Thomas Barnay
email: thomas.barnay@u-pec.fr

Mohamed Ali

Ben Halima

Emmanuel Duguet

Christine Le Clainche

Camille Regaert

The effects of breast cancer on individual labour market outcomes: an evaluation from an administrative panel *

Keywords: I10, J21, J22 breast cancer, labour market participation, difference in differences, matching

Using an administrative data set (Hygie), we apply a difference-in differences with dynamic matching estimation method to the onset of breast cancer. The employment probability decreases by 10 percentage points (pp) one year after the onset of cancer compared to the not-treated group. The detrimental effect of breast cancer on employment increases significantly over time, by up to 12 pp after five years. Another aim of our study is to identify some socio-demographic and work-related protective factors against the adverse effects of breast cancer on labour market outcomes. We stress four potential protective factors related to the negative effect of breast cancer. First, a young age at occurrence reduces this deleterious effect. Second, a high first job wage also appears to be a protective factor. Third, having faced less unemployment in the past is associated with a weaker negative effect of breast cancer on employment in the short run. Finally, we find a moderate "generational effect" after stratification by year of cancer onset.

Introduction

Thanks to advances in organized screening, detection and treatment, cancers may be related to chronic diseases [START_REF] Cutler | Are we finally winning the war on cancer?[END_REF]). Breast cancer imposes a substantial burden on the statutory health insurance in France. This cost burden impacts the long-term disease scheme, which supports all expenditures related to chronic diseases, including cancers. Therefore, the average cost of treatment for breast cancers in France is 10 thousand euros per year. In 2012, the expenditures on all cancers reached 14 billion euros, including 2.3 billion for breast cancer alone. In addition, breast cancer causes indirect costs: lost days of work and productivity losses (Rapport de l'observatoire sociétal des cancers (2014)). 1 In the female population, breast cancer exhibits an earlier onset than that of other cancers 2 and requires treatments associated with functional sequels and therefore raises questions regarding the impact of breast cancer on individual well-being and, especially, labour market outcomes. Furthermore, the net survival rate of women diagnosed with breast cancer in France between 1989 and 2004, according to French cancer registries, is 97% after one year and 86% after five years (Jooste, Grosclaude, Remontet, Launoy, Baldi, Molinie, Arveux, Bossard, Bouvier, and Colonna (2013)), thus allowing the study of the causes of work impairment. initial endowments of human capital, the difficulty of pre-diagnosis working conditions, the type of cancer (site, severity of the disease) and, finally, the nature of the treatment [START_REF] Johnsson | WWork status and life changes in the first year after breast cancer diagnosis[END_REF], Lindbohm, Kuosma, Taskila, Hietanen, Carlsen, Gudbergsson, and Gunnarsdottir (2011), Mujahid, Janz, Hawley, Griggs, Hamilton, Graff, and Katz (2011), [START_REF] Blinder | Return to work in low-income Latina and non-Latina white breast cancer survivors: A 3-year longitudinal study[END_REF]). Past professional biography (unemployment or training episodes) can also lead to stigmatizing effects on the careers of individuals [START_REF] Heckman | Does unemployment cause future unemployment? Definitions, questions and answers from a continuous time model of heterogeneity and state dependence[END_REF], [START_REF] Gregg | The wage scar from youth unemployment[END_REF]) and, for some social groups, predicts the occurrence of occupationrelated cancers. Feuerstein, Todd, Moskowitz, Bruns, Stoler, Nassif, and Yu (2010) thus stress the importance of improvements in the workplace in terms of schedule flexibility, social support from colleagues, social climate and job stress to protect working ability among cancer survivors.

Focus on Breast cancer. A large international literature (with a particular reliance on US data) is devoted to effect of breast cancer on professional paths (Chirikos, Russell-Jacobs, and Cantor (2002), Chirikos, Russell-Jacobs, and Jacobsen (2002), Drolet, Maunsell, Mondor, Brisson, Brisson, Masse, and Deschenes (2005), Bradley, Oberst, and Schenk (2006), Bradley, Neumark, and Barkowski (2013), [START_REF] Heinesen | Effects of breast and colorectal cancer on labour market outcomes-average effects and educational gradients[END_REF]). For instance, [START_REF] Bradley | Employment-contingent health insurance, illness, and labor supply of women: Evidence from married women with breast cancer[END_REF] show that the negative effect of cancer on employment significantly persists 6 months after the diagnosis but not more. Heinesen and Kolodziejczyk (2013) measure the causal effects of breast and colorectal cancer on labour market outcomes. On the basis of Danish administrative data, they estimate the ATT (average treatment effects on the treated) using propensity score methods with persons with no cancer as a control group. Suffering from breast cancer in year t reduces the probability of being employed by 4.4 percentage points in year t +1, by 5.7 percentage points in year t +2, and 6.7 percentage points in t +3. From Danish data (2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009), Carlsen, Badsberg, and Dalton (2014) stress that women, after a breast cancer diagnosis, who experienced periods of unemployment before the diagnosis have an increased risk of being unemployed thereafter relative to women who worked before diagnosis (79 weeks of unemployment against 26 weeks for working women before diagnosis). Past French studies are more limited. Eichenbaum-Voline, Malavolti, Paraponaris, and Ventelou (2008) and Joutard, Paraponaris, Sagaon-Teyssier, and Ventelou (2012) apply a matching method to survey data that include treatment variables. Marino, Sagaon, Laetitia, and Anne-Gaelle (2013) show that two years after the diagnosis of cancer, the probability of returning to work in the female population is 72% (against 25% six months after the diagnosis).

Many articles underline the role of different health and socioeconomic characteristics that influence the effect of breast cancer on employment. First, a significant body of literature highlights the nature of cancer and types of treatment (Hassett, O'Malley, and Keating (2009), Jagsi, Hawley, Abrahamse, Li, Janz, Griggs, Bradley, Graff, Hamilton, and Kratz (2014)). Treatments require an exit from the labour market, which may be long when women undergo a combination of treatments (surgery, radiotherapy, and chemotherapy). In France, as in other developed countries, facing a combination of treatments (especially chemotherapy before and/or after radiotherapy) has the most detrimental effect on the speed with which a woman with breast cancer can return to work (for a French data set that includes the severity of the disease and the type of treatments, see Duguet and Le Clainche (2016)). Women who have undergone a surgery with partial mastectomy followed by radiotherapy can often return to work in the 6 months following the surgery, if no comorbidity occurs.

Treatment and comorbidity factors and sociodemographic and work-related characteristics may also explain differences in labour outcomes [START_REF] Bradley | Short-term effects of breast cancer on labor market attachment : Results from a longitudinal study[END_REF], Torp, Gudbergsson, Dahl, Fossa, and Fløtten (2011)). Cross-country differences in the delays beyond 6 months may be explained by protective factors such as favourable social protection rules (sick leave legislation, social insurance schemes, work flexibility) and how employers are allowed to arrange working conditions. Using French data, Duguet and Le Clainche (2016) show that the probability of returning to work two years after diagnosis, especially for women diagnosed with breast cancer, increases when appropriate working conditions are implemented. Generally, the onset of cancer affects future investments in human capital (primary or secondary health prevention) due to the difficulty of combining work and cancer treatment (Yarker et 2011) find that, one month after the surgery, 56% of women with breast cancer are on sick leave, the majority full-time. Most of these women are employed at diagnosis, and 91% of them work greater than 75% of full-time. According to Farley, Vasey, and [START_REF] Farley | Long-term effects of cancer survivorship on the employment of older workers[END_REF], in the USA, survivorship affected the probability of working full-time and hours worked for both genders 2-6 years post-diagnosis. In the female population of survivors after a new cancer diagnosis, the negative effects attributable to the cancer are 14 to 17 pp (percentage points) for the employment rate, 14 to 18 pp for full-time employment and 7 to 8 for hours worked per week. [START_REF] Torp | Social support at work and work changes among cancer survivors in Norway[END_REF], using a Norwegian database, highlight that a low socioeconomic position appears to be a risk factor for returning to work. On the basis of Korean data (1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002), after a breast cancer diagnosis, working women are more frequently unemployed if they have low education or a low income (Eunmi, Cho, Shin, Park, Ahn, Noh, Nam, Lee, and Yun ( 2009)).

Using French data, Paraponaris, Teyssier, and Ventelou (2010) study the relationship between cancer occurrence and the type of labour contract. Their findings indicate that fixedterm contracts exhibit greater risk of job loss for workers in the female population (-8 pp relative to permanent contracts).

From a theoretical perspective, the return to work depends on economic incentives. Bradley, Neumark, and Barkowski (2013) demonstrate that the negative effect of breast cancer on employment is reduced if the patient's health insurance is dependent on the job. This result refers to the "job lock" assumption, i.e., workers remain in their current job to maintain their health insurance. In contrast to the USA, in France, for particular diseases that require intensive, expensive and long-term care (such as cancer), the long-term disease scheme has been implemented to pay for the related treatment costs and provide equal access to health care.

In this study, we focus on the sociodemographic and work-related aspects of cancer. Given the French legal framework, we cannot assume a job lock effect. We expect that the negative effect of breast cancer will persist over time even if the woman does not suffer from a recurrence of the disease, as in [START_REF] Heinesen | Effects of breast and colorectal cancer on labour market outcomes-average effects and educational gradients[END_REF] in the Danish case.

We examine four potential types of socioeconomic factors that could influence employment in the short and long term. First, breast cancer occurring at a younger age could be less disadvantageous than at a later age [START_REF] Petersson | Work situation and sickness absence in the initial period after breast cancer surgery[END_REF]). Second, a strong socioeconomic position is associated with appropriate working conditions following diagnosis and with a better return to work [START_REF] Eunmi | Impact of breast cancer diagnosis and treatment on work-related life and factors affecting them[END_REF]). Third, we account for the stability of the past career before the onset of cancer [START_REF] Heckman | Does unemployment cause future unemployment? Definitions, questions and answers from a continuous time model of heterogeneity and state dependence[END_REF], [START_REF] Gregg | The wage scar from youth unemployment[END_REF]). Finally, we also examine a generational effect. Advances in medicine coud lead to a better return to work for the most recent generation of cancer survivors, but other factors, such as discrimination 3 in the labour market, could play a more ambiguous role.

Using panel data from the National Pension Fund and the National Health Insurance Fund, we examine two issues. First, we estimate, for the first time in France, the effects of breast cancer on employment outcomes up to five years after its onset. We perform a difference-indifferences analysis combined with a dynamic matching algorithm. Second, we highlight the role of protective factors that attenuate the adverse effects of cancer on labour market outcomes. The paper is organized as follows. Section 2 is devoted to the methodology and the econometric model specifications. In section 3, we present our main findings, and a discussion is provided in section 4. The final section presents our conclusions.

Data and methodology

Data. The HYGIE data set was constructed from two nationwide administrative sources. The HYGIE data were extracted from the National Pension Fund (CNAV) and the National Health Insurance Fund for Salaried Workers (CNAMTS) administrative databases. The resulting database contains individual information on beneficiaries, their professional careers, medical consumption, sick leave, employees' professional context and a few characteristics concerning employer establishments. The HYGIE database makes it possible to study the relationships among health, work, professional career and firm characteristics. HYGIE is representative of private sector employees in France.

Specifically, the files were extracted from the National Career Management System (SNGC), which groups all private sector employees in France, and the National Statistical Beneficiaries System (SNSP), which groups all private sector retirees in France, matched with sickness benefit data taken from the National Health Insurance Inter-regime Information System (SNIIR-AM). The CNAV data constituted the point of entry and included a random sample of beneficiaries aged 22 to 70 years who contributed to the general pension fund at least once in their lives. The CNAMTS data concern both primary and secondary beneficiaries of the National Health Insurance scheme who received sickness benefits for at least one health service in 2003, 2004 or 2005. The linkage of the CNAV and CNAMTS data enabled us to construct the HYGIE database panel of 538,870 beneficiaries from 2005 to 2010.

There are 552,048 working people in these data. Restricting to women leaves 225,340 observations. For this study, we use a data set that includes the entire career of the workers from their first job to the year 2008. We drop the retirement years and retain only the active period to evaluate the impact of cancer on employment. The demographic data include gender and birth year. The HYGIE data include the wage from the first job. We take the ratio of this first job wage to the yearly median wage and compute four equally sized initial wage classes. This approach corrects for inflation and indicates a worker's position in the income distribution.

The medical data include a sick leave dummy and the International Statistical Classification of Diseases and Related Health Problems (henceforth, ICD). We identify breast cancer with ICD code C50. To implement the dynamic matching methodology, we retain all the women in the sample, that is, women with breast cancer, women with no disease and women with another disease. These other diseases include all other cancer types and the French definition of longterm diseases (such as diabetes).

These diseases are on an official list and must last at least 6 months. They are fully financed by the statutory health insurance. Our approach also guarantees that the estimation results obtained for all of the cancers and diseases in this data set are comparable, as the same data set can be used for all of them.

Three labour market statuses were identified in the HYGIE data: employment, regular employment and unemployment. The outcome variables indicating the individual's employment status were identified in the HYGIE database on the basis of contributions paid for qualifying periods (unemployment, sickness/occupational accident) and quarters of national social security scheme contributions. Individuals were thus identified as follows. Regular employment means at least one quarter of contribution to the national social security system and the absence of unemployment spells. Employment means at least one quarter of contribution, with or without unemployment spells.

We use the following indicator of career stability. We divide the number of years with an unemployment spell by the total number of years spent in the labour market. We match women on the value of this indicator one year before cancer onset to control for their past labour market performance. We use the same type of indicator for past health: the number of years with sickness leave divided by the total number of years spent in the labour market. We also use the lagged value of this indicator (one year before cancer) when we match women.

To implement the dynamic matching methodology, we retain all the women in the sample, again, women with breast cancer, women with no disease and women with a disease other than breast cancer. These other diseases include all other types of cancer and all of the diseases that correspond to the French definition of long-term diseases (such as diabetes). These diseases are on an official list, and must last at least 6 months. They are fully financed by the compulsory health insurance scheme. This approach also guarantees that the results obtained for all of the cancers and diseases in the data set are comparable, as the same data set can be used for all of them. Methodology. Our estimator is similar to that of [START_REF] Heckman | Matching as an econometric evaluation estimator: Evidence from evaluating a job training programme[END_REF]. We wish to explore the impact of breast cancers on the employment history of women. We account for four estimation issues. First, cancer is likely to cause a break in individual employment histories. In this case, only a dynamic approach can identify the break with a before-after analysis, where the break date is the date of the cancer and is specific to each individual. A static approach can only compare individuals with different health status at the date of the survey and cannot directly analyse the impact of cancer for each individual. Second, the women in the data set have different ages, and hence the observation window differs across women. Therefore we need to match women by age to ensure that we compare women who had a health event (or did not) at the same age and during the same year. Third, we account for the fact that the panel data model includes correlated individual and time effects. We also allow for heterogeneous time trends by adding matching to the usual difference-in-differences estimator. Fourth, we perform a non parametric estimation, and hence no specific distributional assumption is made.

In the standard difference-in-difference approach (henceforth, DiD; see [START_REF] Lee | Micro-Econometrics for Policy, Program and Treatment Effects[END_REF] for details), the variables that are constant over time should not alter the estimations, which should allow this approach to control for individual-level unobserved heterogeneity. Our approach generalizes DiD in two directions. First, we match on the lagged endogenous variables such that we compare women who had comparable labour market situations before the health event. This allows for a better assessment of the health-labour causality because the women in our cancer group have the same occupation as the women in the control group before the health event occurs. Therefore, we can rule out reverse causality from labour to health, as the women who work more often are matched with similar women in the control group. Second, the standard DiD method assumes that the time effects share parallel trends in the health-event group and in the control group. By matching on individual characteristics, we allow for the time trends to differ between the health-event and control groups, to the extent that the slopes of the trends depend on the individual characteristics and the lagged endogenous data. For these reasons, our application of DiD should provide a more robust estimation than in the standard case.

The Hygie data set provides a detailed, dynamic account of two main variables: cancer occurrence and occupational status. In this section, we analyse how it is possible to evaluate the impact of cancer on subsequent labour market history. To successfully identify the impact of health events, we need to account for two types of quantities: on one hand, the difference in histories between those women who experienced cancer and other women and, on the other hand, the variations in the labour history of one woman before and after cancer. In this section, we show that the DiD method with matching allows us to estimate the effect of cancer by controlling both for the observable individual variables and unobservable individual heterogeneity, including when the latter is correlated with the observable individual variables.

The outcome variables are the annual activity dummies corresponding to the three following occupational statuses: employment, unemployment and sick leave. One can interpret our analysis as an assessment of the impact of breast cancer on these occupational dummies.

We consider all the women with breast cancer, denoted i ∈ I , where I stands both for the index set of the women with breast cancer and their number. A woman i ∈ I is observed between the years t - i and t + i , and a breast cancer diagnosis occurs in year t i ∈ (t - i , t + i ). To evaluate the effect of the cancer, we compare the occupation of woman i in t i -1 to the occupational choice k years after the health event, in t i +k ≤ t + i . In what follows, we take employment as example, but any other occupational status can be treated in the same manner. The employment probability of woman i during year t , denoted p i ,t , depends on a vector of individual explanatory variables X i , an unobservable individual effect α i , potentially correlated with X i , a time effect β 0,t and a joint effect of the explanatory variables with the time effect β 1,t (X i ). The employment dummy variable d i ,t follows a Bernoulli process with mean p i ,t given by

d i ,t = p i ,t + i ,t p i ,t = f i (X i ) + α i + β 0t + β 1,t (X i ) + γ i (t -t i ) × T i ,t
where f i (.) is an unknown function relating X i to the employment probability p i ,t , γ i is the effect of the breast cancer on the probability of being employed, and T i ,t a dummy variable equal to 1 if there is a breast cancer (t ≥ t i ), 0 otherwise (t < t i ). The γ i terms depend on how much time passed since the breast cancer occurred tt i . The i ,t terms are idiosyncratic error terms with E i ,t X i , α i , β 0t , β 1t (X i ), T i ,t = 0. Henceforth, we consider the effect of the cancer between t i -1 and t i + k, and thus, we wish to estimate an average value for γ i (k), k ≥ 1.

The estimation proceeds through the elimination of all the components except γ i (k). The techniques used to achieve this goal are based on differencing (for α i and β 0,t ), matching (for X i and β 1,t (X i )) and averaging (for i ,t ). In the first step, we will match the women experiencing a cancer (i ∈ I ) with their (statistical) twins as follows:

J (i ) = j : t - j ≤ t i -1, t i + k ≤ t + j , t j > t i + k and X j = X i
the two first inequalities simply require that the twins be present over the same period as woman i . The third inequality defines dynamic matching, the twins in J (i ) should experience their cancers (or another long-term disease, if any) after the end of the comparison period for woman i . This implies that we match i with, on one hand, women who will neither experience cancer nor another long-term disease and, on the other hand, women who will experience either cancer or another long-term disease at a later date. When someone does not experience cancer (or another long-term disease), we use the convention t j = {+∞}. Ultimately, the matched individuals should have the same individual characteristics. The notation J (i ) will also be used to indicate the number of twins matched with woman i . Note that two women can share common twins, as we make use of all potential matches for each woman. By definition, the outcome variable of the pair does not include the effect of the cancer, and hence the pair's outcome variable is given by

d j ,t = p j ,t + j ,t p j ,t = f j (X j ) + α j + β 0t + β 1,t (X j )
and the average outcome of i 's twins is given by 1

J (i ) j ∈J (i ) d j ,t = β 0t + β 1,t (X i ) + 1
J (i ) j ∈J (i ) f j (X i ) + α j + j ,t
Consider first the difference between woman i and all her matches j ∈ J (i ) before the health event; we eliminate the terms in β 0,t i -1 and β 1,t i -1 (X i ) and obtain the following:

D i ,t i -1 = d i ,t i -1 - 1
J (i ) j ∈J (i ) d j ,t i -1 (1)
= f i (X i ) + α i + i ,t i -1 - 1
J (i ) j ∈J (i ) f j (X i ) + α j + j ,t i -1
and when we take the difference after the cancer date, we also eliminate the β components:

D i ,t i +k = = d i ,t i +k - 1
J (i ) j ∈J (i ) d j ,t i +k (2) = f i (X i ) + α i + γ i (k) + i ,t i +k - 1
J (i ) j ∈J (i ) f j (X i ) + α j + j ,t i +k
the DiD (2) and (1) therefore leads to

DD i (k) = D i ,t i +k -D i ,t i -1 = γ i (k) + i ,t i +k -i ,t i -1 - 1
J (i ) j ∈J (i ) j ,t i +k -j ,t i -1 such that E(DD i (k)) = γ i (k) ∀i , k.
Our estimator is simply the average of these individual health effects. We define

γ(k) = 1
I i ∈I DD i (k) such that E γ(k) = 1
I i ∈I γ i (k).
It remains to compute the variance of γ(k). The method is explained in the appendix. We performed the estimation with SAS 9.4.

Results

Our sample includes 2,547 women who suffer from a breast cancer for enough years to compute a before-after difference. These women will be matched with 203,392 women with no long-term (six months) disease and 19,059 women with another long-term disease. Overall, women who experienced breast cancer have similar incomes to the others. They are somewhat less represented in the highest income class. They are also significantly older, as expected. In 2008, 60.4% of women with no disease were less than 45 years old, while only 13.5% of women with a breast cancer were under 45 (Table 1).

We use a matching method to eliminate the effect of the confounding variables on this naive average difference in outcome. We then perform five estimations (full sample; by age of cancer onset (more or less than the median age of 48); by first wage classes (in quartiles); by the stability of the professional career (measured with the indicator (c), which represents the share of the number of years worked with at least one unemployment spell over the number of years worked, and three classes are selected c = 0, low unemployment (0 < c ≤ 0.16) and high unemployment (c > 0.16)); and by onset year with three classes (t ≤ 1990, 1991 ≤ t ≤ 1999 and t ≥ 2000), thus allowing us to measure short-(one year) and long-term (2 to 5 year) breast cancer effects on two labour market outcomes (employment and regular employment, e.g., without unemployment spells).

Effects on employment.

We use two indicators of employment. The first indicator is a dummy variable equal to one when a worker has validated at least one quarter in employment. We call this "employment" because it is representative of a job drawn at random from the total population. This definition includes all types of jobs: stable jobs but also insecure jobs and some quarters of unemployment or disease in the same year. To better assess the effect of breast cancer, we also use a more restrictive definition of employment "regular employment". Here, we require that there is no unemployment during the same year. Note that both definitions of employment are compatible with fixed-term contracts, and hence the "regular employment" jobs need not be especially stable over time. This indicator simply identifies workers who have been working during the year without experiencing unemployment.

The employment probability decreases by 9.8 pp one year after the onset of cancer relative to the not-treated group (Table 2). We also observe that the detrimental effect of breast cancer on employment significantly increases over time, reaching 12.1 pp five years later. If we consider regular employment (without unemployment spells), the adverse effect remains but is less pronounced (from 6.3 pp in t + 1 to 10.5 in t + 5). At this stage, we can assume that the regular employment variable selects a protected population because such individuals better cope with the negative effects of a breast cancer on the career path.

A young age of occurrence. We compare the effect of cancer depending on the age of the worker at the onset of breast cancer. For workers under 48 (the median age at cancer onset in our data), the negative impact of a breast cancer remains constant over time (between 8 and 10 pp for irregular employment, 7 pp for permanent employment), whereas this impact increases linearly in the older female population (above 48), from 8.4 pp to 15.2 pp for employment and from 5.2 to 13 pp for regular employment. These patterns suggest that late cancer onset is more detrimental to the employment of breast cancer survivors, with a negative effect that increases over time. This pattern should be explored in depth and is likely due to opposite effects. Indeed, from a medical perspective, breast cancers at a younger age (before menopause) are known to be more aggressive, everything being equal, than a cancer onset occurring after menopause (after the mean age of 51 in France). 4Early cancers are often diagnosed with delay due to the absence of screening programmes and the low prevalence of the desease in that age range (less than 5% of breast cancers occur before 40). Moreover, early cancers usually receive a worse prognosis. Furthermore, we can assume that the young women may have more "energy" to return to work earlier than do older women, who also tend to be more discriminated against. However, several alternative explanations might account for the consequences of a patient's age at cancer onset. First, the nature of comorbidities and treatments might differ depending on the age of occurrence. Second, breast cancer occurring at older ages can be especially disabling and prevent women from maintaining a job (undergoing hormonotherapy, which is more frequently applied for post-menopausal women, is given for five years and often entails unpleasant side effects). Third, as more women approach retirement age, the opportunity cost of exiting employment decreases. Fourth, the decline in the probability of employment for older women may be explained by the "double penalty" phenomenon that can lead to amplified effects of exiting the labour market. Traditional analyses of investments in human and health capital can be enhanced by accounting for changes in preferences or age-related discrimination, which has been particularly noted for older workers (e.g., Datta Gupta and Larsen (2010)).

High income effect. We test the protective role of initial income on the negative impact of breast cancer on employment by dividing the sample into four classes (relative wage at the date of labour market entry). 5 Our findings show that workers in the upper relative wage class (above the third quartile of the relative wage) are especially protected (Table 3). The negative effect of cancer varies from 3 to 12 pp only, while it is considerably larger for other workers (under both employment measures). The decrease in the employment rate in the lowest relative wage class tends to increase over time from 10 pp (t + 1) to 18 pp (t + 5). This detrimental effect of cancer increases over time for all wage classes. The effect of cancer is strictly decreasing with the relative wage class from the first to the third year after the onset of cancer. For instance, for employment, the effects after one year are -15 pp, -12 pp, -9.3 pp and -4.7 pp. Several arguments can explain these results. First, the French social protection system is highly redistributive and allows lower income workers to secure a better income replacement rate than higher income workers. This may create an inactivity trap that would explain the long term effect of -17 pp for the first quartile. Second, the lower wage class workers are more often affected by comorbidities associated with poor working conditions and tend to exit the labour market permanently. Third, the lowest quartile workers cannot easily benefit from work arrangements because the nature of their jobs makes doing so difficult.

Career stability. We test another assumption related to the nature of the past career (measured by the occurrence of unemployment spells in past years). The results show that employment regular employment both decrease by 5 pp one year after breast cancer for women without any past unemployment spells. This short-term effect is greater for individuals who were unemployment in the past. Workers with a low past unemployment rate face a 11 pp decrease in employment, while those with a high past unemployment rate (above 16%) face a 16.6 pp decrease. However, this result does not extend to regular employment, as all workers with past unemployment spells face a 8 pp decrease one year after the onset of cancer. When comparing the outcome over a longer horizon, we obtain different findings with respect to stability class. People with a low past unemployment rate face a stronger decrease in their regular employment rate than do workers with the highest past unemployment rates. This could indicate the following dynamics: people with a low past unemployment rate would be more frequently driven toward unemployment and transition into a higher unemployment class in the future. Our findings indicate that for individuals who had previous unemployment spells, it is more difficult to remain in regular employment after a cancer shock.

A generational effect.

The last test concerns a potential generational effect. We consider the impact of expected medical advances on the time to return to work for young. To test this hypothesis, we perform the estimation by onset year and consider short and medium terms (from t + 1 to t + 3). In the year following the diagnosis, the decline in irregular employment is very similar whatever the onset year (about 9-10 pp), which appears to contradict the innovation assumption. Nevertheless, we stress the clear generational effect we observe concerning the effect on regular employment. For a cancer diagnosis that occurred after 2000, the decrease in employment is 5.2 pp versus 6.2 pp between 1990 and 2000 and 9.3 pp if the cancer appears before 1990. The gap between generations is corroborated for two and three years after cancer onset. The decline in employment for the new generation can be explained by several factors. The improvement in medical treatments for cancer in recent years could support this effect. The incidence of in situ cancers increased significantly from 1990 to 2005 in all age groups but particularly among 50 to 74-year-old women. This trend reversed after 2005. The incidence of invasive cancers grew slowly from 1990 to 1996 and then more sharply beginning in 1996, primarily among women aged 50-74 years, before declining in 2004. Finally, cancers in an advanced stage at diagnosis decreased after a peak in the early 2000s. Changes in incidence likely reflect the combined influence of several factors (risk factors, screening and diagnostic techniques). In addition, organized mammography screening for breast cancer was widespread in France in 2004. This programme allows all women aged 50 to 74 to have a mammogram and a clinical breast exam once every two years. Thus, for breast cancer, the 5-year survival increased from 81% in 1990 to 89% in 2002.

Discussion and conclusion

Our study is original in several respects. First, the estimation of the effect of cancer on professional situations covers a long-term period from one to five years after diagnosis. In addition, it relies on administrative data to identify the careers of a large sample of private sector employees. Moreover, the sample size permits performing a DiD analysis with exact matching and defining a rigorous control group that exploits the panel dimension of the data. Finally, we examine a relatively young female population (in our sample, 43% of the women with a breast cancer are less than 45 years of age), which reinforces the relevance of the analysis of career paths of this population.

Our main findings confirm, for the first time in France using this econometric method, the detrimental effect of breast cancer on employment. The proportion of individuals who have completed at least one quarter of employment decreases substantially after the onset of cancer: by 10 pp after one year, and the effect is long-lasting because it remains at 12 pp five years after cancer onset.

It is obviously difficult to compare our results with those of other studies because of the differences in data and methods and cross-country differences in labour market structure, public financing of cancer's cost and sick leave. Bearing this in mind, our findings are similar to those of Moran, Short, and Hollenbeak (2011) and [START_REF] Heinesen | Effects of breast and colorectal cancer on labour market outcomes-average effects and educational gradients[END_REF], who use similar econometric methods by combining DiD and propensity score matching. Using Danish administrative register data, Heinesen and Kolodziejczyk (2013) estimate the effects of breast cancer on labour market outcomes for three-year survivors. Using two different control groups, they find following probabilities of decreased employment: 4.4 pp in t + 1, 5.3-5.8 pp in t + 2 and 6.2-6.7 pp in t + 3.

Using US data, Moran, Short, and Hollenbeak (2011) focus on a young population, as our study does (cancer at younger ages, 28-54 years) and estimate the effect of surviving cancer on long-term employment outcomes (2-6 years post-diagnosis). Breast cancer survivors had employment rates that were 7-8 pp lower than the control group. These findings are significantly lower than ours. In general, the generosity of the French health insurance system can explain the more common occurrence of inactivity.

The originality of our study is to focus on many demographic and professional characteristics as protective factors against the deleterious effects of breast cancer on employment. As expected, a young age of occurrence, a high initial wage class and past employment stability promote a better return to work (similar to the results of Heinesen and Kolodziejczyk in 2013). Many studies identify social gradients in breast cancer survivors (e.g., [START_REF] Carlsen | Cancer and the risk for taking early retirement pension: a Danish cohort study[END_REF]).

Nevertheless, the most interesting findings concern the potential generational effect related to the medical advances. This result should be explored in greater depth.

Finally, several limitations of our study should be noted. In addition to that related to the definition of cancer (specific to this study), the data do not allow the identification of the cancer stage, the type of treatment or the severity. Before the Cancer Plan, which was adopted in France in 2004, that is during the common period of our study, an hypothesis can be made regarding the existence of a link between the aggressiveness of treatments (that is with significant pain and potentially adverse long-term side effects) and social gradients. Indeed, due to improved prevention behaviours, upper-income-class women are often better diagnosed and at an earlier stage of their cancer and can therefore recover without as much recurrence. However, this holds only for the older women (who are also benefiting from organized cancer screening programmes). TABLE 2. Effect of a breast cancer (C50), time-related conditioning. Difference-in-differences with dynamic matching estimates. Matching variables: lagged outcome dummies (employment, unemployment, disease), year of birth (exact), first job relative income class (r , 4 levels), past disease class (3 levels), past unemployment class (c,3 levels).

A Estimation of the standard errors

This section presents the computation of the standard errors without using bootstrapping, to save computing time. The ATT estimator, denoted ĉ, can be written as follows: ĉ = 1

I i ∈I ĉi with ĉi = ∆y i - 1
J (i ) j ∈J (i ) ∆y j
where I is the treated set (and their number), y i is the outcome variable of individual i , ∆y i is the before-after difference and J (i ) is the set of individual i 's twins (and their number). The previous formula defines the DiD estimator. If two treated individuals have the same matching variables, and if their treatment occurs on the same date, they will be matched with exactly the same twins, and hence the same mean will be subtracted from their outcome variable. We regroup the treated according to their matching variables and treatment date. Let k ∈ K be a specific vector regrouping the matching variables and the treatment date; the set of all the treated individuals in the matching group k (and their number) is defined by

I (k) = {i ∈ I : (X i , t i ) = k} , k ∈ K
and we let J (k) denote the common matching twins' set of the treated in group k (and their number). By definition, the I (k) sets define a partition of the treated set I = k I (k), I (k) I (k ) = ∀k = k . Therefore, the ATT can be rewritten as follows: ĉ = 1

I k∈K i ∈I (k) ∆y i -m k
where m k = J (k) -1 j ∈J (k) ∆y j is the matching twins' common mean within group k. Simplifying within the sum, we obtain ĉ = 1

I k∈K i ∈I (k) ∆y i -I (k)m k
To compute the variance of this estimator, we make the standard independence assumption between the y i 's. First, the groups k are independent of one another because they have neither a treated nor a twin in common. We obtain

V ( ĉ) = 1 I 2 k∈K V i ∈I (k) ∆y i -I (k)m k
Second, the y i 's are independent of the m k 's because they are computed from different individuals. We obtain

V ( ĉ) = 1 I 2 k∈K V i ∈I (k) ∆y i + I (k) 2 V (m k ) = k∈K I (k) I 2 V m T k + V (m k ) with m T k = I (k) -1
i ∈I (k) ∆y i the mean outcome of the treated within group k. The estimator is obtained by replacing the theoretical statistics with their empirical counterparts. TABLE 4. Effect of a breast cancer, complementary variables (C50), time-related conditioning. Difference-in-differences with dynamic matching estimates. Matching variables: lagged outcome dummies (employment, unemployment, disease), year of birth (exact), first job relative income class (r , 4 levels), past disease class (3 levels), past unemployment class (c,3 levels). Note: ATT: Average effect of the treatment on the treated. ASE: Asymptotic standard error. * : significant at 5%. †: significant at 10%. TABLE 5. Effect of a breast cancer (C50), complementary variables, socioeconomic conditioning. Difference-in-differences with dynamic matching estimates. Matching variables: lagged outcome dummies (employment, unemployment, disease), year of birth (exact), first job relative income class (r , 4 levels), past disease class (3 levels), past unemployment class (c, 3 levels).

  al. (2010), Johnsson et al. (2011)). The onset of cancer can also modify the nature of the labour contract (e.g., full-time/part-time, working hours). Many studies shed light on the relationship between cancer occurrence and work duration (Farley, Vasey, and Moran (2008), Paraponaris, Teyssier, and Ventelou (2010), Petersson et al. (2011), Torp et al. (2011)). From a Swedish sample of 756 working women who have undergone breast cancer surgery, Petersson et al. (

TABLE 1 .

 1 Sample statistics.

	Variable	No chronic disease Breast cancer Difference
	a ≤ 35	32.6%	2.1%	+30.5%
	35 < a ≤ 45	27.8%	11.4%	+16.4%
	45 < a ≤ 55	19.0%	30.4%	-11.4%
	a > 55	20.6%	56.1%	-35.5%
	r ≤ Q 1	23.9%	25.5%	-1.6%
	Q 1 < r ≤ Me	25.2%	20.1%	+5.1%
	Me < r ≤ Q 3	25.4%	23.6%	+1.8%
	r > Q 3	25.5%	30.9%	-5.4%

Note: a: age in 2008. r : first job relative income class.

TABLE 3 .

 3 Effect of a breast cancer (C50), socioeconomic conditioning. Difference-indifferences with dynamic matching estimates. Matching variables: lagged outcome dummies (employment, unemployment, disease), year of birth (exact), first job relative income class (r , 4 levels), past disease class (3 levels), past unemployment class (c, 3 levels).

	Time				Employment		Without unemployment
	after event	Treated Matched # twins	Treated lagged	ATT	ASE	Treated lagged	ATT	ASE
	r ≤ Q 1 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	615 534 465 381 330	100.0% 100.0% 100.0% 100.0% 100.0%	563 533 513 488 459	t i -1 0.795 0.799 0.804 0.811 0821	-0.152 * -0.132 * -0.137 * -0.122 * -0.176 *	0.012 0.013 0.014 0.015 0.017	t i -1 0.676 0.677 0.685 0.690 0.694	-0.101 * -0.107 * -0.126 * -0.110 * -0.165 *	0.013 0.013 0.015 0.016 0.018
	Q 1 < r ≤ Me t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	499 440 377 329 289	100.0% 100.0% 100.0% 100.0% 100.0%	509 479 456 412 381	t i -1 0.840 0.848 0.859 0.857 0.848	-0.119 * -0.097 * -0.081 * -0.079 * -0.083 *	0.010 0.011 0.012 0.014 0.014	t i -1 0.697 0.702 0.713 0.705 0.685	-0.071 * -0.075 * -0.063 * -0.067 * -0.077 *	0.011 0.013 0.015 0016 0.017
	Me < r ≤ Q 3 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	649 567 488 406 341	100.0% 100.0% 100.0% 99.8% 100.0%	653 621 600 558 517	t i -1 0.892 0.899 0.902 0.901 0.903	-0.093 * -0.082 * -0.081 * -0.082 * -0.103 *	0.010 0.010 0.011 0.013 0.013	t i -1 0.807 0.811 0.807 0.820 0.827	-0.060 * -0.060 * -0.066 * -0.089 * -0.084 *	0.010 0.011 0.012 0.014 0.015
	r > Q 3 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	784 680 604 528 450	100.0% 100.0% 100.0% 100.0% 100.0%	660 635 620 598 574	t i -1 0.899 0.907 0.911 0.917 0.920	-0.047 * -0.051 * -0.061 * -0.092 * -0.119 *	0.007 0.008 0.010 0.011 0.014	t i -1 0.843 0.854 0.864 0.877 0.878	-0.031 * -0.049 * -0.054 * -0.090 * -0.095 *	0.008 0.010 0.011 0.013 0.015
	c = 0 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	1217 1082 967 848 739	100.0% 100.0% 100.0% 99.9% 100.0%	351 352 352 348 342	t i -1 0.917 0.920 0.918 0.920 0.924	-0.054 * -0.057 * -0.080 * -0.089 * -0.120 *	0.007 0.008 0.009 0.010 0.012	t i -1 0.917 0.920 0.918 0.920 0.924	-0.047 * -0.063 * -0.077 * -0.099 * -0.110 *	0.008 0.009 0.010 0.011 0.012
	0 < c ≤ 0.16 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	743 634 533 446 367	100.0% 100.0% 100.0% 100.0% 100.0%	163 149 142 130 114	t i -1 0.846 0.853 0.862 0.871 0.874	-0.114 * -0.102 * -0.099 * -0.109 * -0.125 *	0.008 0.009 0.010 0.010 0.011	t i -1 0.704 0.710 0.720 0.727 0.717	-0.077 * -0.079 * -0.096 * -0.109 * -0.111 *	0.009 0.010 0.011 0.012 0.012
	c > 0.16 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	587 505 434 350 304	100.0% 100.0% 100.0% 100.0% 100.0%	283 261 247 229 215	t i -1 0.764 0.777 0.789 0.787 0.781	-0.166 * -0.127 * -0.093 * -0.083 * -0.119 *	0.012 0.012 0.011 0.013 0.013	t i -1 0.533 0.531 0.539 0.543 0.530	-0.079 * -0.082 * -0.053 * -0.043 * -0.087 *	0.012 0.013 0.014 0.015 0.016
	Note: ATT: Average effect of the treatment on the treated. ASE: Asymptotic standard error. * : significant at 5%.
	†: significant at 10%.

  Note: ATT: Average effect of the treatment on the treated. ASE: Asymptotic standard error. * : significant at 5%.

	Time					Disease		Unemployment
	after event	Treated Matched # twins	Treated lagged	ATT	ASE	Treated lagged	ATT	ASE
	r ≤ Q 1 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	615 534 465 381 330	100.0% 100.0% 100.0% 100.0% 100.0%	563 533 513 488 459	t i -1 0.039 0.039 0.039 0.037 0.030	0.510 * 0.295 * 0.151 * 0.044 * 0.029 *	0.014 0.014 0.013 0.010 0.010	t i -1 0.208 0.212 0.192 0.189 0.185	-0.048 * -0.013 0.017 0.011 0.043 *	0.010 0.011 0.013 0.014 0.015
	Q 1 < r ≤ Me t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	499 440 377 329 289	100.0% 100.0% 100.0% 100.0% 100.0%	509 479 456 412 381	t i -1 0.074 0.073 0.074 0.076 0.076	0.486 * 0.215 * 0.122 * 0.035 * 0.060 *	0.015 0.014 0.013 0.011 0.011	t i -1 0.214 0.209 0.202 0.210 0.221	-0.043 * 0002 0.016 0.029 * 0.009	0.008 0.011 0.014 0.014 0.014
	Me < r ≤ Q 3 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	649 567 488 406 341	100.0% 100.0% 100.0% 99.8% 100.0%	653 621 600 558 517	t i -1 0.052 0.060 0.061 0.064 0.067	0588 * 0.323 * 0.172 * 0.045 * 0.046 *	0.014 0.015 0.014 0.010 0.011	t i -1 0134 0.136 0.139 0.123 0.120	-0.039 * -0.013 0.012 0.025 * 0.004	0.007 0.009 0.011 0.012 0.013
	r > Q 3 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	784 680 604 528 450	1000% 100.0% 100.0% 100.0% 100.0%	660 635 620 598 574	t i -1 0.061 0.059 0.060 0.059 0.062	0.518 * 0.256 * 0.139 * 0.058 * 0.059 *	0.014 0.014 0.012 0.010 0.012	t i -1 0.120 0.109 0.099 0.083 0.082	-0.023 * -0.002 -0.011 0.003 -0.004	0.006 0.009 0.009 0.011 0.013
	c = 0 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	1217 1082 967 848 739	100.0% 100.0% 100.0% 99.9% 100.0%	351 352 352 348 342	t i -1 0.044 0.047 0.050 0.050 0.050	0.526 * 0.277 * 0.143 * 0.051 * 0.056 *	0.012 0.012 0.011 0.008 0.009	t i -1 0.000 0.000 0.000 0.000 0.000	-0.008 0.015 * 0.022 * 0.035 * 0.035 *	0.005 0.007 0.008 0.010 0.010
	0 < c ≤ 0.16 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	743 634 533 446 367	100.0% 100.0% 100.0% 100.0% 100.0%	163 149 142 130 114	t i -1 0.057 0.055 0.051 0.050 0.052	0.514 * 0.249 * 0.135 * 0.031 * 0.035 *	0.011 0.010 0.010 0.008 0.008	t i -1 0.239 0.230 0.214 0.205 0.206	-0.034 * -0.013 0.007 0.019 † 0.009	0.007 0.008 0.010 0.011 0.011
	c > 0.16 t i + 1 t i + 2 t i + 3 t i + 4 t i + 5	587 505 434 350 304	100.0% 100.0% 100.0% 100.0% 100.0%	283 261 247 229 215	t i -1 0.075 0.074 0.076 0.080 0.076	0.551 * 0.307 * 0.170 * 0.056 * 0.047 *	0.013 0.014 0.013 0.010 0.010	t i -1 0.406 0.417 0.414 0.408 0.417	-0.099 * -0.044 * -0.030 * -0.037 * -0.044 *	0.010 0.013 0.014 0.014 0.015

†: significant at 10%.

The median age at breast cancer onset was approximately 65 years in France in 2012 (INCA (2015)). Cancer before age 40 represents 5% of the cases diagnosed. Diagnoses of new cases are generally made after 55 years of age, partly due to systematic screening from 50 years onward (INCA (2014)).

[START_REF] Paraponaris | Job tenure and self-reported workplace discrimination for cancer survivors 2 years after diagnosis: Does employment legislation matter?[END_REF] highlight the endogeneity of discrimination for female survivors after cancer. They underline that productivity or number of children could contribute to a decrease in the return to work through the effect of perceived discrimination.

Cancers at a young age are more often "triple negative," which prevents the use of efficient treatments. Hormonotherapy for hormonodependent cancer treatments is more effective after menopause than before, and the neo-adjuvant therapy for non-hormonodependent cancers is not considered effective (see, e.g.,[START_REF] Pourquier | Le traitement des cancers du sein : ses progrès et ses limites[END_REF])

.[START_REF] Bradley | Short-term effects of breast cancer on labor market attachment : Results from a longitudinal study[END_REF] The relative wage is equal to the ratio of the entry wage to the median entry wage in the same year. Four classes are defined by the quartiles of this variable, which is defined for all workers. Notice that the quartiles are not computed based on the cancer population but on the total population.


D Christian Glattli
email: christian.glattli@cea.fr

Preden S Roulleau

Levitons for electron quantum optics

Keywords: single electron source, quantum shot noise, levitons

Levitons for electron quantum optics

1 Single electron sources In this introduction, we will distinguish single charge sources from coherent single electrons sources. The former have been developed for quantum metrology where the goal is to transfer an integer charge at high frequency f through a conductor with good accuracy to realize a quantized current source whose current I = ef shows metrological accuracy. The latter, the coherent single electrons source, aims at emitting (injecting) a single electron whose wave-function is well defined and controlled to realize further single electron coherent manipulation via quantum gates. The gates are provided by electronic beam-splitters made with Quantum Point Contacts or provided by electronic Mach-Zehnder and Fabry-Prot interferometers. Here it is important that the injected single electron is the only excitation created in the conductor. The frequency f of injection is not chosen to have a large current, as current accuracy is not the goal, but only to get sufficient statistics on the electron transfer events to extract physical information.

single charge sources for current standards

The first manipulation of single charges trace back to the early 90's where physicists took advantage of charge quantization of a submicronic metallic island nearly isolated from leads by tunnel barriers. The finite energy E C = e 2 /2C to charge the small capacitor C with a single charge being larger than temperature (typically one kelvin for C =1fF), Coulomb Blockade of tunneling occurs allowing for single charge manipulations. An architecture made of several island in series lead to single electron pumps and turnstiles [1,2]. Pumps and turnstiles were dedicated to quantum metrology with the hope to realize a current standard where the current I = ef is related to the frequency of single charge transfer through the device. A 7-junction pumps lead soon to 15.10 -8 current accuracy but limited to too low currents [3] for metrology. Later hybrid superconducting normal junctions single electron pumps lead to higher operating frequencies to reach 10 -6 accuracy [4].

In parallel, electron pumps have been realized using 2D electrons confined in GaAs/GaAlAs semiconductors heterojunctions where the metallic island is replaced by a quantum dot obtained by surface gate depletion. Here pairs of split gates were used to form a narrow constriction which selects the number of electronic modes transmitted between the dot and the leads and ultimately leads to a 1D tunnel barrier. This Quantum Point Contact (QPC) offers tunability of the tunnel barrier whose fast manipulation was exploited to capture a single electron in the dot from the left contact and release it to the right contact providing fast accurate electron pumping [5]. Similar double modulated barrier pumps at GHz frequencies lead to average pump current of several hundred picaoamperes [6] more suitable for metrology. Single gate pumps have been implemented using silicon nanowires [7] and etched GaAl/GaAsAl nanowires. Recent improvements gave accuracy better than 1.2.10 -6 for a 150pA generated current [8] making these devices promising for current standard in quantum metrology while similar devices combined with quantum Hall effect may give a short path to metrology triangle closure [9].

Another approach to single charge sources is based on moving quantum dots carrying integer charges using of Surface Acoustic Waves (SAWs) [10]. In a piezo-electric material like GaAs, a mechanical wave can be generated by the electric field created by interdigitated gates deposited on top of the device. An electrical potential well moving at the surface phonon velocity, a few km/s, can trap electrons confined in a 1D wire and the moving quantum dot so realized can transfer them through the conductor at GHz frequencies leading to good current quantization accuracy [11].

single electron sources for electron quantum

optics In all the examples cited above, the goal was a good accuracy in the charge transfer. In the following we will focus on quantum experiments appropriate for electron quantum optics where the emitted single electrons play the role of flying charge qubits [12]. Depending on the system used the charge or the spin may code a binary qubit information. The flying qubit approach is fundamentally different from the standard qubit realization which are based on a static localized two-level system (nuclear or electron spin, atom, charge state of a superconducting metallic island, photonic state of resonator circuits, etc.).

Here the information is coded by the presence or not of a particle (electron, photon) propagating or delocalized in spatial modes.

Experiments in this direction using the SAW technique described above have been done recently. SAW assisted transfer of single electrons through depleted 1D channels have demonstrated single electron transfer over several µm between two distant quantum dots [13,14] and the spin state of one and two electrons have been transferred [15]. Using quantum dot sources, single electrons have been emitted at high energy above the conductor Fermi level. The energy and time properties of the wavepacket have been analyzed [16]. The partitioning of spin entangled electron pairs emitted that way has been recently realized [17]. These examples are promising for further quantum experiments where only the spin coherence of electrons is concerned, but regarding orbital quantum coherence, appropriate for charge flying qubits, different approaches are needed.

The first coherent single electron source has been proposed and realized by one of the present authors with the ENS Paris team [18]. It is based on a mesocopic capacitor which has been precedently realized [19] to check the universal quantization of the charge relaxation resistance (called Büttiker's resistance h/2e 2 ) predicted by M. Büttiker [20]. Here no dc current is produced but only an ac current made from the periodic injection of single electrons followed by single holes above and below the Fermi energy E F of the target quantum conductor. For ease of operation and further use in electron quantum optics, the conductor is in the integer quantum Hall effect regime where a strong magnetic field quantizes the electron cyclotron orbits and only 1D chiral modes propagating along the sample edges, called edge channels, ensure electron conduction. Here, the capacitor is made so small that energy level quantization makes it behave as a quantum dot. To ensure energy level and charge quantization, the capacitor is weakly connected to the leads, the chiral edge channels, via a quantum point contact which controls the tunnel coupling. The operating principle is as follows, see Fig. 1. Starting from a situation where the last occupied energy level is below the Fermi energy, a sudden rise of the voltage applied on a top gate capacitively coupled to the mesoscopic capacitor rises the occupied energy level above the Fermi energy. After a time equal to the life time ≃ h/D∆ of the energy level and controlled by the barrier transmission D, an electron is emitted at a tunable energy ε e above the Fermi level (∆ is the energy level spacing). Then restoring the top gate voltage to its initial value pulls down the energy level below the Fermi energy: an electron is captured or equivalently a hole is emitted with a definite energy -ε h below the lead Fermi energy.

The mesoscopic capacitor single electron source has been used for electron quantum optics experiments like the partitioning of single electrons in an electronic beam splitter [21], an Hanbury Brown-Twiss experimental set-up commonly used in optics, and the Hong Ou mandel (HOM) electron interferometry where two electron emitted with a controlled relative time delay are sent to an electronic beam-splitter and second order interference revealing particle indistinguishability is observed in the detection statistics at the beam-splitter output [22]. Similar experiments have been repeated with the leviton source discussed below [23,24]. The 2007 first realization of a single electron source has triggered many theoretical works [25][26][27][29][30][31][32][33][34] as it opened the possibility to test dynamical quantum transport at the single electron level, investigating single and two-electron coherence, non-local entanglement, waiting time statistics or quantum heat fluctuations. However the mesocopic capacitor source is delicate to operate and to fabricate (tuning two identical energy levels and emission time to demonstrate electron undistinguishability in [22] was a tour de force). Like single photon sources it displays a fundamental quantum uncertainty in emission time, a quantum jitter measured by high frequency shot noise in [35]. A simpler complementary approach to single electron sources for electron quantum optics came more recently with the realization of voltage pulse sources. [18]).

D V exc D Time 2eV (t) exc V G 1 2 3 1 2 3 a) b) V exc V exc V exc 8
The voltage pulse source is based on a simple principle. A voltage pulse V (t) is applied on the contact of a quantum conductor where electrons can propagate in few modes. According to ac quantum transport laws, if all other contacts are grounded, a current I(t) = e 2 /hV (t) is injected form the contact in each mode emitted from this contact (disregarding spin). Then, by tuning the amplitude and duration of the voltage pulse such that ∫ ∞ -∞ I(t)dt = e a single charge is injected. The procedure seems too simple to think it can give correct results. Indeed, if a single charge is actually injected, it is in general accompanied by unwanted neutral excitations in the form of electron-hole pairs resulting from the perturbation of the electrons already present in the conductor (the Fermi Sea). However, nearly twenty years ago, in their quest for a clean electron generation model for studying the Full Counting Statistics in quantum conductors, L. Levitov and collaborators discovered that, if the voltage pulse has the shape of a Lorentzian, a clean single electron injection (without extra neutral excitations) can be realized [36][37][38][39]. The Lorentzian voltage pulse single electron source has been experimentally realized in 2013 by the present authors and their team at CEA Paris-Saclay and the predicted minimal excitations states, now called levitons, have been carefully characterized by their minimal shot noise [23]. The single electron leviton source is presented in more detailed and reviewed in the next parts of the present paper.

In order to conclude this part I will refer the reader to Fig. 2 which shows a comparison of the Mesoscopic capacitor and Leviton single electron sources, both being suitable to perform electron quantum optics. The figure shows the complementary aspects of the two sources. The mesoscopic capacitor source emits energy resolved electrons while the leviton source is time-resolved. There is a duality of their properties if we interchange time and energy: Lorentzian energy distribution versus Lorentzian time wavepacket, semi-exponential time wavepacket versus semi exponential energy distribution above the Fermi energy. Being energy-resolved, the mesoscopic capacitor source allows to make nice fundamental tests of the energy relaxation and decoherence of a Landau like quasiparticle above the Fermi sea [40]. For the same reason the electrons injected well above the Fermi sea are more prone to decoherence than the levitons whose energy distribution is as close as possible to the Fermi energy [40]. Due to charging effects complicating electron injection, the mesoscopic capacitor source is limited to single charge injection. The voltage pulse source lacks of exact charge quantization (as the charge is tunable) but it can simultaneously injects any number of electrons while keeping the levitonic minimal excitation property. Interestingly, it was theoretically shown that if the mesoscopic capacitor is driven adiabatically (i.e. the energy level not suddenly risen above the Fermi energy but slowly risen at constant speed), the Lorentzian energy shape of the level gives rise 2 Levitons: time-resolved single electron with minimal excitation property.

E F ε t mesoscopic capacitor electron source: t e t Γ - ∝ ) ( ψ 2 2 0 2 ) ( 1 ) ( Γ + - ∝ ε ε ε ψ t iw t t + ∝ 1 ) ( ψ ) (t V E F w e .

principle

We consider a single mode conductor, spin disregarded. The following results can be directly extended to multiple modes including spin. The single mode can be an edge channel of the integer quantum Hall effect for filling factor one. It can also be a single electronic mode of a 2D conductor spatially filtered by a narrow constriction like a Quantum Point Contact inserted in the middle of the 2D conductor. When a time dependent voltage V (t) is applied, say on the left contact of conductor while the opposite contact is grounded, electrons emitted at energy ε below the Fermi level of the contact, and experiencing the voltage, acquire a time dependent phase

ϕ(t) = e ∫ t -∞ V (t ′ )dt ′ /h.
Because of the time dependency, energy is not conserved and the electron are scattered in a superposition of quantum states of different possible energies. The amplitude of probability to have the energy displaced from ε to ε + δε is :

p(δε) = ∫ ∞ -∞ e -iϕ(t) e iδεt/h dt (1)
and the probability P (δε) = |p(δε)| 2 . Eq. 1 is the basis of the Floquet scattering theory developed by M. Büttiker and M. Moskalets for periodically driven conductors [28]. For V (t) a voltage pulse, the injected charge q is q = ∫ ∞ -∞ I(t)dt. For a spinless single mode, for which I(t) = e 2 V (t)/h, this gives the Faraday flux (or action):

∫ ∞ -∞ eV (t)dt = (q/e)h (2)
and the phase increment:

∆ϕ = ϕ(-∞) -ϕ(∞) = 2π(q/e) (3)
In general p(δε) have non zero values for both positive and negative δε. Consequently all electrons of the left contact Fermi sea are displaced up and down in energy. Compared with the reference Fermi sea of the grounded right contact, this gives both electron and hole like excitations. At zero temperature, with sharp Fermi sea distribution, one can quantify the number of electrons N e and hole N h excitations created by the pulse. They are given by:

N e = ∫ ∞ 0 δεP (δε)d(δε) (4)
N h = ∫ 0 -∞ (-δε)P (δε)d(δε) (5)
The charge introduced by the pulse in the conductor is q = e(N e -N h ) and the total number of excitations is

N exc = N e + N h .
For a single electron source injecting electrons in the form of a minimal state called leviton, it is mandatory to have no hole excitations creation, i.e. N exc = N e = 1 such that only excitation is the injected electron, as schematically shown in Fig 3 . To do that, the Fourier transform of the phase term e -iϕ (t) , which in Eq. 1 gives p(δε), must be zero for all negative δε. According to standard complex integration technique, this implies the phase term e -iϕ(t) , prolongated in the complex plane must have no pole in the lower half plane and at least one pole in the upper half complex plane. The simplest expression for the phase term with only one pole, at say t = iw, is:

e -iϕ(t) = t + iw t -iw (6)
With this form, the phase increment is ∆ϕ = 2π and, from Eq.3, the associated charge is q = e while the phase derivative is a Lorentzian with 2w full width at mid-height. The corresponding voltage pulse is therefore a Lorentzian:

V (t) = h e 2w t 2 + w 2 (7)
Eqs. 6,7 uniquely define the type of voltage pulse needed to generate a single charge leviton. Generalization Copyright line will be provided by the publisher to multiple electron injection in the form of levitons injected at time t k with width w k is possible by introducing extra poles in the upper complex plane, such that

energy F E ) ( t V ac ε ) ( / t i t i x k i e e e φ ε - - h ε ε ε ∆ + ε ∆ E F ) ( ~ε f ε el. hole energy F E ) ( t V ac ε ) ( / t i t i x k i e e e φ ε - - h ε ε ε ∆ + ε ∆ E F ) ( ~ε f ε el. hole ) (t V ) (t V
e -iϕ(t) = N ∏ k=1 t -t k + iw k t -t k -iw k (8) V (t) = N ∑ k=1 h e 2w k (t -t k ) 2 + w 2 k (9)
The absence of poles in the lower half part of the complex plane in 8, ensures that the multiple electron injection still forms a minimal excitation state. Simultaneous injection of N electrons in the form of levitons is also possible by simply increasing by N the pulse amplitude. The phase term acquires a single pole of order N, in the upper half complex plane e -iϕ N (t) = ( t+iw t-iw ) N . In [41] it was remarked that such phase modulation generates N orthogonal envelope wave-functions of the form ϕ k (u) = (u+iw) k-1 (u-iw) k , with u = t -x/vF , k = 1, N and the resulting quantum state generated from the voltage pulsed Fermi sea is a slater determinant made of these N wave-functions.

The voltage pulse charge injection method also allows to inject arbitrarily a non-integer charge q. Keeping a Lorentzian pulse shape, this gives the phase term:

e -iϕ(t) = ( t + iw t -iw ) q/e (10)
One immediately sees that if q is not integer, the resulting fractional power gives non-analytic properties to the phase term on the real axis. As a consequence, the Fourier transform of e -iϕ(t) acquires non-zero values for positive and negative δε giving both electron and hole excitations. This situation was named in Levitov's original paper [36] "dynamical orthogonality catastrophe" in reference to the Anderson orthogonality catastrophe problem but in the time domain.

To summarize, Lorentzian voltage pulses generate clean minimal excitation states called levitons carrying an integer number of charge only. Other pulse shape introduce extra excitations in the form of neutral electron hole pairs.

Evidence and characterization of levitons 2.2.1 Minimal noise for minimal excitation states

In a practical experiment, as in [23], periodic injection is chosen. For an injection frequency ν, period T = 1/ν, the average current is I = eν. The periodicity implies that P (δε) = Σ ∞ l=-∞ P l δ(δε-lhν), the discrete P l being the lphoto-absorption probabilities. To experimentally demonstrate that the right pulse shape, the Lorentzian, generates a minimal excitation state, the expected levitons were sent to an artificial quantum impurity, a Quantum Point Contact, which partially transmits a single electronic mode. Its mode transmission D was tunable by gates and measurable by dc conductance. The idea is to take advantage of the partitioning of electron and hole excitations by the QPC into transmitted and reflected modes. This fundamentally quantum probabilistic effect generates what is known as quantum shot noise,where shot noise means current noise. It can be shown that the shot noise at zero temperature is a direct measure of the total number of excitations N exc = N e + N h [36,39,42]. The spectral density of the current noise S I is given by:

S I = 2e 2 νD(1 -D)(N e + N h ) (11)
By comparing the noise generated with sinewave, square wave and Lorentzian pulses, ref. [42] showed that for integer charge, thermal corrections included, the shot noise of Lorentzian pulses is indeed minimal and corresponds exactly to the noise of a clean single charge. Thus minimal excitation states, levitons, generate minimal noise.

energy spectroscopy: absence of hole excitations

Energy spectroscopy is another way to characterize the levitonic state. Indeed, contrary to sinewave or square wave pulses, it is expected that Lorentzian pulses carrying charge e generate no hole excitation. It is thus important to show the absence of hole excitations for energies below the Fermi energy. Again, the partition shot The number of holes is minimized to zero and the shot noise spectroscopy is asymmetric. The figures are adapted from [23] noise of a Quantum Point Contact can be used to demonstrate this. The idea is to add a dc voltage V R on the opposite right contact. Under negative bias, the hole emitted from the periodically driven left contact in the energy range -eV R < < 0 will no longer contribute to noise and the corresponding shot noise variation will measure their number. For positive V R , electrons emitted in the energy range 0 > ε > V R will anti-bunch with electron excitations coming from the driven left contact and no noise is produced. Again, the noise variation with V R will give a measure of the electron excitations. With a finite V R , the shot noise is still given by expression Eq. 11 but for V R > 0, N e is replaced by

N e,VR = ∫ ∞ EVR δεP (δε)d(δε) and N h un- changed while for V R < 0, N e is unchanged by N h relaced by N h,VR = ∫ -e|VR| -∞
(-δε)P (δε)d(δε). Fig. 4 shows shot noise spectroscopy comparison of sine wave and Lorenztian pulses and comparison with predictions. For a sine wave, a symmetric noise variation with V R is observed due to symmetric electron and hole contribution which strikingly differs from the asymmetric noise variation for Lorentzian due to the absence of holes.

Time domain characterization: Hong Ou

Mandel correlations In the celebrated Hong Ou Mandel photonic experiment, the authors were at first interested to determine the time shape of the photon pairs emitted by parametric down conversion of a Laser pulse sent to a non-linear crystal. To do this, they took advantage of the photon bunching effect which is expected to occur when two undistinguishable photons arrive at the same time and mix on a beam-splitter. According to Bose statistics, both photons prefer to exit at the same beamsplitter output and experimentalists repeating the experiment will find fluctuations ⟨(∆N ) 2 ⟩ in the detection photon number N due to the partitioning by the beamsplitter of two bunched photons. This is twice the particle noise expected if the photons were separately arriving and partitioned by the beam-splitter. By varying the delay τ between the photon arrival, one progressively goes from the first to the second situation and the variation is simply given by the overlap of the photon wave-functions ψ which partially mix for finite τ in the beam-splitter, i.e. ⟨(∆N

) 2 ⟩ τ = ⟨(∆N ) 2 ⟩ τ =∞ (1 + |⟨ψ(0)|ψ(τ )⟩| 2 )
. This provided a measure of the photon wave-packet extension and a nice evidence of bosonic quantum statistics. A similar trick can be done with electrons which, being fermions, anti-bunch. When in coincidence (τ = 0) they will always take different outputs to satisfy Pauli exclusion and the particle noise is zero.

Hong Ou Mandel (HOM) correlations can be realized using levitons periodically sent with a relative delay τ from opposite contacts toward a Quantum Point Contact playing the role of the electronic beam-splitter while measuring the cross correlation shot noise between output leads. One expects

S I = 2e 2 νD(1 -D)2(1 -|⟨ψ(x)|ψ(x -v F τ )⟩| 2 ) (12)
where ψ is the leviton wavefunction. If doubly charge levitons are sent, one expects:

S I = 2e 2 νD(1 -D)2(2 -|⟨ψ 1 (x)|ψ 1 (x -v F τ )⟩| 2 - |⟨ψ 2 (x)|ψ 2 (x -v F τ )⟩| 2 ) (13)
where ψ 1,2 are the first two orthogonal levitonic wave functions [41] carrying the two incoming electrons. Remarkably, the voltage pulse electron injection technique allows the generation of an arbitrary number of electrons and for performing N-electrons HOM correlation. In this case, Eq. 13 generalizes easily. Because of constraints due to Fermi statistics, a direct comparison between HOM correlations with N-photon Fock states [59] and those with N-electron minimal excitation states cannot be done. This situation opens new perspectives in quantum mechanics. Fig. 5 shows the HOM shot noise experiment done with single and doubly charged levitons. The comparison with expected HOM noise variation is excellent without adjustable parameters. This provides a good check of the levitonic states with one or two electrons in the time domain. The HOM shot noise of colliding sine-wave pulses is also shown for comparison. Here, because sine-wave pulses give rise to charge pulses accompanied with a cloud of electron-hole pairs and all these excitations interfere in the electron beam-splitter, the physical information to extract is not clear (except to confirm the good agreement with HOM shot noise theoretical modeling [42]).

The leviton HOM correlations give remarkably 100% noise suppression at τ = 0. This contrasts with the small HOM dip observed in ref [22] where electron are injected from a mesocopic capacitor. In the latter case, the reason for a weak HOM noise suppression is not due to the different nature of the source but is due to the propagation in the leads between the source and the beam-splitter. In [22] the leads are made of two chiral edge channels (Integer Quantum Hall regime at filling factor 2) and electrons are injected in the outer edge channel. A (pseudo)spin charge separation occurs where the incoming electron fractionalizes. This induces decoherence and energy relaxation which spoils the otherwise naively expected full HOM noise suppression. Working with a single edge or putting the source closer to the beam-splitter would produce a better HOM shot noise dip.

In [42], it was remarked that the HOM noise shape versus τ does not depends on temperature. This was actually experimentally observed [41]. This remarkable decoupling occurs only for single charge levitons but not for

LORENTZIAN SINE-WAVE

Figure 6 Temperature dependence of the HOM shot noise for lorentzian (left) and sinewave (right) pulses. By plotting the ratio of the HOM noise to the theoretical zero temperature HOM noise versus the HOM delay τ , one observes that, contrary to the sinewave pulses, for levitons the HOM noise shape versus τ is not affected by thermal excitations. Figures adapted from [41] sine wave pulses nor for doubly charged levitons. The robustness of the 100% HOM dip to temperature for any voltage pulse shape and of the HOM leviton noise curve versus τ to temperature effects and decoherence has been recently theoretically studied in [60]. Fig 6 shows experimental HOM noise measurements demonstrating the decoupling between the temperature and the delay τ for the case of a leviton and shows the absence of decoupling for sinewave pulses.

Full characterization: Wigner function of a leviton

In the last two sections levitons have been characterized in time or in energy. But a wave function is a more complex object. Because of the fundamental quantum uncertainty of conjugated variables ( energy versus time or momentum versus position), standard quantum measurements gives in general information on the wavefunction versus one variable only. Having all information on the wave function requires a lot of different measurements to 'see' the wavefunction under all aspects. This is done using Quantum State Tomography (QST). It consists in measuring the energy density matrix ϱ(ε (p, x)) .

′ , ε) =< ψ † (ε ′ )|ψ(ε) > or the coherence < ψ † (t ′ )|ψ(t) > to
W (t, ε) = ∫ ∞ -∞ dδϱ(ε + δ/2, ε -δ/2)e (-iδt/h) (14)
For periodic injection, only the non-diagonal elements of ρ differing by a multiple of the fundamental frequency ε ′ -ε = lhν are non zero. This implies that the Wigner function is a periodic function of time and this conveniently restricts the number of measurements to be done. A Quantum State Tomography procedure for Measurements of ϱ(ε ′ , ε) has been theoretically proposed by C. Grenier in [44] and has been experimentally implemented and reported by the present authors in [24]. First let us remark that the Shot noise spectroscopy makes a measurement of the diagonal term of the energy density matrix as one can deduce from the measurements the electron distribution function f (ε) = ϱ(ε, ε). In order to access the off-diagonal elements where the energies differ by khν, the same shot noise spectroscopy technique is used but, following [44], instead of only applying a dc voltage V R on the right reservoir, a small amplitude V ac sine-wave voltage is added at frequency kν. Electrons emitted by the right reservoir at energy ε are thus is weak amplitude superposition of states of energies ε ± khν. Mixed in the beam-splitter, they antibunch with the ε and ε ± khν leviton energy components. Finally, Introducing a delay τ between the small ac sinewave signal and the leviton pulses, like in the HOM experiment, provides a way to give a convincing signature of the interference of the ε and ε + khν terms in the resulting shot noise variation. In [24] only the first two harmonic terms were measured (k = 0, 1, 2) but the third one could have been measured as well. The expected shot noise difference ∆S I obtained when the noise is measured with V ac on and off is given by:

∆S I =S 0 I 2kη cos(2πkντ ).
. (15) from which the off diagonal terms can be extracted. state topology. As an example, we propose the following CNOT quantum gate coupling two flying qubits (a) and (b). Starting with the sate |1a⟩|1b⟩ by injecting levitons on the appropriate contacts, the output is expected to be |1a⟩|1b⟩ + |0a⟩|0b⟩ a Bell state. Here we have used a coulomb coupling which provides a conditional π phase shift when an electron on the lower edge of qubit (a) performs a distant scattering with an electron propagating on the upper edge of qubit (b). Fig. 9 schematically displays such a quantum circuit with chiral ballistic electrons to perform the CNOT quantum gate.

∫ eV R 0 (ϱ(ε, ε + khν) -ϱ(ε, ε -khν))dε/hν
Leviton single electron sources have stimulated many theoretical works. The time domain control and the special minimal excitation state property leads to well posed problems not accessible to dc voltage electron sources. Here we will cite only a few examples of many theoretical predictions concerning levitons. For example, the non-local entanglement of electrons emitted by two independent dc voltage sources, as first considered in [45] in a two particle Hanbury Brown Twiss set-up, was difficult to characterize and found very fragile with respect to thermal fluctuations [46]. The same problem has been recently recently revisited using leviton sources and it has been shown that non local characterization of entanglement is possible without the need of post-selection [61]. Another example is the generation of neutral levitons pairs a controlled time variation of the transmission amplitude of a punctual scat-ter like a QPC which has have been first predicted in [48]. The principle is based on a controlled variation of the QPC transmission d(t) and reflection r(t) amplitudes. Writing the scattering amplitudes in the form of d(t) = sin ϕ(t) and r(t) = i cos(ϕ(t), to ensures unitarity of the QPC scattering matrix, and choosing a phase variation ϕ(t) corresponding to that used to produce levitons with voltage pulses, see Eq. 6, neutral leviton anti-leviton pairs are predicted to be emitted in the adjacent leads from the modulated scatter. These pairs are expected to be entangled [49]. Embedding, these neutral leviton pairs in a Mach-Zehnder electronic interferometer, their entanglement has been theoretically demonstrated and the entanglement detection theoretically shown possible in [50].

Levitons can also be useful for new quantum detection schemes. For example, the Full Counting Statistics (FCS) of levitons partitioned by a quantum conductor has been theoretically shown detectable by sending synchronized levitons in a coupled Mach-Zehnder interferometer [61]. Here, remarkably, no noise measurements are needed, only dc current measurements are necessary to access the electron FCS.

Regarding interaction, it has been predicted that levitons can also be generated in an interacting 1D electron system [39] called a Luttinger Liquid (LL). A nice example is given by the chiral edge states of the Fractional quantum Hall effect (FQHE), whose best understood state, the Laughlin state, occurs at filling factor 1/3 [51]. Here the current is carried by a single edge channel. This is a chiral Luttinger liquid which is expected to generate elementary excitations carrying fractional charge e * = e/3 [52] and showing anyonic statistics [53] in full correspondence with the bulk Laughlin excitations. This regime has been considered in [54] and more recently in [55]. Counterintuitively and contrary to what was suggested in [39], no fractionally charged levitons can be generated in a fractional edge but only integer charge leviton. Indeed, it is mandatory that the total phase variation resulting from the voltage pulse be not a fraction of 2π in order to have only electron like excitations with no holes and thus a minimal excitation property. This implies that the Faraday flux, which in the 1/3 FQHE regime writes ∫ e * V (t)dt be equal to nh. As I(t) = e * (e/h)V (t) the total injected charge for 2nπ phase shift is thus q = ne, where n an integer. Can these integer charge minimal excitation states be used to probe the fascinating fractionally charge excitations? Having the time domain control of e * excitations could open new way to probe their anyonic statistics via Hong Ou Mandel correlations or other two-particle interferometry protocol. In Fig. 10 we propose a possible source of on-demand anyons: integer charge levitons are first generated and then sent to a QPC which is tuned in the weak backscattering regime (i.e. shows a weak reflection probability). In this regime, shot noise measurements, using dc voltage source to create an incoming flux of integer charges, have shown that a Poissonian emission of fractional charges e * can be emitted to contribute to the backscattering current [56], [57]. We conjecture that, similarly, an integer charge leviton would be partitioned into an e/3 backscattered charge and a 2e/3 transmitted charge. We think this situation would lead to the generation of levitonic like charge pulses carrying fractional charge. Here, the deterministic emission of fractionally charge levitons would be lost because of the Poissonian backscattering statistics but the time resolution and the time domain control inherited from the initial integer charge levitons will be kept and could be used for further experiments like Hong Ou Mandel collision of anyons to probe their quantum statistics.

In another perspective it is interesting to address the physics of non-integer charge pulses. Only voltage pulses can realize this situation while quantum dot based charge sources can only emit integer charge particles. Indeed by continuously tuning the voltage pulse amplitude on a contact one can generate any value of the injected charge. It was shown from Eq. 10 that only an integer q gives rise to a minimal excitation state. Injecting fractional charges, even using Lorentzian pulses can not be viewed as pure levitons. However non-integer charge pulses created by Lorentzian voltage pulses have interesting properties which deserve attention. In [58] it was shown that half-levitons, i.e. charge e/2 created by Lorentzian voltage pulses, minimize the shot noise of a Superconducting/Normal conductor junction the same way as integer charge levitons minimize the noise in a purely normal conductor. Here this occurs in the energy sub-gap regime where, while the normal current is suppressed by the superconducting gap, a small current is allowed via the so-called Andre'ev reflection mechanism. In Ref. [60], the nature of half integer Lorentzian charge pulses have been studied and it has been shown that they can form remarkable fractionally charged zeroenergy single-particle excitations states. Separating the half-leviton into a e/2 charge part and its accompanying neutral electron-hole cloud, [60] showed that a half-leviton and a anti-half leviton mixed in a semi-transparent electronic beam-splitter can elastically annihilate, a property not shared with ordinary distinguishable electron and hole excitations.

Non-integer charge pulses behave differently than integer pulses in electronic Mach-Zehnder (MZI) or Fabry-Prot (FPI) interferometers. This was first noticed in the work of [62] where, from numerical dynamical simulations, it was remarked that, for pulses having an extension smaller than the MZI arm difference or the FPI perimeter, the transmission was dependent on the charge. For example, the Fabry-Prot transmission shows oscillations with the charge q modulo e. The explanation was found in the phase difference 2π(q/e), see Eq. 3, between the front and the back of the charge wave-packet which combines with the orbital phase accumulated in the interferometer. Striking effect on the interference visibility of the current in an electronic Mach-Zehnder interferometer have been also found in [63]. For example, while the visibility as a function of an external magnetic flux vanishes when the injection period corresponds to have exactly one electron in the MZI for charge q = e, Ref. [63] found that for q = e/3 the visibility vanishes when there are exactly three fractional charge pulses in the MZI. More generally robust visibility cancelation occurs when there are p charges for fractional charges q = ek/p. Similar results are found in a FPI interferometer. This intriguing properties call for looking at possible anyonic or parafermionic quantum statistics that may obey these charge pulses. The above mentioned perspectives are only a few direction and do not pretend to give a fair exhaustive account of the impressive literature that levitons have triggered and we apologize for missing references and topics. We think that we are only walking on the surface of a Fermi Sea Iceberg.

Figure 2

 2 Figure 2 Top figure: the mesoscopic capacitor source (energyresolved source). Its energy distribution is lorentzian and the time wavepacket envelope is semi-exponential. Bottom Figure: the voltage pulse leviton source. Its energy distribution is semi exponential and the time wavepacket is lorentzian. The two sources are suitable for electron quantum optics. They show a duality when interchanging time and energy

Figure 3

 3 Figure 3 Top figure: effect of an arbitrary pulse on the Fermi sea. Electrons are scattered to different energies above and below their initial energy. Electron hole excitations are created, as schematically represented in the energy distribution f (ε). Bottom figure: applying a Lorentzian shape voltage pulse generates a minimal excitation state called Leviton , no hole excitation is created.

Figure 4

 4 Figure 4 Top figure: shot noise spectroscopy of a sine wave pulse. Since an equal number of electron and hole excitations is created the noise spectroscopy is symmetric with respect to VR = 0.The schematic zero temperature energy distribution f (ε) is shown on the left. Bottom figure: shot noise spectroscopy of a lorentzian pulse.The number of holes is minimized to zero and the shot noise spectroscopy is asymmetric. The figures are adapted from[23]

6 TFigure 5

 65 Figure 5 Top figure: HOM experiment for one and two electrons injected in the separate input of a beam-splitter. The HOM shot noise resulting from partitioning of antibunched electrons is plotted versus the arrival time delay τ giving access to the wavefunction overlaps. Note the excellent agreement with the expected theory (solid line). Bottom figure: HOM experiment with two different pulses. The sine pulse creating a large amount of electron hole excitations, the HOM shot noise is enhanced. The figures are adapted from [42].

Figure 7

 7 Figure 7The off diagonal terms of the energy density matrix of a leviton (bottom figures) are extracted from the shot noise (middle figures) measured when performing the HOM interference of a leviton mixing (colliding) in a beam-splitter with a small sinewave pulse at frequency ν (left figure) and 2ν (right figure). The top figures show the noise variation with the time delay τ between the small sinewave pulse and the periodic levitons. The oscillations, and the doubling of the period at frequency 2ν with respect to ν, prove the existence of non-diagonal coherence terms in the energy density matrix. The figures are adapted from[24].

Figure 8

 8 Figure 8Experimental Wigner function extracted from the offdiagonal components of the density matrix (adapted from[24]).

Fig 7 shows the noise variations for k = 1 , 2 .

 12 The characteristic doubling of the period with τ ensures that interference terms are indeed measured. This demonstrates the fully coherent nature of the levitonic quantum state. The deduced off-diagonal components (bottom of the figure) are then used to extract from Eq. 14 the leviton Wigner function truncated to the first two harmonic temporal components as shown in Fig.8

Figure 9

 9 Figure 9 Schematic representation of a CNOT quantum gate with Levitons. Two Levitons are injected in rail 1a and 1b. The output state is a maximally entangled state. The vertical dashed line schematically indicates the Coulomb interaction which could provide a conditional π phase shift in the two particle (two-qubit) wavefunction.

Figure 10

 10 Figure 10Proposal for a time-resolved anyonic source in the Fractional Quantum Hall regime (filling factor 1/3). Integer charge levitons are first created with a charge e Lorentzian voltage pulse applied on the contact of a 1/3 fractional edge channel. Then, the leviton arrives at a Quantum Point Contact tuned in the weak backscattering (WB) regime (weak electronic reflection probability). It is conjectured that the leviton will break into a e/3 backscattered and 2e/3 transmitted fractionally charged pulses. From the backscattered side, this realizes a non-deterministic source of e/3 excitations with Poissonian statistics. The time resolved properties inherited from the original integer charge leviton may open the way for new quantum experiments to probe the anyonic statistics.

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Perspectives Levitons seem to be the ideal source for electron quantum optics and offers wide perspectives. The excellent ability to synchronize leviton sources could be exploited to realize leviton flying qubits using the edge Copyright line will be provided by the publisher

Acknowledgements Support from the ERC Advanced Grant 228273 MeQuaNo is acknowledged.