Samanvitha Kunigal

Vijaya Shankar

Yann Claveau

Tojo Rasoanarivo

Chris Ewels

Jean Le Bideau

Impact of Li, Na and Zn metal cation concentration in EMIM-TFSI ionic liquids on ion clustering, structure and dynamics †

We use molecular dynamics calculations to investigate the behavior of metal cations (Li, Na and Zn) within ionic liquids (ILs), specifically EMIM-TFSI, and their impact on key properties, particularly focusing on ion-ion correlations and their influence on diffusion and conductivity. The study explores the competition between metal cations and EMIM ions for binding to TFSI and analyzes ion pair dynamics, revealing that metal cation-TFSI pairs exhibit significantly longer lifetimes compared to TFSI-EMIM pairs. This competitive interaction and the increased stability of metal cation-TFSI pairs at higher concentrations leads to reduced ion exchange, resulting in decreased diffusion and conductivity. The observations underscore the importance of ion size and charge in determining their behavior regarding IL dynamics. Overall, this work provides valuable insights for designing ILs with customized properties, particularly in the context of optimizing conductivity and addressing energy storage challenges.

Introduction

Room-temperature ionic liquids (ILs) have emerged as useful electrolytes in the field of electrochemistry, offering safer alternative to traditional solvents. [1][2][3] ILs have found applications in batteries and supercapacitors due to their low flammability, wide electrochemical window, and low volatility. 4,5 In addition, their viscosity decreases with increasing temperature. This makes them particularly suitable for higher operating temperature ranges, as ILs demonstrate good ionic conductivities under such conditions. 6,7 Within this context, there has been a significant surge of interest in developing metal-ion IL-based batteries as potential strategic replacements for conventional Li-ion batteries electrolytes. The ongoing quest for sustainable alternatives to Li + includes notably monovalent ions such as Na + and divalent ions such as Zn 2+ . [8][9][10][START_REF] Moreno | Appetecchi[END_REF][12] Exploration of ILs as electrolytes relies on understanding their dynamical properties such as diffusivity and conductivity. Previous studies have shown that adding metal cations greatly influences the dynamics of IL ions. 13,14 Those metal cations interact with the anions of the IL and this, in turn, affects the dynamics of the system. In particular, several experimental and simulation studies have consistently shown that the introduction of metal cations to ILs significantly influences their viscosity and, consequently, their conductivity: [15][16][17] experiments, e.g. by Han et al. 18 and Martinelli et al., 19 have demonstrated that the conductivities of various Li based ILs decrease due to a combination of factors including ionic association, viscosity, anion size, and thermal properties; molecular dynamics simulations by Kubisiak et al. 20 observed a decrease in IL diffusivity and conductivity upon the introduction of lithium or sodium cations, with the effect varying with metal concentration; Borodin et al. 21 further explored the structure and transport of monovalent (Li, Na) and divalent (Mg, Zn) metal cations as a function of temperature in PYR 14 TFSI using both molecular dynamics and quantum mechanics.

While these studies highlight the significant impact of cona Nantes Université, CNRS, Institut des Matériaux de Nantes Jean Rouxel, IMN, F-44000

Nantes, France * E-mail: yann.claveau@cnrs-imn.fr † Electronic Supplementary Information (ESI) available centration, nature (monovalent or divalent), size, and electronic structure of the metal ion on the dynamics of the ions in the system, a comprehensive comparative study examining the distinct effects on IL behaviour with the addition of various metallic ions is lacking. Numerical values from molecular dynamics are force field dependent, and it is important to compare results generated using consistent force fields and parameterisations. This current study explores dynamic aspects of the interplay between different metal cations, i.e.

Li + , Na + or Zn 2+ and a host IL, ethyl,methylimidazolium (EMIM) and bis(trifluoromethylsulfonyl)imide (TFSI) (Fig. 1). The EMIM-TFSI ionic liquid is chosen as it is well-known and well-studied, 22 with a stable electrochemical potential window up to 4.6 V. 23 It also shows a high conductivity well adapted to supercapacitors operating above room temperature. The addition of metal salt is desirable for pseudosupercapacitors and hybrid systems, in order to provide higher power and higher energy density systems. 24 Using classical molecular dynamics simulations, our aim is to understand the interactions between the different species through a thorough comparison of structural and dynamic properties as a function of concentration, temperature and metal cation type.

Computational method

Simulations were conducted using the LAMMPS molecular dynamics package. 25 Input files were prepared with the help of fftool 26 and packmol. 27 Resultant atom trajectories were analysed using the TRAVIS package, 28 MDAnalysis, 29,[START_REF] Gowers | Proceedings of the 15th Python in Science Conference[END_REF] and in-house Python scripts.

We first studied neat EMIM-TFSI as a reference, before mixing EMIM-TFSI with varying concentrations of monovalent (Li(TFSI), Na(TFSI)) and divalent (Zn(TFSI) 2 ) salts. The non-polarisable CL&P force field [START_REF] Lopes | [END_REF] was used for EMIM + , TFSI -, Li + , and Na + . For Zn 2+ , which is not currently included in CL&P, we used Stote parameters, 32 which are also based on standard 12-6 Lennard-Jones and Coulomb potentials.

An infinite liquid is simulated using a periodic box. We ensured that the cubic box is large enough to avoid finite size effects (typically a width w at least four times the van der Waals radius cut-off, i.e. w ∼ 50 Å), through a convergence study of diffusion coefficient in NVT simulations on EMIM-TFSI, from w = 40 to 60 Å. As a result, our neat EMIM-TFSI system contains 300 ion pairs contained in a cubic box with side length 51.47 Å. The concentration of the added salt was varied from n Me + /n Me + +EMIM + = 0.04 to 0.24 cationic molar fraction. The resultant compositions of all the systems are detailed in Table 1. A time-step of 1 fs was used throughout, with atomic coordinates stored every 2 ps. Long range electrostatic interactions were solved using the PPPM solver. The cutoff for long range interactions was set at 12 Å. The Shake algorithm was used to constrain H bonds. The systems were equilibrated in the NPT ensemble at P = 1 atm for 10 ns until the density was fully converged. The systems were then further equilibrated in the NVT ensemble for 10 ns. Production runs were carried out in the NVT ensemble for 50 to 70 ns depending on the temperature. All runs at 393 K were carried out for 50 ns except the ion-pair lifetimes for the Zn system which ran for 180 ns, due to the long stability lifetimes of Zn-TFSI pairs. In principle one should avoid using NVT for production runs since thermostats can affect the dynamics of the system. However, NVT simulations are computationally less expensive and more stable than NVE simulations. To ensure that the NVT ensemble is an acceptable approximation, we verified that the use of a Nose-Hoover thermostat does not affect significantly the dynamics of our systems (more details in ESI).

Representative convergence studies are given in ESI, including densities that are in good agreement with experiment (less than 2% deviation where experimental data available).

Results and Discussion

Analysis of Self Diffusion Coefficients

The average self diffusion coefficients of all the ions in the system are obtained through the Einstein relation from mean squared displacement (MSD) calculations: 33

D = 1 2d lim t→∞ d dt 1 N N ∑ i=1 |⃗ r i (t) -⃗ r i (t 0 )| 2 t 0 ( 1
)
where d is the dimensionality of the system, N is the number of ions, ⃗ r i (t) is the position of ion i at time t, and ⟨•⟩ t 0 denotes an average over all starting times t 0 . The calculated values for the Li + system are shown in Figure 2. In general the EMIM self-diffusion coefficients are approximately twice that of the TFSI, which are themselves roughly twice that of the Li. The diffusion coefficients of all the ions in the system increase continuously with temperature, as expected. This allows us to focus on the system at high temperature, i.e. 393 K, to ensure faster dynamics and reduce the computational cost. The self diffusion coefficients in the Na + and Zn 2+ systems as a function of temperature behave very similarly, and are given in ESI.

We next explore the variation in self-diffusion coefficient as a function of metal cation concentration at 393K. Figure 3 shows that while, in general, all the ion self-diffusion coefficients decrease with increasing metal cation concentration, there are some ion-specific features. This is in good agreement with previous simulations performed with different parameters for Na and Li. 34,35 We note that these self-diffusion coefficients are around one order of magnitude lower than those reported in experiments, 36 a discrepancy commonly seen in studies of ILs when using nonpolarisable force fields. However, it has been shown that this does not impede the analysis of relative trends. 37 As the concentration of metal cations increases, the diffusion coefficients of both TFSI and metal cations decrease. This is consistent with an increasing number of TFSI anions bound to the metal cations as the metal concentration increases and is discussed further in Section 3.3.

Turning to the metal cations, both Na and Li show very similar self-diffusion coefficients (although D Na decreases slower than D Li with concentration), significantly larger than that of Zn. This suggests that ion charge state is a key driver in determining selfdiffusion. This is despite the possibility of Na + ions potentially being able to pair with more TFSI due to their size (see Section 3.2 below). In contrast, the divalent cation Zn 2+ , due to its double valency and size comparable to Li + , is expected to interact more strongly with the TFSI ions in the system, resulting in the lowest diffusion coefficients.

Interestingly, we also note a reduction in the self-diffusion coefficients of the EMIM ions concurrent with an increase in metal cation concentration, even though we do not expect them to be directly coupled. However, analysis of the underlying mechanisms resulting in these trends in diffusion coefficient, in particular for D EMIM , is difficult based only on system averaged data. In order to obtain more insight into the processes at play, it is necessary to examine the local structure, as well as the diffusive behaviour of individual ions within the overall population.

Structure Analysis

The diffusion process is intimately tied to the structure. Therefore, a thorough understanding of the structure will help in comprehending the interactions among the different species. For this reason, radial distribution functions (RDF) are plotted as a function of concentration and nature of metal cation. Figures 4a and4c show the RDF of TFSI oxygen atoms (O TFSI ) and centers of mass of TFSI ions themselves around the metal cations. Interestingly the local structure seems to be largely independent of the metal cation concentration. The first solvation shells of O TFSI around Zn 2+ and Li + are both situated at 2.1 Å, whilst for Na + it is further away at 2.5 Å (Fig. 4a). This largely reflects their respective ionic radii: Zn 2+ (0.88 Å) and Li + (0.90 Å), being smaller than Na + (1.16 Å), can be in closer proximity to the oxygen atoms. In each case there is a single O-cation distance, with a sharp peak for Zn, slight broadening for Li, and significant broadening for Na. This broadening reflects the binding strength to the ion (the ion's "softness"), again primarily due to the cation charge state and the distance of the oxygen from the cation nucleus. A broader secondary peak around 2.2 Å further from the cation represents the oxygen atoms attached to the second sulphur of TFSI. The integrated RDF for O TFSI (Fig. 4b) around Me + shows clearly six oxygen atoms around Zn, corresponding to an octahedral configuration (Fig 5). The Na and Li form a much less pronounced plateau, converging to between five and six oxygen neighbours, suggesting similar oxygen octahedral coordination but with significantly more structural variation.

Unlike the single peak for individual O TFSI , the distribution of TFSI molecules around Li and Na show two distinct peaks, again with the TFSI slightly further from the Na (Fig. 4c). From the integrated RDF (Fig. 4d) the closer peak corresponds to around 2 TFSI molecules for Li and Na, with 2 further TFSI molecules described by the second peak. Since 5-6 oxygens sit around the Li and Na, together these results suggest that Li and Na have, on average, two more tightly bound TFSI and two slightly further away, and of these, typically two are bound via a single oxygen atom and two with two oxygens. Previous experimental results obtained by decomposition of Raman spectra show a signature associating 2 TFSI per Li + , 38 suggesting the more weakly bound pair seen in the RDF are not detectable in Raman.

The Zn-TFSI RDF is more complex since the second peak also has a slight shoulder at about 4.2 Å. The integrated RDF (Fig. 4d) shows that there are on average 0.5 -0.8 TFSI most closely bound to Zn, and around 5.5 -5.2 TFSI in total when all three peaks are integrated. These fractional values can either come from an average over different configurations, or could be linked to time averaging of the RDF over the whole production run (see Section 3.4). Considering the 0.24 molar cation concentrations where the total TFSI in the coordination shell converges cleanly to 5.2, these results suggest that in the majority of cases, TFSI are arranging in a distorted octahedron around the Zn. The integrated RDF shows that there is one closer TFSI site, one intermediate and four slightly further away, however the closer TFSI site is not always occupied. Given the oxygen coordination is stable at six, this suggests that in the cases where the closer TFSI is not occupied, one of the other TFSI has two oxygen atoms bound to Zn, likely associated with the observed peak shoulder in the Zn-TFSI RDF. These configurations are represented by two snapshots in Fig. 5 where the octahedron cages of 6 oxygens is made by 6 or 5 TFSI.

Similar results have been obtained regarding the coordination of oxygen of TFSI in PYR14-TFSI with metal cations. 21 However, they found that most of Zn 2+ solvates was dominated by bidentate coordination configurations (3 TFSI sharing 2 oxygens). These differences could be due to the different cation of the IL (PYR14 instead of EMIM) or more likely to the use of a polarisable force field. Indeed, another study on EMIM-TFSI 20 showed that the use of non-polarisable force field tend to increase the fraction of monodentate interactions between the Li or Na cation and TFSI anions, which in turn decreases the diffusivity. However, they have also showed that the non polarisable force field better reproduce experimental trends than Drude-particle force field, although the latter performs better quantitative estimates of diffusion coefficients and conductivities.

We note that at the highest 0.24 cation molar concentration, if each of the 72 Zn was surrounded by 5.2 unique TFSI molecules this would require 374 TFSI molecules, that is to say all the TFSI (see Tab. 1). This implies that either some TFSI molecules must be shared between Zn atoms, or that the system no longer has any TFSI interacting only with EMIM. The same conclusion can be drawn for cations in the other systems, for example in 0.24 Li and Na, if each cation had 4 unique TFSI neighbours, that would require 288 of the 300 TFSI molecules in the system.

The above structural analysis is consistent with the observed diffusivity (Fig. 3), where Zn 2+ diffuse slower due its stronger interaction and its greater number of tightly bound TFSI, and although Na + ions have more oxygen around them than Li + , their soft interaction (broad peak, "soft" solvation shell) allows them to diffuse faster than Li + . However, while the structural analysis can explain the varying diffusion speeds of the metal cations, it does not provide an explanation for the observed decrease in EMIM cation diffusion with increasing metal-cation concentration. To address this, dimer lifetime analysis is required.

Lifetime Analysis of Ion Pairs

The dynamics of ILs is predominantly governed by ion-ion correlations, which can be described via correlation functions between different ion types. Here, we use dimer existence auto correlation functions (DACF) as implemented in TRAVIS. 28 A DACF for a given set of particles is defined as a function that retains a value of 1 as long as the specified criteria are met, and transitions to 0 once the criteria are no longer satisfied, i.e.

c(τ) = 1 N 2 N ∑ i=1 N ∑ j=1 ∞ 0 β i j (t)β i j (t + τ)dt (2)
where,

β i j = 1 if criteria is satisfied 0 otherwise
. In this study, threshold distance criteria have been chosen as follows. For EMIM-TFSI pairs, they are considered to be paired if the distance between the geometric centre of the two species is less than the sum of the maximal molecular radii r EMIM + r TFSI =7.80 Å. The cut-off distances for metal cation-TFSI are chosen from the end of the first solvation peak in the cation-O TFSI RDF (Fig. 4a), and set to 3.35 Å for Li-TFSI, 3.56 Å for Na-TFSI and 2.80 Å for Zn-TFSI.

Using the above criteria, the lifetime T of the aggregate is then twice the total integral of this auto-correlation function c(τ):

T = 2 ∞ 0 c(τ)dτ (3)
Figure 6 shows the lifetime of different TFSI-cation complexes as a function of metal cation concentration. The metal cation-TFSI species have lifetimes one to two orders of magnitude longer than TFSI-EMIM, consistent with their stronger binding. The lifetime ordering for metal cation-TFSI is in agreement with the structural data in Section 3.2 and the calculated ion self-diffusion coefficients (Fig. 2). The shortest lived species are Na-TFSI with average lifetime from 2.3-3.3 ns, while Li-TFSI is around three times longer (around 9ns). This explains why, despite having more oxygen in its first solvation shell, Na + maintains a diffusion coefficient similar to that of Li + (Fig. 3) due to this higher exchange rate of TFSI ions. Zn-TFSI lifetimes are over 50 times longer than Na (around 120 ns), consistent with the calculated strong Zn-TFSI binding, and its low self-diffusion coefficient. During the calculation runs of 50 ns for Na and Li, there is therefore extensive breaking and reforming of cation-TFSI pairing (notably for Na), whereas for Zn-TFSI the complexes are extremely stable, even over the extended 180 ns calculation period chosen.

As the concentration of metal cation increases, the interaction between the metal cation and TFSI becomes more pronounced and persists for a longer duration, partly also because there is then less "free" TFSI available for exchange. This is manifested by the extended lifetime of metal cation-TFSI dimers at higher concentrations, in turn resulting in slower diffusion of the metal cations. These longer metal cation-TFSI interactions clearly dominate the briefer EMIM-TFSI encounters, thereby influence the dynamics of the entire system. This can be seen by the increase in EMIM-TFSI lifetime with increasing metal cation concentration, and explains the decrease in diffusion rate of EMIM and TFSI ions in metal cation-EMIM TFSI system in comparison to neat EMIM-TFSI (Fig. 3). Since these are the majority carriers in the system, this is also linked to experimentally observed viscosity increases. The lifetime increase is most evident in Li-TFSI (increase of 60% going from 0.04 to 0.24 molar concentration), compared to 45% for Na and only 11% for Zn.

Considering the lifetime data as a whole we can start to build an overall picture of ion behaviour in these systems. At lower metal cation concentration, TFSI-Na and TFSI-Li clusters are shorter lived, with the metal cations more mobile consistently. The remaining TFSI-EMIM form short-lived, relatively weakly bound couples. As the metal cation concentration increases, there is an increased trend to forming more stable, longer-lived TFSImetal clusters, which slow the metal cations down. This also structures and "rigidifies" TFSI-EMIM exchange with less overall TFSI mobility, leading to a slight increase in lifetime of TFSI-EMIM clusters as a result. Within this picture there are differences for the different metal cations. Na + is larger and less tightly bound to TFSI, hence maintaining shorter lived clusters and a higher diffusivity as a result. In stark contrast Zn 2+ is tightly bound to its TFSI neighbours, forming extremely long-lived clusters with a very low metal cation diffusivity.

Distribution of diffusion coefficients

The above analysis is primarily focused on trends exhibited by averaged observables. While this approach provides a broad overview of the behaviour, it may mask discrete behaviours within the ILs. By plotting the distribution of these quantities, it is possible to reveal the existence of multiple distinct populations.

To achieve this, the entire MSD curve for each individual ion in the system was fitted. Even though such a fit does not represent the diffusive regime as per standard MSD curves (lim t→∞ dMSD dt ), it still yields a value (the slope) that we can term as the "overall diffusion coefficient" for that ion. This value accounts for the entire path traversed by the given ion throughout the simulation run, thereby eliminating the occurrence of negative diffusion coefficients due to poor fits for some ions. To enhance statistical robustness, nine sets of different trajectories have been accumulated.

The distribution of diffusion coefficients of EMIM, TFSI and metal ions are plotted for two molar ratios of metal cations (0.04 and 0.24) in Fig. 7. In general the metal cations show the smallest full width half maximum (FWHM) in the diffusion coefficient values, indicating they show relatively uniform behaviour. The distribution in TFSI values is roughly twice that of the metal ions. EMIM shows the highest range in values, with FWHM 3-4 times that of the TFSI, more consistent with conventional liquid behaviour. In all cases the distribution appears monomodal, although we cannot exclude the possibility of two peaks that are partially overlapping due to the use of window algorithm. This algorithm increases the statistics by iterating over the reference time. However, this approach could potentially mask a bimodal distribution. The reason is that the window algorithm averages over different time windows, which could blend two distinct peaks into a single one if these peaks occur in different time windows. Therefore, while the window algorithm enhances the overall statistics, it might also mask the presence of multiple distinct behaviors in the diffusion coefficients. Even when we analyze the mean square displacement (MSD) of each ion separately, this sum over t 0 could still potentially hide a bimodal distribution. Hence, while our results suggest a monomodal distribution, further analysis with a different method might be necessary to confirm this.

The most notable change occurs with increased metal cation concentration, which greatly decreases the spread in distribution of diffusion coefficients, as well as decreasing the fitted maximum. Notably this behaviour is also seen in the EMIM population, as well as the TFSI and metal cations. This clearly shows that increasing the metal cation concentration renders the whole system more homogenous, reducing the spread of faster diffusing ions. This suggests the metal cations are reducing variability in the system, rendering it more ordered and "rigid". This is consistent with the calculated increase in ion-pair lifetimes seen above.

Comparing the metal cations, the distribution width of metal cation diffusion coefficients for Li, Na and Zn reflect the strength of the metal-cation binding. At 0.24 molar concentration the broadest distribution in cation diffusion coefficients is for Na (FWHM of 7 × 10 8 cm 2 /s), with Li approximately half this, and Zn nearly 20 times smaller. This is fully consistent with the RDF data and local structure discussion presented above.

Building on the analysis of diffusion coefficients and dimer lifetimes, we have shown that the lifetime of metal-TFSI pairs significantly influences the overall diffusion. Increasing the quantity of metal-TFSI pairs orders the whole system, reducing the amount of TFSI available to pair to EMIM, and it is these more weakly interacting species that are dominating the system diffusion. The 1-11 effect of adding more metal cations on EMIM diffusivity therefore suggests that the primary impact of metal cations on overall conductivity is indirect rather than direct. To explore this in more detail, we next turn to conductivity calculations.

(a) (b) (c) (d) (e) (f) (g) (h) (i)

Conductivity

Conductivity of a system is the collective dynamic property obtained due to the contribution of all the interactions between different pairs of ions in the system. It is derived from ion-ion correlation function while accounting for the charge of the interacting species.

The ionic conductivity of the systems is derived from the Green-Kubo relation and reads as follows:

σ = 1 6k B TV lim t→∞ d dt N ∑ i=1 N ∑ j=1 q i q j ⃗ r i (t) -⃗ r i (t 0 )] • [⃗ r j (t) -⃗ r j (t 0 ) (4
) Where t 0 refers to initial time , q i , q j ⃗ r i , ⃗ r j refer to the charge and position of the i th and j th particles and ⟨•⟩ denotes the ensemble average of the system.

Correlation between particles implies that their movements are not independent; the displacement of one particle is associated with the displacement of another. In Eq. 4, the dot product of the displacement vectors of the i-th and j-th particles,

[⃗ r i (t) -⃗ r i (t 0 )] • [⃗ r j (t) -⃗ r j (t 0 )]
, quantifies this correlation. A positive dot product indicates that the particles move in the same direction, while a negative dot product means movement in opposite directions. This leads to four possible scenarios:

• When q i and q j have the same sign (q i q j > 0):

-If the particles are correlated and move in the same direction (forming a charged aggregate), their contribution to the conductivity is positive.

-If they move in opposite directions, their contribution is negative.

• When q i and q j have opposite signs (q i q j < 0):

-If the particles are correlated and move in the same direction (forming a neutral aggregate), their contribution to the conductivity is negative.

-If they move in opposite directions, their contribution is positive.

The formula for ionic conductivity derived from the Green-Kubo relations, as shown in Eq. 4, offers several advantages over the Nernst-Einstein equation σ = N ions V k B T ∑ N species i=1 q 2 i D i . Firstly, it takes into account the correlations between the movements of different ions, which the Nernst-Einstein equation does not (although other groups have successfully proposed ways to exploit the Nernst-Einstein equation 39 ). This makes it a more accurate representation of the actual behavior of ions in a real system, where the movements of ions are often correlated due to their strong electrostatic interactions with each other. Secondly, the Green-Kubo formula for ionic conductivity is directly related to values that can be measured with dielectric spectroscopy. Dielectric spectroscopy measures the response of a material to an applied electric field, which is directly related to the movements of charged particles within the material. Therefore, the Green-Kubo formula provides a theoretical basis for understanding the results of dielectric spectroscopy measurements.

Using the above formula, the conductivity of the different systems is represented in Fig. 8. It shows that the conductivity decreases with increase in concentration of metal cation like diffusivities and dimer lifetimes.

Although Zn systems possess the lowest diffusion coefficients, relatively more TFSI around Zn 2+ , longest lifetime for Zn-TFSI ion pairs due to the divalency of Zn 2+ ions and their small sizes, the ionic conductivities of the three systems remain quite close, similar to EMIM-TFSI dimer lifetimes.

Similarly to Ref. 20 we have calculated the different correlations in Eq. 4: Me-TFSI, Me-EMIM, EMIM-TFSI, EMIM-EMIM, TFSI-TFSI, Me-Me. Figure 9 represents these contributions for a Zn concentration of 0.04. Others systems can be found in ESI.

Firstly, it is important to acknowledge the substantial uncertainty surrounding the conductivities. The conductivity can fluctuate significantly (ranging from half to double), particularly at lower metal concentrations, depending on the time range over which the charge correlations have been fitted. This aligns with the diffusion coefficient histograms (Fig. 7), which also have a wider distribution at lower metal concentrations. We see the same pattern here: at lower concentrations, charge correlations vary greatly from one pair to another, resulting in these erratic curves. Therefore, it's risky to compare different systems.

However, one commonality is that the primary contributors to conductivity are, firstly, EMIM-EMIM pairs and, secondly, EMIM-TFSI pairs. For the same reason, despite the Zn-TFSI dimer having a very long lifetime, 50 ns runs are adequate for analyzing correlations since metal-ion pairs don't significantly contribute to conductivity.

This observation is consistent with the observed dimer lifetime: metal-cations are stuck to TFSI while EMIM cations can exchange their counterions. However, by increasing the concentration, EMIM becomes more strongly paired to TFSI. To increase the conductivity it is then crucial to decrease correlations between the metal ion and its counterion.

Due to the poor statistics it is impossible to discuss the correlation trends in greater detail. The use of a window algorithm (iterating over initial time to artificially increase the number of trajectories) or additional trajectories should be considered. however, the use of window algorithm, as in the case of the diffusion coefficient distribution, could potentially mask certain features of the data. Specifically, if there are multiple distinct correlation trends present, the window algorithm could blend these into a single trend due to its averaging over different time windows. This could result in the loss of information about individual trends and make it appear as though there is only a single, uniform correlation trend. Therefore, while the window algorithm can enhance the overall statistics and allow for a broader analysis, it may also obscure the presence of multiple distinct behaviors in the correlation trends. Consequently, while it is a useful tool, its use should be complemented with other methods to ensure a comprehensive analysis.

Conclusions

The study shows that all dynamical properties exhibit a decrease with increasing concentration of metal-cations. A notable competition is observed between metal-cation and EMIM in their propensity to form a bind with TFSI. At high concentration, the metal cation-TFSI ion pairs are more stable and persist for a longer duration, which in turn leads to a decrease in the number of free TFSI ions. This results in a decrease in TFSI ions to interact with EMIM ions, which in turn leads to a decrease in the number of "free" EMIM ions, hence to diffusivity and conductivity.

The diffusion of Li + and Na + ions is similar, with Na + ions diffusing slightly faster than Li + ions. This is due to the larger size of Na + , which allows it to be less strongly paired to TFSI than Li + , and thus to exchange its counterions more easily. The influence of metal cations on conductivity is not direct, but rather indirect. Decreasing their correlations with their counterions should lead to an higher conductivity by increasing degrees of freedom of EMIM and TFSI. In order to further investigate charge correlations and related conductivity, it would be necessary to increase the number of trajectories.

Moreover, the soft character and larger radius of Na + ions leads to a higher mobility, especially at low concentration when there are more "free" TFSI. On the other hand, Zn, with its two charges and small size form a strong and stable dimer with TFSI.

In the future design of ionic liquids with high conductivity, soft ions should be preferred. This is because soft ions can more easily form and break connections, leading to higher conductivity.

For applications in energy storage, the design of the IL could be such that the metal cation is less tightly bound to TFSI than EMIM. This points towards a strategy of confinement for reducing metal ion correlations, which could open roads for the development of high-performance ILs for all solid-like energy storage devices. The less tightly paired the metal cation is with TFSI, the more freely it can move, potentially leading to higher conductivity.

Fig. 1

 1 Fig.1Ball and stick images for EMIM + and TFSI -ions. In all our simulations, atoms are free to rotate around bonds. Hence, in case of TFSI, distances between carbons at the end vary (example distance shown correspond to the configuration of TFSI in the figure).

Fig. 2

 2 Fig.2Calculated self-diffusion coefficients (10 -7 cm 2 s -1 ) of the component ions in bulk Li-EMIM-TFSI, as a function of temperature (K), with Li concentrations of 0.04 (rose), 0.12 (pink), and 0.24 (magenta). Symbols represent EMIM (circles), TFSI (triangles) and Li (squares).

Fig. 3

 3 Fig. 3 Calculated ion self-diffusion coefficients in Me x -EMIM 1-x -TFSI systems (Me=Li, Na, Zn), as a function of concentration x at T = 393 K. Red dotted lines indicate Li based IL, blue lines are for Na and green dashed lines for Zn. Circles show EMIM cations, triangles for TFSI anions and squares for metal cations.

  Figures 4b and 4d show the corresponding integrated RDFs, which represent the associated coordination numbers for oxygen atoms and TFSI centers of mass around metal cations respectively.

Fig. 4

 4 Fig. 4 Radial distribution functions for (a) TFSI oxygen atoms and (c) TFSI ions, around metal cations (Li, Na and Zn). Coordination number of (b) TFSI oxygen atoms and (d) TFSI ions around metal cations. Magenta indicates Li systems, blue for Na and green for Zn. Light colors (solid lines) are for low metal cation concentration (0.04), dark colors (dashed lines) for high concentration (0.24). System temperature is 393 K, all distances in Å.

Fig. 5

 5 Fig. 5 Snapshots of (a) 6 TFSI in octahedral configuration around Zn 2+ and (b) 5 TFSI in octahedral configuration around Zn 2+ with one of the TFSI sharing 2 oxygens with TFSI.

Fig. 6

 6 Fig. 6 Lifetime of different ion pairs in the system as a function of metal ion concentration (Eq. 3), at 393K. Circles are for EMIM-TFSI pairs, triangles for TFSI-metal ones. Black represents neat IL, red is for Li based ILs, blue for Na based ones and green for Zn based ion pairs.

Fig. 7

 7 Fig. 7 Histograms of diffusion coefficients of individual ions in the system as a function of metal cation molar concentration. Pink, blue and green are Li, Na and Zn based ILs respectively. First, second and third rows show distribution of diffusion coefficient of EMIM, TFSI and metal cations respectively. Densities have been normalized by their maximum to make reading easier. Full width half maximum (FWHM) and average density coefficients are indicated each plot based on a Gaussian fit.

Fig. 8

 8 Fig. 8 Total conductivity using Green-kubo formula in function of metal concentration.

Fig. 9 (

 9 Fig. 9 (a) Correlations between ion pairs and (b) their corresponding conductivity, for Zn-EMIM-TFSI with a Zn concentration of 0.04. 15 trajectories have been used at 0.04, for 50 ns and 9 for others concentrations. The conductivity is proportional to the slope of the correlation at large time.

Table 1

 1 Composition of the simulated systems for metal cations Li, Na and Zn. Cationic molar fractions are given in parentheses

	Nb of Metal cations Nb of EMIM cations (concentration) (concentration)	Nb of TFSI anions with Li/Na with Zn
	12 (0.04) 36 (0.12) 72 (0.24)	288 (0.96) 264 (0.88) 228 (0.76)	300 300 300	312 336 372

[1][2][3][4][5][6][7][8][9][10][START_REF] Moreno | Appetecchi[END_REF]

| 1-11

Acknowledgements

We thank the GLiCID regional computing mesocentre, supported by Pays de la Loire, for computational support.

Notes and references

Author Contributions

JLB conceived the study. SKV performed MD simulations, TR performed calibrating MD simulations. YC supervised the simulations. SKV and YC analyzed the results. SKV, YC, CE and JLB contributed the overall interpretation of the results. The manuscript was written by SKV, YC, CE and edited by YC, CE and JLB.

Conflicts of interest

There are no conflicts to declare.


Mikhail Luginin

Nikolay Ignatiev

Anna Fedorova

Alexey Grigoriev

Alexey Shakun

Aleksander Trokhimovsky

Franck Montmessin

Oleg Korablev

Modeling of aerosols from TIRVIM solar occultations onboard ExoMars/TGO

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

Modeling of aerosols from TIRVIM solar occultations onboard ExoMars/TGO

Mikhail Luginin [START_REF] Korablev | The Atmospheric Chemistry Suite (ACS) of Three Spectrometers for the ExoMars 2016 Trace Gas Orbiter[END_REF], N. Ignatiev [START_REF] Korablev | The Atmospheric Chemistry Suite (ACS) of Three Spectrometers for the ExoMars 2016 Trace Gas Orbiter[END_REF] The main fraction of aerosols on Mars consists of mineral dust, while H2O ice and CO2 ice crystals are also encountered depending on the season and location. TIRVIM with its wide spectral range permits a spectral separation between dust and H2O ice clouds particles. This work is dedicated to aerosols modeling in the solar occultation mode. Preliminary analysis of the transmission spectra retrieved from the TIRVIM solar occultation data shows presence of both dust and H2O ice aerosol particles on some altitudes.

Acknowledgements

ExoMars is the space mission of ESA and Roscosmos. The ACS experiment is led by IKI Space Research Institute in Moscow. The project acknowledges funding by Roscosmos and CNES. Science operations of ACS are funded by Roscosmos and ESA. M. Luginin acknowledges the support from the Ministry of Education and Science of the Russian Federation, grant №14.W03.31.0017.


Arnaud Chéron

Jérôme Ronchetti

Pierre-Jean Messe

Employer-provided health insurance and equilibrium wages with two-sided heterogeneity

come

Introduction

How does health insurance interact with the labor market and especially wages? This issue is obviously not new, and has already given to interesting developments in the literature. At the start, most studies were mainly empirically-oriented and related to the United States experience (see for instance Gruber and Madrian (2002) for a survey). A distinctive feature of the US is indeed that health insurance is available almost exclusively through the workplace. 1 On the theoretical side, potential impacts on wages and job mobility were basically examined by referring to compensating differential models à la Rosen (1986) with competitive labor market. This notably suggests that health insurance leads to lower wages for those that value that insurance (Summers, 1989; Gruber and Krueger, 1990), and some job lock can arise when heterogeneity of health insurance packages makes job switching unattractive.2 Then, following Dey et Flinn (2005), a recent strand of the literature has grown to assess general equilibrium effects of health insurance, in the context of a labor market with search frictions. This approach is particularly well suited to look at the quantitative implications of some reforms of the health care system, as for instance proposed by Aizawa and Fang (2013) who focus on the Affordable Care Act in the US.

As regards to this existing literature, our paper aims to develop a more stylized equilibrium search model with both firms' and workers' heterogeneity. The firm has an imperfect information on the health status of the job seekers, which may imply some adverse selection. This two-sided heterogeneity implies that wages and health expenditures policies -which are designed to address this adverse selection issue -depend on the firm's type. Our goal is therefore to derive an analytical characterization of the model properties, as a way to identify the key factors explaining the relationship between firm's type and health expenditure. We therefore highlight that this relationship turns out to be ambiguous. To show this result, we first look at employer's wage and health insurance policies of the firm, then introduce labor supply decision through endogenous reservation wages and lastly determine the search equilibrium.

Model environment and employer's behavior

The continuous time model is featured by heterogenous employers according to productivity , who choose a wage posting policy w for the job (one job is one firm). belongs to [ , ] and we define G( ) the distribution function of firms' types. Workers also differ regarding their health endowment y, that determines the overall productivity of the job-worker pair + y. For a sake of simplicity, we consider two types of workers, the healthy workers with health status denoted by y h and unhealthy workers with health status y l < y h . 3 The point is that this health endowment is not observed by the employers, but the latter can choose to ex-ante invest in health with intensity c = c( ). If the worker is healthy, health expenditures have no impact on productivity, whereas if the worker is unhealthy, health investment allows them to improve their health status from y l to y h during the period with a probability µ(c). 4 We set µ(c) ≡ 1 -exp(-γc) that satisfies µ ∈ [0, 1] and µ > 0, µ < 0 with γ > 0 and we only consider steady-state. Let α denote the share of workers in good health, which corresponds to the probability that firm's employee is healthy. We denote by δ the exogenous job's destruction rate. We do not consider on-the-job search. For each job with productivity the employer decides what package (w( ), c( )) to offer. Labor supply decision of the workers determines reservation wages, but as a preliminary step we first consider two exogenous wages according to worker's type, w h > w l .

The problem of the firm is to maximize the intertemporal value of the job, Π( ), by choosing c and w. This value function satisfies:

Π( ) = αΠ h ( ) + (1 -α)Π l ( ) if w ≥ w h Π l ( ) if w ∈ [w l , w h [
where:

rΠ h ( ) = y h -w( ) -c -δΠ h ( ) rΠ l ( ) = [µ(c)y h + (1 -µ(c))y l ] -w -c -δΠ l ( )
This shows that if the employer sets a wage below the reservation wage of workers in good health, then he knows with certainty that the employee will be unhealthy; in such a case, health expenditures c could help increase productivity from y l to y h with probability µ(c).

Otherwise, that is for w ≥ w h , worker's type is unknown. So health expenditures have no incidence on productivity if the worker is in good health (with probability α).

Accordingly, the value of the job rewrites as follows:

(r + δ)Π( ) = [αy h + (1 -α)µ(c)y h + (1 -α)(1 -µ(c))y l ] -w -c if w ≥ w h [µ(c)y h + (1 -µ(c))y l ] -w -c if w ∈ [w l , w h [
and the optimization problem of the firm is to max Π( ) {c,w} .

On the one hand, it is straightforward to see that the wage policy of the firm is characterized by a productivity threshold ˜ so that w( ) = w h ∀ ≥ ˜ and w( ) = w l ∀ < ˜ . This means that at this stage, firms have to choose between two reservation wages, because for instance paying more than w h would not allow to improve intertemporal productivity of the job. 5 The threshold ˜ satisfies the condition αΠ h

(˜ |w = w h ) + (1 -α)Π l (˜ |w = w h ) = Π l (˜ |w = w l )
, where Π l (˜ |w = w h ) defines the value of the job occupied by an unhealthy worker that earns the highest wage w h . This implies:

˜ [αy h + (1 -α)µ(c)y h + (1 -α)(1 -µ(c))y l ] -w h - c r + δ = ˜ [µ(c)y h + (1 -µ(c))y l ] -w l - c r + δ
where c ≡ c(˜ ), and it can be rewritten as follows:

α˜ = w h -w l y h -y l 1 1 -µ(c) (1)
Equation ( 1) has several implications. First, ˜ is increasing with µ. Indeed, if firm's health expenditures have a high incidence on the productivity of unhealthy workers, the employer has less need to attract healthy workers. Consequently, the probability of offering a high wage is lower. Second, ˜ is positively related to the level of health investment c( ). A high amount of health expenditures allows the firm to improve the productivity of unhealthy workers, overcoming therefore the adverse selection problem. 6 On the other hand, the condition that determines health expenditures shows that c depends on firm's type, and hence on related wage policy. First order condition with respect to c indeed implies:

1 + ∂w ∂c = µ (c) (1 -α)(y h -y l ) ∀ ≥ ˜ (y h -y l ) ∀ < ˜ (2)
This implies c = c( ). The expected return of health expenditures therefore not only depends on µ (the marginal impact on the probability to switch from productivity y l to y h ) but also on the firm's productivity and related wage policy. On the one hand if < ˜ , wages are so low that firms only contact unhealthy workers. The marginal impact of health expenditures on the probability to improve worker's component of productivity from y l to y h is definitely given by µ . On the other hand if ≥ ˜ a share α of workers is already in good health. So the net marginal impact on expected productivity is only (1 -α)µ < µ . At this stage, equations ( 1) and (2) already show that there exists a discontinuity of health expenditures around = ˜ . Taking w h and w l as given, ∂w ∂c = 0, it comes that c ( ) 7 This means that in some cases firms with the highest productivities spend less for workers' health than some firms with lower productivities. Accordingly, this also implies that switching from low wage w l to high wage w h is not necessarily associated with higher health expenditures. This typically could occur for those who benefit from w h but who are in firms with productivity above but close to ˜ .8 6 Introducing on-the-job search, Aizawa and Fang (2013) argue that this effect concerns mainly large firms. Indeed, small firms can not retain their workers for a long time and can not benefit from the positive effects of health investment on productivity of the unhealthy workers.

> 0 ∀ ∈ [ , ˜ [ and ∀ ∈ [˜ , ], but c(˜ -) > c(˜ ).
7 This result is straightforward from (2). Assume in addition γ = 1, so µ(c) = 1 -exp(-c), we get for instance:

c = log + log(y h -y l ) + log(1 -α) ∀ ≥ ˜ 0 ∀ < ˜
3 Endogenous reservation wages of the workers

The next step is to determine endogenous reservation wages and to check whether the reservation wage of healthy workers w h is always higher than the reservation wage of unhealthy workers w l . Due to asymmetric information issue, we need to distinguish the situation of a worker in poor health according to the firm's productivity: if ≥ ˜ this unhealthy worker will indeed earn the same wage as healthy ones, w h ( ). Conversely if < ˜ employees in those jobs are all in poor health, and earn w l ( ). We denote Γ(˜ ) the probability for an unemployed worker to contact a firm with a productivity component lower than ˜ . We also introduce ψ i an indicator of health-dependent satisfaction i = {l, h}, with ψ h > ψ l .9

Actually, for workers in firms with productivity ≥ ˜ , it is straightforward to see that the optimal wage offer is equal to the lowest accepted wage by workers in good health. As firms do not observe ex-ante health status, this is the only strategy that avoids job rejection for sure. The point is therefore that this lowest accepted wage does not depend on health expenditures of the firm since healthy workers already get the highest satisfaction ψ h and productivity y h . This implies that there is only one reservation wage, w h ( ) = w h ∀ ≥ ˜ . Let E h denote the value function for a healthy employed worker and let U h be the value for a healthy unemployed individual. The latter only accepts the wage offer if ≥ ˜ . The value functions are given by:

rE h = w h + ψ h -δ [E h -U h ] rU h = b + ψ h + λ [1 -Γ(˜ )] [E h -U h ]
where λ stands for the contact rate of unemployed workers. It is obvious that the reservation wage w h = b is the unique solution of E h = U h .

Then, turning to the workers in poor health, we need to consider the fact that they can earn either w h or w l ( ). The latter wage should indeed depend on through the health expenditure policy of the firm. Value functions for the workers with the low types can then be written as follows:

rE h l ( ) = w h + µ(c( ))ψ h + [1 -µ(c( ))]ψ l -δ E h l ( ) -U l ; ∀ ≥ ˜ rE l l ( ) = w l ( ) + µ(c( ))ψ h + [1 -µ(c( ))]ψ l -δ E l l ( ) -U l ; ∀ < ˜ rU l = b + ψ l + λ ˜ E h l (x) -U l dΓ(x) + λ ˜ E l l (x) -U l dΓ(x)
where E h l defines the expected value of unhealthy individuals who work in firms with productivity greater than ˜ , and therefore earn the wage corresponding to the reservation wage of workers in good health. In addition, with a probability µ(c) such individuals expect to improve their health and related satisfaction. So their value functions depend on the type of the firm, through the level of health expenditures provided, i.e. c = c( ). Then, E l l defines the expected value for unhealthy individuals working in firms with productivity ∈ [ , ˜ [. They receive the reservation wage w l ( ) that depends on productivity. Indeed, the value of being employed is now increasing with the firms' health expenditures, through the probability µ of being healthier. This health investment decision depends on the type of firms, which will influence the wage. Let U l be the value of unemployment for workers in poor health.

The reservation wage w l ( ) that characterizes the wage policy of firms with productivity < ˜ is the solution of E l l ( ) = U l ∀ . It is straightforward to see that this reservation wage satisfies:

w l ( ) = b -µ(c( ))(ψ h -ψ l ) + λ ˜ E h l (x) -U l dΓ(x)
Furthermore, it comes also that

˜ E h l (x) -U l dΓ(x) = µ(c( ))(ψ h -ψ l ) 1-Γ(˜ ) r+δ+λ[1-Γ(˜ )]
, from which we find:

w l ( ) = b -µ(c( ))(ψ h -ψ l ) r + δ r + δ + λ[1 -Γ(˜ )] < w h = b
This means that the wage earned by unhealthy workers is negatively correlated with the firm's health investment. In addition, this relationship is stronger if the employee's valuation of the benefit is high (high value of ψ h -ψ l ).10 So we find here the standard wage-health insurance trade-off in line with the theory of compensating differentials (Summers, 1989; Gruber and Krueger, 1991).

However, this holds only if the productivity of the firm is lower than the threshold ˜ . If ≥ ˜ , the unique reservation wage of the workers in good health is offered, because the value functions of these workers no longer depend on health expenditures. This means formally that the wage policy of the firm is characterized by: ∂w( )

∂c = 0 ; ∀ ≥ ˜ ∂w( ) ∂c = -µ (c)(ψ h -ψ l ) r+δ r+δ+λ[1-Γ(˜ )] < 0 ; ∀ < ˜

Equilibrium wages and health expenditures

To derive the explicit solution of the equilibrium we need to proceed in two steps. First, we have to find the solution of a system that jointly determines (i) the productivity threshold ˜ of the firm's type below which the wage policy is so low that only unhealthy individuals agree to work within the firm, and (ii) the health expenditure of this critical firm's type, denoted by c. The corresponding system is given by:

α˜ = µ(c) 1 -µ(c) ψ h -ψ l y h -y l 1 1 + k[1 -Γ(˜ )] 1 = µ (c)(1 -α)(y h -y l )˜
where k ≡ λ/[r + δ]. Then, with µ(c) ≡ 1 -exp(-γc), it should be noticed that this rewrites as:

exp(γc) = γ(1 -α)(y h -y l )˜ ⇐⇒ c = Φ(˜ ) (3) 1 1 -exp(-γc) = γ 1 -α α ψ h -ψ l 1 + k[1 -Γ(˜ )] ⇐⇒ c = Ψ(˜ ) (4) with Φ( ) = 1 γ log[γ(1 -α)(y h -y l ) ]
, Φ > 0, Φ < 0, and Ψ( ) ≡ f (ψ h -ψ l ) > 0, Ψ < 0. Accordingly, provided that ψ h -ψ l satisfies Ψ( ) > Φ( ), there always exists a unique solution for {c, ˜ }.

Next, we can characterize wage and health expenditure policies according to firm's productivity with respect to the critical type ˜ ; {w( ), c( )} satisfies:

w( ) = b ∀ ≥ ˜ b -[1 -exp(-γc( ))](ψ h -ψ l ) r+δ r+δ+λ[1-Γ(˜ )] ∀ < ˜ (5) exp[γc( )] = (1 -α)(y h -y l ) ∀ ≥ ˜ (y h -y l ) + (ψ h -ψ l ) r+δ r+δ+λ[1-Γ(˜ )] ∀ < ˜ (6)
Figure 1 shows a graphical representation of wage and health expenditure policies of the firms according to productivity , as defined by equations ( 3)-( 6). This highlights a key implication of the model: due to discontinuity of health expenditures around ˜ , at search equilibrium we do not have a monotonous relationship between firm's type and health expenditures c, that is: c ( ) > 0 ∀ < ˜ , c(˜ -) > c(˜ ) and c ( ) > 0 ∀ ≥ ˜ .

On the one hand, below the productivity threshold ˜ we obtain a positive relationship between c and . In turn, this leads wage and health insurance to be negatively related, as predicted by the theory of compensating differentials. As the productivity is rising, employers have some incentives to increase health expenditures. Firms indeed do not observe ex ante the health status of workers and the marginal gain from improving health of unhealthy workers, i.e. (y h -y l ) , is increasing with . We refer to this channel as a productivity effect.

On the other hand, if the productivity exceeds ˜ , the firm attracts healthy workers who do not value the health insurance. High wages attract healthier workers, which reduces therefore the value of investing in worker's health; this relates to an adverse selection effect. This explains the discontinuity around ˜ where c(˜ -) > c(˜ ). Nevertheless, as high productivity firms also hire unhealthy workers, the productivity effect remains, so c( ) is increasing over the interval [˜ , ]. All in all, it is unclear whether an increase in firm's productivity from < ˜ to ≥ ˜ is associated with higher or lower average health expenditures.

Moreover, it is worth emphasizing that this ambiguity between health expenditure and productivity of the firm relies on adverse selection issue. Figure 2 indeed shows what would be the optimal wages and health investment decision by the employer in the context of perfect information. If the employer knew perfectly the health status of the workers, she would offer to the latter a contract with high wage and no complementary insurance (c h ( ) ≡ 0 ∀ ). In turn, for the unhealthy workers, we would have health expenditures as much higher as the productivity of the firm is high (c l ( ) > 0).

Conclusion

The main contribution of this paper is to explicitly show that in a context of a search equilibrium with two-sided heterogeneity, the relationship between health insurance provided by the employer and firm's type (productivity) is unclear. We highlight how it relates to adverse selection issue.

One could ask whether this result still holds if we consider a continuum of health status. If workers' health affects the value they place on health investment, we will still obtain the two offsetting effects regarding the health insurance-productivity relationship. Indeed, a high wage policy may attract healthier workers and reduces therefore the employers' incentives to invest in their workers' health. As the productivity effect still remains, given that firms do not know ex ante the workers' health, health expenditures could increase or decrease.

It is worth noting that our model does not account for on-the-job search and considers only transitory effects of health investment on health. The introduction of job-to-job mobility implies between-firm competition so employers may have interest to offer a wage higher than the reservation wage. It may make our model less tractable without changing the main mechanism at work: higher wages attract healthier workers which reduces the firms' incentives to invest in health.

Figure 1 :

 1 Figure 1: The relationship between wage and firm's health expenditures according to the productivity level w(ε)

Figure

  Figure 2: The perfect information case

As emphasized by[START_REF] Aizawa | Equilibrium Labor Market Search and Health Insurance Reform, NBER working[END_REF], it is the case for 90% of the workers.

See Madrian (1994) for a seminal paper.

We will discuss in the end of the paper the potential implications of considering a continuum of health status rather than only two.

In other words, we consider that health expenditures of the firms have only transitory effect on the worker's health status. Considering persistence adds complexity that no longer allows to derive closed form solutions and analytical results.

The fact that distribution of wages collapses to the distribution of reservation wages of unemployed workers is a well known result in a context where there is no "on-the-job" search.

In contrast, if firms were able to observe and discriminate according to health status of the worker, optimal choices would be zero health expenditure for healthy workers and a strictly increasing profile of those expenditures with for the unhealthy ones.

Here we follow Grossman's seminal model(1972) considering health as a consumption commodity.

Note that if ψ h = ψ l , we would have w( ) = b ∀ and this would imply ˜ = .


Ketan Kulkarni

Thita Chiasakul

Nicoletta Riva

Renée Eslick

Caterina Casari

Renée Eslick

Renee Eslick

Exciting times for the ISTH early career professionals

Exciting times for the ISTH Early Career professionals

This is an exciting time for early career professionals (ECP) in thrombosis and hemostasis.

ECPs are the fastest growing sub-group within the International Society on Thrombosis and Hemostasis (ISTH), with a desire to actively contribute and participate in the Society. A key priority of the ISTH leadership and senior membership group is to support, engage and nurture early career members. There are unique challenges faced by ECP, such as balancing the need to finish specialized clinical or scientific training whilst establishing a future independent career. These difficulties can be particularly striking for ECPs training in small institutions or developing countries.

In order to objectively assess challenges, needs and expectations of ECPs, questions tailored to ECPs were included in a membership survey conducted in 2018 by the Society.

The results of the survey were striking, indicating significant differences between the younger and the more established professionals:  Annual meetings and journals were identified as the most important benefits of ISTH membership for the vast majority of respondents, regardless of career stage.

However, a significantly higher proportion of ECPs were drawn to the ISTH's education, training and professional development resources, online webinars and speaking/publishing opportunities.

 ECPs reported that obtaining funding/grants (59%) and getting published (45%) were the major barriers hampering their professional career. Other identified barriers were dealing with work-life balance (31%), connecting with mentors abroad (26%), finding a permanent position (27%) and obtaining fellowships for mobility (23%).

 ECPs highly valued the networking and presenting opportunities given at ISTH congresses. However, the financial costs of conference attendance were cited as the primary barrier preventing in-person attendance in 78% of respondents.

 Interestingly, ECPs indicated that ISTH could improve their congress experience and their career by providing more networking opportunities with senior mentors (41.5%), organizing more educational sessions (41.5%), creating more opportunities for presenting their work (37.5%), increasing the number of fellowships (42%) and allowing more flexibility for fellowship applications (38.5%).

These findings were instrumental in guiding the work of the Early Career Committee (ECC), whose mission is to engage and represent the voice and needs of ECPs within ISTH. In a collaborative and enthusiastic environment, the ECC has been working in close collaboration with ISTH staff and leadership, to promote and answer ECP needs. Several projects have stemmed from these fruitful collaborations. We welcome and encourage ideas from all ISTH members to support ECPs.

In conclusion, we believe there has never been a more exciting time to be an ECP in thrombosis and hemostasis and to become an ISTH member. There is great enthusiasm within the ISTH society and leadership to improve the experience of ECPs, to develop opportunities to address their needs and ultimately to assist them in launching independent and successful careers. All ECPs have great value in contributing to this mission and will not be derailed by unforeseen hurdles, such as the COVID-19 pandemic. As members of the ECC, we would be thrilled to have you join us in these efforts.

First

  , a univocal definition of ECP has been approved and implemented within the society and ECPs are now eligible for tier-based deeply discounted or free membership and discounted congress registration fees. Secondly, to meet the needs of networking and ECP dedicated initiatives, a Trainee Session and Career Development Sessions are included in the annual conference program. Even this year, when we could not meet in person due to the COVID-19 pandemic, ECP dedicated virtual sessions have been organized shortly after the first Virtual Congress. Platforms that encourage networking and communications among ECPs and with senior mentors, both in person and online, have been endorsed. For the first time at the 2019 Melbourne Congress, ECPs were welcomed into an Early Career Lounge. On social media, resources and opportunities for ECP can be found on the ISTH website at the dedicated Early Career opportunities page (https://www.isth.org/page/earlycareer/) and by following the #ISTHEarlyCareer hashtag. In addition, a biannual Career Development Webinar series has recently been successfully launched. Efforts are currently underway to create more speaking opportunities for ECP at ISTH events & meetings and improving mentoring possibilities within the society. Moreover, the current Reach-the-World and Training fellowships are being considered for revision aiming to facilitate flexible learning opportunities and establishing lifelong connections by easing geographic and financial barriers.

Ketan

  the ISTH Early Career Committee *


Sandrine Guerlet
email: sandrine.guerlet@lmd.jussieu.fr

N Ignatiev

T Fouchet

F Forget

E Millour

R M B Young

Thermal structure and aerosol content in the martian atmosphere from ACS-TIRVIM on board ExoMars/TGO

), the climatology dataset obtained from TIRVIM will be assimilated into the LMD Mars General Circulation Model to improve our understanding of the Martian atmospheric system.

The ACS-TIRVIM instrument

The Atmospheric Chemistry Suite (ACS) is a set of three spectrometers including a thermal-infrared channel, TIRVIM [START_REF] Korablev | The Atmospheric Chemistry Suite (ACS) of Three Spectrometers for the ExoMars 2016 Trace Gas Orbiter[END_REF]. It is primary dedicated to monitoring the thermal structure and aerosol content in the Martian atmosphere by acquiring spectra in nadir geometry. ACS-TIRVIM is a Fourier-transform spectrometer covering the range 600-6000 cm -1 (1.7-17µm) with a spectral resolution of 1.2 cm -1 . We focus here on the range 600-1300 cm -1 , which covers absorption by CO2 (centered at 667 cm -1 ), water ice clouds (centered at 820 cm -1 ) and dust (centered at 1100 cm -1 ). TIRVIM is similar to the Thermal Emission Spectrometer (TES) on board Mars Global Surveyor or the Planetary Fourier Spectrometer (PFS) on board Mars Express, also operating in nadir geometry. The advantage of the TIRVIM data set over previous instruments comes from the TGO orbit, which was designed to sample a complete daily cycle every two months. Hence, TIRVIM has the capacity to uniquely study both the diurnal and seasonal variability of the thermal structure, dust and ice cloud opacity, while previous instruments mainly sample(d) the atmosphere at ~midday and ~midnight.

Retrieval algorithm

We have developed a line-by-line radiative transfer model coupled to a bayesian retrieval algorithm to retrieve vertical profiles of the temperature from ~5 to ~45 km, surface temperature, and integrated optical depth of dust and water ice clouds. Following the method of [START_REF] Conrath | Mars Global Surveyor Thermal Emission Spectrometer (TES) observations: Atmospheric temperatures during aerobraking and science phasing[END_REF], a priori temperature profiles are built from the TIRVIM spectra themselves. As nadirviewing spectra cannot constrain the dust vertical profile, we assume that it is well mixed and retrieve a scaling factor to an a priori profile. Surface emissivity is taken from previously-derived TES emissivity maps. This algorithm will be described in more detail in a future publication.

Synthetic retrievals

In order to evaluate the performance of our algorithm and identify challenging cases, we performed synthetic retrievals for a great diversity of scenes. To start with, we extracted the surface and atmospheric state of the Martian atmosphere from the Mars Climate Database (MCD) for various locations (sampling different elevations), seasons, local times and aerosol scenarios. For each of these scenes, we computed a synthetic TIRVIM spectrum and added realistic noise. We then run our retrieval algorithm, starting from the aforementioned a priori profiles that are independent of the MCD.

Overall, the retrieved temperature profiles are close to the "true" profiles used to compute synthetic spectra, with a typical error of 2-3 K. The main

EPSC

European Planetary Science Congress challenges lie in the retrieval of the temperature in the first scale height above the surface and close to the uppermost level probed by the core of the CO2 band at 667 cm -1 (near the 1-Pa level). In these altitude ranges, the contribution functions are quite broad, leading to significant degeneracy of the inverse problem. As a consequence, we are able to retrieve the mean temperature in these altitude ranges, but not the slope of the temperature profile.

Regarding dust and water ice clouds, our algorithm performs well in retrieving their integrated opacity during daytime (~9am -~5pm), even if the assumed vertical distribution of aerosols is very different from the "true" profiles used to generate synthetic spectra. However, there are special cases for which the TIRVIM nadir spectra are insensitive to changes in the aerosol load, which implies that we cannot retrieve aerosol opacity. This occurs when the temperature contrast between the surface and the atmospheric layer with strong aerosol opacity is low (typically in the morning and evening). Examples of synthetic spectra at different local times and aerosol content are shown in Figure 1 to illustrate this issue. We highlight two special cases, for a given season and location: an example of a spectrum insensitive to cloud opacity (here at midnight) and an example of spectrum not sensitive to dust opacity (here at 20h).

Application to ACS-TIRVIM

We have applied our algorithm to the TIRVIM data acquired during the two Mars Capture Orbits in November, 2016 and March, 2017. To validate the retrieved temperature profiles, we searched for colocated measurements by the Mars Climate Sounder (MCS) on board Mars Reconnaissance Orbiter. This instrument operates in limb viewing geometry such that it measures the temperature with a much greater vertical resolution. It is however on a sunsynchronous orbit and acquires spectra only near 3am and 3pm. For these first TGO orbits, 6 ACS-TIRVIM data were colocated with MCS data, and the retrieved temperature profiles agreed well (typically within ~5K) except in one case (systematic difference of 10K). This work will be continued soon with the TIRVIM data acquired during the operational orbits, still being calibrated at IKI.

Summary and Conclusions

We have developed a retrieval algorithm to analyse ACS-TIRVIM spectra acquired in nadir-viewing geometry. It is used to measure surface and atmospheric temperature (~5-45 km) as well as the integrated opacity of dust and water ice clouds. We have performed synthetic retrievals to evaluate its performance, identify challenging scenes and have applied it to TIRVIM data acquired during the Mars Capture Orbits. We validated the results using colocated MCS data. At the time of the EPSC meeting in September, we expect to have analyzed TIRVIM data acquired during the first few months of its operational phase and will discuss the results especially in term of the diurnal cycle of the temperature, aerosols, and comparisons with predictions from the LMD Mars GCM. Future efforts will focus on error characterization, a crucial aspect for upcoming data assimilation.

Figure 1 :

 1 Figure 1: Examples of synthetic TIRVIM spectra computed at different local times, as labeled, with or without aerosol.We highlight two special cases, for a given season and location: an example of a spectrum insensitive to cloud opacity (here at midnight) and an example of spectrum not sensitive to dust opacity (here at 20h).

Acknowledgements

The ACS experiment is led by IKI in Moscow. We acknowledge support by CNES and Roscosmos.


Eva Zakka-Bajjani

J Dufouleur

N Coulombel

P Roche

D C Glattli

F Portier
email: fabien.portier@cea.fr

Experimental determination of the statistics of photons emitted by a tunnel junction

Keywords: numbers: 73.23.-b, 73.50.Td, 42.50.-p, 42.50.Ar

We report on a microwave Hanbury-Brown Twiss experiment probing the statistics of GHz photons emitted by a tunnel junction in the shot noise regime at low temperature. By measuring the crosscorrelated fluctuations of the occupation numbers of the photon modes of both detection branches we show that, while the statistics of electrons is Poissonian, the photons obey chaotic statistics. This is observed even for low photon occupation number when the voltage across the junction is close to hν/e.

What is the statistics of microwave photons radiated by the current fluctuations of a quantum conductor? When the conductor is at equilibrium at temperature T , the statistics is that of a black body radiation [1]. However little is known about the statistics of the photons emitted in the non-equilibrium case where the conductor is biased by a voltage V ≫ k B T /e and the current fluctuations are due to quantum shot noise. An intriguing question is the link between the statistics of electrons and that of the emitted photons. This problem has recently attracted theoretical interest and the full range of photon statistics, from chaotic to non-classical, has been predicted. The result depends on the competition between the fermionic and bosonic statistics of electrons and photons respectively [2]. On the one hand, the Pauli principle makes electrons emitted by a contact essentially noiseless and the current noise only results from electron scattering with sub-Poissonian statistics. On the other hand photons emitted by electrons may show bunching effect transforming their initial statistics from sub-Poissonian to super-Poissonian. More generally, this rich physics relates to the problem of the electron Full Counting Statistics [3], as the second moment of the photon noise directly links to a fourth-moment of the current fluctuations [2,4,13].

In this work, we present the first measurements of the statistics of photons radiated by a quantum conductor in the shot noise regime. For the simplest quantum conductor studied here, a tunnel junction, although the statistics of electrons crossing the conductor is Poissonian, the photon statistics is shown to be chaotic. This is found even in the regime of vanishing electron shot noise where the voltage is close to the photon energy (eV hν), so that the photon population is small, in agreement with the prediction of [2]. As a by-product, our experimental method based on Hanbury-Brown Twiss (HBT) microwave photon correlation is found to provide a direct measurement of the non symmetrized current noise power, called emission noise. Here, the experiments linearly amplifies the field amplitude to a classical level and further detects the microwave power and its fluctuations. This contrasts with experiments measuring the average power with on-chip quantum detectors such as quantum dots or superconducting tunnel junctions [5].

The relation between current noise and photon emission can be understood following Nyquist's approach [1]. Consider a conductor of resistance R connected to a circuit made of a lossless transmission line of characteristic impedance Z c terminated by a matched resistor. To simplify, let's assume that Z c ≪ R. When electrons in the conductor generate a current fluctuation I(t), a voltage V (t) = Z c I(t) builds up at the input of the transmission line exciting an electromagnetic mode which propagates and is finally absorbed in the resistive load. Introducing the spectral density of the current fluctuation S I (ν) = 2 +∞ -∞ I(0)I(τ ) e i2πντ dτ , we can express the electromagnetic power radiated by the conductor in frequency range ν, ν + dν as dP = Z c S I (ν)dν = N (ν)hνdν, where N (ν) is the mean photon population of the electromagnetic mode at frequency ν. This establishes a direct link between S I (ν) and N (ν). Let us take a further step and consider the (low frequency) fluctuations δN 2 of the photon population. They originate from the intrinsic fluctuations of the current noise in the conductor (the 'noise' of the noise). However the problem of the connection between the statistics of the photons and the electrons is complicated by the bunching effect occurring when several photons are simultaneously emitted into the transmission line. The photon distribution emitted by a classical current was first addressed by Glauber who showed that the photon statistics is Poissonian [6]. Solving the same problem in the case of quantum electronic shot noise requires a model that treats electrons and the detecting environment on the same quantum footing. Such a treatment was recently developed by Schomerus and Beenakker [2]. In particular they have shown that a tunnel junction emits photons with chaotic statistics. This occurs even in the regime of small photon number when the applied voltage on the junction is close to hν/e. This is the regime addressed by our experiments.

The experimental set-up, shown in figure 1, is similar to the one described in [7]. An Al/AlO x /Al tunnel junction of resistance R t = 502Ω ± 1Ω was cooled to ∼ 30 MA in a dilution fridge. A 0.1 T magnetic field sup- presses the Al superconductivity. The two sides of the tunnel junction are separately connected to 50 Ω coaxial transmission lines via two quarter wave length impedance adapters, raising the effective input impedance of the detection lines to Z eff = 200Ω over a one octave bandwidth centered at 6 GHz. Two rf-circulators, thermalized at mixing chamber temperature ensure a circuit environment at base temperature. We note δI 1,2 the fluctuating currents in either detection branches resulting from the fluctuations of the current through the tunnel junction. S I1 , S I2 , S I1,I2 stand for the autocorrelated and cross-correlated spectral densities. From the equivalent circuit represented in figure 1, one easily sees that:

1 V 30 mK 500 Ω 200 / 50 λ/4 adapter Ω Ω circulator 4.2K LNA r.t. filters lo ADC1 2 V ADC2 a b ds V I S 1 ( ) I t δ 2 ( ) I t δ t R eff Z eff Z t r 2 2 2 , P P δ < > 2 1 1 , P P δ < > em ( , ) ds P V v
S I1 (ν, T, V ds ) = S I2 (ν, T, V ds ) = -S I1I2 (ν, T, V ds ) = R t 2Z eff + R t 2 S I (ν, T, V ds ), (1)
where the two first equalities result from current conservation. The noise power detected in each detection line in a frequency range ∆ν reads

P 1,2 = Z eff δI 2 1,2 = 4Z eff R t (2Z eff + R t ) 2 P em , (2)
where P em = R t S I (ν, T, V ds )∆ν/4 is the emitted power. This can be expressed as an excess noise temperature

∆T n1,2 = P 1,2 /[k B ∆ν 1,2 ].
The two emitted signals are then amplified by two cryogenic Low Noise Amplifiers. Up to a calculable gain factor, the detected noise power contains the weak excess noise ∆T n1,2 on top of a large additional noise generated by the cryogenic amplifiers T n1,2 ≃ 5K. After fur-ther room temperature amplification and eventually narrow bandpass filtering, current fluctuations are detected using three alternative techniques. First (fig. 1.a), we implemented the measurement scheme described in [7], using two calibrated quadratic detectors whose output voltage is proportional to noise power. Secondly (fig.

1.b), current fluctuations are digitized, after down conversion achieved by mixing with a suited local microwave signal, using an AP240 Acqiris Acquisition Card able to sample at 1Gsample/s. A quantitative comparison with the well established first method has qualified this new method. The third method, dedicated to the study of photon noise, is a hybridization of the two previous ones: the outputs of the two quadratic detectors are digitized to perform the photon HBT cross and auto correlations fluctuations of P 1,2 .

First experiment: mean photon occupation number. We measure the increase in noise temperature due do the photon emission by shot noise, as a function of V ds and the measuring frequency ν, using the quadratic detectors. In order to remove the background noise of the amplifiers, we measure the excess noise, ∆S I1,2 (ν, T, V ds ) = S I1,2 (ν, T, V ds )-S I1,2 (ν, T, 0). Practically, this is done by applying a 93Hz 0-V ds square-wave bias voltage on the sample through the DC input of a bias-Tee, and detecting the first harmonic of the square-wave noise response of the detectors using lock-in techniques. The results are quite similar to the one reported in ref. [7], and lead to an electron temperature T e ∼ 70 MA. Although T e is significantly higher than the mixing chamber temperature, it is low enough to make the thermal population of photons negligible in the 4-8 GHz frequency range, where all our measurements are done. In the high bias limit (eV ds ≫ k B T, hν), ∆S I ∼ 2eI. Equation 2then yields an excess noise temperature ∆T n1,2 = eV ds /4k B in both detection branches in the case of ideal coupling R t = 2Z eff . In practice, we get excess noise temperatures ∼ 2dB lower than expected from the independently measured attenuation of the various microwave components connecting the sample to both amplifiers. A ∼ 100fF capacitance for the junction, shunting part of the microwave signal, accounts for this discrepancy. This value is quite reasonable given the area of our tunnel junction (1.4 µm 2 ).

Second experiment: auto and cross-correlated electronic noise. We record the current fluctuations with the acquisition card using a 5 ns sampling time, chosen large enough to avoid any correlation between successive points, thus maximizing the effective bandwidth. Here again, we eliminate background noise and parasitic correlation between the two inputs of the acquisition card by measuring excess fluctuations . ∆δV 2 1 and ∆δV 2 2 are proportional to the excess noise power:

∆δV 2 1,2 = G 1,2 Z 0 P 1,2 = G 1,2 Z 0 Z eff ∆S I1,2 ∆ν,
where Z 0 = 50 Ω is the input impedance of the acquisition card, ∆ν is the bandpass of the two filters, centered around the same frequency ν, and G 1,2 stands for the gain of chain 1,2. The benefit of this method is that it also gives access to the cross-correlation term:

∆δV 1 δV 2 = G 1 G 2 Z 0 Z eff ∆ν cos(2πντ )∆S I1 I2 dν (3)
where τ is the difference of propagation time of electromagnetic waves between the sample and detectors 1 and 2. One easily gets:

∆δV 1 δV 2norm = ∆δV1δV2 q ∆δV 2 1 ∆δV 2 2 = -sinc(π∆ντ ) cos(2πντ )
≃cos(2πντ ) for ∆ντ ≪ 1.

(4) Equation 4expresses the anticorrelation of current fluctuations δI 1 and δI 2 , modified by the phase difference of the microwave signals induced by τ . This is illustrated in Figure 2 for ν = 6 GHz with ∆ν=100 MHz. Here, τ is varied using two calibrated phase shifters inserted in both detection lines, around a value τ 0 which is a priori not known.

As photons emitted at times differing by more than ∼ 1/∆ν do not show correlations, one needs to minimize τ before measuring the cross-correlated power fluctuations. As ∆ν ≪ ν, the most sensitive way to do so is to ensure that ∆δV 1 δV 2norm is constant for various values of ν. The result of such an adjustment is shown in the inset of figure 2. Although small parasitic microwaves reflections introduce extra modulations, ∆δV 1 δV 2norm doesn't change sign for 4GHz ≤ ν ≤ 8GHz. This implies that τ ≤ 125ps, so that ∆ντ ≪ 1 and the delay between the two lines doesn't affect the power correlations.

Third experiment: auto and cross-correlated photon noise. We now adopt the original HBT set-up [8], using quadratic detectors connected to the digitizer. This gives access to S P out . In the case of a tunnel junction consisting of many weakly transmitted electronic modes, one would expect [9] the emitted photons to follow a negative binomial distribution, as the emitted power results from the incoherent superposition of a large number of sources (the electronic modes of the tunnel junction). This is confirmed by the rigorous treatment of ref. [2], with corrections corresponding to emission of two photons at the same energy by the same electron. In the case of a low impedance (compared to the resistance quantum h/e 2 ), these corrections are small and one expects the photons emitted by the shot noise power of a tunnel junction to have the same counting statistics as thermal photons, although their origin is quite different and the frequency dependence of photon occupation number N (ν) doesn't correspond to a Bose-Einstein -0,5 0,0 0,5 -1,0 -0,5 0,0 0,5 1,0 thermal distribution. The cross-correlated power fluctuations are then expected to be positive, reflecting the bosonic nature of the emitted photons, and proportional to the products of the power emitted in both detection branches. The autocorrelated power fluctuations are expected to be enhanced by the contribution of the noise of the amplifiers [10]. Let's note ∆ν min (∆ν max ) the smaller (the bigger) of the bandpasses, both centered around the same frequency ν. One thus expects:

ν(τ-τ 0 )/2π 5 6 7 -1 0 1 ν (GHz) τ 1 τ 2 ( ) 1/ 2 2 2 1 2 1 2 / V V V V δ δ δ δ ∆ ∆ ∆
S P out 1 P out 1 = 2∆ν 1 G 2 1 k 2 B (T n1 + ∆T n1 ) 2 + G 1 hνk B (T n1 + ∆T n1 )] ≃ 2∆ν 1 G 2 1 k 2 B (T n1 + ∆T n1 ) 2 S P out 1 P out 2 = 2G 1 G 2 ∆ν min k 2 B ∆T n1 ∆T n2 , (5)
Equation 5 shows the benefit of HBT cross-correlation to study photon statistics, as it suppresses the term due to the mixing of the input power with the input noise of the amplifier. In order to get rid of imperfectly known gains and attenuation we normalize the excess autocorrelated power fluctuations by their zero voltage bias values and the cross-correlated power fluctuations by their geometric mean. Figure 3 range show similar agreement, with a maximum systematic deviation of roughly 3%. As shown by figure 4, the cross-correlated power fluctuations are positive, showing the bosonic character of the emitted excitations. On a quantitative level, one should observe:

∆S P out 1 P out 2 S P out 1 P out 1 S P out 2 P out 2 1/2 V ds =0 = ∆T n1 T n1 ∆T n2 T n2
∆ν min ∆ν max ,

Figure 4 shows both the cross-correlated power fluctuations, normalized to the equilibrium auto-correlated power fluctuations, and the product of the excess noise temperatures normalized by (∆ν min /∆ν max ) 1/2 . They are found to coincide within 4%. As the 'photon noise' is related to a fourth order-correlator of the electronic current [2,4], this constitutes, to the best of our knowledge, the first measurement of such a correlator in the quantum regime. It shows that the emitted radiation remains chaotic, even when the occupation number of the emitted photon modes tends to zero (eV ds , k B T ≪ hν).

We would like to add a short note to the question of the symmetrization of the current correlator involved in this shot noise measurement. The treatment of Beenakker and Schomerus [2] yields power fluctuations proportional to the square of the electronic emission noise density [12] S I (ν) = 2 +∞ -∞ I(0)I(τ ) e i2πντ dτ . Using a symmetrized correlator S I,sym = 1 2 (S I (ν)+S I (-ν)) increases the emitted power per unit bandwidth by hν/2. As the difference doesn't depend on bias voltage, excess noise measurements cannot distinguish between the two definitions. However, the quadratic dependence of the power fluctuations with the emitted power allows to distinguish them. As shown by figure 4 I,sym . The right axis gives the corresponding N (ν). The dotted line represent the onset of finite frequency excess shot noise [5,7,11].

our observations.

In conclusion, we have performed the first experiment probing the statistical properties of photons emitted by a phase coherent conductor. The data are found in perfect agreement with the predictions of Beenakker and Schomerus, showing that a biased low impedance tunnel junction emits a chaotic radiation. The crosscorrelated power fluctuations are found to be proportional to the square of the emission electronic noise density. This opens the way to the investigation of the statistical properties of photons emitted by mesoscopic conductors where electronic correlations might have a stronger impact, such as quantum point contacts, for which it is expected that the sub-Poissonian statistics of electronic shot-noise could be 'imprinted' on the corresponding emitted photons [2,13].

It's a pleasure to acknowledge precious help from Q. Le Masne, P. Bertet and D. Vion with the sample fabrication. We greatly benefited from help during the measurements from B. Dubost. This work was supported by the ANR contract 2e-BQT, and by the C'Nano Idf contract QPC-SinPS.

FIG. 1 :

 1 FIG. 1: Schematic diagram of the measurement set-up corresponding to a) detection of the amplified microwave power by quadratic detectors b) fast digitization of the down-converted current fluctuations. c) Equivalent microwave circuit.

FIG. 2 :

 2 FIG. 2: Normalized cross-correlation spectrum ∆δV1δV2 norm measured at ν=6 GHz ± 50MHz as a function of ν∆τ . The solid line represents the sinusoidal prediction of eq. 4. Inset: ∆δV1δV2 norm averaged over 200 MHz, as a function of ν for two values of τ . The solid lines correspond to equation 4 for τ =83 ps (upper line) and τ =12 ps (lower line).

1 (FIG. 3 :

 13 FIG.3: Normalized auto-correlation spectrum of power fluctuations vs normalized excess noise temperature. Symbols: experimental data. Line: Theoretical prediction. The top axis gives the corresponding N (ν).

FIG. 4 :

 4 FIG.4: Open squares: Normalized cross-correlation spectrum of power fluctuations. Solid line: theoretical prediction, assuming SP P ∝ S 2 I . Dashed line theoretical prediction, assuming SP P ∝ S 2 I,sym . The right axis gives the corresponding N (ν). The dotted line represent the onset of finite frequency excess shot noise[5,7,11].