Martin Tanguy

come

An assessment approach of maritime supply chain of energy vulnerability to piracy risk by simulation of spatial behavior

INTRODUCTION

Maritime network of energy

The maritime supply chain of energy concerns all travels realized between ports on the maritime space. The use of this space has increase since 1970. This increase is due to the globalization which required a strong need of energy [START_REF] Rodrigue | The Geography of Transport Systems[END_REF]. In 2007, 62% of the petroleum was shipped by maritime transportation, as 27% of GNL, and 14% of coal. [START_REF] Mérenne-Schoumaker | Géographie de l'énergie: acteurs, lieux[END_REF]. Main energy supply regions are located in Middle-East, Russia, Venezuela. They are linked by the maritime network of energy to major consumption regions which are North America, Europe and East Asia. This network is vital for the worldwide economy and used by oil tankers which link the different infrastructures (ports, refineries and offshore platforms) of the energy supply chain. According to the 2014 annual report of IEA, Oil&Gas represent more than an half of the worldwide energy demand. This demand tends to increase on the 2015-2035 period. Even if the part of fossil energy will decrease (82 to 76%) the demand in Oil&Gas will increase to 7027 Mt in 2012 at 8793 mt in 2035. Oil&Gas is transported from production to consumption area by the energy supply chain. The supply can be resumed into three steps: extraction, transportation and consumption. Between the extraction and consumption, energy is transported by oil tankers within maritime space. [START_REF] Kaluza | The Complex Network of Global Cargo Ship Movements[END_REF] study the maritime network by type of vessel and characterize for each type the network specificities. In the case of oil tankers, the number of vessels is lower than container ships and bulk dry carriers according to the AIS data set of the study. Second, an oil tanker changes their paths depending of the supply and demand in energy and are less predictable that container ships. Third, the network is more limited due to the fact they only transport energy and they link specific ports and refineries. The network also includes nodes inside the maritime space due to the presence of offshore platforms. Maritime transportation knows many risks due to the specificity of maritime space. [START_REF] Elliott | A Typology of Marine and Estuarine Hazards and Risks as Vectors of Change: A Review for Vulnerable Coasts and Their Management[END_REF] presents a typology of risks can meet the ships during transportation. These risks have different spatiotemporal features. For example, some risks are static, like reefs, but some are more complex and dynamic. Storms are dynamic and move through space but also have a dynamic shape due to many atmospheric causes. Piracy is a different type of risks due to the fact the dynamic of piracy is dependent of the dynamic of maritime traffic on network. In this paper, we focus on this specific risk and propose and approach to measure the impact of piracy on the maritime network.

An assessment approach of maritime supply chain of energy vulnerability to piracy risk by simulation of spatial behavior M. Tanguy & A. Napoli

MINES ParisTech, PSL Research University, CRC -Centre for research on risks and crises, France

ABSTRACT: Nowadays, oil tankers represent 25% of the targets of maritime piracy attacks. The main goal of this paper is to propose an approach to evaluate the vulnerability of the maritime supply chain to piracy risk. This approach, based on spatial reasoning and modelling tend to measure the impact of piracy on the network by the changes on maritime network realized by disruptions. To achieve it, we use multi-agent system, which can be defined by a set of agent within a shared environment. Agents are entity which has their own goal, capacity and resources which induce a specific behavior. By using the sum of these behaviors, the main objective of this paper is to measure disruptions by a conceptual approach to qualify and quantify the piracy risk.

Piracy at sea

The International Maritime Organization (IMO) distinguishes two types of dangers at sea: safety and security. If security refer to risk relative to activates (e.g. storm, oil spill), safety refer to malevolent acts. It include terrorist attacks but also piracy risk. Maritime piracy is a major issue for the safety and security at sea, despite a decreased of attacks since 5 years, with 245 in 2010 and 445 in 2014 according to IMB in 2015. This trend is due to the improvement of controls at sea especially on Somalia coast (139 attacks in 2010 and 3 in 2014). But this trend should not hide those new risky zones increase for maritime transport like Indonesia and Malacca strait with 40 attacks in 2010 to 100 in 2014. Due to these attacks, vessels can change their way to find safer maritime route.

There is several factors explaining the piracy is the contemporary era [START_REF] Chalk | Maritime piracy: Reasons, dangers and solutions[END_REF] : the reduction of crew member by vessels for cost-cutting reasons, the political focus on homeland security more than international waters, the nonexistence of maritime police for all the maritime space, the corruption between some crew members and pirate, the anarchic situation in Somalia and the lack of sovereignty of some states and the pay of increasing sum of ransoms and the proliferation of arms into the world due to the conflicts. All of these factors eased the risk of piracy. This map illustrates (Figure 1) the maritime space use for transportation and the different area of piracy (Somalia Coast, Golf of Guinea, South East Asia and some specific ports in South America). Piracy has an impact on maritime transport: first for lives of crewmember, with an important risk of death and injury, but also on economy. The annual cost of piracy for maritime transport is between 1 and 16 billions of dollars. Attacks on ships carrying energy represent a significant percentage. In 2006, they represent 12 % of piracy attacks and more than 24 % in 2007. [START_REF] Bouejla | A Bayesian Network to Manage Risks of Maritime Piracy against Offshore Oil Fields[END_REF]. According to ICC IMB Piracy report this trend is stable with 25% of piracy attacks on oil tanker in 2014. There are several respond from governmental and industry to piracy which makes decrease the number of attacks, nay eradicate attacks, like in Somalia coast zone between 2010 and 2015.

This decrease is due to counter-measure like recommended transit corridors, transit by groups, escorted vessels, coordinated patrol and on-board security crewmembers [START_REF] Vaněk | Using Multi-Agent Simulation to Improve the Security of Maritime Transit[END_REF]. All these measures make decrease the piracy in the coast of Somalia and reduce the vulnerability of the maritime network of energy. Meanwhile, new risks zone can appear like in the Gulf of Guinea.

Network modelling and disruptions

These risks have an impact by causing disruptions on the network and produce an additionally cost for energy transportation, due to the additional time and distance needed to reach port of destination. Disruptions are defined by a loss of performance for a spatial structure to link different parts of the spatial structure [START_REF] Zavitsas | The Vulnerability of the Petroleum Supply Chain[END_REF]. Loss of performance can be characterized by size, scale and time of the loss of accessibility which increase the vulnerability of the maritime network. Vulnerability can be defined as an impact, as the degree of degradation of the disruption, or as a capacity to respond and reorganize the spatial structure after the disruption [START_REF] Veyret | Aléas et Risques Dans L'analyse Géographique[END_REF]. Several approaches exist to study the vulnerability; all are related to the concept of accessibility which is the capacity of a location to be reached from another one [START_REF] Rodrigue | The Geography of Transport Systems[END_REF]. We aim to measure this cost by the use of spatial reasoning and modelling by an evaluation of piracy risk for energy transportation. Indeed, maritime space is a spatial mediator and its efficiency of is linked to accessibility [START_REF] Gleyze | La Vulnérabilité Structurelle Des Réseaux de Transport Dans Un Contexte de Risques[END_REF]. To measure this accessibility, we tend to use the shortest paths and evaluate the accessibility for vessels depending of maritime barriers (e.g Reefs, mandatory zones, straits) and risks. This paper focuses on spatial modelling of piracy risks and proposes a conceptual approach to evaluate the impact of piracy risks on the maritime network of energy.

To achieve it, we use multi-agent paradigm which highlight the relationship between vessels and maritime space. Spatial constraints have an influence on vessels behaviors, and the sum of these behaviors allows us the measure the impact on maritime network during disruption (by deviation of vessels) and after, by the possible new configuration of network with a changed accessibility level

STATE OF ART

Multi-agent modelling

Agent-based modelling is a method of modelling and simulation where agents are modeled in a shared environment to interact each other's. They are used to find emerging behaviors and structures over simulations through the complexity of relationships existing between agents and environment. An agent is an entity which has its own goals and capacities in a shared environment [START_REF] Ferber | Les Systèmes Multi-Agents: Un Aperçu Général[END_REF]. [START_REF] Crooks | Introduction to Agent-Based Modelling[END_REF] listed the different features of agent. First is autonomy, which means agents are autonomous units, able to take their own decision depending of the environment, the goal and capacity of the agent. Second is heterogeneity which means agents can have different attributes (e.g. age, nationality) which can induce different behavior. The third is active meaning they are an independent influence on the simulation. This makes some advantage for our problematic and the approach is able to capture emergent phenomena, providing a natural environment for the study and is particularity flexible for the relation in geographical model [START_REF] Crooks | Introduction to Agent-Based Modelling[END_REF].

The main go al of agent-based modelling is to model complex systems adopting a bottom-up approach by an individual-centric modelling. Ferber (1995) classifies multi-agent modelling in 3 categories: coexistence, cooperation and competition depending of the goa and capacity of agent as well as the resources of the environment. Coexistence model need a same goal and capacity for agents and sufficient resources of environment. Cooperation model works when agents have same goal but resources or capacity of agents are limited, this category is used for congestion models or collaborative model. The last category, competition model are used in case of conflicted goal (E.g. predator-prey model). These different model are use for specific problems which is unable to solve with a

Multi-agent and space

Multi-agent systems are spatially explicit, agents have a location within the environment, and agentbased systems are particularity adapted for spatial modelling and simulation. For [START_REF] Pumain | Une approche de la complexité en géographie[END_REF] multiagent systems have 3 major assets in geography: the role of actors in the territories of transformations, on the place that take the representations in these processes and on the mechanisms of adaptation and speciation geographic features. Indeed, geography complexity helps to "understand the intervention of intermediate levels (emergence) between individual representations and the formation or maintenance of large social systems" [START_REF] Daudé | Systèmes Multi-Agents Pour La Simulation En Géographie: Vers Une Géographie Artificielle[END_REF]. Thus, the complexity helps to highlight "emerging mesogeographical structures" as well as a dynamic multi-scale, between the individual entities and comprehensiveness. To model in geographical agent-based [START_REF] Langlois | Simulation des systèmes complexes en géographie[END_REF] proposes a paradigm based on Agent/Organization/Behavior to study the relationship existing between social agents (vessels) and spatial agents (maritime space configurations). The main interest of this formalization is the differentiation between social agent and spatial agent. Social agents are used like any agent in agent-based systems and spatial agent are used for the localization of agents. This localization can influence the behavior of agent and agents can influence the spatial properties of the locations. These localization agents are used to inform the location of agent within the shared environment. Three category of spatial structure can be distinguished for localization agents: regular structure [START_REF] Langlois | Simulation des systèmes complexes en géographie[END_REF])(e.g. square, hexagon.), non-topological irregular structure (e.g. points, lines and surface in a continuous space.), and irregular topological structure (e.g networks, Delauney triangulation.).

Agent-based modelling and piracy

Many case of multi-agent modelling exist in geography: in transportation, road and rail have been study since many years with multi-agent modelling, but study on maritime transport is quite limited [START_REF] Vaněk | Using Multi-Agent Simulation to Improve the Security of Maritime Transit[END_REF]). Nevertheless, [START_REF] Marchione | Spatial, Temporal and Spatio-Temporal Patterns of Maritime Piracy[END_REF] demonstrate that at a macro level, incident of piracy is clustered into the maritime space and thus, the importance of space for the safety at sea. [START_REF] Jakob | Agents vs. Pirates: Multi-Agent Simulation and Optimization to Fight Maritime Piracy[END_REF] propose a model to the optimization of fight for the risks of piracy, by a competition agent-based modelling for merchant vessel, piracy vessel and army vessel and test several solutions to piracy attack. Maritime space is model according of two characteristics: bathymetry and weather. They used a continuous space modelling where vessels move depending of an origin/destination matrix. Geography and bathymetry take account of the possi-bilities of move and weather as a factor on speed for vessels.

If the system doesn't take account of the constraints of space, the results of their study are interesting and show the consequences on the counter-measures on piracy attacks. The study concludes that the most accurate answer to attack according to the simulation is to increase the number of navy vessels into restricted corridors. The IRTC (International Recommended Transit Corridor) establish in 2009 demonstrate the efficacy of this measure to decrease the number of attacks.

APPROACH PROPOSED

As seen before, we consider in our work the maritime network of energy as the sum of the trips made by vessels. Multi-agent modelling allows us to understand the vulnerability at a global scale by the behavior of each vessel depending of their objective and capacity. Its objective can be resumed in their environment by accessibility to evaluate the impact of disruptions on the maritime network. In this paper, we propose an approach to measure the impact of area by the accessibility of space. We don't aim to model the piracy ships but we tend to simulate the influence of piracy risk on the maritime network and assess its impact.

Modelling objectives

The first objective of the modeling approach is to define a level of accessibility for each point of the maritime area. Accessibility is a relative concept, this level of accessibility is related to several factors: geographic, economic, but also safety. The second objective is to describe individual behaviors related to accessibility levels of maritime space. These behaviors are described in two stages: first, choosing a travel destination from a starting point in the network. Third, this method aimed to assess the vulnerability in a context of piracy risk, the goal is to simulate space-time disturbances to test the influence of these on the maritime energy network. Finally, the last objective of this method is to evaluate through indicators vulnerability of maritime network through different spatial behavior of ships and function disturbances scenarios in the maritime space.

An agent-based approach

We choice multi-agent modelling and simulation in our work for several reasons: first, we consider the maritime network as the result of the sum of all the paths. Second, we consider multi-agent modelling handle to detect the friction of space caused by disruptions. Third, these disruptions can make emerge new paths and network configuration into the maritime network. We consider the maritime network like a spatial network, with topological property (degree distribution, clustering) but also geometrical properties by taking account of distance between the different nodes of the graph. The model is composed of three different elements in interactions: the ports and offshore structure, the maritime space used for transportation and the vessel which are agent with a spatial behavior. These elements are describe in the following table : Agent which transport a quantity of energy within the maritime space _____________________________________________ 1. Ports/Offshore elements : They correspond to the different locations of the maritime network. Each port has its own demand and/or supply and a gravity model can be used to identify the trips to realize. If the trip is necessary, a vessel is created which a quantity of energy from the supply location to the demand location.

Spatial elements :

The maritime space corresponds to the distance to cross between the origin and destination. The space is divided into cells which have all an accessibility rate. The accessibility level depends of the distance but also to the friction link to the elements of maritime space. Piracy is a friction which induces losses of accessibility.

Vessel elements :

Vessels are agent with the goal to transport an amount of energy to a destination. Vessels have a spatial behavior link to their goal, capacity and environment (maritime space). These spatial behaviors form a trip between ports and thus a link between two locations. The sum of the trips forms the maritime supply chain of energy.

Spatial modelling, distance and accessibility

In our case of study, maritime network isn't restricted to an infrastructure like road and rail network. (Rodrigue,2013). Nevertheless, if maritime space is a mostly free space, several level of accessibility explains the shape of maritime network. To represent the space, we use a squared grid who provide for each cell of space topological and geo-metric information. For the spatial behavior of vessels, we use Euclidian distance into the spatial modelling to find the shortest path according to the accessibility level. The accessibility level corresponds to the relative ease to access a place in space comparing to the neighbors place. This accessibility level depends of several factors:

(1) Ports and platform proximity matrix: Accessibility is measured by the Euclidian distance to the port. A shortest path between two ports is the sum of the accessibility between two locations. (Figure 3) (2) Piracy risk matrix: For each travel of energy, we check the distance of the trips to insecure coast. This makes decrease the accessibility of cells. Each constraint has an impact of accessibility. Elements of space are classified depending of the friction of space they produce. In the case of piracy, the risk increases as it is near the insecure coast (Figure 4). The total accessibility of each cell is the sum of the three matrixes. These matrixes are dynamic, and depend of the supply and demand of locations, evolution of piracy zones and ever realized trips over the maritime. In our case, three matrixes are needed for the accessibility modelling, but more matrixes can be added depending of the case of study (Figure 6). The overall accessibility is equal of the sum of all accessibility and their relative weight and can be describe under the formula pnxn where x represent a matrix, p the weight of the matrix and n the number of matrixes. This accessibility is used by vessels to transport the energy and the main factor of the spatial behavior of vessels.

Spatial behavior modelling

The accessibility matrix is use in a second time to determine the spatial behavior of vessels. A spatial behavior can be defined by a goal (a location to reach) and a capacity to reach this destination (e.g. ports, offshore platform) by means of the accessibility matrix. However, several method of distance can be used for vessels spatial behavior. The first one is topological distance(Figure 7), based on the number of cells a vessel has to cross through the spatial modeling to reach the goal of destination. The agent moves among the eight adjacent squares depending on the lowest topological distance. The topological distance corresponds to the number of moves to reach a place. The topological distance can be used for the most basic spatial behavior but can't account of the constraints of space. To solve it, spatial behavior has to take account of accessibility level. Two types of dis-tance corrected by weigh is possible: absolute and relative. The first one, absolute weighted distance (Figure 8) correspond to the path with the best accessibility overall, taking account of the sum of all the locations crossed within the accessibility matrix. For a vessel, this means to choice the most accessible path between its actual position in the matrix and the destination to reach. Each distance can be used for the vessel to build its spatial behavior. In this table, a comparison of distance form our example is made by several indicators. T represent the topological distance or the number of moves within the matrix, MA the average accessibility of the location of the matrix crossed and OA the sum of the accessibility of the path. The selection of distance for the spatial behavior depends of the vessel capacity and the environment. Indeed, theses distances are specific and measure different possibilities of move. The first one, topological distance measure the spatial shortest path due to the fact we use Euclidien distance in the matrix (This doesn't works in case of Von Numann or Moore distance). The second, weighted absolute distance, is used to taking account the effect of accessi-bility of space on the spatial behavior and measure the temporal shortest path. The last one, weighted relative distance can be use of case of unpredictable case like disruption and measure the temporal shortest path in case of modification of accessibility within maritime space. The figure below (Figure 10) represents the different distance, their relationship and the associate measurement.

Simulation on NetLogo

To create and test our model, we choice NetLogo which is a multi-agent modelling and simulation platform. NetLogo works with Logo language and is a platform use for the exploration of emergent phenomena. We took a simple example to illustrate our problematic. A number of ports and offshore platforms are connected by vessels which draw a transportation network by the number of trips realize by zone (squared cells). At each iteration in the simulation, vessels move forward their goal destination depending of the accessibility for one cell. Accessibility is function of the shortest distance and visited paths and cannot cross piracy impacted zone in orange. Vessels use the spatial and gather quickly into paths forming a maritime network. At 4057 iterations (Figure 12) emerge a new path on the left of the network. This is due to the gathering of the vessels in the center of the network. This gathering induces new spatial behaviors to reach ports on the left and create a new link within the network. This configuration increases the risk of piracy due to the proximity with impacted piracy area. This emerging phenomenon demonstrates the interest in the use of multi-agent systems. This example illustrates the influence of accessibility over the spatial properties of maritime network. We tend to measure the impact of piracy by adding an accessibility level due to the presence of piracy risks.

Evaluation of disruption due to piracy risk

The sum of these spatial behaviors forms the maritime network and especially the maritime supply chain of energy for the oil tanker. These spatial behaviors take account of accessibility level in space. The evaluation of vulnerability can be done by the evolution of shortest paths between the ports of the network. Indeed, vessels moves among the eight adjacent squares depending of accessibility rate to reach their own goal. The increase of shortest paths is an indirect impact of accessibility losses. Several measurements are possible for the measure of impact of disruptions. The first measure of vulnerability is the shortest path evolution within space. Indeed, accessibility can increase the distance between two ports due to the additional distance due to the disruption. The global disruption impact can be measured by the sum of the shortest paths before and after the disruption. The second measure is the shortest path in time: accessibility can also increase the temporal duration for reach a port form another one. The global disrup-tion impact can be measured by the sum of the shortest paths before and after the disruption.

Finally, the last measure is cost. From the losses of temporal and spatial distance, the cost of transportation can be measure (fuel, decrease of energy delivered by the maritime network) and link to accessibility.

FURTHER WORKS AND CONCLUSION

All of these measures would be used for quantify and qualify the vulnerability of the maritime network of energy. We tend to test the proposed model and measure the effect of piracy on the global maritime network of energy. To enhance the model we have first to specify the dynamic of piracy risk, like the presence of mother ship used by pirates to enlarge piracy area. After this improvement, we aim to test our system on real geographical area and test the impact of piracy within several ocean and sea. Indeed, as seen before, piracy zone constantly appear and disappear due to geopolitical and economic factors. Multiagent modelling may help to understand this impact and are particularity adapted to transportation vulnerability study. By the study of impact of disruption on network, we can observe the evolution of the maritime network thought different structures emerging of the evolution of accessibility within maritime space. This issue is crucial for the economy and especially energy. To discover new spatial knowledge on the transportation of energy may bring new highlights on maritime network vulnerability. Nevertheless, our approach tends to be generic and we aim to test several type of scenario of disruptions on the maritime network (E.g. storms, oil spill, and canal closure).

Figure 1 .

 1 Figure 1. UNITAR Map of the Piracy zone and maritime network in Indian Ocean.

  . The figure below (Figure 2) describes our general framework.

Figure 2 .

 2 Figure 2. Impacts of disruptions on the accessibility of maritime network.

Figure 3 .

 3 Figure 3. Accessibility matrix from a specific location

Figure 4 .

 4 Figure 4. Accessibility losses due to the piracy zone

Figure 5 .

 5 Figure 5. Accessibility due to the already visited paths

Figure 6 .

 6 Figure 6. Overall accessibility matrix

Figure 7 .

 7 Figure 7. Topological distance

Figure 8 .

 8 Figure 8. Absolute distance

Figure

  Figure 9. Relative distance

Figure 10 .

 10 Figure 10. Distance and associated shortest paths

Figure 11 .

 11 Figure 11. Random maritime network simulation at 2470 iteration (NetLogo)

Figure 12 .

 12 Figure 12. Random maritime network simulation at 4057 iteration (NetLogo)

Table 1

 1

	. Elements of the model
	Element of the model ______________________________________________ Properties
	Ports/Offshore platform	Location of origins and destina-
		tions of the maritime network with
		demand and/or supply of energy
	Space	Gridded space subdivided by cells
		with an accessibility level
	Vessel

Table 2 .

 2 Comparison of the different distance

	Distance type ______________________________________________ T MA OA
	Topological distance	2	-	-
	Weighted absolute distance	2	2.33	7
	Weighted relative distance _____________________________________________ 3 2.25 9


Unpublished. Revised version of « De la généalogie à la génétique. Enjeux et contradictions de l'identité islandaise », in Morel A., Gil R., Gaussot L., Burger M., Identité(s), Presses de la MSHS, Poitiers, 2004 : 51-61.

IDENTITY THROUGH DNA AND SPIRITUALISM IN ICELAND

Christophe Pons IDEMEC -CNRS -AMU cpons@mmsh.univ-aix.fr « Contemporary Icelanders defined themselves through fidelity to their own early history. While other peoples invent traditions to match a new historical situation, the Icelanders reproduced the celebrated images of another epoch to invent themselves. Far from contributing to their survival, this cultural reproduction may have been directly destructive to their social reproduction. In comparison with the old, 'real' Icelanders, the contemporary people were non-distinct, even invisible. By inventing the Icelanders from past images, the present people was defined out. » Kirsten Hastrup, Nature and Policy in Iceland 1400-1800, 1990, p. 194.

My intention here is to examine two historical events that took place in Iceland about a century apart, both of which provoked a great deal of popular interest. The first is the influence of spiritualism i that arrived on the island at the turn of the nineteenth and twentieth centuries, and that never ceased to increase throughout the development of the independence movement that would free Iceland from the Danish Crown between 1918 and 1944. The second event, closer to us, marks the end of the 1990s, and is still unfolding today. It has to do with the place that research on human genome took in this country, and the craze it generated. Seemingly, these two events do not have much in common. Generally, in the West, spiritualism in both protestant and catholic countries has mostly lost its influence, and the trend is rather to classify it among "beliefs" that have become historically obsolete. On the other hand, genomic research projects us into the future, a synonym of either progress or ethical dangersit dependsbut nevertheless always a manifestation of an evolving science.

However, even though spiritualist theories are now relegated to the most remote fringes of science, it is worth recalling that, at the time, they also unfolded in learned circles; far from rejecting these theories outright, these circles, on the contrary, did their outmost to test their validity in the pure positive tradition of experimentation. The staggering number of experiments conducted by leading universities on both sides of the Atlantic are proof of the expectations that a large number of enlightened people had for spiritualism, as a sign of some new science towards which they must aim. In the end, the present distancing of the scientific universe from these theories results more from abandonment than a concerted invalidation [START_REF] Bensaude-Vincent | Des savants face à l'occulte : 1870-1940[END_REF]. In providing this historical reminder, I do not intend in any way to compare spiritualism and research on the human genome from a scientific point of view, but rather to underline the fact that a nineteenth century Icelander could credit spiritualism with the same level of scientific validity as his grandchild, today, would attach to research on the human genome. Therefore, it is clearly between these popular interests, at a century's interval, that a parallel must be drawn, because the craze for spiritualism in the nineteenth century can help us understand the collective enthusiasm for research on the human genome in the twentieth. Indeed, a similar underlying symbolic thought seems to run through both of these interests that, when the events arose, triggered analogies between specific cultural principles. This "symbolic thought" is certainly not the cause of these historical events, but it is probably at the root of the shape and shade they took. The "cultural principles" in question that, in a more modern language we would call "conceptual operators," are categories of identity and ancestrality. In Iceland, there is a strong homology between the living and the dead, both in the living community's perception of the dead, which is in a way similar to their image of the living community, but also in the personalized relations that each individual maintains with some of the ancestors he carries within. Empirically, this homological rapport is expressed through extremely tenuous relations: not only do Icelanders keep track of their ancestors, they also meet them and send them messages, thereby maintaining a long-standing intimate partnership with them. Within the context of this "Icelandic symbolic thought," identity and ancestrality categories form an inextricably-linked pair that certain "markers" of events in the community of the living cause to powerfully resurface.

Spiritualism and Independence in the Nineteenth Century

No doubt the desire for independence existed in Iceland throughout the seven centuries of colonization between 1262 and 1944. But it is during the second part of the nineteenth century that a real "nationalist awakening" occurred. For the small number of Icelanders at that time, this "awakening" manifested itself less as a fierce battle against a foreign power than as an internal need to recreate a national spirit ii . Within this laborious task of reinventing a distinct identity, the migratory movement of a handful of Icelanders who left for Manitoba, on the banks of Lake Winnipeg, played an important role. They were indeed the first to recreate a free Icelandic community, called the Vesturheimur, the Western World. In 1891, these thousand or so Icelanders were granted a treaty of exception by the Canadian government, which provided them not only with a territory but also with the recognition of their Icelandic nationality and language while allowing them to benefit from the same rights as all Canadian nationals. This community, liberated from Danish custody, would in turn nourish independence in its native country. This will become apparent in particular through a crop of magazines, edited in the Vesturheimur and distributed in Iceland. Grosso modo, this budding independence movement would be structured around two major themes, strangely linked together. First, as can be expected, there was a political reflection regarding the terms and conditions upon which an original constitution should be restored. It should assert Icelandic identity by distancing it from Danish monarchism. From this perspective, inspiration came mainly from the archaic Icelandic Republic created in 930, shortly before its conversion to Christianity. But this political reflection was also based on a passionate interest in spiritualist theories and, in particular, in those focusing on the return of ancestors. It is obvious that in most Reformed countries, including Denmark, this movement was extremely popular at the end of the nineteenth century. Yet it takes on a special dimension in Iceland, because the process of national reconstruction is based not only on the need to establish a republican constitution, but also on a "new religion," one's own that, while remaining Christian, would distinguish itself from colonial Christianity. This spiritualist movement made it possible to conjure up ancestors and rely on them to create this new religion. According to the words of Haraldur Nielsson, theologian and nephew of the bishop of Iceland, the aim was to "reconstruct a rationalist Lutheranism based on positive faith that the spiritualist way will strengthen and ennoble" (Jónsson, 1968: 72-73). Therefore, at the turn of that century, the unofficial histories of religious spiritualism and political independence merged in the livingrooms of Reykjavík where politicians, essayists, poets and clergy met around mediums giving voice to "ancestors". These experiences, still private, would then rapidly enjoy a popular success. Of course, it stands to reason that all of Iceland did not become spiritualist, nor did all the political parties and interest groups become unanimously focused on this dialogue with the dead. On the contrary, as of 1905 the sometimes-violent stands taken against what was called pejoratively the association of ghosts (draugafélagið) demonstrate opposite feelings (Gissurarson & Swatos, 1997: 107-108). But, at the same time, after a local government was recognized in 1904, the dynamic forces of spiritualism made their solemn declaration of national independence by presenting, on the ancient plains of the Parliament of Þingvellir, the very young but oh so popular medium Indriði Indriðason astride a fiery white stallion, mimicking as such the revived image of Óðin, the shaman-god, on his mythical mount Sleipnir. During the same period, Guðmundur Jónsson, another young medium, was giving voice to illustrious ancestors, among whom was the famous Snorri Sturluson (thirteenth century), by drafting, through automatic writing, articles that would then be published under the name of these presumed authors (Guðnarsson & Ásgeirsson, 1996: 73-82). On December 1, 1918, a treaty finally granted autonomy and, eighteen days later, one of the first associations of the autonomous Iceland to be created was the Sálarransóknarfélags Íslands, the spiritualist society of Iceland. Relations between this movement and the Lutheran Church of Iceland would remain ambiguous. Though the Church would not recognize spiritualism, it did not oppose it either and, until recently, several clergymen demonstrated their interest in it, even their affiliation to it.

If this rapid overview is too incomplete to have real historical value, it demonstrates nevertheless the process by which ancestors supported the issue of shaping a national identity. There was, so to speak, a symbolic shift in the political debate that, in stating the identity issue, released that of ancestrality. What is remarkable is really not so much that these ancestors appeared on the occasion of an important event, but that they appeared suddenly in the public arena. The return of the dead coming to "solve" the affairs of the living is not new in the Nordic mentality. Icelanders (and more broadly Scandinavians) did not wait for spiritualist theories before calling on their dead. But these contacts usually took place within family units. This time, the role these ancestors were playing in establishing the new spirit of national consciousness had an original collective dimension. The influence of the spiritualist circle in Iceland declined little by little as of the 1970s, with ancestors returning to the more intimate spaces of their lineage groups. This didn't mean that their public role would not resurface.

Ancestors and DNA in the Twentieth Century

Iceland now enjoys a certain prestige among Western media. Indeed, this small country at the boundaries of Northern Europe is often presented in laudatory terms: one of the oldest republics in the world; now focusing on its social system; demonstrating a very high standard of living; without conflict or unemployment; where one can live to a very old age thanks to the contact with "wilderness". But the enchantment with this Nordic postcard was suddenly shattered in 1998 when the Icelandic government allowed a biomedical research company to undertake the systematic recording of its 280,000 citizens' DNA. The company involved is called deCode Genetics, founded and directed by the Icelander Kári Stefánsson but financed and registered in Delaware, USA. Through this agreement, Iceland became the first nation in the world to identify it genetic pool by recording the individual genetic maps of its entire population. One of the main interesting elements in this event is the profoundly divergent reactions it created, in particular between the Icelandic "passivity" (since, with the exception of a few groups of opponents, public opinion was rather favorable) and the astonishment of foreign media, especially European, which often presented the event in apocalyptic terms. The overall feeling was one of alarm towards Icelandic public opinion. What seemed especially shocking was the fact that this country was part of the European West! In short, there was a non-exotic element that prevented people from transferring this affair to some distant alterity, and this proximity made it that much more shocking. How could Icelanders, who are modern people, morally and ethically agree to reveal and sell their DNA, that sacred and indivisible value of the twentieth century man? This is a summary of the essence of the lack of understanding, which was very clearly encapsulated in these introductory sentences of a French article: "This story is not taking place in Papua, nor the Solomon Island. It is unfolding near us, in Europe, in the northern part of the continent, in Iceland" [START_REF] Piquart | Islande ; ceux qui ont dit non ![END_REF]. As for the Icelanders, their reasons for supporting this agreement were multiple. First, without dwelling on it, one must mention a series of economic reasons. Indeed, deCode soon turned out to be very lucrative by attracting important funds, creating several high-level jobs and providing unforeseen prospects in the field of scientific research. But the CEO Kári Stefánsson also evoked other arguments to which Icelanders were particularly responsive. It is these arguments that made the identity and ancestrality categories resurface, raising the national debate to another level. Kári Stefánsson probably knowingly brandished them, in a premeditated manner we might say, well aware that they would certainly please. But this certainly went beyond his desires: his arguments had the impact of reactants in stimulating the underlying symbolic thought, making public opinion "vibrate" in favor of this project. The genetic venture was then propelled by this symbolic thought that had suddenly been reactivated, to such a point that Kári Stefánsson became a sort of "prophet" in his protestant country --a new emblematic and charismatic figure of Iceland that headlined national news for many months. The arguments involved were twofold: international influence and a necessary exhaustiveness.

First, the international influence, since Iceland was to gain a major advance in the field of very high-level research from all this. deCode was indeed announcing that the precise documentation of the genetic identity of this small nation would bring about huge progress in medical research and create an understanding of the process of hereditary transmission of a great number of diseases, to such a point that an Icelandic opponent to this project noted that Kári Stefánsson's leitmotif was basically to tell his people: "We Icelanders will save the world!" [START_REF] Reverchon | Steindor Erlingsson, réfugié à Manchester, fuit le fichage génétique islandais[END_REF]. However, what can the idea of "saving the world" mean for a small remote country that has only 280,000 inhabitants, and was a colony for seven centuries? It provided it with an unexpected projection to the front and centre of the international scene and, furthermore, a projection into modernity! This concept of modernity is very important in Iceland. Indeed, according to popular perception, the country's economic and social expansion is due to the American presence which, during the Cold War, allowed it to develop along the lines of the Western model. In less than twenty years, without having to experience the intermediate step of the industrial revolution, Iceland moved from a quasi-autarkic society, marked by withdrawal into clan unity, to a tertiary society at the forefront of the richest countries. If this simplified depiction requires qualification, it nevertheless corresponds to the most widely accepted popular understanding of the Icelandic common sense. From this angle, the corollary to this stunning propulsion was a strong ambiguous dual feeling of shame and pride. A feeling of shame because, according to Icelanders themselves, "until the Second World War, we were primitives, living in stone huts, covered with grass, built directly on the ground. We led a miserable life; we were blood relations!" But also a feeling of pride, because this degrading and primitive past is also the mythical and mythicized past of their ancestors, the Vikings, a people recognized and respected by all, whose history they can read in the prestigious literature of the great medieval sagas. This dual feeling was then translated into what could be called an evolutionary complex, that is to say, in the sort of seemingly impossible task of remaining profoundly archaic while aspiring to the greatest possible modernity. This is of course the great challenge of this kind of national identity that rests on the double bind of the evolutionary complexdetermining "how to be archaic and modern at the same time". On the one hand, they cannot build a positive image of themselves without linking it to the notion of ancestrality; on the other, they are haunted by the idea of having been (fewer than forty years ago) at the bottom of the ladder according to nineteenth century evolutionary theories which, even though they no longer prevail in learned circles, are largely dominant in common representations. At another level, it is once again this evolutionary complex that partly explains the Icelandic frenzy for consuming everything that is perceived as providing an external sign of modernity, and especially for recording in comparison to other Western countries. Therefore, we regularly learn about the most recent record-setting through the Icelandic media: the record number of food processors per household; the record number of Internet connections per inhabitant; the record number of subscriptions to cellular phones, and so on.

So, what deCode Genetics was making possible was really a gratifying redefinition of this identity! With deCode, Icelandic archaism became the subject of international interest; in addition, this took place within what is perceived as the high-tech of scientific modernity: research on the human genome. Furthermore, Kári Stefánsson did not cease to boast of the genealogical merits of his nation, the only one in all the Western World to have families capable of retracing their ancestors as far back as thirty-eight generations; of identifying the mythical Vikings as the founding ancestors of their lineage, authors of the medieval literature of which Icelanders are nowadays pathologically proud, but that links them to this notion of archaism and primitive representations among Westerners. By thrusting this archaic past into ultra-modernity, Kári Stefánsson met the dual challenge of the Icelandic evolutionist shame. But this feat that allowed Icelanders to value once again the representation of their identity was not perceived in the same way by foreign media. In the Old World, the idea of touching the human genome conjured up different ghosts, those of genetic manipulation and the modified man. We became sensitive to the dangers of attacks on private property, to the disclosure of personal information, and especially to the manipulation of lineages iii . And, on this last point, the fact that a Nordic nation was involved, wrapped in this common imagery of the tall blue-eyed blond, probably had an impact on the demonization process of the deCode project. Therefore, it is remarkable that Icelanders did not become worried about the arguments surrounding genetic manipulation of lineages and were able to become filled with enthusiasm for this project, without having to also manage ethical and moral issues too difficult to imagine. This is where the second argument mentioned earlier came into play: the necessary exhaustiveness.

Indeed, the deCode project insisted on the importance of the collective character of its venture. But Kári Stefánsson's genius was to declare that by being exhaustive, the project could identify a genetic pool of 650,000 people when there were only 280,000 Icelanders! The aim here is not to discuss the deCode arguments, but to underline the fact that, for the average person, this arithmetic seemed at the least surprising. How could you obtain a genetic pool of 650,000 individuals by adding up the DNA maps of 280,000 individuals? In fact, deCode was proposing a multiple research combining the genetic data of the living, the collection of genealogies recorded since at least 1703, and the medical files of Icelanders who had died in the past twenty years iv . In this country where the practice of genealogy is a national sport, it was possible to meet the challenge. Through genetic-genealogical crosschecking, it became achievable to trace the genetic maps of every Icelander who had died since the first decades of the twentieth century! Such an argument meant therefore that the research on the human genome could trace the presence of 370,000 ancestors concealed among the 280,000 living, which also meant it could perform the arithmetical magic of a pars pro toto, where everything is superior to the sum of its parts. Icelanderswho by the way are very good at countingwere very receptive to this argument which provided a total of 650,000 by adding the 280,000, because it concluded the identity debate with a process of unification of the living and the dead within a unique genealogical continuum.

*

The paralleling of these historical events is supported on two levels.

On the one hand, both events provide evidence regarding the structure of the categories of identity and ancestrality within the Icelandic symbolic thought. We should evidently note that the entrenchment of these categories is not specific to Iceland. It is in fact a classic in ethnography, a number of societies having made us familiar with this semantic shift from the notion of the individual that becomes loaded with ancestral affiliations. To repeat a now famous expression, we are facing a context in which "to be is to participate" [START_REF] Lévy-Bruhl | L'âme primitive[END_REF]. In the first case, it seems obvious that the ancestrality category was set in motion by the identity category: it was the historical and social context of the nineteenth century, revolving around the reconstruction of a national identity that called upon ancestrality by providing spiritualism with a founding role and place. On the other hand, the structure is reversed since the public debate was transformed into a national identity issue because, from the outset, it relied on the archaic and primitive past of Icelandic ancestors.

On the other hand, these events also bear witness to an original but certainly recurrent process throughout the history of Iceland (these two examples serving simply as illustrations) that is separate from the usual model of the reappearance of the dead. As mentioned earlier, ancestors relate first of all to their lineage groups. Even if they frequently make incursions into other families because of the social networks and relations that the living carry out among themselves, ancestors are nonetheless identified as acting individuals. Moreover, when they manifest themselves, it is always for a specific reasonto announce a death or an upcoming birth; to remind us of symbolic transmissions; to request services; etc. But this is in no way the type of rapport at stake in the two historical events that we have just examined. The dead do not intervene in an active way, as identifiable and acting ancestors, but rather as the passive form of a collective property of the Icelandic people. Therefore, this abstract appropriation of ancestors as markers of a common belonging is perhaps really a sign of the ongoing reinvention of the Icelandic collective identity. As Kirsten Hastrup already noted regarding a previous period, « the Icelanders reproduced the celebrated images of another epoch to invent themselves » (1990: 194).

Translation by Patricia Dumas


O Morizot

J De Lapeyre De Bellair

F Wiotte

O Lopez

P.-E Pottie

H Perrin

Agile low phase noise radio-frequency sine wave generator applied to experiments on ultracold atoms

Keywords: numbers: 39.25.+k, 06.30.Ft, 07.57.Hm

We report on the frequency performance of a low cost (∼500 $) radio-frequency sine wave generator, using direct digital synthesis (DDS) and a field-programmable gate array (FPGA). The output frequency of the device may be changed dynamically to any arbitrary value ranging from DC to 10 MHz without any phase slip. Sampling effects are substantially reduced by a high sample rate, up to 1 MHz, and by a large memory length, more than 2 × 10 5 samples. By using a low noise external oscillator to clock the DDS, we demonstrate a phase noise as low as that of the master clock, that is at the level of -113 dB.rad 2 /Hz at 1 Hz from the carrier for an output frequency of 3.75 MHz. The device is successfully used to confine an ultracold atomic cloud of rubidium 87 in a RF-based trap, and there is no extra heating from the RF source.

I. INTRODUCTION

Radio-frequency (RF) fields are used in cold atom experiments for different purposes: for instance, evaporative cooling performed in a magnetic trap relies on RF field coupling between the different atomic magnetic states [1,2]. This technique led to the first observation of Bose-Einstein condensation (BEC) [3,4]. Also, RF pulses are used for dissociating ultracold molecules produced from ultracold gases through Feshbach resonances [5]. More recently, RF fields have been used together with static magnetic fields for trapping utracold atoms at a temperature of a few µK in unusual geometries [6,7]. There is a growing interest for these "RFbased traps" among atomic physicists, for creating double well traps on atom chips [8,[START_REF] Extavour | Proceedings of the International Conference on Atomic Physics, Atomic Physics[END_REF][START_REF] Jo | [END_REF] or proposing new kinds of confining potentials [11,12,13]. In both cases, a single frequency RF signal must be frequency swept over some range, often larger than the initial frequency, following a precise time function lasting several seconds. Typically the RF frequency is varied between 1 MHz and a few tens of MHz in 0.1 s to 10 s in the ramping stage, and held at the final frequency for seconds in the plateau stage. For cooling purposes, commercial RF generators fit physicists' needs reasonably well, even if a better resolution in arbitrary frequency ramps would be appreciated. However, in the case of RF-based trapping, the requirements are stronger. The main difference between these two situations is as follows: in evaporative cooling the cold atomic sample is located away from the region of efficient coupling, whereas in the RF-based trapping scheme the atoms sit exactly at the point where the RF field has the largest effect. The quality of the RF source is then much more important than for evaporative cooling. In fact, the cloud position is directly related to the value * Electronic address: helene.perrin@galilee.univ-paris13.fr of the RF field frequency, and the trap restoring force, or equivalently the oscillation frequency ν t in the harmonic approximation, is linked to the RF amplitude. As a result, any amplitude noise, frequency noise or phase noise of the RF signal during the ramp or the plateau leads to a heating of the cold atomic cloud. This motivated the construction of a synthesizer fitting our requirements.

This paper is organized as follows. In section II we give explicit expressions for the heating of the cold atom sample for frequency and amplitude noise in the case of RF-based trapping. In section III, we describe our RF synthesizer. Finally, section IV is devoted to experimental results on its performance and comparison between the different RF sources tested on the BEC experiment.

II. REQUIREMENTS ON THE RF SOURCE FOR RF-BASED TRAPPING

In this section, we will focus on the RF-dressed trap that we experimentally produce in the laboratory [7]. The extension of the main conclusions to other RFdressed trap geometries is straightforward.

The trap confines the atoms in all three space dimensions. The trapping force arises from the interaction between the linearly polarized RF field B(t) = B RF cos(2πν RF t) and the atoms in the presence of an inhomogeneous magnetic field. This interaction results in a transverse confinement of the atoms to the surface of an ellipsoid. The atoms are free to move along the confining surface, resulting in a kind of "bubble trap" [6]. Due to gravity, however, the atoms are concentrated at the bottom of the ellipsoid. Their motion is pendulumlike in the horizontal directions, and imposed by the RF interaction along the vertical z axis. This last direction is thus the most sensitive to the RF field properties (frequency ν RF , amplitude B RF ) and we will concentrate on the vertical motion in the following. Along this direction, heating or atomic losses may arise from frequency or amplitude noise, phase hops or sudden frequency hops during the RF ramp.

A. Dipolar excitation heating

Very generally, for atoms in a one dimensional harmonic trap with a trapping frequency ν z , any effect producing a jitter in the trap position z results in linear heating through dipolar excitation. The average energy of the cold atomic cloud E increases linearly as [14]:

Ė = 1 4 M ω 4 z S z (ν z ) (1)
where ω z = 2πν z , M is the atomic mass and S z is the one-sided Power Spectral Density (PSD) of the position fluctuations δz, defined as the Fourier transform of the time correlation function [14]

S z (ν) = 4 ∞ 0 dτ cos(2πντ ) δz(t) δz(t + τ ) . ( 2
)
The time variations of energy, E, and temperature, T , are related by Ṫ = Ė/3k B . The factor 3 arises because only one degree of freedom is responsible for the temperature increase, as is the case in our atom trap. The vertical trap position z is linked to the RF frequency ν RF by z = Z(ν RF ) such that S z is directly proportional to S y , the PSD of relative frequency noise of the RF source, through:

S z (ν) = ν RF dZ dν RF 2 S y (ν). (3)
The function Z depends on the geometry of the static magnetic field. In a quadrupolar field, for instance, Z is linear with ν RF and its derivative is simply a constant. From Eqs. ( 1) and (3), we infer that the linear heating rate is proportional to S y (ν z ).

To fix orders of magnitude, within the static magnetic field of our Ioffe-Pritchard trap [7], ν z may be adjusted between 600 and 1500 Hz and the typical temperature of the cold rubidium 87 atoms ranges from 0.5 to 5 µK. To maintain a temperature below condensation threshold for a few seconds, a linear temperature increase below 0.1 µK/s is necessary. This rate corresponds to S z (ν z ) = 0.27 nm/ √ Hz for an intermediate trap frequency of 1000 Hz and ν RF = 3 MHz, which in turn corresponds to a one-sided PSD of relative frequency fluctuations of the RF source S y (ν z ) = -118 dB/Hz.

B. Parametric heating

Fluctuations of the RF field amplitude B RF are responsible for parametric heating in the vertical direction. The trapping frequency ν z is inversely proportional to

√

B RF [6]:

ν z = dZ dν RF -1 2F h M γB RF . (4)
Here, γ is the gyromagnetic ratio of the atom and F is the total atomic spin (F = 2 for rubidium 87 in its upper hyperfine state). The atoms are assumed to be polarized in their extreme m F = F substate. The cloud temperature increases exponentially due to amplitude noise with a rate Γ, where

Γ = π 2 ν 2 z S a (2ν z ) (5)
and S a is the PSD of the relative RF amplitude noise [14].

In order to perform experiments with the BEC within a time scale of a few seconds, Γ should not exceed 10 -2 s -1 . Again, for a typical oscillation frequency of 1000 Hz, this corresponds to S a < -90 dB/Hz. This requirement is rather easy to match and does not limit the choice of the RF source, as -110 dB/Hz is commonly reached. However, particular care must be taken in the choice and installation of the RF amplifier usually used after the source.

C. Phase hops

Controlling the phase of the RF source is not a crucial point for evaporative cooling, but is an issue in the case of RF-based traps, where it is associated with trap losses. In the latter situation, the atomic spin follows an effective magnetic field oscillating at the RF frequency. A phase hop results in a sudden flip of this effective field, the atomic spin being then misaligned with the new direction of the field. Some of the atoms end up with a spin oriented incorrectly and escape the trap.

For this reason, phase hops should be avoided. This is difficult to achieve with an analog synthesizer over a wide frequency sweep. By contrast, Direct Digital Synthesis (DDS) technology is well adapted to this requirement [START_REF] Goldberg | Digital Techniques in Frequency Synthesis[END_REF].

D. Frequency steps

The drawback of DDS technology is that, although the phase is continuous, the frequency is increased by N successive discrete steps δν. A sudden change in the RF frequency also results in atomic losses, through the same mechanism as for phase hops. The effective magnetic field rotates, at most, by the small angle δθ = 2π δν/(γB RF /2) [START_REF]The factor BRF/2 arises from the fact that only half of the power of the linearly polarized RF has an effect on the atoms[END_REF]. For a linear ramp over a frequency range ∆ν = N δν, the fraction of atoms remaining after the full ramp is of order (1 -F δθ 2 /2) N . Given the expression for δθ, this reads:

1 - F 2 4π ∆ν N γB RF 2 N ≃ 1 - F 2N 4π ∆ν γB RF 2 . (6)
Thus, for the remaining fraction to be larger than 95%, the number of frequency steps should be larger than 10F (4π ∆ν/γB RF ) 2 . For example, for a 2 MHz ramp with a typical RF amplitude of 200 mG, N should be larger than 16,000. In addition to this loss effect, a sudden change in the RF frequency results in a sudden shift of the position of the RF-dressed trap. This may cause dipolar heating of the atoms, especially if this frequency change occurs every trap period. The frequency steps should thus be as small as possible, a few tens Hz to a hundred Hz typically.

III. DEVICE DESCRIPTION

Our experiment has the following requirements. First, during the ramp the gap between two successive frequencies must fulfil the criterion discussed in section II D. Second, adiabaticity criteria require a controlled, optimized ramp. Third, the ramp duration should be tunable from one experiment to the other on a time scale ranging from 50 ms to 10 s. Finally, frequency and amplitude noise must be small enough, as discussed in previous section.

Given the amplitude of the frequency sweep we need to perform in our experiment, DDS technology appears to be an ideal solution. We previously used a commercial DDS-based RF generator, the Stanford DS-345. Its memory length is limited to 1,500 frequency points for each waveform with an adjustable step duration of 40 Msample/s/N , with N =1 to 2 34 -1. The major inconvenience of this device is that it is unable to hold the final frequency at the end of the ramp. Instead, the frequency sweep is looped indefinitely. It forced us to sacrifice either frequency resolution during the ramp or duration of the plateau. To benefit from both a low noise RF spectrum during the ramp and a very small frequency step, and to improve the possibilities of the RF source, we designed a digital RF synthesizer with a > 200, 000 memory length and great agility, fitted to our experimental requirements.

The main features of the RF synthesizer are as follows. It is able to generate 262,144 sine waves in a row in the radio frequency band (DC -10 MHz), owing to its 1 M-byte fast asynchronous Static Random Access Memory (SRAM). Each frequency is an integer chosen by the user. A key feature of the device is a variable sample frequency over the sequence, as the duration of each generated frequency can be tuned from 1 µs to 1 hour.

The general architecture of the device is sketched in Fig. 1. It is made up of one evaluation kit DDS board, and a "starter kit" Field Programmable Gate Array (FPGA) board. The device is managed by a Personal Computer (PC). The DDS is clocked by an ultra-stable external reference signal. The output of the DDS is a sine wave, filtered through a 10 MHz low-pass filter. The frequency ramp synthesis starts when a TTL signal is sent to the device. The DDS board combines digital parameters and an analog reference clock frequency to generate a sine wave [START_REF] Goldberg | Digital Techniques in Frequency Synthesis[END_REF]. The heart of the DDS board is a digitally programmable device using DDS technology, the AD9851. It has a 32-bit phase accumulator, a 14-bit digital phase-toamplitude converter and a 10-bit Digital-to-Analog Converter (DAC). Its maximum clock frequency is 180 MHz, and its maximum output frequency is 70 MHz. The phase, relative to the clock signal, is encoded in 5 bits, and is adjustable to any value from 0 to 2π. This results in a rather poor phase resolution of 196 mrad.

The FPGA board manages the 1 Mb memory, the time settings and the input/output of the device through serial port. The FPGA is a Xilinx spartan-3 XC3S200, providing 200, 000 logic gates. These logic gates are designed with VHDL [START_REF]Very High Speed Integration Circuit Hardware Description Language[END_REF]. A Universal Asynchronous Receiver Transmitter(UART) and a Picoblaze microcontroller are loaded into the FPGA, in order to communicate through serial port to the PC and to load the on-board memory. We wrote our own VHDL scripts to manage the DDS board and the FPGA in-board memory. The FPGA board is clocked internally at 50 MHz. The output sample rate of the device depends on the number of clock cycles n cycle between frequency data transferred to the DDS board. We set n cycle = 50 so that n cycle /50 MHz = 1 µs, large enough thus ensuring safe operation of the frequency data.

Software was developed in C with CVI Labwindows in order to configure the device. The user writes a plain text file ordering all the frequencies of the desired frequency ramp. The set of frequencies is separated into 10 groups of adjustable length, with a given sample rate for each. The group lengths and the corresponding sample rate are each translated by the software into 4 bytes. In addition, each frequency in a given group, an integer written as a decimal number, is translated into 4 bytes (32 bits). The software sends these bytes by serial port to the FPGA board.

The clock frequency f c and its phase noise level are the key points for setting the frequency performance of the device. The lower the phase noise of the clock signal, the lower is the minimum phase noise of the output frequency of the device (see section IV A).

The clock signal used for the experiment, see next section, is the 10 MHz clock signal from an ultra-stable Oven Controlled Crystal Oscillator (OCXO) BVA-8600.

Its phase noise PSD is -115 dB.rad 2 /Hz at 1 Hz. As this clock frequency is very close to the desired maximum output frequency (≃ 10 MHz, see next section), and to fulfil Shannon's theorem [START_REF] Nyquist | Proc. IEEE[END_REF][START_REF] Shannon | Proc. Institute of Radio Engineers[END_REF], we generate a higher clock frequency by using the internal frequency clock multiplier, at ×6, of the DDS board.

IV. RESULTS

The device presented in section III was first tested for its frequency stability performance, as described in section IV A and summarized in Table I. It was then integrated into a Bose-Einstein condensation experiment, see section IV B.

A. Device frequency performance 1. Quantization error, Phase Accumulator truncation and "magic" frequencies

By construction, digitization yields to inaccuracies in frequency synthesis. The output frequency of 32-bit resolution DDS is given by

f RF = f c × w 2 32 (7)
where w is a binary 32-bit tuning word. The output frequency can thus differ slightly from the desired frequency.

As f c = 60 MHz, the maximum digitization error δf is 6 × 10 7 /2 32 = 0.014 Hz.

As our software only takes integer frequencies as input, a given frequency f RF will be synthesized without sampling error if it may be written exactly as an integer in the form given at Eq. ( 7). This condition is written as:

w = n × 2 32-p (8)
where n is a positive integer and p is the power of 2 in the prime factorization of the clock frequency f c . In our case, f c = 2 8 × 3 × 5 7 Hz and p = 8 so that every frequency verifying

f RF = n × 234375 Hz (9)
will yield to no digitization error. n should be less than 2 p /3 for the desired frequency to be in the synthesizer range (f c /3).

In addition, when the AD9851 converts the calculated phase to an effective output amplitude, only the first most significant 14 bits are used, even though the AD9851 is a 32-bit synthesizer, in order to handle practicable number of entries in a lookup table. Truncating the phase results in errors in amplitude that are periodic in the time domain. These errors will be seen as spurs in the frequency domain. However, for particular frequencies which are exactly encoded by the first 14 bits (The last 18 bits are 0.), the phase is not truncated at all, yielding no spurious effects and the best PSD of phase noise. This occurs for every frequency satisfying

f RF = n × f c /2 14 (10)
with n a positive integer. As f c = 60 MHz, we have f RF = n × 3662.109375 Hz. The most stringent condition being the first one, we will denote the frequencies satisfying Eq. ( 9) as "magic frequencies". In order to illustrate the difference between a "magic frequency" and another one, we performed a set of noise PSD measurements for two frequencies: a first set for f RF = 3 MHz which is not a "magic" frequency, and a second set for f RF = 3.75 MHz which is a "magic" frequency.

Spectral density of noise

We recorded the spectral density of noise of our synthesizer at a given frequency f RF by FFT analysis of the beat note at 0 Hz with a second synthesized reference signal. The measurement bench is sketched in Fig. 2. In order to generate a tunable reference signal in the RF range we used an analog synthesizer, Rhode & Schwartz SML-01 (R&S), for synthesizing a signal at a high frequency, and then divided by 100 to give f RF for subsequent mixing. The beat note was recorded and analyzed by a digital FFT analyzer HP 3562A sampling at 256 kHz. The R&S, as all the measurement devices, was clocked at 10 MHz by the ultra-stable BVA-8600 quartz oscillator. All the plots shown on Fig. 3 are raw spectra of the beat note. The reference signal (R&S) itself was characterized by making a beat note with a second identical R&S based synthesis. This corresponds to the line labelled as "Reference" in Fig. 3.

By tuning the phase difference φ between the RF signal and the reference signal to π/2, we recorded phase noise. At f RF = 3 MHz, the PSD of phase noise is -78 dB.rad 2 .Hz -1 at 1 Hz, which corresponds to a PSD of relative frequency noise of -207 dB.Hz -1 . From the data reported on Fig. 3, and assuming a Lorentzian lineshape for the beat note, the linewidth δf of the RF signal, given by

δf = πν 2 RF 1 ∆f ∆f df S y (f ) , (11)
is as small as 30 mHz over a bandwidth ∆f = 100 kHz for an output frequency of 3 MHz [START_REF] Halford | Frequency Standards Metrology Seminar[END_REF]. We found similar results for output frequencies from 1 to 5 MHz. At a "magic" frequency, as for example at 3.75 MHz, where truncation effects cancel out, the results are even better, with a PSD of phase noise as low as -113 dB.rad 2 .Hz -1 at 1 Hz, only 2 dB higher than the phase noise of the BVA 8600. The observed value corresponds to the ultimate phase noise of the DDS chip specified by the manufacturer. The relative frequency noise is then -244 dB.Hz -1 .

The frequency noise performance is naturally linked to the quality of the master oscillator. To illustrate this fact, the same measurements were repeated with the improved OCXO of a DS-345, the ERC EROS-750-SBR-4, as master oscillator. No significant change in the frequency performances were noticed at non magic frequencies. At a magic frequency, the PSD of phase noise increased to -100 dB.rad 2 /Hz, which is consistent with the phase noise specifications of this quartz.

The PSD of relative amplitude noise S a was recorded with the same measurement bench, by tuning the phase φ to 0. The reference signal was also delivered by the R&S. The recorded spectrum is very close to the "Reference" line itself at the level of -120 dB.Hz -1 at 1 Hz, close to the input noise of the FFT analyzer. Note that for relevant excitation frequencies ν = 2ν z (larger than 1.2 kHz), the PSD of amplitude noise is lower than -130 dB.

In practice, we use in our BEC experiment a programmable RF attenuator Minicircuit ZAS-3, driven by an analog output channel of a National Instrument PC card PCI-6713, in order to vary the RF amplitude sent to the RF coil. We repeated the measurements described above with this attenuator and found an increase of S a , starting at -115 dB.Hz -1 at 1 Hz, which corresponds to a noise figure of the programmable attenuator of 5.1 dB. At the output of the attenuator, the RF signal is amplified by a class-A amplifier Kalmus 505 F. Its gain is 37 dB and its noise figure is typically +10 dB according to the manufacturer specifications.

B. Integration into the experiment

In this section, we present the main experimental results concerning the heating rate of the atomic sample.

As described in section II, the RF signal is used for producing the bubble-like trap, where ultracold rubidium atoms are accumulated at the bottom of the surface. The trap is very anisotropic with stronger confinement in the vertical direction [7]. For the RF-dressed trap to be efficiently loaded from the standard Ioffe-Pritchard static magnetic trap, described in [START_REF] Colombe | [END_REF], the RF frequency f RF has to be ramped up from 1 MHz to a final fixed frequency f end RF ranging from 2 to 10 MHz. The static magnetic field, necessary both for magnetic trapping and RF-induced trapping, is always present. A typical ramp is shown on Fig. 4. The frequency is ramped more slowly around 1.3 MHz, corresponding to the resonant frequency at the center of the magnetic trap where adiabaticity of spin rotation is more difficult to obtain. At the end of this ramp, which may last between 75 ms and 500 ms, the RF frequency is held between 0.1 and 10 s for testing the lifetime and heating rate of the atoms in the RF-based trap.

The RF signal is amplified by 37 dB with a single stage amplifier, and the RF field is produced by a small circular antenna. The RF field is linearly polarized and its amplitude B RF may be adjusted between 70 and 700 mG. We FIG. 4: Typical shape of a radio-frequency ramp applied to the ultracold atomic sample. In the present example fRF is increased from 1 to 3 MHz within 150 ms. At the end of this ramp, the RF frequency is maintained at its final value for some holding time in the RF-based trap, dashed line.

record the atomic temperature after the RF ramp as a function of time while the atoms are confined in the RFbased trap. The temperature is deduced from the cloud size along z after a 7 ms of ballistic expansion. The same measurement was repeated with different RF sources. First, we used a Agilent 33250A synthesizer with a RF amplitude larger than 500 mG for both the frequency ramp and the final holding frequency. Such RF analog synthesizers operated at fixed frequency exhibit very good relative frequency noise in most cases, typically at the -180 dB/Hz level or better. However, as mentioned by Colombe et al. [7] and confirmed by White et al. [21], the relative frequency noise increases by a few decades if the output frequency is driven with an external analog voltage. The frequency was indeed tuned through an external voltage control provided by the PC analog board (NI 6713), such that the modulation depth was ±1 MHz on a central frequency of 2 MHz. We obtained both a short lifetime, typically 400 ms at 1/e, and a strong linear heating, as shown on Fig. 5 full circles. The heating rate is measured to be 5.0 µK/s. This rate, given the RF amplitude, corresponds to a relative frequency noise of S y = -100 dB/Hz at the trap frequency of 600 Hz. This noise is quite high and is related to voltage noise on the external frequency control of the synthesizer. This effect is strong in our case as the frequency is varied with a large modulation depth (∆f /f = 1).

We also tested a two step scheme, with a first ramp performed by a Stanford DS-345 DDS (1,500 frequency points), followed by the R&S maintained at fixed frequency for the full holding time. This scheme allows one to benefit from the excellent frequency stability of the second device used at fixed frequency. With this setup, We observe a heating rate of 5.0 µK/s in the first case and 0.47 µK/s the second one, as given by a linear fit, full lines.

heating during the plateau stage seemed to be completely suppressed [7]. However, a large dispersion was observed in the atom number data after the switching, which prevented us from characterizing the heating very precisely. This dispersion is due to a random phase hop at the switching time between the two synthesizers, resulting in atomic losses. We studied the effect of the random phase hop on the atom number by recording, for each experiment, the phase difference at the switching time with a control oscilloscope. The results are presented on Fig. 6, full circles. For the maximum phase hop, π, 80% of the atoms are lost. This figure depends on the atomic temperature, the losses being higher at lower temperature, and is well reproduced by theory, as shown on Fig. 6, black line. The theoretical curve is calculated for an RF amplitude of 470 mG by averaging the loss probability over the positions of the atoms, as deduced from a thermal distribution at a temperature of 4 µK. The fact that the trap is able to hold two of the five spin components of the F = 2 hyperfine state is taken into account.

Finally, we performed the heating rate and lifetime measurements with our DDS device and an amplitude of 70 mG. We found an increased lifetime, up to 9.6 s. During the first second, the vertical "kinetic temperature", deduced from the vertical cloud size after time of flight, decreases due to thermalization with the horizontal degrees of freedom which initially have a lower energy due to the loading procedure: Fig. 5, open diamonds. After 1 second, a linear heating rate of 0.47 µK.s -1 is clearly observable. No exponential parametric heating is measurable. We believe that the remaining linear heating is due to residual excitation by scattered light, which was not present in the previous case. The same heating rate was indeed observed within the static magnetic trap, when the RF source was off. Heating directly linked to the RF source is not observable.

V. CONCLUSION

We built a low cost DDS RF generator, based on a DDS chip and its evaluation kit. It is extremely agile since we have the ability of using up to 10 different time steps during the sequence, allowing an efficient use of the FPGA on-board memory. In contrast to Stanford DS-345 RF synthesizers, the frequency ramp is not repeated at the end of the sequence. The last frequency is kept until a new TTL pulse is proceeded to end the sequence. Its large memory length, up to 262,144 frequency samples, allows very small frequency steps, thus strongly reducing the atomic losses in our RF-dressed trap experiment. The DDS technology ensures that the phase is continuous all over the frequency ramp, which is essential for RF trapping applications.

Apart from these high resolution features, the device presents excellent spectral properties. The power spectral density of phase noise is as low as -113 dB.rad 2 .Hz -1 at well chosen "magic frequencies", and in any case remains below than -78 dB.rad 2 .Hz -1 at worst, dominated by phase truncation. This last value yields a linewidth of 30 mHz over a 100 kHz bandwidth. The corresponding relative frequency noise level is 90 dB below the requirements of our RF-dressed trapping experiment.

The amplitude noise of the DDS synthesizer is measured to be below -130 dB.Hz -1 at frequencies above 1 kHz. Amplitude noise turns out to be the limiting noise present in our RF chain at the present time due to the several dB noise figures of the attenuators and amplifiers in use in our experiment. They actually increase the RF amplitude noise to -115 dB.Hz -1 at 1 kHz, which still remains 25 dB below our experiment requirements.

All these features make our DDS apparatus very wellsuited for cold atom experiments. It may be used for optimized evaporative cooling implementation, with arbitrary frequency ramp profiles including a final "RFknife", in a standard BEC experiment. It is of particular importance in the case of atom chip experiments [22], where the cooling ramp is usually done very quickly (1 s typically). It is also ideal for developing new RF-based traps. In particular, the possibility of configuring the time step, that is the duration of each frequency point, is very attractive for our application. This feature is all the more interesting in that the memory can be separated in several zones, each being allocated a chosen number of points and a chosen time step, so we can dilate or compress in time the whole or a part of the ramp, while keeping the same frequency series. For instance, if we want to accomplish a ramp of N frequency values with a small time resolution, say 10 µs, and then keep the final frequency constant over ten seconds, we have the ability to address a first memory zone with the first N -1 points and a 10 µs time step, and a second memory zone with the last frequency value and a 10 s time step. A single time step over the whole sequence would have forced us to sacrifice either the resolution of the ramp or the duration of the final plateau.

The device was successfully implemented in our ultracold atom experiment. First, we set boundary specifications on the RF source performances for trapping experiments regarding heating rate and lifetime. In the same manner, we set conditions on maximum phase hops and frequency step magnitude. With our home made device, we observed no heating due to the RF source during the plateau step. A remaining heating rate of 0.47µK/s was identical to the one obtained in the magnetic trap and was limited by other noise sources, presumably scattered photons. Despite the very large number of frequency points, heating along the vertical axis is still present during the loading stage of the RF-dressed trap. We attribute this heating to non-adiabatic deformation of the confining potential during the loading process, which we could not make slower due to the other noise sources.

While the actual performances of the device are already very good, a few improvements are still possible. First, the memory length of the DDS is basically determined by the FPGA in-board memory size. A larger sample size may easily be implemented by replacing the board, provided that the data transfer rate is improved through the use of an Ethernet port or a USB port. Second, the sample rate is 1 MHz, limited by the data transfer rate from the FPGA board. Higher FPGA clock frequency, or smaller clock counts n cycle , would allow higher sample rates. The frequency limit due to the AD-9851 DDS chip itself is 3 MHz. This modification would improve the temporal resolution by a factor of 3. Finally, the maximum output frequency of 10 MHz may be very easily increased to 20 MHz by replacing the low-pass filter. By clocking the DDS with a 180 MHz clock, one could even reach output frequencies up to 60 MHz. Note that with a similar DDS chip, the AD 9858, frequency clocks as high as 1 GHz are allowed, which makes possible the synthesis of RF frequencies up to 330 MHz.

FIG. 1 :

 1 FIG.1: Layout of the system.

3 FIG. 2 :

 32 FIG.2: Noise spectrum measurement bench. The gain in the mixer was fitted to the output signal, depending on the phase between the two beating signals.

1 )FIG. 3 :

 13 FIG.3: PSD of relative amplitude noise, upper plot, and phase noise,lower plot. Averaging = 100. FFT's sampling frequency = 256 kHz. The low pass filter cutting frequency of the mixer was set to 200 kHz. The reference curve corresponds to the beat note of two identical R&S synthesizers. Plotted for comparison is the phase noise specifications of the BVA 8600 quartz oscillator, as given by the manufacturer.

FIG. 5 :

 5 FIG.5: Comparison of heating of the atomic cloud in the bubble trap: Agilent 33250A synthesizer, full circles, or present device, open diamonds, is used for creating the RF ramp and the final radio-frequency f end RF . We observe a heating rate of 5.0 µK/s in the first case and 0.47 µK/s the second one, as given by a linear fit, full lines.

FIG. 6 :

 6 FIG.6: Number of atoms remaining after switching between the two synthesizers as a function of the phase hop. Experimental data, full circles, are compared to a calculation, full line, assuming a RF amplitude of 470 mG and a temperature of 4 µK.

TABLE I :

 I Performance of the device with fc = 60 MHz. The relative frequency noise is computed from the phase noise data, and given for 3.75 MHz ("magic" frequency) and 3 MHz (larger noise value).

	Parameters	Min.	Max.	Units
	Dynamic	0	10	MHz
	Line-width	-	30	mHz
	Digitalization error	0	14	mHz
	Sample rate	adjustable	1	MHz
	Memory length	1	262,144	pts
	Phase noise@1Hz	-113	-78	dB.rad 2 /Hz
	Rel. freq. noise@1Hz	-244	-207	dB/Hz
	Rel. ampl. noise@1Hz	-	-120	dB/Hz

Acknowledgments

We are indebted to R. J. Butcher for a critical reading of the manuscript. This work was supported by the Région Ile-de-France (contract number E1213) and by the European Community through the Research Training Network "FASTNet" under contract No. HPRN-CT-2002-00304 and Marie Curie Training Network "Atom Chips" under contract No. MRTN-CT-2003-505032. Laboratoire de physique des lasers is UMR 7538 of CNRS and Paris 13 University. The LPL is a member of the Institut Francilien de Recherche sur les Atomes Froids.