Frederic Martinez

Valérie Le Floch

Bernard Gaffié

Gaëlle Villejoubert

Keywords: gambling, illusion of control, risk taking, social reference

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

Reports of Wins and Risk Taking 2

Reports of Wins and Risk Taking: An Investigation of the Mediating Effect of the Illusion of Control Imagine that two gamblers, Alice and Kelly, are playing on video lottery machines.

Alice is playing on her own in a public house whereas Kelly is in a busy casino. Both have just lost their last Euro. Kelly then witnesses another gambler win with the usual eye-catching sight and sound display. Who will be more likely to gamble again, Alice or Kelly? To answer this question from a traditional perspective on gambling behavior, one would seek to ascertain who among the two protagonists has the longest history of gambling, what social norms govern public house gambling behavior, or what are their individual differences (e.g., family history, intelligence, extraversion, substance abuse, etc.). In the present study, however, we address this issue from the perspective of social cognition. As suggested by [START_REF] Rockloff | An experiment on the social facilitation of gambling behavior[END_REF], co-actors or an audience may facilitate performance for a variety of behaviors. Building upon this observation, this study aims to gain a better understanding of the effect of knowing that someone else has won on one's own gambling behavior.

Stories about gambling wins are widely publicized by the media yet losses are much more frequent than wins [START_REF] Hill | Choose six numbers, any numbers[END_REF][START_REF] Mcmullan | What makes gambling news[END_REF].

These practices may have important consequences on gambling behavior. For instance, knowing that someone else has won has been shown to increase individuals' risk taking (Kearnay & Drabman, 1992;[START_REF] Floch | L'annonce du résultat d'autrui : un second point de référence[END_REF]. However, little is known about the mechanisms that underpin the effect of reports of wins on gambling behavior. There is evidence showing that the perception of other gamblers' characteristics can influence an individual's illusion of control [START_REF] Dykstra | Model competence, depression, and the illusion of control[END_REF][START_REF] Langer | The illusion of control[END_REF] and this illusion has been suggested to be a key factor in increasing risk taking (e.g., [START_REF] Walker | The psychology of gambling[END_REF]. This suggests a causal model within which the illusion Reports of Wins and Risk Taking 3 of control could be a mediator of the effect of the knowledge of another person's win on one's risk-taking behavior (see Figure 1). In the next section, we critically review evidence from the literature supporting this model. We then present two experiments that provide a direct empirical test of this model. In the first experiment, we sought to discover if reports of wins increased beliefs in the winner's ability to control the outcome of the gamble. In the second experiment we sought to test whether the suppression of the belief that the previous winner had exerted control over the gamble's outcome could cancel the effect of reports of wins on risk taking.

Linking Reports of Wins, Risk taking and the Illusion of Control

Why and how reports of wins increase gambling behaviors and risk taking?

Studies anchored in social learning theory [START_REF] Bandura | Social learning theory[END_REF] have shown that peer modeling frequently induced risk-taking behavior in children and adolescents [START_REF] Browne | Predictors of lottery gambling among American college students[END_REF][START_REF] Hardoon | Social influences in children's gambling behavior[END_REF]. For example, preschool children were shown to be more likely to engage in risk-taking behavior in a game of pure chance after seeing a peer having just ostensibly won this game [START_REF] Kearney | Risk Taking/Gambling-like behavior in preschool children[END_REF].With adults, witnessing the win seems unnecessary to increase risk taking. [START_REF] Mushquash | An examination of the cue-reactivity of Gambling wins[END_REF] indeed reported that about half of 182 respondents from the general public felt the urge to gamble after learning that someone else had won. A recent study [START_REF] Rockloff | An experiment on the social facilitation of gambling behavior[END_REF] investigated the existence of a social facilitation effect [START_REF] Zajonc | Social Facilitation[END_REF] for gambling behavior on electronic gaming machines. Some participants received false feedback from the computer designed to suggest that other gamers in adjacent rooms were playing and sometimes winning at the same game. This experiment demonstrated that the implied presence and wins of other gamblers intensified gambling, measured by greater persistence and lower final payouts.

Reports of Wins and Risk Taking 4

Martinez, [START_REF] Martinez | Lien entre perception de contrôle et prise de risque dans un jeu de hasard: Quand l'annonce d'un gain d'autrui intervient[END_REF] directly tested whether reporting another person's win would affect subsequent gambling behavior under laboratory conditions. This study did not test the effects of the presence of co-actors on gambling behavior in real time. However, it suggested that information about the performance of others is likely to intensify gambling behavior as measured by risk taking. In this study, participants individually played a computer-simulated "French roulette game"1 and were falsely told that a previous participant had a large or a small win during their play.

Results indicated that the knowledge alone of another person's large win was enough to increase risk taking. In the following section, we begin by reviewing evidence suggesting that reports of others' wins may indeed lead to an increase in illusion of control.

Why and how reports of wins lead to an increase in illusion of control? [START_REF] Langer | The illusion of control[END_REF] defined the illusion of control as the "expectancy of a personal success probability inappropriately higher than the objective probability would warrant." (p.

313

). According to Langer, the more skill-related factors (choice, stimulus familiarity, active involvement, and competition) are present in a game of chance, the more an illusion of control is likely to occur. In her first study, [START_REF] Langer | The illusion of control[END_REF] examined the effect of competition in a game of pure chance where the player selecting the highest card would win the betting round. She demonstrated that participants bet significantly more money when playing against an incompetent competitor than when playing against a competent one. Langer interpreted these results as showing that, in direct competition, participants disregard the uncontrollable and random nature of chance.

Instead they misconceive the game of pure chance as a skilled task where their probability of winning depend on their level of competence relative to that of other players and where they can apply strategies to increase their wins. Direct competition, however, is not necessary to generate such an illusion of control. [START_REF] Dykstra | Model competence, depression, and the illusion of control[END_REF] did not use a directly competitive task: they merely showed to their participants a video of another player that was either presented as a competent gambler or as an incompetent one. Results showed that participants who had watched an incompetent-looking model gave higher estimates of their own success at the game. These results suggest that participants evaluate how well they could perform in the chance task based on their perception of the other gambler's competence as if it were a skilled task. [START_REF] Martinez | Lien entre perception de contrôle et prise de risque dans un jeu de hasard: Quand l'annonce d'un gain d'autrui intervient[END_REF] reported that the knowledge of another person's win significantly increased the time taken by participants to select bets. General decisions result from a compromise between the desire to make a correct choice and the desire to minimize effort [START_REF] Lipman | How to Decide How to Decide How to[END_REF]; see also [START_REF] Payne | The adaptative decision maker[END_REF].

Additionally, the amount of effort spent on a task is regulated as a function of the expected effects of the actions undertaken [START_REF] Bandura | Social foundations of thought and action: A social cognitive[END_REF]). Thus, one may expect that individuals will only "invest" in making the effort of thinking when they perceive the task to be controllable. In fact research has already established that individuals who believe they have little control over subsequent outcomes do not persevere as much as those who believe they can exert control over these outcomes [START_REF] Testa | The impact of social comparisons after failure: The moderating effects of perceived control[END_REF].

Thus the longer time taken to place bets under the knowledge of another gambler's win in Martinez et al.'s (2005) study could be interpreted as additional, albeit indirect, evidence for the hypothesis that knowledge of another gambler's win entails an increase in illusion of control.

Reports of Wins and Risk Taking 6

On the basis of these results, we expect that knowing how much another gambler has won will be interpreted as a skill-related cue. Specifically, reports of wins should increase the belief that gambling is a skilled task and thereby induce an illusion of control, which in turn would lead to an unrealistic anticipation of personal success. To validate our proposed causal model (see Figure 1), the link between illusion of control and risk taking should also be established. We review evidence for such a link in the following section. Do we take more risks when we believe we can control the outcome of a situation?

The illusion of control has often been suggested to be a key factor in increasing risk taking [START_REF] Dixon | Manipulating the illusion of control: Variations in gambling as a function of perceived control over chances outcomes[END_REF][START_REF] Griffiths | The role of cognitive bias and skill in fruit machine gambling[END_REF][START_REF] Ladouceur | A cognitive perspective on gambling[END_REF][START_REF] Langer | The illusion of control[END_REF][START_REF] Walker | The psychology of gambling[END_REF]. Although illusory control does appear to encourage risk-taking in other domains [START_REF] Horswill | The effect of perceived control on risk taking[END_REF], the link between taking risks and overlooking the uncontrollable and random nature of chance has not been firmly established in gambling games. Delfabbro and Winefield (2000) found that players who staked more money also verbalized more thoughts, which in turn would suggest they erroneously assessed their ability to control the game. However, the authors themselves acknowledged that participants' spontaneous verbalizations may represent a post facto justification for their risk-taking behavior rather than a causal determinant of such behavior. Similarly, [START_REF] Caron | Erroneous verbalizations and risk taking at video lotteries[END_REF] showed that participants exposed to an accomplice's deceptive verbalizations (e.g., "now, I played really badly!") took more risks than participants who heard adequate verbalizations (e.g., "it's too bad that we don't control chance!") or no verbalization at all. The authors, however, did not measure the effect of the accomplice's verbalizations on participants' own beliefs.

Reports of Wins and Risk Taking 7

Therefore, a direct causal link between erroneous beliefs on controllability and risktaking behaviors remains to be empirically established.

The Present Study

The present study aimed to test a causal model where the illusion of control is a mediator of the effect of reports of others' wins on one's risk-taking behavior (see Figure 1). In order to test this model, we first manipulated whether or not gamblers knew about another person's win and assessed the effect of this manipulation on gamblers' illusion of control and risk-taking behavior. In addition, to test for the hypothesis that the illusion of control was a mediator of the causal relation between knowledge of another's win and risk taking, we also directly assessed whether the degree of illusory control affected risk taking. Thus, building on Martinez et al.'s (2005) findings, we expected that knowing how much a previous player had won in a gamble would increase risk taking. Specifically, we anticipated that the knowledge of another person's win would strengthen the belief that Roulette is a game of skill, which requires strategic, effortful thinking. Consequently, knowledge of another person's win was expected to result in increased gains expectancy and increased betting times. Finally, we hypothesized that an increase in the illusion of control would also lead to an increase in risk taking. We report two experiments. The first experiment examined whether the illusion of control is a mediator of the effect of knowing another gambler's win on risk-taking. The second experiment examined whether discrediting the belief that the other gambler may have won by exerting control over the gamble's outcome is sufficient to eliminate the effect of knowing about this gambler's wins.

Reports of Wins and Risk Taking 8

Experiment 1

Method

Participants

An opportunity sample of 24 female students was selected in the Psychology building of the University of Toulouse. Ages ranged from 18 to 27 years. None of the participants in either this or the following experiment had any history of gambling, nor were they familiar with French roulette.

Design

Participants were randomly assigned to one of two groups (equal number per cell).In the experimental group, participants were informed of the amount won by the previous gambler. In the control group, this information was omitted. Each participant took part in 15 game trials2 .

The illusion of control, the hypothesized mediating psychological construct (see Figure 1), was measured with three dependent variables. From a statistical point of view, mediators are defined as variables that "transmit the effect of one variable to another variable" (MacKinnon, Fairchild, & Fritz, 2007, p. 594) whereby a cause only indirectly affect an outcome through a mediator. The construct validity of the variable conceptualized as the statistical mediator is an important concern in testing mediation.

To remove the biasing effect of measurement error, [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations[END_REF] advised that multiple independent and converging measurements should be collected to tap a latent variable. Therefore, we used two classical subjective measurements of illusion of control [START_REF] Langer | The illusion of control[END_REF]: the belief that gambling is a skilled task (assessed with selfreport scales) and their gain expectancy (measured as the expected number of chips at the end of the gambling session). Additionally, akin to [START_REF] Martinez | Lien entre perception de contrôle et prise de risque dans un jeu de hasard: Quand l'annonce d'un gain d'autrui intervient[END_REF], we measured the time taken to place bets as a potential additional measurement of illusion of control.

Finally, risk taking, the outcome variable, was measured using the RT index [START_REF] Martinez | Lien entre perception de contrôle et prise de risque dans un jeu de hasard: Quand l'annonce d'un gain d'autrui intervient[END_REF]; see Appendix 2) adapted from [START_REF] Ladouceur | Caractéritiques psychologiques de la prise de risque monétaire des joueurs et des non-joueurs à la roulette[END_REF] which takes into account the number of chips placed on a bet as well as the probability of losing one's stakes, and the current resources of the gambler at the time of betting.

Materials and Procedure

Participants were invited to play a computer-simulated French roulette game individually for 45 minutes. In fact, the experiment stopped at the 15 th round and lasted around 25 minutes. This was to ensure that participants would not suspect that the experiment was to stop when it did and as a result would not be tempted to place all their chips on the last bet. The experimenter explained the rules of the French roulette to the participants (See Appendix 1).

The different types of bets possible, their placement on the layout table (reproduced in Figure 2), the probability of a win and their potential payouts (summarized in Table 1) were then presented to the participants. A booklet summarizing this information was available throughout the course of the experiment.

Following this presentation, the experimenter explained the functioning of the software.

Participants were given a credit of a 100 chips. At any point in the game, they were able to see the number of chips they had left on the computer screen. They were warned that the program would stop automatically if they lost all their chips. For each game trial, the software recorded the number chips placed on any given bet, the probability of losing this bet, and the current resources of the gambler at the time of placing the bet(s). These measures were later used to calculate the risk taking index.

Reports of Wins and Risk Taking 10

In addition, participants in the experimental condition were given the following information before the start of the game (translated from the French):

"At the end of the game, I will write down the number of chips you have left on the sheet under the keyboard. As you can see on this sheet, Dominique Fabre3 who played before you has won 750 chips." To avoid raising the suspicions of participants, the experimenter then drew their attention to the disorderly state of the room and said: "Why is there a sheet of paper under the keyboard? I'll remove it later". The reminder of another person's win thus remained visible throughout the duration of the experiment. In the control group, the experimenter indicated a blank sheet and simply said: "At the end of the game, I will write down the number of chips you have left on the sheet under the keyboard."

Participants completed their familiarization with the task with five practice trials4 .

Before the game session began, participants were asked to report how many chips they expected to have at the end of the game (expectancy measure) as well as to rate their agreement with two statements ("There are strategies that increase the probability of winning"; "The more I think about strategies to place my bets, the more I increase my probability of success") on a 7-point scale from "totally disagree" to "totally agree" (subjective illusion of control measure). In addition, the time they took to place a bet was recorded on each betting trial.

Results and Discussion

Does knowing about another gambler's win affects risk taking and illusion of control?

Participants' risk taking (RT) on each given trial was calculated by taking into account the number of chips placed on any given bet, the objective probability of winning with such a bet as well as the total number of chips owned by the gambler at the time of placing the bet(s) (see Appendix 2 for the exact formula). The resulting score was greater than 0 and less than 1 where a higher score was indicative of greater risk taking. Those RT scores were averaged across the fifteen trials to provide one risktaking index per participant. Overall, participants' RT index was higher when they knew the previous gambler had won 750 chips; M = .12 (SD = .08) vs. M = .03 (SD = .01) in the control group. A one-way analysis of variance (ANOVA) confirmed this difference was statistically significant; F (1, 22) = 13.62, p < .001 partial η² = .38.

Ratings for the two statements about personal control were significantly and highly correlated; r(24) = .84, p <.001; and were averaged to form an index of subjective personal control. Participants' index of personal control was again higher when they knew the previous gambler had won 750 chips; M = 5.92 (SD = .79) vs. M = 3.17 (SD = 1.47) in the control group. This difference was reliable; F (1, 22) = 32.64, p < .001, partialη² = .59. Finally, learning about someone else's win also resulted in

Reports of Wins and Risk Taking 12

Does illusion of control mediates the effect of knowing about another gambler's win on risk taking?

Although the results presented above are consistent with our hypotheses, they do not, in themselves, attest to the mediating effect of the illusion of control. To test for our causal model, we followed [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations[END_REF] regression-based procedure to test for mediation effects. In line with Martinez et al.'s (2005) findings, the longer the betting times the greater the score on self-report measures of the illusion of control, r (24) = .72, p <.001, and the greater the number of chips the participants expected to retain at the end of the game; r (24) = .70, p <.001. We therefore standardized these three variables and computed an average score, which provided an estimate of the latent illusion of control construct for each participant.

Regression analyses showed that knowledge of another person's win was a significant predictor of risk-taking index, β = .62, t(24) = 3.69, p < .01, and a significant predictor of illusion of control, β = .82, t(24) = 6.74, p < .001. However, when the risktaking index was regressed on both knowledge of another person's win and illusion of control, the effect of knowledge of another's win became non-significant, β = .15, t(24) = .54, p = .59, while the illusion of control remained a significant predictor of risktaking, β = .57, t(24) = 2.11, p < .05 (adjusted R 2 = .44 for the full model). A Sobel test confirmed that the effect of this mediation was significant (z = 1.98, p < .05).

Altogether, these results, summarized in Figure 3, establish what Baron and Kenny refer to as a "perfect mediation".

The purpose of this first experiment was to test a causal model of risk-taking behavior in gambling. The present results confirmed that the effect on risk taking of knowing about previous players' wins was mediated by players' increased illusion of control. As such, these results suggest that, in the absence of any other information, the Reports of Wins and Risk Taking 13 previous gambler's win was misinterpreted as a sign that this previous player was skilful, rather than merely lucky. Previous research has established that the perceived level of competence of other gamblers may affect one's own illusion of control (e.g., [START_REF] Dykstra | Model competence, depression, and the illusion of control[END_REF]. It is thus plausible to assume that the misattribution of the previous gambler's win to their skill amplified participants' illusion of control, which, in turn, resulted in an increase in risk taking. Experiment 2 was designed to provide an empirical test for this explanation.

Experiment 2

The second experiment aimed to further examine what motivated participants' increase in illusory control upon learning the (successful) outcome of a previous player.

We hypothesized that this resulted from the misattribution of the previous' player outcome to their skilfulness. If this explanation is correct, the extent of gamblers' illusion of control and as well as the extent to which they place risky bets should be positively correlated with the degree of personal control they attribute to the previous gambler. Furthermore, players should no longer be influenced by the knowledge of the previous gambler's win if they are made aware that this win was fortuitous. Such information should prevent the misinterpretation of the previous gambler's win as evidence of their skillfulness. Thus, we anticipated that knowing that a previous player had fortuitously won should have no effect on participants' perceived personal control and hence, no longer affect their risk taking. We tested these predictions in Experiment 2.

Reports of Wins and Risk Taking 14

Method

Participants

An opportunity sample of 36 female students was selected in the Psychology building of the University of Toulouse. Ages ranged from 19 to 26 years. None of these students had been participants in Experiment 1.

Design

Participants were randomly assigned to one of three groups. The first experimental condition (hereafter the "win" condition) and the control condition were identical to the conditions used in Experiment 1. In a second experimental condition, (hereafter the "fortuitous win" condition), participants also learnt that a previous gambler had won a large amount; however, they were also informed that this win had been acknowledged as fortuitous. The dependent measures were identical to those used in Experiment 1. In addition, at the end of the experiment, all participants were asked to rate the extent to which the previous gambler controlled the situation on a scale ranging from 0 (not at all) to 20 (completely).

Materials and Procedure

The materials and procedure were identical to those used in Experiment 1. Participants in the "win" and in the "fortuitous win" condition learnt that the previous player had won 750 chips. In the fortuitous win condition, however, the experimenter added: "but she told me that she did not control the situation. She played randomly and unexpectedly won 750 chips".

Results and Discussion

As expected, gamblers' illusion of control and risk taking was related to the degree of control they attributed to the previous winner. A significant positive correlation was observed between the degree of control attributed to the previous Reports of Wins and Risk Taking 15 winner and participants' subjective estimates of personal control, r(24) = .72, p <.001, their gain expectancy, r(24) = .46 p < .05, and their betting times, r(24) = .43, p < .05.

Attributed control also was also strongly correlated with participants' own risk taking index, r(24) = .59, p < .001.

As predicted, risk-taking indices were affected by the way in which the wins of the previous player were specified; F (2, 33) = 10.5, p < .001, η² = .39. Participants who were simply told that the previous player had won 750 chips took higher risks when betting compared to those had no information, M = .09 (SD = .04) vs. M = .03 (SD = .03). Planned comparisons revealed that this difference was reliable, thus replicating our findings from Experiment 1, Dunnett's F(1, 22) = 13.62, p <.01. By contrast, participants who were told that the previous gambler's win was fortuitous exhibited similar levels of risk taking than those in the control condition, M = .03 (SD = .03) vs. M = .03 (SD = .03), Dunnett's F(1, 22) = .01, p = .63.

A similar pattern of results was observed for the measures of illusory control. All measures were affected by the type of report of wins, F(2, 33) = 5.7, p< .0, η² = .26, F(2, 33) = 4.64, p < .05, η² = .22, and F(2, 33) = 11.66, p < .001, η² = .41 for the subjective estimates of personal control, expectancy, and betting times, respectively.

Planned contrasts revealed that being told that the previous gambler won 750 chips increased participants' perceived personal control, M = 5.33 (SD = 1.45) vs. M = 3.63

Reports of Wins and Risk Taking 16

In keeping with our current predictions, however, participants in the "fortuitous win" condition behaved no differently from those in the control group. They exhibited the same level of personal control: M = 3.83 (SD = 1.27) vs. M = 3.63 (SD = 1.33), Dunnett's F(1, 22) = .16, p = .91; provided similar expectancy estimates, M = 74.58 (SD = 42.39) vs. M = 65.92 (SD = 47.74, Dunnett's F(1, 22) = .22, p = .98 and took similar amounts of time to place their bets, M = 20.75 (SD = 5.34) vs. M = 19.8 (SD = 5.9), Dunnett's F(1, 22) = .17, p = .59.

Finally, we sought to replicate the analysis for mediation effect of illusion of control when participants were simply told another player had won 750 chips compared to the control group. Results showed all criteria for mediation were satisfied. Specifically, knowledge of the previous player's win was a significant predictor of risktaking, β = .62, t(24) = 3.68, p < .01, and a significant predictor of the illusion of control, β = .6, t(24) = 3.51, p < .01. When, however, the risk-taking index was regressed on both knowledge and illusion of control, the latter remained a significant predictor, β = .63, t(24) = 3.82, p < .01 while the effect of the knowledge of another person's win on the same index was no longer a significant predictor, β = .24, t(24) = 1.47, p = .16, (adjusted R 2 = .61 for the full model). These analyses thus replicated the "perfect mediation" [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations[END_REF] observed in Experiment 1, Sobel test z = 2.53, p < .05. Figure 4 summarizes these results.

To summarize, this second experiment demonstrated that the increase in risk taking subsequent to the report of a previous player's win was contingent on the belief that this player had exerted control over the outcome of his game session. The degree of control assigned to another person was strongly and positively correlated with participants' own illusion of control and risk taking. When participants believed that the previous winner had no control over the outcome of his gamble, learning the amount he Reports of Wins and Risk Taking 17 had won no longer affected participants' illusory control or risk taking. When, however, participants were simply told about the amount won by the previous player, such knowledge significantly increased their risk taking, and their level of illusion of control was a significant mediator of this relationship, thereby replicating the results obtained in Experiment 1.

General discussion

The literature on the psychology of gambling suggests that gambling behavior can be facilitated by several determining factors, such as the physiological arousal that occurs when an individual is in a casino [START_REF] Anderson | Real and laboratory gambling, sensationseeking and arousal[END_REF][START_REF] Dickerson | On the determinants of persistent gambling behaviour. High-frequency poker machine players[END_REF], the presence of other gamblers who recognize gambling as socially acceptable [START_REF] Wood | The acquisition, development and maintenance of lottery and scratchcard gambling in adolescence[END_REF], the imitation of other gamblers (Gupta& Derevensky1997; [START_REF] Hardoon | Social influences in children's gambling behavior[END_REF], or the transmission of erroneous verbal information about the concept of chance [START_REF] Caron | Erroneous verbalizations and risk taking at video lotteries[END_REF]. All these accounts suggest that Kelly, the fictitious character introduced earlier, would be more likely to gamble again because she is playing in a casino, among other gamblers, rather than on her own in a public house. Kelly would also witness other players gambling and, sometimes, winning.

Our objective in this article was to test an alternative explanatory model of the urge to gamble where knowing that another gambler has won increases illusion of control, which in turn encourages risk taking. [START_REF] Martinez | Lien entre perception de contrôle et prise de risque dans un jeu de hasard: Quand l'annonce d'un gain d'autrui intervient[END_REF] found that betting times was a significant mediator of the effect of reports of wins on risk taking.

Increased betting times under the knowledge of another player's outcome, however, could have been a reflection of many other concerns such as distractibility or peerpressure. The present study used multiple independent and converging measures of the mediator to improve the illusory control construct validity. As such it extended Reports of Wins and Risk Taking 18 [START_REF] Martinez | Lien entre perception de contrôle et prise de risque dans un jeu de hasard: Quand l'annonce d'un gain d'autrui intervient[END_REF] work by using a sounder methodology to assess the mediating effect of illusory control.

Moreover, this study provided the first demonstration that when the win of the previous gambler was reported to be fortuitous, it not longer affected players' own illusion of control or risk-taking behaviors (Experiment 2). This finding confirms that it is not the knowledge of another person's win in itself that increases one's risk taking, but rather the belief that this win came about because the other player was able to exert control over the outcome of the gamble. Together, these findings clearly reveal how social influences such as reports of another player's win may affect gambling behavior.

Limitations and Future Research

Participants in this study knew they would not lose any real money, possibly limiting the generalizability of our findings. Although [START_REF] Kühberger | Framing decisions: Hypothetical and real[END_REF] found that "hypothetical" decisions (without real payoffs) appeared to match "real" decisions in laboratory tasks. They also added that this may not hold true for decisions where emotions and motivated needs play an important role, as it would be difficult to elicit these emotional states with hypothetical decisions. Given that real gambling decisions give rise to higher emotions than artificial ones [START_REF] Anderson | Real and laboratory gambling, sensationseeking and arousal[END_REF], the behaviors of the participants may have differed from those observed in real casinos. Anecdotal evidence suggests that our participants were indeed emotionally involved in the computer-based lottery game, as they tended to express great pleasure upon winning and great dissatisfaction upon losing. More importantly, higher levels of emotional involvement should be unlikely to favor rational thinking and if anything should lead to an increase in the illusion of control. As such, we would expect those results are likely to hold outside the laboratory.

Reports of Wins and Risk Taking 19

The fact that only one type of game (French roulette) was used could be seen as another limitation of this research. Indeed, the numerous types of possible bets at each round may have given participants the impression that their probability of winning could be increased through the use of strategies. The mere act of choosing a lottery ticket has been shown to inflate perceptions of winning odds [START_REF] Langer | The illusion of control[END_REF]Wohl & Enzlze, 2002). In addition, [START_REF] Gilovich | Biased evaluations of randomly determined gambling outcomes[END_REF] found that subjects who personally made a series of choices exhibited a bias in their outcome evaluations.

However, [START_REF] Wohl | The deployment of personal luck: Sympathetic magic and illusory control in games of pure chance[END_REF] found that the mediating variable for such biases is not the belief in one's ability to select the "winning ticket", but rather the belief that luck is a personal quality that can be used intentionally. In our study, however, participants selecting bets in the "fortuitous win" condition (Experiment 2) did not show any increase in illusion of control. This result demonstrates that the key determinant in generating an increase in illusion of control is the belief that the previous winner controlled the gamble rather the mere opportunity to choose among different bets.

Finally, an interesting direction for future research would be to investigate individual characteristics, which may affect players' sensitivity to the report of other's wins. [START_REF] Abrams | Behavioral Understanding in Pathological gambling: A clinical guide to treatment[END_REF], for example, emphasized the importance of dispositional traits (e.g., impulsiveness, sensation seeking and extraversion) in the development of pathological gambling. Similarly, individuals with varying scores on the PGS index (Problem Gambling Severity index, [START_REF] Ferris | The Canadian problem gambling index: Final report: Canadian[END_REF]) may be more or less sensitive to the report of others' wins.

Conclusions

Our results illustrated the importance of studying social factors in understanding gambling behaviors. We were able to demonstrate that the knowledge of the amount won by another gambler increased the new player's illusion of control over the outcome Reports of Wins and Risk Taking 20 of the gamble, which in turn increased the risks this player was willing to take, unless the reported win was qualified as fortuitous. The present findings have potentially important implications for the study of risky decision making in general. For example, a well-known consequence of Prospect theory [START_REF] Kahneman | Prospect theory: An analysis of decision under risk[END_REF][START_REF] Tversky | Advances in propect theory. Cumulative representation of uncertainty[END_REF] is that individuals tend to be risk averse in the domain of wins: they would rather choose to win $3, 000 for sure than take part in a gamble for an 80% chance of winning $4, 000. Our findings suggest, however, that this classic pattern of choice could be inverted simply by letting the decision-maker know that another person who chose to gamble has won $4, 000. Finally, those results also have important implications for the prevention of excessive gambling behaviors: they suggest that an efficient measure to inhibit the effect of others' wins on one's own gambling in games of pure chance such as the French roulette could be to remind players that those wins are fortuitous.

Figure captions

  greater gain expectancies; M = 235 (SD = 134.06) vs. M = 61.75 (SD = 57.96) in the control group; and longer times to place a bet; M = 31.62s (SD = 7.19) vs. M = 19.67s (SD = 5.89) in the control group. Once again, both these differences were statistically reliable; F (1, 22) = 16.89, p < .001 partial η² = .43 and F(1, 22) = 19.83, p < .001, partialη² = .47 respectively.

(

  SD = 1.33) in the control condition, Dunnett's F(1, 22) = 9.02, p <.01. It also significantly increased expectancy, M = 195.67 (SD = 191.61) vs. M = 65.92 (SD = 47.74) in the control condition, Dunnett's F(1, 22) = 5.18, p <.05. Finally, it also significantly increased betting times, M = 41.91 (SD = 8.46) vs. M = 19.8 (SD = 5.9), Dunnett's F (1, 22) = 12.92, p < .01. These results therefore replicated Experiment 1's findings.

Figure 1 :

 1 Figure 1: Proposed overall model of causal relationships between the knowledge of

Figure 2 :

 2 Figure 2: The placement of different types of bets on the layout table

Figure 3 :Figure 4 :

 34 Figure 3: Causal model between the knowledge of another person's win, betting times

There are several variants of the «roulette» game; the French roulette is the most popular version in European casinos in general, and in French casinos, in particular.

Note that although the number of participants per cell was relatively small, each of these participants participated in fifteen trials (see procedure) thereby increasing the power to detect treatment effects.

This name was selected because it is gender neutral in French.

Since participants were not familiar with the Roulette game, the experimenter enquired several times about the participants' understanding of the rules of this game and offered to re-explain the rules. All participants confirmed they had understood the rules. This was also confirmed by their gambling behaviours which were also suggestive of a good understanding of the game rules.

Appendix 1: The rules of French roulette "French roulette is a game of pure chance based on the random selection of a number between 0 and 36 inclusive. The roulette wheel is divided into 37 evenly spaced slots numbered 0, and 1 through 36. The zero slot is green and the other numbers are either red or black. These numbers are reproduced in a grid on the layout table (displayed on the computer screen). Several other boxes are represented outside this grid of numbers. Before the wheel is spun, players make bets against the House by placing chips in various boxes or on the lines between the boxes on the layout table. A small ivory ball is then released into the outer edge of the spinning wheel, circling many times before slowing down. As the ball loses momentum it falls and drops into one of the slots to end that round of the game. There are three main categories of bets, distinguished by their winning odds and payouts. All combinations of these bets are possible in the same attempt; you can win or lose in each different bet you place."

Appendix 2: Risk taking index [START_REF] Martinez | Lien entre perception de contrôle et prise de risque dans un jeu de hasard: Quand l'annonce d'un gain d'autrui intervient[END_REF]

RT represents the amount of risk taken; NC, the number of chips placed on any given bet; PL, the probability of losing of this bet; and CT, the resources of the gambler at the time of placing the bet(s). The value of this measure varies in theory on a scale from 0 to 1. In practice, it can neither be equal to 0 (since NC> 0 and PL> 0), nor 1 (since PL < 1).

Reports of Wins and Risk Taking 27


Bruno Macke

Bernard Ségard
email: bernard.segard@univ-lille1.fr

From fast to slow light in a resonantly driven absorbing medium

Keywords: numbers: 42.25.Bs, 42.50.Hz, 42.50 Gy

We theoretically study the propagation through a resonant absorbing medium of a time-dependent perturbation modulating the amplitude of a continuous wave (cw). Modeling the medium as a twolevel system and linearizing the Maxwell-Bloch equations for the perturbation, we establish an exact analytical expression of the transfer function relating the Fourier transforms of the incident and transmitted perturbations. It directly gives the gain and the phase shift undergone in the medium by a harmonic modulation. For the case of a pulse modulation, it enables us to determine the transmission time of the pulse center-of-mass (group delay), evidencing the relative contributions of the coherent and incoherent (population) relaxation. We show that the group delay has a negative value (fast light) fixed by the coherent effects when the cw intensity is small compared to the saturation intensity and becomes positive (slow light) when this intensity increases, before attaining a maximum that cannot exceed the population relaxation time. The analytical results are completed by numerical determinations of the shape of the transmitted pulses in the different regimes.

I. INTRODUCTION

Convincing demonstrations of fast light in a linear homogeneous medium were performed in the 1980s by exploiting the steep anomalous dispersion associated with a well-isolated, narrow and strong absorption line of the medium, leading to negative values of the group velocity [1,2]. Ideally bell-shaped pulses can then propagate moderately distorted in such a way that the maximum of the transmitted pulse occurs before that of the incident pulse. Time-advances exceeding 0.5 times the halfduration at half-maximum of the pulse-envelope have been so evidenced in experiments involving a true shape detection of the latter [2]. See also [3,4]. More generally, due to the causality principle (originating the Kramers-Kronig relations), fast light is expected every time that the carrier frequency of the pulses coincides with the bottom of a well-marked dip in the medium transmission and the pulse distortion resulting from the first order variations of the transmission and of the group velocity versus frequency cancels when this coincidence is exact [5]. The dip in the transmission may be natural or created in various arrangements involving electromagnetically induced absorption [6,7], stimulated Raman [8] or Brillouin [9] scattering, etc. The literature on fast light is abundant (for reviews, see, e.g., [10][START_REF] Milonni | Fast Light, Slow Light and Left-Handed Light[END_REF][START_REF] Boyd | [END_REF][13]) but it seems that the fractional pulse-advance with moderate distortion reported in [2] has not been overtaken. This is easily explained by remarking that, since the transmission dip at the carrier frequency must be well pronounced, there are spectral regions where the overall transmission of the system is much larger. For obvious reasons of noise (no matter its origin), instability and hypersensitiv-ity to small defects in the incident pulse-shape, the usable transmission-dynamics cannot be too large and this limits the observable fast-light effects in any linear system [14]. The dynamics actually involved in the fast-light experiments does not exceed 45 dB and is often much lower. The situation is quite different in the slow-light experiments where the medium transmission is maximal at the carrier frequency and can be very low on both sides of this frequency, with transmission-dynamics exceeding 1000 dB [15,16].

In the present article, we study the propagation in an absorbing medium of a continuous wave (cw) whose amplitude is pulse-modulated with a low modulation index. The medium is modeled as a homogeneously-broadened two-level system (TLS) [START_REF] Allen | Optical Resonance and Twolevel Atoms[END_REF] with a resonance frequency ω 0 and a relaxation time T 1 (T 2 ) for the population difference (the coherence). We assume that the cw is on exact resonance and we are interested in the propagation of the pulsed part of the wave (the pulse) whose optical spectrum is centered on ω 0 and maximum at this frequency. The pulse may thus be seen as a continuous superposition of fields symmetrically detuned from the cw field (sidebands) that probe the changes of the TLS properties induced by the cw. The propagation in a strongly driven TLS of two symmetrically detuned fields is a basic problem in nonlinear optics [START_REF] Sargent | [END_REF]19]. When the sidebands are associated with an amplitude modulation, it has been experimentally evidenced [20] and theoretically demonstrated [START_REF] Sargent | [END_REF]21] that the transmission of the modulation presents a maximum on resonance for strong enough cw intensities. According to our general analysis, the pulse propagation may be slow in such conditions. On the other hand, when the cw intensity is small compared to the saturation intensity, the TLS behaves linearly both for the cw and the pulse, the propagation of which is not modified by the presence of the cw (fast light regime). Our main purpose is to analyze how takes place the transition from a fast to a slow light regime when the cw intensity increases. Our calculations take full account of the depletion of the cw intensity during the propagation and are made for arbitrary values of the relaxation times T 1 and T 2 . The corresponding results (mainly analytical) are new and it seems that the possibility of switching from fast to slow light in a TLS by simply adding a resonant cw to the pulses has not been previously considered.

The arrangement of our paper is as follows. In Sec. II, we specify the system under consideration and we establish an exact analytical expression of its transfer function for the time-dependent part of the field envelope. This expression directly gives the gain and the phase shift undergone in the medium by a harmonic modulation, examined in Sec.III. The propagation of pulses is studied in Sec. IV, with a special attention paid to the transmission delay of their center-of-mass (identified to the group delay). The analytical results are completed by numerical determination of the envelope of the transmitted pulses. We finally conclude in Sec. V by summarizing the main results.

II. TRANSFER FUNCTION OF THE MEDIUM

As above mentioned we consider the case where the cw is exactly resonant. The amplitude modulation is then an eigenmode of modulation in the sense that that it is conserved during the propagation [22][23][24]. We denote ℓ the thickness of the medium, α its unsaturated absorption coefficient on resonance for the intensity and z (0 < z < ℓ) the direction of propagation of the wave assumed to be plane and polarized along the x-axis. We write the x-component of the electric field as :

E x (z, t) = Re e iω0t E(z, t) (1)
where E(z, t) is the slowly varying field-envelope. As in all the following, t is a local time, that is the real time minus n 0 z/c, c being the velocity of light in vacuum and n 0 the refractive-index around ω 0 of the eventual host medium. Denoting µ the dipole matrix element for the transition (chosen real), R(z, t) = µ E(z,t) the Rabi frequency, N (z, t) the population difference per volume unit (N 0 its value at equilibrium) and P (z, t) the envelope of the electric polarization induced in the medium, it is convenient to introduce the dimensionless quantities

D = N N0 , P = i P N0µ T1 T2 and E = µ E √ T 1 T 2 = R √ T 1 T 2 .
Note that all these quantities are real for the resonant case considered here and that I = E 2 is the intensity normalized to the saturation intensity. In such conditions, the Maxwell-Bloch (MB) equations governing the system evolution take the simple form :

∂E ∂z = - α 2 P (2)
T 2 ∂P ∂t = DE -P (3)
T 1 ∂D ∂t = -P E + (1 -D) (4)
These equations are easily solved when only the cw is present. In the following we denote the corresponding values of E, P, D and I by the index cw. Combining Eqs. ( 3) and (4), we find D cw (z) = 1/ [1 + I cw (z)] and

P cw (z) = E cw (z)/ [1 + I cw (z)].
Replacing this result in Eq. ( 2), we get :

∂ ∂z [I cw (z) + ln I cw (z) + αz] = 0 (5)
I out + ln I out = I in + ln I in -αℓ (6)
where I out and I in are short hand notations of I cw (ℓ) and I cw (0). When a small time-dependent perturbation δE in (t) (also real for an amplitude modulation) is added to the constant amplitude E in = √ I in of the field of the incident cw, we search for solutions of the MB equations under the form E(z, t) = E cw (z) + δE(z, t), D(z, t) = D cw (z)+δD(z, t) and P (z, t) = P cw (z)+δP (z, t). Taking into account the cw solution, we get at the first order of perturbation

∂(δE) ∂z = - α 2 δP (7)
T 2 ∂(δP ) ∂t = E cw δD + D cw δE -δP (8)
T 1 ∂(δD) ∂t = -E cw δP -P cw δE -δD (9)
These linear equations can be solved by introducing the Fourier transforms ∆E(z, Ω), ∆P (z, Ω) and ∆D(z, Ω) of δE(z, t), δP (z, t) and δD(z, t) [START_REF]Throughout this paper we use the definitions, sign con[END_REF]. From Eqs ( 8) and ( 9), we get :

∆P (z, Ω) = [1 + iΩT 1 -I cw (z)] ∆E(z, Ω) [1 + I cw (z)] [(1 + iΩT 1 ) (1 + iΩT 2 ) + I cw (z)]
(10) Replacing this result in Eq.( 7) and integrating on z, we finally obtain the transfer function H(Ω) relating ∆E out (Ω) and ∆E in (Ω), the Fourier transforms of δE out (t) = δE(ℓ, t) and δE in (t). It reads as :

H(Ω) = exp   ℓ 0 f (z, Ω) dz   (11)
where

f (z, Ω) = α 2 I cw (z) -(1 + iΩT 1 ) [1 + I cw (z)] [(1 + iΩT 1 ) (1 + iΩT 2 ) + I cw (z)] (12)
is the complex gain factor at the abscissa z. By means of Eq.( 5), the integration on z in Eq.( 11) can be transformed in an integration on I cw . We get :

H(Ω) = exp   1 2 Iout Î in 1 + iΩT 1 -I cw I cw [(1 + iΩT 1 ) (1 + iΩT 2 ) + I cw ] dI cw   (13)
and finally

H(Ω) = exp 1 + i ΩT2 2 1 + iΩT 2 ln I in + (1 + iΩT 1 ) (1 + iΩT 2 ) I out + (1 + iΩT 1 ) (1 + iΩT 2 ) - ln I in /I out (1 + iΩT 2 ) ( 14
)
where I out and I in are related by Eq.( 6). This expression of H(Ω) is the central result of our paper. In the limit

I in ≪ 1, I out ≈ I in exp(-αℓ) and H(Ω) is reduced to exp(- αℓ 2(1+iΩT2)
) which, as expected, is the transfer function of a linear medium with a Lorentzian absorption line [5]. Quite generally, we see that the poles of H(Ω) are in the half-plane Im (Ω) > 0 and that H (-Ω) = H ⋆ (Ω). This ensures that the impulse response h(t), inverse Fourier transform of H(Ω), cancels for t < 0 and is real, confirming that the system is causal [START_REF]Throughout this paper we use the definitions, sign con[END_REF] and that the amplitude modulation is an eigenmode of modulation. At the first order of perturbation considered here, we finally remark that

I(z, t) = [E cw (z) + δE(z, t)] 2 ≈ I cw (z)+2δE(z, t) I cw (z) (15)
This means in particular that a quadratic detection, as currently used in optics, will deliver a time dependent signal δI out (t) proportional to δE out (t) and that the instantaneous modulation indices for the intensity and for the field are such that δI(z,t) Icw(z) = 2 δE(z,t) Ecw(z) in every point.

III. TRANSMISSION OF A HARMONIC MODULATION

System of two-level atoms When the amplitude-modulation is harmonic of frequency Ω (Ω > 0) , the optical field has two sidebands at ω 0 ± Ω that both act as probes of the system driven at ω 0 (Fig. 1). The modulus G(Ω) and the argument Φ(Ω) of H(Ω) are then respectively the gain and the phase-shift of the output amplitude-modulation with respect to the input one. The gain coefficient g(z, Ω) = Re [f (z, Ω)] and the phase-shift coefficient ϕ(z, Ω) = Im [f (z, Ω)] are the corresponding quantities per unit length. Due to the depletion of the cw intensity I cw during the propagation, these latter quantities depend on z. Note that the expression of g(z, Ω) derived from Eq.( 12) is consistent with the results given in [START_REF] Sargent | [END_REF]21], also as in [20] for the particular case T 1 = T 2 . Figures 2 and3 show a set of profiles g(Ω) and ϕ(Ω) obtained for T 2 = 2T 1 (purely radiative relaxation) with various cw intensities. For I cw ≪ 1 (fully linear case), we obviously retrieve the gain and dispersion profiles g(Ω) = -

ω 0 ω 0 ω 0 + Ω ω 0 -Ω ω 0 + Ω ω 0 -Ω
α 2(1+Ω 2 T 2 2 ) and ϕ(Ω) = αΩT2 2(1+Ω 2 T 2 2 )
, associated with a Lorentzian absorption line. The gain coefficient (Fig. 2) keeps negative as long as I cw < 1. It cancels at Ω = 0 when I cw = 1 (absolute cw intensity equal to the saturation intensity). Beyond, it becomes positive in the frequency range

Ω < Ω g = (I 2 cw -1) / (I cw T 1 T 2 + T 2 1 )
, attaining an absolute maximum α/16 for I cw = 3. When I cw ≫ 1, Ω g ≈ R cw , the Rabi frequency associated with the cw field. We also note that, g(Ω) being an even function, g(0) = α(Icw-1) 2(Icw +1) 2 is always an extremum of the gain-coefficient. This extremum is flat for

I cw = b ± b 2 -T 2 /T 1 with b = 3/2 + T 1 /T 2 + T 2 /2T 1 , that is I cw = 3 ± √ 7 when T 2 = 2T 1 .
Despite some common points, the profiles g(Ω) significantly differ from those obtained by using a probe independent of the driving field [START_REF] Sargent | [END_REF][START_REF] Haroche | [END_REF][27][28][29]. They are a bit simpler, probably because the amplitude modulation is an eigenmode of modulation (contrary to the single sideband modulation). On the other hand, the refractive index experienced by the optical probe field, well defined for the single sideband case [30], is not defined for the case of an amplitude modulation where the probe field has two frequency-components at ω 0 ± Ω that are not independent.

The phase-shift coefficient ϕ(Ω) (Fig. 3) has a maximum ϕ max = α/4 (phase advance) at Ω = 1/T 2 for I cw ≪ 1 and an absolute minimum ϕ min ≈ -0.078 α (phase lag) at Ω ≈ 1.14/T 1 for I cw ≈ 3.3. On the other hand, the slope dϕ dΩ | Ω=0 at the origin is also maximal for I cw ≪ 1. This slope is half-maximum, cancels and attains its (negative) minimum, for I cw respectively equal to 0.135, 1/2 and 5/4. Representative profiles of the (overall) gain G(Ω) and phase-shift Φ(Ω) are shown Fig. 4 and Fig. 5 for αℓ = 10, with various intensities I in chosen according to criteria analogue to those used Fig. 2 and Fig. 3. The optical thickness retained is of the order of that actually used in [2]. When

I in ≪ 1, G(Ω) = exp - αℓ 2(1+Ω 2 T 2 2 ) and Φ(Ω) = αℓΩT2 2(1+Ω 2 T 2 2 )
. Except for this fully linear case, Eqs. [START_REF] Milonni | Fast Light, Slow Light and Left-Handed Light[END_REF] and [START_REF] Boyd | [END_REF] show that G(Ω) and Φ(Ω) are strongly affected by the depletion of the cw intensity in the medium, often not taken into account in the literature. This point is illustrated Fig. 6 where we give the profile G(Ω) = exp [ℓg(Ω)] which would be obtained by neglecting the cw depletion, i.e. by taking I cw (z) = I in everywhere. The depletion obviously forces one to use larger incident cw intensities to obtain similar gain pro- files and the maximum gain is smaller. The difference is more important when the optical thickness αℓ is very large. From Eqs. ( 6) and ( 14), we find that the maximum gain then tends to √ αℓ/2 whereas it equals exp [αℓ/16] when the depletion is neglected. There are obviously similar effects on the phase-shift Φ(Ω). The previous results take a very simple form in the limit case where T 2 is negligible compared to T 1 . The MB equations are then reduced to the rate equations [START_REF] Allen | Optical Resonance and Twolevel Atoms[END_REF] that provide a good description of the propagation of light in numbers of saturable absorbers [31][32][33][34][35][36]. Putting T 2 ≈ 0 in Eqs. ( 12) and ( 14), we get

f (z, Ω) = - α 2 (1 + I cw ) + αI cw (1 + I cw ) 2 1 1 + i ΩT1 1+Icw (16) H(Ω) = I out I in 1 + I in + iΩT 1 1 + I out + iΩT 1 (17)
Apart from the term -α 2(1+Icw) independent of Ω, the complex gain factor f (z, Ω) is identical to that of a Lorentzian gain line of half-width at half maximum (1 + I cw ) /T 1 with a gain-coefficient on resonance αI cw / (1 + I cw )

2 for the amplitude. During the propagation in the medium, the modulation index is magnified whereas the phase of the modulation is delayed [35][36][37][38]. The magnification K(Ω) of the modulation index and the phase lag L(Ω) = -Φ(Ω) for both the amplitude and the intensity modulations are easily deduced from Eq.( 17). We get :

K(Ω) = G(Ω) I in I out = (1 + I in ) 2 + Ω 2 T 2 1 (1 + I out ) 2 + Ω 2 T 2 1 ( 18
)
L(Ω) = tan -1 ΩT 1 (I in -I out ) (1 + I in ) (1 + I out ) + Ω 2 T 2 1 (19)
The phase lag, always positive for this "incoherent" case, attains its maximum

L max = tan -1 (I in -I out ) /2 (1 + I in ) (1 + I out )
for

ΩT 1 = (1 + I in ) (1 + I out ).
We remark that L max < π/2, the upper limit being approached when I in ≫ 1 and I out ≪ 1, that is for αℓ → ∞. Consequently the time delay of the output modulation can never exceed a quarter of the modulation period T = 2π/Ω (about an eighth for αℓ = 10). Since the work reported in [39], numerous slow-light experiments performed in saturable media have been analyzed by invoking hole burning via coherent population oscillations (CPO), resulting in a reduction of the group velocity. In fact, a more direct analysis is provided by the basic model of saturable absorber [33] and, as shown in detail in [36], the signals observed in most CPO experiments can be perfectly reproduced by means of Eqs. ( 18) and (19), eventually extended to take into account inhomogeneous effects.

IV. PULSE PROPAGATION

Strictly speaking a harmonic modulation does not contain any information and, e.g., a time-delay of T /8 as considered at the end of the previous section can also be seen as a time-advance of 7T /8. Unambiguous demonstrations of fast or slow light require to use pulses of finite duration and energy. An important parameter to characterize the propagation is then the transmission-time of the centerof-mass of the pulse envelope. Following the use in signal theory [START_REF]Throughout this paper we use the definitions, sign con[END_REF], we define the center-of-mass of a signal y(t) of normalized area as ´∞ -∞ ty(t)dt. A direct application of the moment theorem [START_REF]Throughout this paper we use the definitions, sign con[END_REF] shows then that the transmission-time of the center-of-mass of the pulse envelope is equal to the group delay τ g = -dΦ dΩ | Ω=0 , whatever the pulse distortion may be [5]. On the other hand dϕ dΩ | Ω=0 appears as the transmission delay or group delay per unit length, equal in our local time picture to 1/v g -n 0 /c ≈ 1/v g , where v g is the group velocity. We emphasize that, due to the cw depletion, v g is not uniform. From Eq.( 14), we get τ g = τ 1 + τ 2 with

τ 1 = T 1 1 I out + 1 - 1 I in + 1 ( 20
)
τ 2 = T 2 1 I out + 1 - 1 I in + 1 -ln 1 + 1/I out 1 + 1/I in (21)
τ 1 (τ 2 ), proportional to T 1 (T 2 ), may be considered as the contribution to the group delay of the incoherent (coherent) effects. We see that the incoherent or population effects always lead to a delay whereas the coherent ones mainly lead to a much larger time-advance (Fig. 7), at least when T 2 is comparable to T 1 (the coherent effects obviously disappear in a saturable absorber where T 2 ≈ 0). Before commenting further on Eqs. ( 20) and ( 21), consider the simpler results obtained when the depletion of the driving wave is neglected (I cw = I in everywhere). From Eq.( 12), we then get :

τ 1 = T 1 αℓI in (1 + I in ) 3 (22)
τ 2 = -T 2 αℓ (1 -I in ) 2 (1 + I in ) 3 ( 23
)
τ 1 /T 1 , negligible for I in ≪ 1, is maximum for I in = 1/2 whereas τ 2 /T 2 starts from the large negative value -αℓ/2 before attaining a small positive maximum for I in = 2. The total group delay cancels for

I in = T 2 / (T 2 + 2T 1 )
and is maximum for I in = (2T 2 + T 1 ) / (T 2 + 2T 1 ) that is respectively for I in = 1/2 and I in = 5/4 when the relaxation is purely radiative (see Fig. 3). As expected, taking into account the depletion of the cw intensity does not modify the delays for I in ≪ 1 but shifts the curves τ 1 (I in ) and τ 2 (I in ) to larger I in , dramatically spreads them and reduces the amplitude of their maximum (Fig. 7). Equation (20) shows that τ 1 cannot overtake T 1 , this value being approached when I in ≫ 1 and I out ≪ 1, that is for αℓ extremely large [35]. An estimate of the maximum of τ 1 for large but realistic values of αℓ can be obtain by anticipating that it is attained for I out ≪ 1. From Eq.( 6) we get then I out ≈ I in exp (I in -αℓ) and

τ 1 ≈ T 1 1 I in exp (I in -αℓ) + 1 - 1 I in + 1 ( 24
)
The maximum is obtained for (I in + 1) 3 ≈ exp (αℓ -I in ). For αℓ = 10 (Fig. 7), we find I in ≈ 4.8 and, putting this value in Eq.( 24), max (τ 1 /T 1 ) ≈ 0.80 in perfect agreement with the exact value. By the same method we find max (τ 1 /T 1 ) ≈ 0.995 for αℓ = 200. An upper bound to the total group delay can be obtained by using the method of the Lagrange multipliers, perfectly adapted to the search of a maximum under constraint, here fixed by Eq.( 6). Following the procedure exactly as exposed in the original work of Lagrange [START_REF] De La Grange | Méchanique Analitique[END_REF], we find an extra relation between I in and I out at the maximum of τ g , namely

I in I out = I in + 1 I out + 1 3 2T 1 + T 2 (1 -1/I out ) 2T 1 + T 2 (1 -1/I in ) (25)
A numerical exploration shows that the upper bound to the group delay is attained when I in is extremely large whereas I out keeps finite. Equation ( 25) takes then the form

(1 + I out ) 3 ≈ I 2 in [I out -T 2 / (2T 1 + T 2 )] (26)
which can be verified if and only if

I out ≈ T 2 / (2T 1 + T 2 ).
Injecting this result in the general expression of τ g , we finally find :

sup [max (τ g )] = T 1 + T 2 2 -T 2 ln 2 1 + T 1 T 2 (27)
When T 2 ≈ 0 (saturable absorber), we obviously retrieve the upper bound T 1 for τ g whereas this upper bound is (2ln 3) T 1 ≈ 0.90T 1 when the relaxation is purely radiative (T 2 = 2T 1 ). For the latter case max (τ g /T 1 ) is only 0.68 for αℓ = 10, significantly below its upper limit, and raises to 0.89 for αℓ = 200. As before mentioned, the group delay τ g is the transmission-delay of the pulse center-of-mass, as large as the pulse distortion may be. On the other hand, the envelope δE out (t) of the transmitted pulse is simply the inverse Fourier transform of H(Ω)∆E in (Ω). When the duration τ p of the incident pulse is long enough, ∆E in (Ω) is concentrated around Ω = 0 where H(Ω) ≈ I in /I out exp(-iΩτ g ). We then get δE out (t) = I in /I out δE in (t -τ g ). The whole envelope is multiplied by I in /I out and time shifted by τ g , without any distortion. Strictly speaking, this solution is only valid for a very long pulse and is not really interesting insofar as the time delay is then negligible compared to the pulse duration. For finite τ p , there is always some pulse-distortion whose importance depends in particular on the transmission dynamics (as defined in the introduction). If, as usual, the incident pulse is bell-shaped and symmetric with a maximum at t = 0, the envelope of the transmitted pulse will be dissymmetric with a maximum at a time τ m such that τ m /τ g < 1, its center-of-mass keeping exactly at the time τ g no matter the pulse duration. The challenge in the slow or fast light experiments is to obtain a fractional delay or advance |τ m | /τ p as large as possible, with moderate distortion. τ m and τ g are then comparable. The following figures are obtained by using standard techniques of Fast Fourier Transform. The incident pulse is Gaussian and τ p is its half-duration at half-maximum. Time (in τ p -units)

Figure 8: Normalized pulse envelopes obtained for αℓ = 10 and T2 = 2T1, when the group advance -τg is maximum (solid line) and half-maximum (dashed line). τp is chosen such that the fractional advance of the pulse maximum -τm/τp is significant whereas the distortion remains moderate (τp/T1 = 12.5). For the first case, τm/τg = 0.88 and -τm/τp = 0.71. For the second case τm/τg = 0.91 and -τm/τp = 0.37. The envelope of the incident pulse is given for reference (dotted line).

Figures 8 and 9 are obtained for I in = 0.001 and I in = 1.9 leading respectively to a group advance -τ g nearly equals to its maximum αℓT 2 /2 and the half of this value. The corresponding transmission dynamics is respectively 43 dB and 26 dB. On Fig. 8, τ p has been chosen in order that the fractional advance -τ m /τ p is significant (respectively 0.71 and 0.37) and the distortion keeps moderate (τ m /τ g respectively equals 0.88 and 0.91). When the pulse duration is shortened, the fractional advance increases but not as much as one could expect (respectively up to 1.1 and 0.68) and this is paid by a dramatic distortion of the transmitted pulse (Fig. 9). Figure 10 is obtained for I in ≈ 8.0, leading to the largest normalized group delay (τ g /T 1 ≈ 0.68). The cor-

-1 0 1 Pulse Envelope -3 -2 -1 0 1 2 3
Time (in τ p -units) responding transmission dynamics is small (7 dB). This explains in part that, even when the pulse duration is optimized (dashed line), the fractional delay τ m /τ p does not exceed 0.27. Another reason is that the coherent relaxation negatively contributes to the group delay (see Fig. 7). For a saturable absorber (T 2 ≈ 0) of same optical thickness αℓ, the maximal group delay, now attained for I in ≈ 4.8, is upgraded to 0.80T 1 (see Fig. 7) whereas the transmission dynamics attains 15 dB. Figure 11 shows the pulse envelopes obtained for two different pulse durations. As expected, the maximum fractional delay, obtained for τ p ≈ 0.44T 1 (dashed line), is significantly larger than for the case of radiative relaxation (more than two times larger). We however remark that the corresponding fall of the pulse is considerably lengthened. An extensive study of the pulse distortion in this particular system can be found in [35].

V. CONCLUSION

In their letter appeared in 1982 and soberly entitled "Linear Pulse Propagation in an Absorbing Medium", Chu and Wong [1] paved the way for the now-called fastlight experiments. We have examined in the present article how the propagation is modified when the absorbing medium is driven by a cw. Modeling the absorbing medium as an ensemble of two-level atoms, we have more specifically considered the case where both pulses and cw are on exact resonance, a condition simply realized by pulse-modulating the cw amplitude, with a low modulation index. This particular arrangement eliminates the pulse-distortion associated with the first order variations of the absorption and of the group delay versus frequency. The basic result of our paper is the exact analytical expression of the transfer function relating the Fourier transforms of the incident and transmitted modulations for arbitrary values of the coherent and incoherent relaxation times (Eq.14). It shows the importance of the effects resulting from the depletion of the cw intensity along the propagation. When the modulation is only harmonic (as in numerous experiments), it directly gives the gain and the phase shift undergone by the modulation. They significantly depart from those obtained with a unique probe field (single sideband modulation). When the modulation is actually pulsed, the transmission delay of the pulse center-of-mass, identified to the group delay, is deduced from the transfer function by a simple calculation of derivative. A remarkable point is that the group delay is the sum of two terms, respectively proportional to the coherent and incoherent relaxation times. These two terms being mainly of opposite sign and depending differently on the cw intensity, this explains why the transmission delay, strongly negative when the cw intensity is low, may become (slightly) positive when the latter increases. Finally the numerical determination of the pulse shapes confirms a general property of the fast and slow light systems, namely that significant advances or delays with moderate distortion can only be obtained in media with a large transmission dynamics. Though our study is only theoretical, it is illustrated for a realistic value of the optical thickness, comparable to that actually used in the microwave experiment reported in [2]. In the optical domain, suitable optical thickness and time-scale could probably be obtained by using an ensemble of cold atoms. We expect that our theoretical work will stimulate such an experiment.

Figure 1 :

 1 Figure 1: The driving wave (cw) at ω0 can modify the propagation of the pair of sidebands at ω0 ± Ω.

Figure 2 :

 2 Figure 2: Gain coefficient g(Ω) in α/2 units as a function of Ω in 1/T1 units for T2 = 2T1. The different curves are labeled by the corresponding value of the normalized intensity Icw .

Figure 3 :

 3 Figure 3: Same as Fig.2 for the phase-shift coefficient ϕ(Ω) in α/2 units.

Figure 4 :

 4 Figure 4: Gain G(Ω) as a function of Ω in 1/T1 units for T2 = 2T1 and αℓ = 10. The different curves are labeled by the corresponding value of the normalized intensity Iin .

Figure 5 :

 5 Figure 5: Same as Fig.4 for the phase-shift Φ(Ω). The maximum and minimum phase-shift are respectively αℓ/4 = 2.5 Rd at Ω = 1/T2 for Iin ≪ 1 and -0.56 Rd at Ω = 1.68/T1 for Iin ≈ 11.5.

Figure 6 :

 6 Figure 6: Same as Fig.4 when the depletion of the driving wave is not taken into account.

  T 1 with depletion τ 1 /T 1 without depletion τ 2 /T 2 with depletion τ 2 /T 2 without depletion

Figure 7 :

 7 Figure 7: Normalized coherent and incoherent contributions to the group delay as a function of the incident driving intensity Iin, depending whether the cw depletion (αℓ = 10) is taken into account or not.

Figures 8 ,

 8 9 and 10 show typical shapes of the transmitted pulse for T 2 = 2T 1 (radiative relaxation) and αℓ = 10.

Figure 9 :

 9 Figure 9: Same as Fig.8 when the pulse durations are those maximizing the fractional advance -τm/τp . For the solid line τp/T1 = 4.1 , τm/τg = 0.44 and -τm/τp = 1.1. For the dashed line τp/T1 = 2.8 , τm/τg = 0.38 and -τm/τp = 0.68.

Figure 10 :

 10 Figure 10: Normalized pulse envelopes obtained for αℓ = 10 and T2 = 2T1, when the group delay τg is maximum. For the solid line τp/T1 = 2.27 , τm/τg = 0.77 and τm/τp = 0.23. For the dashed line τp/T1 = 1.44 , τm/τg = 0.56 and τm/τp = 0.27. The envelope of the incident pulse is given for reference (dotted line).

  τ p -units)

Figure 11 :

 11 Figure 11: Same as Fig 10 when T2 ≈ 0 (saturable absorber). For the solid line τp/T1 = 2.0 , τm/τg = 0.79 and τm/τp = 0.32. For the dashed line τp/T1 = 0.44 , τm/τg = 0.31 and τm/τp = 0.57.