shows that average momentum evolution is, for arbitrary φ and Θ, a mix of two qualitatively different behaviors: ballisticity, corresponding to the first term in the brackets, and oscillation, described by the second term. This contrasts with SQRs, where the dynamics is either ballistic or oscillatory.

For long times and φ = 0, the ballistic term dominates and one gets

P β=0 (t) = P β=0 (t = 0) + Dt with D = K sin X 0 sin 2 φ 1 + sin 2 φ . (39)
being the rate of change of the mean momentum. The behavior of D as a function of X 0 is displayed in Fig. 3 for different values of κ. It is interesting to compare Eq. ( 39) with its counterpart for SQRs, which is given by Eq. ( 26): note in particular that in the present case D = 0 if X 0 = π/2, whereas for SQR, the maximum of D occurs when the force is maximum (i.e at X 0 = π/2).

The coefficient D depends periodically on the kick intensity κ via φ = κ cos X 0 , as shown in Fig. 4. In contrast to the simple resonance case, increasing the kick force does not necessarily increase the diffusion, an effect that persists if the momentum is averaged over the quasimomentum distribution.

Analogous results describing the ballistic behavior of an initial state which is an eigenvector of the momentum have been obtained via a quite different approach [START_REF] Sokolov | [END_REF]. Our localized-packet approach provides a clearer picture of the underlying physics.

For β = 0, the occurrence of ballisticity depends on the periodicity of M t and on the initial conditions. More precisely, ballisticity will emerge if the relation M t+tr = M t is fulfilled for some (integer) recurrence time t r . This happens for any rational value of quasimomentum. For example, if β = 1/2, M t [see Eq. ( 33)] has a period of 4 kicks: M t+4 = M t . One can apply the above reasoning to M = M 4 M 3 M 2 M 1 . To illustrate the resulting behavior, Fig. 5 shows the time-evolution of average momentum obtained by direct integration of the Schrödinger equation. 39)] as a function of κ for two initial positions X0=π/4 (dots) and X0 = π/8 (full line). One sees that increasing κ (that is, the kick intensity) does not necessarily increase the momentum slope. One observes essentially the same kind of behavior.

For a general rational quasimomentum β = p/q, the ballistic diffusion rate is roughly proportional to 1/q. Irrational quasimomenta, that may be consider as the limit q → +∞, do not produce ballistic behavior.

V. AVERAGING OVER QUASIMOMENTUM

In experiments performed with laser-cooled atoms (not with ultracold atoms -BECs) the initial momentum distribution is larger than the Brillouin zone unless velocity selection is performed. In position space, this corresponds to an initial wavepacket which is localized on a single potential well, around a position X 0 . In this simple case, and for k = 2πℓ, we are able to give an analytical expression for any value of t, of the momentum and kinetic energy averaged on quasimomentum β.

Starting from Eq. (9a):

P (t) = P (t = 0) + K t n=1 1/2 -1/2 dβ π -π dX sin(X + nv) |ψ β (X, 0)| 2 .
Assuming that ψ β (X, 0) is β-independent (ψ β (X, 0) = ϕ(X)and π -π dx |ϕ(X)| 2 = 1 ), and recalling that v = 2πℓ(β + 1/2), we easily see that averaging on β lead to:

P (t) = P (t = 0).
A similar reasoning can be applied for kinetic energy. From Eqs. (9b) and ( 22), we have:

E (t) = E (t = 0) + K 2 2 1/2 -1/2 dβ π -π dX t n=1 sin(X + nv) 2 |ϕ(X)| 2 + K t n=1 1/2 -1/2 dβ π -π
dx sin(X + nv)J(X, t = 0)

When we integrate over β, the term of the last term cancels out, and the only contributions come from

1/2 -1/2 dβ t n=1 sin(X + nv) 2 = 1/2 -1/2 dβ t n=1 sin 2 (X + πℓn + 2πℓnβ) = t 2
We finally get the diffusive behavior

E (t) = E (t = 0) + K 2 t 4 . ( 40
)
This result evokes the classical kicked rotor in chaotic regime, whose kinetic energy grows linearly with time with the same rate. This shows that the ballistic behavior, which corresponds to a "null-measure ensemble" of rational quasimomenta, is very hard to detect by measuring quantities averaged over quasimomentum.

Experimentally, the optimum situation for observing QRs is to perform a quasimomentum selection, either by using stimulated Raman transitions [29,46,47] or by using a Bose-Einstein condensate [14,27,28]. However, it is possible to detect QRs with atoms issued of a magnetooptical trap if one can measure the full momentum distribution with enough precision to see the ballistic parts of the wavefunction separating out of the diffusive part for long enough times, as experimentally evidenced by d'Arcy et al. [23].

VI. CONCLUSION

In the present work, we presented a description of quantum resonances of the kicked rotor in position space, both for simple and for high-order quantum resonances. We have shown that the dynamics can be understood by considering that the spatial wavepacket comes back to its in ital form after a finite number of kicks, according to the (rational) value of the quasimomentum. For a localized wavepacket (or an finite ensemble of localized wavepackets, in the case of HQRs), one can interpret the dynamical behavior in terms of the action of a finite number of successive kicks. This picture, inspired of the atomoptics analog of the Talbot effect, proves very useful both as it providing a intuitive understanding of the underlying physics ans as it leads to analytical developments for experimentally-relevant quantities. and can be related to P β in the following way. One has from the definition of section II:

ψ β (n) = 1 √ 2π π -π ψ β (X) e -i(β+n)X dx therefore: (n+β) ψ β (n) = i √ 2π π -π dXψ β (X) ∂ ∂X e -i(β+n)X = i √ 2π ψ β (X) e -i(β+n)X π -π - i √ 2π π -π dXe -i(β+n)X ∂ ∂X ψ β (X)
The first term on the RHS vanishes (u β (X) = ψ β (X) e -iβX is 2π-periodic). One then has:

P = - ik √ 2π n dβ ψ * β (n) π -π dXe -i(β+n)X ∂ ∂X ψ β (X) = - ik 2π dβ π -π π -π dXdX ′ e -iβ(X-X ′ ) ψ β (X ′ ) ∂ ∂X ψ β (X) n e -in(X-X ′ )
Finally, using n e -in(X-X ′ ) = k δ(X -X ′ -2kπ)

P = -ik dβ π -π dXψ * β (X) ∂ ∂X ψ β (X) = 1/2 -1/2 dβ P β where P β = π -π dXψ * β (X) [P ψ β (X)]
For the kinetic energy, the same developments lead to:

P 2 = n dβ(n + β) 2 ψ β (n) 2 = dβ P 2 β where P 2 β = π -π dX [P ψ β (X)] * [P ψ β (X)]
(note that the same rules can be obtained for P k = dβ P k β ). The free-propagation factor exp(-i π 2 n 2 ) in Eq. ( 27) has two different values according to the parity of n:

exp -i π 2 n 2 = 1 (n even) = -i (n odd).
Replacing

ψ β (n, t) = 1 √ 2π π -π dXe -inX u β (X, t) = 1 √ 2π π -π dXe -i(n+β)X ψ β (X, t)
in Eq. ( 27), one finds

ψ β (X, t -) = exp -iπβ 2 /2 2π n π -π dX ′ e -i(n+β)(X-X ′ ) ψ β (X ′ , t -1)e -iπn 2 /2 e -iπnβ
one can separate even and odd terms ; this leads to:

ψ β (X, t -) = exp -iπβ 2 /2 2π π -π dX ′ ψ β (X ′ , t -1)e -iβ(X-X ′ ) p e -i2πpβ e -i2p(X-X ′ ) 1 -ie -iπβ e -i(X-X ′ )
Using the relation p e -i2πpβ e -i2p(X-X ′ ) = (1/2) n δ(X -X ′ -πβ + nπ) and integrating with respect to X ′ , gives Eq. ( 29) (note that only n = 0, 1 contributes for β > 0 and n = 0, -1 for β<0).

"Two-level" system

The coupled equation for the amplitudes c 1,2 (X, t) are obtained in the following way. Insertion of Eq. ( 32) at time (t -1) ψ β (X + w(t -1), t -1) = c 1 (X, t -1)ψ β (X, 0) + c 2 (X, t -1)ψ β (X -π, 0), (B1) in Eq. ( 29) gives ψ β (X + wt, t) = e -iφ(X,t) √ 2 e -iπ/4 [c 1 (X, t -1)ψ β (X, 0) + c 2 (X, t -1)ψ β (X -π, 0)] + e iπ/4 e iβπ [c 1 (X -π, t -1)ψ β (X -π, 0) + c 2 (X -π, t -1)ψ β (X -2π, 0)] and can be put in a simpler form [using ψ β (X -2π, 0) = e -i2πβ ψ β (X, 0)]:

ψ β (X + wt, t) = e -iφ(X,t) √ 2 e -iπ/4 c 1 (X, t -1) + e iπ/4 e -iβπ c 2 (X -π, t -1) ψ β (X, 0)+ e iπ/4 e iβπ c 1 (X -π, t -1) + e -iπ/4 c 2 (X, t -1) ψ β (X -π, 0)

Comparing to Eq. ( 32) one obtains: c 1 (X, t) = e -iφ(X,t) √ 2 e -iπ/4 c 1 (X, t -1) + e iπ/4 e -iβπ c 2 (X -π, t -1) c 2 (X, t) = e -iφ(X,t) √ 2 e iπ/4 e iβπ c 1 (X -π, t -1) + e -iπ/4 c 2 (X, t -1) .

This last expression can be put into the form c 2 (X -π, t) = e iφ(X,t) √ 2 e iπ/4 e iβπ c 1 (X -2π, t -1) + e -iπ/4 c 2 (X -π, t -1) .

One can show that the amplitudes are periodic functions, i.e c 1,2 (X, t)= c 1,2 (X +2π, t) by combining Eq. ( 32) combined with the equality ψ β (X -2π, t)= e -i2πβ ψ β (X, t). This property leads to the matrix expression

c 1 (X, t) c 2 (X + π, t) = 1 √ 2
e -iφ e -iπ/4 e -iφ e iπ/4 e -iβπ e iφ e iπ/4 e iβπ e iφ e -iπ/4 c 1 (X, t -1) c 2 (X + π, t -1)

The case β = 0

The explicit expression for the amplitudes c 1,2 (X, t) is obtained for β = 0 as follows. The diagonalization matrix P is formed with the eigenvectors of M t . For λ = e ±iΘ they are given by (-ie -iφ , e -iφ -√ 2e ±iΘ ) T . P is then obtained as P = -ie -iφ -ie -iφ e -iφ -√ 2e iΘ e -iφ -√ 2e -iΘ .

In order to obtain the amplitudes c 1 and c 2 at time t, one performs explicitly the development corresponding to Eq. [START_REF] Meystre | Atom Optics[END_REF]. The algebra is simple, although rather long, and the final result is Eq. (35).

For the second term p 2 , on obtains [in the last equality we replaced (X -w)by X, as the integration is over one period, the integration limits can be kept the same]. One recognizes P β (t -1) in the first term while the second term cancels out. The recursion relation for the momentum thus reads

p 2 =
P β (t) = P β (t -1) + K π -π dX sin X |ψ β (X, t)| 2 . ( C2
)
We now use the decomposition of ψ β (X, t) proposed in Eq. [START_REF] Cohen-Tannoudji | Fundamental Systems in Quantum Optics[END_REF]. Eq. (C2) then transforms into:

P β (t) = P β (t -1) + iK cos(βπ) π -π
dX sin(X + wt) c * 1 (X, t)ψ * β (X, 0) + c * 2 (X, t)ψ * β (X -π, 0) × (c 1 (X, t)ψ β (X, 0) + c 2 (X, t)ψ β (X -π, 0)) .

If the initial wavefunction has a narrow distribution centered around position X 0 , all terms involving overlaps of functions ψ β (X, 0) and ψ β (X + π) tend to zero [48]. This expression therefore simplifies to

Figure 1 :

 1 Figure 1: Free evolution of a localized wavepacket (β = 0)between two successive kicks for k = 4π. (a) Initial wavepacket centered at X = π/2 shown in the interval X ∈ [-π, π], (b) wavepacket at t = 0.115, showing complete delocalization, (c) wavepacket at t = 0.25, showing two replicas of the initial shape, (d) "reconstructed" wavepacket at t = 1 -, identical to the initial one.

Figure 2 :

 2 Figure 2: Wavepacket plotted at different integer times, for k = 4π and an irrational quasimomentum (β = π 50 ≈ 0.063). (a) Initial wavepacket (|ψ β | 2 ) centered at X = 0, (b) t = 1, (c) t = 2, (d) t = 5. The shape of the potential is represented in dotted lines. The wavepacket interacts with the potential at different positions, and the average effect tends to zero.

Figure 3 :

 3 Figure 3: Momentum slope D [Eq.(39) ] as a function of X0 for 3 values of κ: 0.5 (full line), 1.0 (dots) and 2.0 (dashed).

8 Figure 4 :

 84 Figure 4: Momentum slope D [Eq.(39)] as a function of κ for two initial positions X0=π/4 (dots) and X0 = π/8 (full line). One sees that increasing κ (that is, the kick intensity) does not necessarily increase the momentum slope.

π / 8 Figure 5 :

 85 Figure5: Evolution of the average momentum for β = 1/2 obtained by numerical integration of the Schrödinger equation. The initial initial wavefunction has the same Gaussian shape as in Fig.1and is centered at the positions X0 = π/4 (dark line), which displays a dominant oscillatory behavior, or X0 = π/8 (light line), displaying dominant ballisticity.

  Appendix B: BLOCH WAVE EVOLUTION FOR THE k = π RESONANCE 1. Bloch wave evolution

P 2 (

 2 β (t) = P β (t -1) + K π -π dX sin(X + wt) |c 1 (X, t)| 2 -|c 2 (X -π, t)| 2 |ψ β (X, 0)| C3)Finally use the fact that ψ β (X, 0) is much narrower then any other factor, and assumingπ -π dX |ψ β (X, 0)| 2 = 1 [49] gives P β (t) = P β (t -1) + K sin(X 0 + wt) 1 -2 |c 2 (X 0 -π, t)| 2where we used the normalization |c 1 (X, t)| 2 + |c 2 (X -π, t)| 2 = 1.

  -w, t -1) + e -i π 4 e -iβπ ψ * β (X -x -π, t -1) (X -w, t -1) + e i π 4 e iβπ ∂ ∂X ψ β (X -w -π, t -1)

	-β (X × e -i π ik 2 π -π dX e i π 4 ψ * 4 ∂ ∂X ψ β = -ik 2 π -π dX ψ * β (X, t -1)	∂ ∂X	ψ β (X, t -1) + ψ * β (X -π, t -1)	∂ ∂X	ψ β (X -π, t -1)
	+	k 2	π -π	dX e iβπ ψ * β (X, t -1)	∂ ∂X	ψ β (X -π, t -1) -e -iβπ ψ * β (X -π, t -1)	∂ ∂X	ψ β (X, t -1)

Appendix A: AVERAGE MOMENTUM The average momentum reads:

Starting from Eq. ( 28), recursion relation for ψ β (X, t), we easily obtain a recursion relation for its derivative

Using these expressions into for calculating the average momentum produces two terms; let us call them p 1 and p 2 . The first one is given by:

dX sin X |ψ β (X, t)| 2