M Van Camp

O De Viron

Gwendoline Pajot-Métivier

Fabien Casenave

Arnaud Watlet

Alain Dassargues

Marnik Vanclooster

Direct measurement of evapotranspiration from a forest using a superconducting gravimeter

 Continuous gravity measurements reveal the evapotranspiration of a forested ecosystem at the mesoscale (~50 ha).

 An oak-beech forest evaporates 1.7 mm of water during sunny summer days.

 Ground gravity is now able to isolate regular signals at the level of a few tenths of mm of 28 water

Introduction

Improving the assessment of the different components of the terrestrial hydrological cycle remains a key challenge for geoscientists. A critical component of this cycle is evapotranspiration (ET), i.e. the process whereby liquid water is converted into water vapor [START_REF] Robinson | Studies of the impact of forests on peak flows and baseflows: a European perspective[END_REF][START_REF] Sun | Potential Water Yield Reduction due to Reforestation Across China[END_REF]. ET encompasses the evaporation flux from soil and canopy, and the canopy transpiration flux [START_REF] Monteith | Evaporation and environment[END_REF][START_REF] Shukla | Influence of Land-Surface Evapotranspiration on the Earth's Climate[END_REF]. The global ET is more than 85% made up of canopy transpiration, which returns more than 50% of precipitation back to the atmosphere [START_REF] Oki | Global Hydrological Cycles and World Water Resources[END_REF]. In arid to semi-arid environments ET can return more than 95% of the annual precipitation [START_REF] Kurc | Dynamics of evapotranspiration in semiarid grassland and shrubland ecosystems during the summer monsoon season, central New Mexico[END_REF]; hence its assessment is critical to assess recharge. ET plays a major role in terrestrial ecosystems. It controls the amount of green water in the total global water balance and has therefore a major impact on the global fresh water availability [START_REF] Rost | Agricultural green and blue water consumption and its influence on the global water system[END_REF]. Furthermore, ET strongly controls energy transfer between the Earth and the atmosphere and hence within the global climate system.

The direct accurate and precise measurement of diurnal ET is possible at the individual plant scale, using weighing lysimeter devices [START_REF] Verstraeten | Assessment of evapotranspiration and soil moisture content across different scales of observation[END_REF][START_REF] Müller | Forest hydrology research with lysimeter in the northeast German lowlands special methods and results for the forest management[END_REF] and at the mesoscale, i.e. the scale of a small landscape unit, using eddy-covariance flux towers [START_REF] Shi | Comparison of methods for estimating evapotranspiration rate of dry forest canopy: Eddy covariance, Bowen ratio energy balance, and Penman-Monteith equation[END_REF][START_REF] Baldocchi | A Synthesis of Forest Evaporation Fluxes -from Days to Yearsas Measured with Eddy Covariance[END_REF]. Both devices are difficult to implement in heterogeneous, rugged forested ecosystems. Alternatively, a lysimeter type approach was proposed to measure ET at the mesoscale [START_REF] Barr | Monitoring the moisture balance of a boreal aspen forest using a deep groundwater piezometer[END_REF]]. Yet, this latter approach relies on several constraining hypothesis to convert the pressure change obtained by piezometers into ET, such as e.g. the presence of an aquitard, or a negligible runoff. At the regional scale, ET can only be estimated indirectly, e.g. from land surface models [START_REF] Kelliher | Evaporation and canopy characteristics of coniferous forests and grasslands[END_REF]Rana and Katerij, 2000;[START_REF] Wilson | A comparison of methods for determining forest evapotranspiration and its components: Sap-flow, soil water budget, eddy covariance and catchment water balance[END_REF][START_REF] Verstraeten | Comparative analysis of the actual evapotranspiration of Flemish forest and cropland, using the soil water balance model WAVE[END_REF], and remote sensing still requires terrestrial measurements for calibration or parametrization [Jimenez et al., 2011]. Such indirect regional approaches rely also on strong hypotheses, not the least of which is the homogeneity of the terrain and vegetation attributes in land surface model pixels, inducing a considerable uncertainty in regional ET assessments [START_REF] Baldocchi | A Synthesis of Forest Evaporation Fluxes -from Days to Yearsas Measured with Eddy Covariance[END_REF]Hupet et al., 2004]. Reducing regional ET uncertainty therefore relies on the calibration of land surface model parameters, using accurate and precise real ET data at ground level.

We propose a novel way to measure the mesoscale (about 50 ha) ET continuously, based on continuous gravity monitoring. Gravity is the acceleration experienced by a body at rest at the Earth surface, due to the combined gravitational forces of the Earth, the Moon, the Sun, and the planets, and the centrifugal effect from the Earth rotation. Continuous gravity monitoring using up-to-date instruments is precise to 1 part per 100 billion (10 -11 g) [START_REF] Van Camp | Hydrogeological investigations at the Membach station, Belgium, and application to correct long periodic gravity variations[END_REF].

Further, celestial mechanics can provide the relative positions of the Earth, the Moon, and the Sun. The contribution from these bodies can be corrected into the gravity data down to a resolution of at least 10 -11 g [START_REF] Hartmann | The HW95 tidal potential catalogue[END_REF][START_REF] Wenzel | The nanogal software: Earth tide data processing package ETERNA 3.30[END_REF], so that only the signal from the Earth system is left in the gravity data. The Earth rotation is also monitored with high accuracy, which allows a precise correction for the time variation of the centrifugal effect. The atmospheric contribution can be estimated and corrected with precise meteorological data such as the surface pressure. Traditional hydrology-gravity studies remove all these effects to study changes in terrestrial water content, the dominant signal in the corrected gravity time-series coming from the mesoscale terrestrial water content directly around the instrument [START_REF] Creutzfeldt | Simulating the influence of water storage changes on the superconducting gravimeter of the Geodetic Observatory Wettzell, Germany[END_REF] (Figure S1). We go one step further and remove the gravity effect of interflow losses to evaluate losses caused by ET during periods without rainfall, showing that gravity time series can be used as a weighing lysimeter for measuring mesoscale ET over a rain-free period.

Continuous gravity measurements

In 1995, a superconducting (or cryogenic) gravimeter (SG) [START_REF] Goodkind | The superconducting gravimeter[END_REF] was installed in the Membach station, eastern Belgium (50.6°N, 6.0°E, altitude: 250 m), and has been continuously operating ever since. Basic processing of the SG data is done as described by [START_REF] Hinderer | Gravimetric Methods -Superconducting Gravity Meters[END_REF]. The instrumental drift of the SG is corrected using repeated absolute gravity measurements. In this study, tides were removed by computing tidal parameter sets using the ETERNA package [START_REF] Wenzel | The nanogal software: Earth tide data processing package ETERNA 3.30[END_REF] on the gravity time series extending from 2004-06-01 to 2015-01-03 (3825.75 record days). The tidal potential is the Hartmann-Wenzel [START_REF] Hartmann | The HW95 tidal potential catalogue[END_REF] catalog with 7761 waves. The adjusted tidal parameters make it possible to compute a tidal signal which includes both the solid Earth tide and ocean loading effects. The atmospheric loading effects were corrected by using a linear admittance factor also provided in the ETERNA package. It amounts to -3.302 nm.s -2 hPa -1 . An annual modulation effect on the atmospheric S3 tide remained after removing the tidal and atmospheric effects. It was removed by fitting two harmonics with frequencies of (3 cpd + 1 cpy) and (3 cpd -1 cpy) on the gravity residuals [START_REF] Polzer | NDFW Analysis of Gravity, Strain, and Tilt Data from BFO[END_REF].

Soil moisture

The station is located about fifty meters below the surface (Figure S2). The area is one of deciduous forest (Figure S3) and experiences a marine temperate climate (Cfb in the climate scheme of Köppen-Geiger [START_REF] Peel | Updated world map of the Köppen-Geiger climate classification[END_REF]) (Figure S4). Given the position of the nearby river, the low saturated hydraulic conductivity and the low porosity of the bedrock in which the station is excavated [START_REF] Van Camp | Hydrogeological investigations at the Membach station, Belgium, and application to correct long periodic gravity variations[END_REF], the diurnal signal coming from nearby aquifers or river is expected to be smaller than 0.1 nm/s², i.e. much smaller than the ET signature.

Since 2004, three soil moisture time domain reflectometry (TDR) probes at 30, 45 and 60 cm below the surface allow for a continuous monitoring of the gravimetric water content (GWC, the mass of water per unit mass of dry soil) in the partially saturated soil, 48 m above the SG [START_REF] Van Camp | Hydrogeological investigations at the Membach station, Belgium, and application to correct long periodic gravity variations[END_REF]. The corresponding time series, which measures the local-scale water saturation of the soil, shows a seasonal pattern with steps during rainfall events [START_REF] Van Camp | Hydrogeological investigations at the Membach station, Belgium, and application to correct long periodic gravity variations[END_REF][START_REF] Meurers | Correcting gravity time series using rainfall modeling at the Vienna and Membach stations and application to Earth tide analysis[END_REF]. Focusing on a dry period, of which a short sample is plotted in red on Figure S5, we observe that the soil moisture decreases only during the day. This signature is expected from evapotranspiration during dry days [START_REF] Schelde | Comparing Evapotranspiration Rates Estimated from Atmospheric Flux and TDR Soil Moisture Measurements[END_REF].

A small soil moisture increase is seen overnight, probably due to hydraulic lift effects [START_REF] Caldwell | Hydraulic Lift: Consequences of Water Efflux from the Roots of Plants[END_REF], vapor condensation, capillary water redistribution, or imperfect temperature correction of the probes. The soil moisture probes only allow the first 60 cm of the soil to be investigated. For this reason we want to study the potential of gravity measurements to assess ET at the mesoscale around the station.

Modeling the gravity effect of groundwater

Measuring the gravity signature of ET is a metrological challenge, as the expected signal is at the level of a few millimeters of water in a temperate deciduous forest [START_REF] Kosugi | Water flow processes in weathered granitic bedrock and their effects on runoff generation in a small headwater catchment[END_REF].

For the diurnal cycle, the geophysical noise, mostly caused by hydrogeological effects and imperfect correction of atmospheric factors, amounts to 0.4 nm.s -2 , corresponding to 1.0 mm of water. Hence the expected ET signal is at the limit of the precision level of terrestrial measurements [START_REF] Van Camp | Uncertainty of absolute gravity measurements[END_REF], and can only be evidenced at particularly favorable times (Figure 1A). To enhance the signal in the time series, a stacking process over at least a few dozen of days was needed to isolate the signal (Figures 1B and1C). Figure 2 shows the error on the gravity signal calculated as a function of the number of days. This is done for June, July, August and September, during which more than 120 data values are available. The errors were estimated by the Monte-Carlo method using 100,000 random resamples [START_REF] Robert | Monte Carlo Statistical Methods[END_REF]. A precision of 0.2 nm.s -2 , equivalent to 0.5 mm of water, is already obtained after 60 days in June and 70 days in July, while 120 days or more are necessary for August and September, as ET decreases when summer fades to fall (Figure S6). From 2005 to 2014, we selected dry periods, i.e. periods when the soil moisture data exhibits step-like behavior as on Figure S5. This resulted in 139 days in June, i.e. the month experiencing the strongest ET effect (Figure S6). We averaged the gravity values separately for each hour of the day, to obtain an averaged diurnal cycle. The averaged cycle is illustrated for both the soil moisture and the gravity in Figures 1B and1C. On Figure 1C, the stair-like behavior is clear in the stacked soil moisture data, as well as the small increase during the night.

The stair-like behavior in the gravity data is also present. From July to September, the signal decreases and falls within the error bars, making it difficult to extract.

Note that the TDR probes and the gravimeter do not perform the same kind of measurement, otherwise there would be no challenge in obtaining the ET, and it would long be public knowledge. At the Membach station, the TDR probes clearly show fast changes in the shallowest zone above 60 cm. Given the hydrogeological context in the weathered zone, drainage is quite efficient and TDRs cannot reflect the water dynamics in the whole unsaturated weathered zone (1-10 m thick [START_REF] Van Camp | Hydrogeological investigations at the Membach station, Belgium, and application to correct long periodic gravity variations[END_REF]), nor in neighboring zones. We only use the GWC as a measurement of the saturation conditions of the first 60 centimeters, allowing us to identify dry periods. In the gravity signal, there is a residual slope, visible during the night, caused by the subsurface lateral flow. This flow was separated from the diurnal ET signal by removing a linear trend fitted over the nighttime behavior. This flow is caused by an ongoing interflow (subsurface lateral flow) process at the base of the weathered zone lower than the soil moisture probes [START_REF] Kosugi | Water flow processes in weathered granitic bedrock and their effects on runoff generation in a small headwater catchment[END_REF][START_REF] Davies-Smith | Geohydrology and digital simulation of the ground-water flow system in the Umatilla Plateau and Horse Heaven Hills area, Oregon and Washington[END_REF]. This interflow occurs in lower layers of the ground and is consequently not observed by the TDR probes, except just after strong rainfalls.

From those average days, we estimate the daily gravity increase associated to the ET at the level of 0.66±0.14 nm/s². Note that gravity increases because the gravimeter is located underground, below the surface soil moisture. This corresponds to an average of 1.7±0.3 mm of water over the area. Details on the conversion of nm/s² to mm of water are available in the supporting information [START_REF] Okabe | Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies[END_REF]. This value compares with what is expected in such a Cfb humid, temperate climate [START_REF] Baldocchi | A Synthesis of Forest Evaporation Fluxes -from Days to Yearsas Measured with Eddy Covariance[END_REF], and with the 1.9±0.2 mm estimate from an eddy covariance tower in a similar forest in Vielsalm (50.3°N, 6.0°E, altitude: 450 m), 34 km south of Membach [START_REF] Soubie | Evaluation de l'évapotranspiration réelle, de ses composantes et de sa régulation dans un peuplement composé de hêtres et de douglas : analyse comparative de l'effet espèce et des méthodes d'évaluation[END_REF]. Comparing Membach and Vielsalm is only indicative, given that (1) the techniques are different, (2) ET depends on the stand (Douglas fir and beeches in Vielsalm [START_REF] Aubinet | Long term carbon dioxide exchange above a mixed forest in the Belgian Ardennes[END_REF]), topography, altitude, soil conditions, canopy density, age and forest management, (3) ET inferred from the flux tower includes rainy days.

This also compares with the potential ET (PET) of 3.9 mm estimated by using the Penman-Monteith equation, considering PET as the maximum ET (see supporting information, equation [START_REF] Allen | FAO Irrigation and drainage paper No. 56[END_REF][START_REF] Verstraeten | Comparative analysis of the actual evapotranspiration of Flemish forest and cropland, using the soil water balance model WAVE[END_REF]. PET is the amount of evaporation that would occur if sufficient water was available. The PET and the ET inferred from the gravity changes are computed on the same dry days. This differs from the eddy covariance estimates, which are computed taking all the days of the month into account.

Discussions

In the subsurface, evapotranspiration is a major hydrological phenomenon that occurs in dry days, which is well illustrated by the GWC data at the Membach station. Considering that this phenomenon must be linked to changes in the water distribution around the gravimeter, it is theoretically possible to retrieve the associated signature in the gravity signal. Several questions, however, remain concerning those results: First, are those results robust and reproducible elsewhere? We successfully tested the robustness for the Membach station in many ways by taking different years and different times of the year.

It is not possible to infer from the available data if the same can be achieved in other stations. We applied our method to one other station, Walferdange, in Luxembourg, and we obtained similar results. However, based on this single comparison, we cannot conclude that the method would work at every site. Considering the level of the signal, reproducing these results in another station will require working closely with the operator of the station.

Also, can we be sure that the obtained signature is indeed linked to evapotranspiration?

Considering that (1) the amplitude is of the right order of magnitude, (2) the signal cannot be retrieved at winter time when ET is reduced, (3) the signal disappears when it is raining, (4) as soon as we avoid rainy days, the diurnal gravity signal shows this signature that only exists during the day, we can have some confidence in the results.

Finally, what is the scope of this study if it only applies to one or two stations? We consider that, even in this case, this study is useful for three reasons:

(1) From the measurement point of view, it is interesting to show that an appropriate correction of the data enables a signal to be retrieved down to a resolution of a few tenths of nm/s²;

(2) This study provides for the first time a direct measurement of evapotranspiration integrated over a deciduous forest area, a result interesting in its own right;

(3) New methods to estimate evapotranspiration can be validated and calibrated on a site where SG gravity provides an independent estimate. Note that Membach is probably not an ideal case given that the topography and the geology of the subsurface (i.e. silty soil combined with sandstone blocks and gravels) are likely to cause a strong subsurface lateral flow which is likely to disrupt the signal. We think that the ideal site would be in a flat forested area (if possible, installing the gravimeter in a shaft), with lower underground hydraulic conductivity.

Conclusions

We have shown that liquid water losses can be directly inferred from continuous gravity measurements: as water evaporates and transpires from terrestrial ecosystems, the mass distribution varies through the system, changing its gravity field.

Using continuous superconducting gravity measurements, we were able to identify average daily changes in gravity at the level of, or smaller than 10 -9 nm.s -2 (or 10 -10 g) per day.

Considering the topography around the gravimeter, this corresponds in June, from 2005 to 2014, to an average of 1.7 mm of water per sunny day over an area of 50 ha. The strength of this method consists in its ability to ensure a direct, traceable and continuous monitoring of actual ET for years at the mesoscale (~50 ha) with a precision of a few tenths of mm of water.

This can be performed in rugged forested areas with a precision of a few tenths of mm of water. The method is, therefore, complementary to the well-known eddy covariance method which often suffers from methodological uncertainty due to poor footprint modeling. This study provides also a way to assess the quality of the gravity time series and the applied corrections.

Figure 1 .

 1 Figure1. Gravity water content (blue, in %) and inverted gravity signal (red, in nm.s -2 ;

Figure 2 .

 2 Figure 2. The standard deviation of the daily gravity changes as a function of the number

Acknowledgments and Data

We thank B. Meurers (U. Vienna) and W. Zürn (BFO) for their advice about the correction of the tidal effects and M. Vandiepenbeeck (RMI) for providing the weather time series. S. Castelein, M. Hendrickx, G. Rapagnani, P. Bizermana, R. De Dobbeleer, M. De Knijf, B. Frederick and V. Rogge provided invaluable help for the maintenance of the Membach station. We thank the forester A. Delclissar for allowing us to perform hydrological measurements in the forest. We are grateful to O. Francis (U. Luxembourg) for the Walferdange data and ongoing fruitful discussions. We thank two anonymous reviewers and the Editor Valeriy Ivanov for their valuable reviews. Work from OdV was financially supported by CNES through the TOSCA program and the Institut Universitaire de France. Data are available for academic purposes through the website of the Royal Observatory of Belgium:

http://seismo.oma.be/en/gravimetry/observations/online-database.


T K T Nguyen

Thibaut Jonckheere

Adeline Crépieux

A V Nguyen

T Martin

Photo-assisted Andreev reflection as a probe of quantum noise

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Over the last decades, the measurement of noise has become a widely accepted diagnosis in the study of electronic quantum transport 1,2,3 . Indeed, noise provides information on the charge of the carriers in unconventional conductors, when considering the Fano factor -the ratio of the noise to the average current -in the regime where the carriers tunnel independently. Away from this Poissonian regime, noise also contains crucial information on the statistics of the charge carriers 4 . Experimentally, low frequency noise in the kHz-MHz range is more accessible than high frequency (GHz-THz) noise: it can in principle be measured using state of the art time acquisition techniques. For higher frequency measurements, it is becoming necessary to build a noise detector on chip for a specific range of high frequencies. In this work we consider a detector circuit which is capacitively coupled to the mesoscopic device. This circuit is composed of a normal metal -superconductor junction (NS junction). Transport in this circuit occurs when the electrons tunneling between the normal metal and the superconductor are either tunneling elastically, or when they are able to gain or to loose energy via photo-assisted Andreev reflections. The "photon" is provided or absorbed from the mesoscopic circuit which is capacitively coupled to the NS detector. The measurement of a DC current in the detector can thus provide information on the absorption and on the emission component of the current noise correlator.

Several theoretical efforts have been made to describe the high frequency noise measurement process 5 . This is motivated by the fact that in specific transport setups, high frequency noise detection is required in order to fully characterize transport. Examples are Bell inequality tests 6,7 and the detection of the anomalous charges in one dimensional correlated systems 8,9 . The former re-quire information about high frequency noise in order to establish a correspondence between electron number correlators and current noise correlators. The latter requires a high frequency noise measurement in order to obtain a non-zero signal when the nanotube is connected to Fermi liquid leads, which tend to wash out the features of a one dimensional correlated electron system.

In Ref. 10, a detection circuit, which was capacitively coupled to the mesoscopic circuit to be measured, was proposed as a high frequency noise detector. This interesting theoretical idea has so far eluded experimental verification, possibly because in the double dot system, additional (unwanted) sources of inelastic scattering render a precise noise measurement quite difficult. It is therefore necessary to look for detection circuits which are less vulnerable to dissipation, as is the case of superconducting circuits, because of the presence of the superconducting gap. Indeed, the basic idea of Ref. 10 was implemented experimentally recently 11,12 using a superconductor -insulator -superconductor (SIS) junction as a detector, measuring in this case the finite frequency noise characteristics of a Josephson junction.

More recently, the noise of a carbon nanotube/quantum dot was also measured 13 using capacitive coupling to an SIS detector, with the detection of Super-Poissonian noise resulting from inelastic cotunneling processes. The latter proposals, together with their successful experimental implementations, indicate that superconducting detector circuits have advantages over normal detection circuits because of the presence of the superconducting gap.

The purpose of the present work is thus to analyze a similar situation, except that the SIS junction is replaced by an NS circuit which transfers two electrons using Andreev reflection between a normal lead and a superconductor. The present scheme is similar in spirit to the initial proposal of Ref. 10, in the sense that it ex-ploits dynamical Coulomb blockade physics 14 . However, here two electrons need to be transferred from / to the superconductor, and such transitions involve high lying virtual states which are less prone to dissipation because of the superconducting gap, similarly to the SIS detector of Refs. 11,12,13. Andreev reflection 15 typically assumes a good contact between a normal metal and a superconductor, but in general it can be applied to tunneling contacts -it then involves tunneling transitions via virtual states. Consequently, depending on the applied DC bias, two successive "inelastic" electron jumps are required for a current to pass through the measurement circuit. The amplitude of the DC current as a function of bias voltage in the measurement circuit provides an effective readout of the noise power to be measured.

A. Detector consisting of a single NS junction

The detector circuit is depicted in Fig. 1. Two capacitors are placed, respectively, between each side of the mesoscopic device and each side of the NS tunnel junction. This means that a current fluctuation in the mesoscopic device generates, via the capacitors, a voltage fluctuation across the NS junction. In turn, the voltage fluctuations translate into fluctuations of the phase around the junction. The presence of the neighboring mesoscopic circuit acts as a specific electromagnetic environment for this tunnel junction, which is described in the context of dynamical Coulomb blockade 14 for this reason.

Fig. 2 depicts several scenarios for transport through an NS interface. Elastic transfer of single electrons can occur if the voltage applied to the junction is larger than the gap (Fig. 2a). Below the gap, elastic transport can only occur via Andreev reflection 15 , effectively transferring two electrons with opposite energies with respect to the superconductor chemical potential (Fig. 2b). Single electron can be transferred with an initial energy below the gap, provided that a "photon" is provided from the environment in order to create a quasiparticle in the superconductor (Fig. 2c). Similarly, Andreev reflection can be rendered inelastic by the environment: for instance, two electrons on the normal side, with total energy above the superconductor chemical potential, can give away a "photon" to the environment, so that they can be absorbed as a Cooper pair in the superconductor (Fig. 2d).

hω hω ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ µ S µ S µ S µ S µ L µ L µ L µ L b) c) d) a)
As we shall see, such inelastic Andreev processes are particularly useful for noise detection.

B. Detector consisting of an NS junction separated by a quantum dot

After studying the noise detection of the single NS junction, we will turn later on to a double junction consisting of a normal metal lead, a quantum dot operating in the Coulomb blockade regime, and a superconductor connected to the latter (Fig. 3). The charging energy of the dot is assumed to be large enough that double occupancy is prohibited. This setup has the advantage on the previous proposal that additional energy filtering is provided by the quantum dot. Below, we refer to this system as the normal metal-dot-superconductor (NDS) detector circuit. In Fig. 4, the quantum dot level is located above the superconductor chemical potential, and placed well within the gap in order to avoid quasiparticle processes. Because double occupancy is prohibited by the Coulomb blockade, Andreev transport occurs via sequential tunneling of the two electrons. Yet, because of energy conservation, the same energy requirements as in

DEVICE MESOSCOPIC DETECTOR DEVICE V /2 V /2 V g C g C 1 C 2 C c C c C s Z s Z s V d FIG.
3: Schematic description of the NDS set up: the mesoscopic device to be measured is coupled capacitively to the detector circuit. The latter consists of a normal metal leadquantum dot -superconductor junction with a DC bias.

Fig. 2b have to be satisfied for the final states (electrons with opposite energies, see Fig. 4a). In T-matrix terminology, for this transition to occur, virtual states corresponding to the energy of the dot are required, which suppress the Andreev tunneling current because of large energy denominators in the transition rate. Figs. 4 b, c, d, describe the cases where an environment is coupled to the same NDS circuit. Provided that this environment can yield or give some of its energy to the NDS detector, electronic transitions via the dot can become much more likely because electron energies on the normal side can be close to that of the dot level. Such transitions can thus occur even if the chemical potential of the normal lead exceeds that of the superconductor. As we shall see later on, the bias voltage can act as a valve for photo-assisted electron transitions. It is precisely these latter situations which will be exploited in order to measure the noise of the measuring circuit (the "environment").

The paper is organized as follows. The model and the calculation of the photo-assisted current in the NS junction are given in Sec. II, and results for the detection of the noise of a quantum point contact are presented. In Sec. III, we perform the same analysis for the NDS junction and compare the results. The conclusion is point out in Sec. IV.

II. TUNNELING CURRENT THROUGH THE NS JUNCTION

A. Model Hamiltonian

The Hamiltonian which describes the decoupled normal metal lead-superconductor-environment (mesoscopic circuit) system reads

H 0 = H 0L + H 0S + H env , (1)
hω hω hω ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ µ S µ S µ S µ S µ L µ L µ L µ L ǫ D ǫ D ǫ D ǫ D b) c) d) a)
FIG. 4: Andreev reflection in the NDS junction: a) Andreev reflection in the elastic regime. b), c) Photo-assisted Andreev reflection, where a "photon" is provided to or provided by the environment. d) Absorption of a Cooper pair with (reverse) photo-assisted Andreev reflection, where a "photon" is provided by the environment. For cases b), c), d), which require passing through the dot, the tunneling of electrons is sequential.

where

H 0L = k,σ ǫ k c † k,σ c k,σ , (2)
describes the energy states in the lead, with c † k,σ an electron creation operator. The superconductor Hamiltonian has the diagonal form

H 0S -µ S N S = q,σ E q γ † q,σ γ q,σ , (3)
where γ q,σ , γ † q,σ are quasiparticle operators, which relate to the Fermi operators c q,σ , c + q,σ by the Bogoliubov transformation

c -q,↓ = u q γ -q,↓ -v q γ † q,↑ , c † q,↑ = u q γ † q,↑ + v q γ -q,↓ , (4)
and E q = ∆ 2 + ζ 2 q is the quasiparticle energy, ζ q = ǫ q -µ S is the normal state single-electron energy counted from the Fermi level µ S , ∆ is the superconducting gap which will be assumed to be the largest energy scale in these calculations. Hereafter, we also define eV = µ Lµ S and assume µ S = 0.

Here we do not specify the Hamiltonian of the environment because the environment represents an open system: the mesoscopic circuit which represents the environment will only manifest itself via the phase fluctuations φ(0)φ(t) which are induced at the NS junction (or later on for the NDS circuit, at the dot-superconductor junction) because of the location of the capacitor plates. In what follows, we shall assume that the unsymmetrized noise spectral density

S + (ω) = 2 +∞ -∞ dte iωt I(0)I(t) , (5)
corresponding to photon emission (for positive frequency), or alternatively S -(ω) ≡ S + (-ω), the spectral density of noise corresponding to photon absorption, are both specified by the transport properties of the mesoscopic circuit 5,16,17 . Here . . . stands for an irreducible noise correlator, where the product of average currents has been subtracted out.

The tunneling Hamiltonian describing the electron transferring between the superconductor and normal metal lead in the NS junction is

H T = k,q,σ T k,q c † k,σ c q,σ e -iφ , (6)
the indices k, q refer to the normal metal lead, superconductor. We consider for simplicity that T k,q = T 0 . Note that the tunneling Hamiltonian contains a fluctuating phase factor, which represents the coupling to the mesoscopic circuit. Indeed, because of the capacitive coupling between the sides of the NS junction and the mesoscopic circuit, a current fluctuation translates into a voltage fluctuation across the NS junction. Both are related via the trans-impedance of the circuit 10 V (ω) = Z(ω)I(ω).

Next, the voltage fluctuations translate into phase fluctuations across the junction, as the phase is the canonical conjugate of the charge at the junctions 14 : the phase is thus considered as a quantum mechanical operator throughout this paper. For definition purposes, it is convenient to introduce the correlation of the phase operators

J(t) = [φ(t) -φ(0)]φ(0) , (7)
where the phase operator is related to the voltage by the relation

φ(t) = e t -∞ dt ′ V (t ′ ) . ( 8
)
Given a specific circuit (capacitors, resistances,...) the phase correlator is therefore expressed in terms of the trans-impedance of the circuit and the spectral density of noise 10

J(t) = 2π R K ∞ -∞ dω |Z(ω)| 2 ω 2 S I (ω)(e -iωt -1) , (9)
where R K = 2π/e 2 is the quantum of resistance and S I (ω) = S + (-ω).

The present system bears similarities with the study of inelastic Andreev reflection in the case where the superconductor contains phase fluctuations [START_REF] Falci | [END_REF]19 . Such phase fluctuations destroy the symmetry between electrons and holes, and affect the current voltage characteristics of the NS junction.

B. Tunneling current

The tunneling current associated with two electrons is given by the Fermi Golden Rule I = 2eΓ i→f , with the tunneling rate 20

Γ i→f = 2π f | f |T |i | 2 δ(ǫ i -ǫ f ) , (10)
where ǫ i and ǫ f are the tunneling energies of the inital and final states, including the environment, T is the transition operator, which is expressed as

T = H T + H T ∞ n=1 1 iη -H 0 + ǫ i H T n , (11)
with η is a positive infinitesimal.

Throughout this paper, one considers the photoassisted tunneling (PAT) current due to the high frequency current fluctuations of the mesoscopic device, as the difference 14 :

I P AT = I(environment) -I(no environment) , (12)
where, in general, the total current for tunneling of electrons through the junction is I = I → -I ← .

However, experimentally 12 it is difficult to couple capacitively and then to remove the mesoscopic device circuit from the detector circuit. What is in fact often measured is the excess noise, i.e. the difference between current fluctuations at a given bias and zero bias in the mesoscopic circuit. In the latter on, we will calculate the excess noise S + ex (ω) of S + (ω). In this work, we thus measure the difference between the currents through the detector when the device circuit is applied a bias voltage and zero bias, as a function of detector bias voltage, which is defined as

∆I P AT (eV ) = I(eV d = 0, eV ) -I(eV d = 0, eV ) . (13)
This also corresponds to the difference between the PAT currents through the detector when the device circuit is applied a bias voltage and zero bias ∆I P AT (eV ) = I P AT (eV d = 0, eV ) -I P AT (eV d = 0, eV ) because the contributions with no environment cancel out. The difference PAT current thus provides crucial information on the spectral density of excess noise of the mesoscopic device. Notice that our calculation applies to the zero temperature case for convenience, but it can be generalized to finite temperatures.

C. Single electron tunneling

Although our focus of interest concerns photo-assisted Andreev reflection, we need to compute all possible contributions. The current associated with one electron tunneling is given by the Fermi golden rule:

I = 2πe f | f |H T |i | 2 δ(ǫ i -ǫ f ) , (14)
The calculation of the current proceeds in the same way as that of a normal metal junction 14 , except that one has to take into account the superconducting density of states on the right side of the junction which is done by exploiting the Bogoliubov transformation. For the case of electrons tunneling from the normal metal lead to the superconductor (eV > ∆), the current from left to right reads

I → = e ∞ -∞ dte -i(µS -µL)t H T (t)H † T (0) = 4πeT 2 0 N 2 N eV -∞ dǫ ∞ ∆ dE E √ E 2 -∆ 2 1 - 2π R K ∞ -∞ dω |Z(ω)| 2 ω 2 S I (ω) δ(E -ǫ) + 8π 2 eT 2 0 N 2 N R K eV -∞ dǫ ∞ ∆ dE E √ E 2 -∆ 2 |Z(ǫ -E)| 2 (ǫ -E) 2 S I (ǫ -E) , (15)
with R K the quantum of resistance. This current includes both an elastic and an inelastic contribution, the former being renormalized by the presence of the environment. Here, ǫ is the energy of an electron in the normal metal lead and E is the energy of a quasiparticle in the Superconductor lead. Changing variables in the inelastic term to

Ω = ǫ -E, δ = ǫ + E, an using dx(x + a)/ (x + a) 2 -b 2 = (x + a) 2 -b 2 /2
, after computing the current from right to left in a similar way, we obtain

∆I P AT = -C 1e ∞ -∞ dω |Z(ω)| 2 ω 2 S + ex (-ω) K el 1e (eV ) +C 1e eV -∆ -∞ dΩ |Z(Ω)| 2 Ω 2 S + ex (-Ω)K inel 1e (Ω, eV ) ,( 16
)
where the transmission coefficient of the NS junction in the normal state is defined as

T = 4π 2 N L N 0S T 2 0 , C 1e = eT /R K .
The weight functions are defined as

K el 1e (eV ) = (eV ) 2 -∆ 2 , (17)
K inel 1e (Ω, eV ) = (Ω -eV ) 2 -∆ 2 . ( 18
)
Similarly, we obtain the formula of ∆I P AT for the case eV ≤ -∆

∆I P AT = -C 1e ∞ -∞ dω |Z(ω)| 2 ω 2 S + ex (-ω) K el 1e (eV ) +C 1e ∞ eV +∆ dΩ |Z(-Ω)| 2 Ω 2 S + ex (Ω)K inel 1e (Ω, eV ) . ( 19
)
For the case -∆ ≤ eV ≤ ∆: there are no elastic transitions because electrons cannot enter the superconducting gap. Nevertheless, due to the presence of the environment, electrons can absorb/emit energy from/to the environment so that an inelastic quasiparticle current flows.

∆I P AT = C 1e eV -∆ -∞ dΩ |Z(Ω)| 2 Ω 2 S + ex (-Ω)K inel 1e (Ω, eV ) -C 1e ∞ eV +∆ dΩ |Z(-Ω)| 2 Ω 2 S + ex (Ω)K inel 1e (Ω, eV ) . (20)

D. Two electrons tunneling as two quasiparticles

The single electron (quasiparticle) tunneling current is of first order in the tunneling amplitude. We now turn to processes which invoke the tunneling of two electrons through the NS interface. Indeed, because our aim is to show that Andreev reflection can be used to measure noise, we need to examine all two electron processes, we start with the transfer of two electrons as quasiparticles above the gap. Calculations of the matrix element in Eq. ( 10) are then carried out to second order in the tunneling Hamiltonian using the T-matrix:

I = 4πe f | f |H T 1 iη -H 0 + ǫ i H T |i | 2 δ(ǫ i -ǫ f ) , (21)
The initial state is a product:

|i = |G L ⊗ |G S ⊗ |R , (22)
where |G L denotes a ground state, which corresponds to a filled Fermi sea for the normal electrode. |G S is the BCS ground state in superconductor lead. |R denotes the initial state of the environment. On the other hand, our guess for the final state should read

|f = c † k,σ c † k ′ ,σ ′ γ † q,σ γ † q ′ ,σ ′ |G L ⊗ |G S ⊗ |R ′ , (23)
when two electrons are emitted from superconductor.

The "guess" of Eq. ( 23) is an informed one: an electron pair is broken in the superconductor, and one electron tunnels to the normal metal lead, while the other becomes a quasiparticle in the superconductor; the same process for the second electron tunneled to the normal metal lead. When superconductor lead absorbs two electrons

|f = c k,σ c k ′ ,σ ′ γ † q,σ γ † q ′ ,σ ′ |G L ⊗ |G S ⊗ |R ′ , (24)
The "guess" of Eq. and ( 24) is: two electrons can tunnel from the normal metal lead and become quasiparticles in the superconductor. Here, |R ′ is the final state of the environment.

Introducing the closure relation for the eigenstates of the non-connected system {|υ i }, and using the fact that υ|(ǫ i -H 0 ±iη) -1 |υ = ∓i ∞ 0 dte i(ǫi-ǫυ±iη)t , one can exponentiate all the energy denominators. Then, by transforming the time dependent phases into a time dependence of the tunneling Hamiltonian, we can integrate out all final and virtual states. This allows to rewrite the tunneling current in terms of tunneling operators in the interaction representation, to lowest order O(T 4 0 ). For the case of two electrons tunneling from Superconductor lead to normal metal lead, the current reads

I ← = 2e ∞ -∞ dt ∞ 0 dt ′ ∞ 0 dt ′′ e -η(t ′ +t ′′ ) e i(µS -µL)(2t-t ′ -t ′′ ) × H † T (t -t ′′ )H † T (t)H T (t ′ )H T (0) . (25)
Here we consider quasiparticle tunneling so

c † q1σ1 (t -t ′′ )c † q2σ2 (t)c q3σ3 (t ′ )c q4σ4 (0) = -|v q | 2 |v q ′ | 2 e -iEq(t-t ′′ -t ′ ) e -iE q ′ t +|v q | 2 |v q ′ | 2 e -iEq(t-t ′′ ) e -iE q ′ (t-t ′ ) . ( 26
)
The exponentials of phase operators are calculated with the definition of Eq. ( 7), so that 23,24 e iφ(t-t ′′ ) e iφ(t) e -iφ(t ′ ) e -iφ = e J(t-t ′′ -t ′ )+J(t-t ′′ )+J(t-t ′ )+J(t)-J(-t ′′ )-J(t ′ ) ,( 27)

then we further assume that J(t) ≪ 1, which means a low trans-impedance approximation together with the fact that J(t) is well behaved at large times. This allows to expand the exponential of phase correlators e J(t) ≈ 1 + J(t). The result for the current contains both an elastic and an inelastic contribution

I ← = I el ← + I inel ←
, where

I el ← ≃ eT 2 16π 3 R K Ψ 0← - 2π R K ∞ -∞ dω |Z(ω)| 2 ω 2 S I (ω)K el 2e← (ω, eV, η) , (28)
with Ψ 0← , K el 2e← (ω, eV, η) are defined as in Eqs. ( 53) and (54) in Appendix A, and

I inel ← ≃ eT 2 16π 2 R K ∞ 2∆+2eV dΩ |Z(-Ω)| 2 Ω 2 S I (-(Ω))K inel 2e← (Ω, eV, η) , (29)
where K inel 2e← (Ω, eV, η) is defined in Eq. ( 55) in Appendix A. The elastic contribution exists only if eV < -∆. For -∆ < eV only the inelastic part contributes to I ← .

Similarly we calculate for I → . There is a symmetry between the magnitude between the right and left moving current upon bias reversal: the expression for I ← is the same as I → , if we replace -eV by eV .

So in the interval |eV | ≤ ∆, we obtain

∆I P AT (eV )= C 2e ∞ 2∆-2eV dΩ |Z(-Ω)| 2 Ω 2 S + ex (Ω)K inel 2e→ (Ω, eV, η) -C 2e ∞ 2∆+2eV dΩ |Z(-Ω)| 2 Ω 2 S + ex (Ω)K inel 2e← . ( 30
)
with

K inel 2e→ (eV ) = K inel 2e← (-eV ) and C 2e = eT 2 /16π 2 R K .
E. Two electron tunneling as a Cooper pair: Andreev reflection

In this case, we also needs to carry out calculations of the matrix element in Eq. ( 10) to second order in the tunneling Hamiltonian. Typically, the initial state will be as shown in Eq. ( 22). On the other hand our guess for the final state reads:

|f = 2 -1/2 [c † k,σ c † k ′ ,-σ -c † k ′ ,σ c † k,-σ ]|G L ⊗ |G S ⊗ |R ′ , (31)
when a Cooper pair is emitted from superconductor, or

|f = 2 -1/2 [c k,σ c k ′ ,-σ -c k ′ ,σ c k,-σ ]|G L ⊗ |G S ⊗ |R ′ , (32)
when Superconductor lead absorbs a Cooper pair. Here, |R ′ is the final state of the environment. The "guess" of Eqs. ( 31) and ( 32) is again an informed one: indeed, the s-wave symmetry of the superconductor imposes that only singlet pairs of electrons can be emitted or absorbed. This phenomenon has been described in the early work on entanglement in mesoscopic physics 21,22 , and the resulting final state can in principle be detected through a violation of Bell inequalities 6 . For this Andreev process, we can now write the tunneling current as a function of the normal (and anomalous) Green's functions of the normal metal lead, G Lσ , the quantum dot, G Dσ , and the superconductor, F σ (see Appendix B), which is the same as in Ref. 23

I ← = 2eT 4 0 ∞ -∞ dt ∞ 0 dt ′ ∞ 0 dt ′′ e -η(t ′ +t ′′ ) e i(µS -µL)(2t-t ′ -t ′′ ) × k,k ′ ,q,q ′ ,σ -G > Lσ (k, t -t ′′ -t ′ )G > L-σ (k ′ , t)F * σ (q ′ , -t ′′ )F -σ (q, t ′ ) + G > Lσ (k, t -t ′′ )G > L-σ (k ′ , t -t ′ )F * σ (q ′ , -t ′′ )F σ (q, t ′ ) ×e J(t-t ′′ -t ′ )+J(t-t ′′ )+J(t-t ′ )+J(t)-J(-t ′′ )-J(t ′ ) . ( 33
)
The result for the current contains both an elastic and an inelastic contribution:

I ← = I el ← + I inel ← , (34)
where the elastic contribution reads

I el ← ≃ eT 2 2π 3 -eV eV dǫ ∞ ∆ dE ∞ ∆ dE ′ ∆ 2 √ E 2 -∆ 2 E ′ 2 -∆ 2 × 1 - 4π R K +∞ -∞ dω |Z(ω)| 2 ω 2 S I (ω) 1 D 0 ← - 2π R K +∞ -∞ dω |Z(ω)| 2 ω 2 S I (ω) D el ← , (35)
The inelastic contribution to I ← is

I inel ← ≃ eT 2 π 2 R K ∞ eV dǫ ∞ eV dǫ ′ ∞ ∆ dE ∞ ∆ dE ′ × ∆ 2 √ E 2 -∆ 2 E ′ 2 -∆ 2 |Z(-(ǫ + ǫ ′ ))| 2 (ǫ + ǫ ′ ) 2 S I (-(ǫ + ǫ ′ )) D inel ← . ( 36
)
where the denominators are specified in Appendix C. I → is derived in a similar manner, but its expression is omitted here. Nevertheless, it effects will be displayed in the measurement of the noise of a point contact.

The above expressions constitute the central result of this work: one understands now how the current fluctuations in the neighboring mesoscopic circuit give rise to inelastic and elastic contributions in the current between a normal metal and a superconductor.

We find that both current contributions have the same form and the current fluctuations of the mesoscopic device affect the current in the detector at the energy corresponding to the total energy of two electrons in the normal lead. Andreev reflection therefore acts as an energy filter.

Next, we can change variables as in the previous sections. With an arbitrary bias eV , we obtain

∆I P AT (eV ) = -C N S +∞ -∞ dω |Z(ω)| 2 ω 2 S + excess (-ω)K el N S (ω, eV, η) - C N S 2 2eV -∞ dΩ |Z(Ω)| 2 Ω 2 S + excess (-Ω)K inel N S (Ω, eV, η) - C N S 2 ∞ 2eV dΩ |Z(-Ω)| 2 Ω 2 S + excess (Ω)K inel N S (Ω, eV, η) , (37)
with

C N S = eT 2 ∆ 2 /π 2 R K . The kernel functions K el N S , K inel N S are shown in Appendix C.
F. Quantum Point Contact as a source of noise

Spectral density of excess noise

In this section, we illustrate the present results with a simple example. We consider for this purpose a quantum point contact, which is a device whose noise spectral density is well characterized by using the scattering theory 1, 25 . Here however, we consider unsymmetrized noise correlators:

S + (ω) = 2e 2 π T (1 -T )(eV d -hω)θ(eV d -hω) , if ω ≥ 0, 2e 2 π [-2T 2 hω -T (1 -T )(eV d + hω)θ(-eV d -hω) + T (1 -T )(eV d -hω)] , if ω < 0 , ( 38
)
where θ is the Heaviside function and T is the transmission probability, which is assumed to have a weak dependence on energy over the voltage range eV . As pointed out above, we are computing the difference of the PAT currents in the presence and in the absence of the DC bias. This means that we insert the spectral density of excess noise of the mesoscopic device, which for a point contact bears most of its weight near zero frequencies. Excess noise decreases linearly to zero over a range [0, ±eV d ] for positive and negative frequencies:

S + ex (ω) = 2e 2 π T (1 -T )(eV d -|hω|)θ(eV d -|hω|) , (39)

Numerical calculations

We choose a generic form for the transimpedance, similar in spirit to that chosen in Ref. 10. Considering the circuit in Figs. 1 and3, at ω = 0, device and detector are not coupled and the transimpedence should therefore vanish. On the other hand, the transimpedance is predicted to have a constant behavior at large frequency. We therefore choose the following generic form for the transimpedance

|Z(ω)| 2 = (Rω) 2 ω 2 0 + ω 2 , ( 40
)
where R is the typical high frequency impedance and the crossover frequency ω 0 is estimated from the experimental data of Ref. 12, choosing a finite cutoff frequency ω 0 means that at frequencies ω ≪ ω 0 , the mesoscopic circuit has no influence on the detector circuit because low frequencies do not propagate through a capacitor.

We calculate numerically the PAT currents in the three above cases: single and two quasiparticle tunneling, and Andreev reflection. All energies are measured with re- spect to the superconducting gap ∆. In these units we chose ω 0 = 0.3 and η = 0.001. Currents are typically plotted as a function of the DC bias voltage eV of the detector, for several values of the mesoscopic bias voltage eV d (the "environment"). Our motivation is to consider the PAT currents with the condition

|eV | < 1 (|eV | < ∆),
where the effect of the environment on the PAT current is most pronounced, and we shall predict that two electrons tunneling as a Cooper pair (Andreev processes) contributes the most to the PAT current, except close to eV = ∆. Because of the symmetry between negative eV and positive eV , we display the results for eV > 0.

The PAT currents for the three above processes are plotted next to one other in Fig. 5 for comparison. We find in Fig. 5 that if the chemical potential of the normal metal lead is close to the potential of the superconductor lead (eV ≪ ∆), the PAT currents in the three cases are suppressed. For the two cases of quasiparticle tunneling, the PAT currents are equal to zero below a certain threshold. The single quasiparticle current differs from zero at a threshold which is identified as ∆ -eV d , that is quasiparticles are able to tunnel above that superconductor gap only if they can borrow the necessary energy from the mesoscopic device. This explains why the curves associated with different values of the mesoscopic device bias voltage are shifted to the right as eV d decreases. For two quasiparticle particle tunneling, we observe that the PAT current has a similar threshold, which, compared to Fig. 5a, is pushed toward the right in Fig. 5b, because more energy is needed to transfer two electrons above the gap, compared to a single electron. Nor surprisingly, the curves corresponding are once again shifted to the right with decreasing eV d . These curves all have a sharp maximum at eV = ∆.

We turn now to the Andreev PAT difference current, which dominates the two above processes at small and moderate bias. Note that the total Andreev current contains an elastic contribution as well as an inelastic contribution below the gap, contrary to quasiparticle tunneling which have contributions below the gap only because these processes are photo-assisted. Because we are computing the difference between PAT current with and without the mesoscopic bias voltage, we expect that the elastic contribution cancels out. However the first term of Eq. (37) tells us that the presence of the environment also gives rise to an effective elastic contribution to the PAT Andreev difference current. Unfortunately, this elastic correction is not small compared to the true Andreev current.

Looking at Fig. 5, we note that the PAT curve for Andreev processes is shifted to the left when the bias of the mesoscopic device is increased. The environment provides/absorbs energy to pairs of electrons whose energies are not symmetrical with respect to the superconducting chemical potential. At very weak eV , the elastic correction is small compared to the true Andreev current. We expect the PAT current contributions to originate from pairs of electrons in the normal metal below the superconducting chemical potential which can extract a photon from the environment. At the same time, Cooper pairs can be ejected in the normal metal as a reverse Andreev processes provided they borrow a photon to the environment. There is a balance between the right and left currents at very weak bias.

As the bias is increased, electrons pairs whose energies are above this chemical potential will now be able to yield a photon to the environment, giving rise to an increase of the inelastic PAT current. Also, the reverse Andreev process mentioned above becomes more restricted because the available empty states for electrons in the leads lie higher at positive bias. The elastic PAT current (not shown in the figures) increases when we increase the detector bias but it is always smaller than the inelastic contribution. The total PAT current increases gradually (Fig. 5, right panel).

A zoom of this Andreev contribution is made in the region of small bias. There is in fact no threshold for the PAT Andreev current: for small bias, it has a linear behavior (Fig. 6). According to Eq. ( 37), when eV ≥ eV d /2, the inelastic difference current of a Cooper pair tunneling from S to L by absorbing energy from mesoscopic device vanishes. The PAT current now describes two electron tunneling from L to S elastically or inelastically. In this intermediate bias regime, the elastic and inelastic contributions have now a tendency to cancel each other. The current reaches a maximum close to the gap, and then it decreases dramatically at the gap. This is consistent with the fact that for positive bias, the initial state for two quasiparticle tunnel processes and for Andreev reflection is precisely the same: close to the gap two quasiparticle tunneling takes over the Andreev process. It becomes more efficient for electrons to be activated above the gap than to be converted into a Cooper pair because the energy loss needed for the latter is quite large. Unlike a conventional normal metal -superconductor junction with elastic scattering only, where the relative importance of quasiparticle tunneling and Andreev reflection are interchanged precisely at the gap, here the dominance of quasiparticle tunneling manifests itself before the voltage bias reaches the gap. Note also in Fig. 5, that the magnitude of the Andreev current before the two quasiparticle threshold is precisely the same as the magnitude of the two quasiparticle at eV = ∆, which confirms this conversion scenario. A comparison of the two processes is displayed in Fig. 7.

In practice, the different contributions to the PAT current cannot be separated: one measures the sum of the three contributions which are plotted in Fig. 5. However, we claim that for a broad voltage range (from eV = 0 to the two quasiparticle threshold), the main contribution to the current comes from photo-assisted Andreev processes. The confrontation of Eq. ( 37) with an experimental measurement of the PAT current below the gap could thus serve an effective noise measurement, as the weight functions K el N S (ω, eV, η) and K inel N S (Ω, eV, η) are known. Notice that in all our numerical results, the PAT currents are plotted in units of e 3 R 2 T 2 T (1-T )∆/8π 3 h2 R K . We put some tentative numbers in these quantities. Here T = 0.6 is the effective transmission coefficient of the NS interface, ∆ = 240µeV , T = 0.5 is the transmission of the quantum point contact to be measured, R = 0.03R K (R K is the resistance quantum) is the resistance which enters the transimpedance. This implies, e.g. for the PAT current in Fig. 6 at the top of the peak, ∆I P AT ≃ 10 -10 A with the device bias V d = 0.8∆/e = 48µV , which seems an acceptable value which is compatible with current measurement techniques.

III. TUNNELING CURRENT THROUGH A NDS JUNCTION

We now turn to a different setup for noise detection where electrons in a normal metal lead transit through a quantum dot in the Coulomb blockade regime before going into the superconductor. The essential ingredients are the same as the previous section, except that additional energy filtering occurs because of the dot. In this section, we choose the parameters of the device so that only Andreev processes are relevant.

A. Model Hamiltonian

The Hamiltonian which describes the decoupled normal metal lead -dot -superconductor -environment (mesoscopic circuit) system reads

H 0 = H 0L + H 0D + H 0S + H env , (41)
where the Hamiltonian of normal metal lead and superconductor lead are described as above.

The Hamiltonian for the quantum dot reads

H 0D = σ ǫ D c † D,σ c D,σ + U n ↑ n ↓ , (42)
where U will be assumed to be infinite, assuming a small capacitance of the dot. We consider that the dot possesses only a single energy level for simplicity.

The tunneling Hamiltonian includes electron tunneling between the superconductor and the dot, as well as the tunneling between the dot and the normal metal lead

H T = (H T 1 + H T 2 ) + h.c , (43)
H T 1 = q,σ T D,q c † D,σ c q,σ e -iφ , H T 2 = k,σ T k,D c † k,σ c D,σ ,
the indices k, D, q refer to the normal metal lead, quantum dot, superconductor. We consider the simple case T D,q = T 1 , and T k,D = T 2 . For photo-assisted Andreev processes, one needs to carry out calculations of the matrix element in Eq. ( 10) to fourth order in the tunneling Hamiltonian. In what follows, we assume that the dot is initially empty, owing to the asymmetry between the two tunnel barriers. The barrier between the normal metal lead and the dot is supposed to be opaque compared to that between the dot and the superconductor. As a result, the rate of escape of electrons from the dot to the superconductor is substantially larger than the tunneling rate of electrons from the normal lead to the dot (see below for actual numbers).

There are two possibilities for charge transfer processes: a Cooper pair in the superconductor is transmitted to the normal lead or vice versa. The first process involves the electron from a Cooper pair tunneling onto the dot, next this electron escapes in the lead; the other electron from the same Cooper pair then undergoes the same two tunneling processes. Similar transitions, in the opposite direction, are necessary for two electrons from the normal lead to end up as a Cooper pair in the superconductor. Note that this description of events assumes implicity that superconductor lead remains in the ground state in the initial and final states (Andreev process). On the other hand, if normal metal lead is initially in the ground state (filled Fermi sea), it is left in an excited state with two electrons having energies above Fermi energy E F in the final state. The extra energy has been provided by the environment. Typically,

|i = |G L ⊗ |G S ⊗ |0 QD ⊗ |R , (44)
where |G ... denotes a ground state, which corresponds to a filled Fermi sea for the normal electrode. |0 QD is the vacuum of the quantum dot, and |R denotes the initial state of the environment. On the other hand our guess for the final state should read

|f = 2 -1/2 [c † k,σ c † k ′ ,-σ -c † k ′ ,σ c † k,-σ ]|G L ⊗|G S ⊗|0 QD ⊗|R ′ , (45)
when a Cooper pair is emitted from S, or

|f = 2 -1/2 [c k,σ c k ′ ,-σ -c k ′ ,σ c k,-σ ]|G L ⊗|G S ⊗|0 QD ⊗|R ′ ,
(46) when superconductor lead absorbs a Cooper pair. Here, |R ′ is the final state of the environment. The justification for the choice of Eqs. ( 45) and ( 46) is the same as in Sec. II E.

B. General formula for the photo-assisted Andreev current

For the case of two electrons tunneling from superconductor lead to normal metal lead, the current reads

I ← = 2e ∞ -∞ dt ∞ 0 dt 1 ∞ 0 dt 2 ∞ 0 dt 3 ∞ 0 dt ′ 1 ∞ 0 dt ′ 2 ∞ 0 dt ′ 3 e -η(t1+t2+t3+t ′ 1 +t ′ 2 +t ′ 3 ) ×e iµS (2t-t ′ 1 -t ′ 2 -2t ′ 3 -t1-t2) e -iµL(2t-t ′ 2 -t ′ 3 -2t1-t2-t3) × H † T 1 (t -t ′ 1 -t ′ 2 -t ′ 3 )H † T 2 (t -t ′ 2 -t ′ 3 )H † T 1 (t -t ′ 3 )H † T 2 (t)H T 2 (t 1 + t 2 + t 3 )H T 1 (t 1 + t 2 )H T 2 (t 1 )H T 1 (0) . ( 47
)
The problem is thus reduced to the calculation of correlators of the tunneling Hamiltonian in the ground state. Using Wick's theorem, these can be expressed in terms of single particle Green's function because the Hamiltonian of the isolated components is quadratic (except, maybe for the environment, which is dealt separately). We can now write the tunneling current as a function of the normal (and anomalous) Green's functions of the normal metal lead, G Lσ , the quantum dot, G Dσ , and the superconductor, F σ (which are shown in Appendix B)

I ← = 2eT 4 1 T 4 2 ∞ -∞ dt ∞ 0 dt 1 ∞ 0 dt 2 ∞ 0 dt 3 ∞ 0 dt ′ 1 ∞ 0 dt ′ 2 ∞ 0 dt ′ 3 e -η(t1+t2+t3+t ′ 1 +t ′ 2 +t ′ 3 ) ×e iµS (2t-t ′ 1 -t ′ 2 -2t ′ 3 -t1-t2) e -iµL(2t-t ′ 2 -t ′ 3 -2t1-t2-t3) × k,k ′ ,q,q ′ ,σ -F * σ (q ′ , -t ′ 1 -t ′ 2 )F -σ (q, t 1 + t 2 )G > Lσ (k, t -t ′ 2 -t ′ 3 -t 1 -t 2 -t 3 )G > L-σ (k ′ , t -t 1 ) × G t Dσ (-t ′ 1 )G t D-σ (-t ′ 3 )G t Dσ (t 3 )G t D-σ (t 1 ) + F * σ (q ′ , -t ′ 1 -t ′ 2 )F σ (q, t 1 + t 2 ) × G > Lσ (k, t -t ′ 2 -t ′ 3 -t 1 )G > L-σ (k ′ , t -t 1 -t 2 -t 3 )G t Dσ (-t ′ 1 )G t D-σ (-t ′ 3 )G t D-σ (t 3 )G t Dσ (t 1 ) ×e J(t-t ′ 1 -t ′ 2 -t ′ 3 )+J(t-t ′ 3 )+J(t-t ′ 1 -t ′ 2 -t ′ 3 -t1-t2)+J(t-t ′ 3 -t1-t2)-J(-t ′ 1 -t ′ 2 )-J(t1+t2) . ( 48
)
The result for the current contains both an elastic and an inelastic contribution. The elastic contribution reads

I el ← ≃ eγ 2 1 γ 2 2 2π 3 -eV eV dǫ ∞ ∆ dE ∞ ∆ dE ′ ∆ 2 √ E 2 -∆ 2 E ′ 2 -∆ 2 × 1 - 4π R K +∞ -∞ dω |Z(ω)| 2 (ω) 2 S I (ω) 1 D 0 ← - 2π R K +∞ -∞ dω |Z(ω)| 2 (ω) 2 S I (ω) D el ← , (49)
with D 0 ← is the original denominator which is not affected by the environment, which is defined by Eq. ( 62), and D el ← is the denominator product affected by the environment (see Eq. ( 63) of Appendix D). The inelastic contribution reads

I inel ← ≃ eγ 2 1 γ 2 2 π 2 R K ∞ eV dǫ ∞ eV dǫ ′ ∞ ∆ dE ∞ ∆ dE ′ × ∆ 2 √ E 2 -∆ 2 E ′ 2 -∆ 2 |Z(-(ǫ + ǫ ′ ))| 2 (ǫ + ǫ ′ ) 2 S I (-(ǫ + ǫ ′ )) D inel ← , (50)
where D inel ← is the denominator product attributed to the inelastic current, which is defined in Eq. ( 64) of Appendix D.

Here,

γ 1 = 2πN S T 2 1 , γ 2 = 2πN N T 2
2 define the tunneling rates between the superconductor and the dot, between the dot and the normal metal lead, respectively, with N S and N N the density of states per spin of the two metals in the normal state, at the chemical potentials µ S and µ L , respectively. All contributions to the current contain denominators where the infinitesimal η (adiabatic switching parameter) is included in order to avoid divergences. In fact, it has been shown in Refs. 21,26 that a proper resummation of the perturbation series, including all round trips from the dot to the normal leads, leads to a broadening of the dot level. We take into account this broadening by replacing η by γ/2 with γ = γ 1 + γ 2 into our calculations (including only a broadening due to the superconductor). As mentioned above, we have assumed that γ 1 ≫ γ 2 . In order to avoid the excitation of quasiparticles above the gap, these rates need also to fulfill the condition ǫ D + γ < ∆. In what follows, we keep the notation η in our expressions, bearing in mind that it represents the line width associated with the leads. For numerical purposes, it will be sufficient to assume that η is kept very small compared to the superconducting gap, as well as all the relevant level energies (dot level position, bias voltages, etc...).

The above expressions constitute the second main result of this work: one understands now how the current fluctuations in the neighboring mesoscopic circuit give rise to inelastic and elastic contributions in the current in the NDS device.

We find that both right and left current contributions have the same form. The current -current fluctuations of the mesoscopic device affect the detector current at the energy corresponding to the total energy of two electrons exiting in (or entering from) the normal lead. Therefore, we proceed to the same change of variables a for the single NS junction.

For eV > 0, elastic current contributions in I → are present but the same contributions in I ← vanish (the opposite is true for the case of eV < 0).

Changing variables in the inelastic contributions, and defining the kernel functions K el N DS (ω, eV, ǫ D , η), K inel N DS (Ω, eV, ǫ D , η) as in Appendix D, for eV > 0 and eV < 0, we obtain: 51)

∆I P AT (eV ) = -C N DS +∞ -∞ dω |Z(ω)| 2 ω 2 S + ex (-ω)K el N DS (ω, eV, ǫ D , η) - C N DS 2 2eV -∞ dΩ |Z(Ω)| 2 Ω 2 S + ex (-Ω)K inel N DS (Ω, eV, ǫ D , η) - C N DS 2 ∞ 2eV dΩ |Z(-Ω)| 2 Ω 2 S + ex (Ω)K inel N DS (Ω, eV, ǫ D , η) ,(
with C N DS = eγ 2 1 γ 2 2 ∆ 2 /π 2 R K .
The first term in Eq. ( 51) describes the elastic contribution in the PAT current. Although we are less interested in this contribution, we cannot ignore it in practice because it contributes to the total ∆I P AT . The environment affects this current contribution but at the end of the tunneling processes, there is no energy exchanged between the device and the detector circuit. The second term in Eq. ( 51) describes the tunneling of a Cooper pair from the normal lead to the superconductor via the quantum dot, with energy exchange. The electrons can absorb energy (in the case their total energy is smaller than the superconductor chemical potential µ S ) or emit energy (if their total energy is bigger than µ S ). The last term in Eq. ( 51) describes the inverse tunneling process: a Cooper pair absorb energy from the neighboring device, its constituent electrons then tunnel to the normal lead. In this event, the total energy of the outgoing electrons is positive. If on the contrary, this total energy is negative, the Cooper pair has emitted energy to the device.

In order to understand how the detector circuit affects the behavior of the current (in the presence of the environment), we investigate the weight functions 8 as a function of frequency ω, for two values of the bias voltage and two values of the dot level position. This elastic kernel is symmetric under bias voltage reversal (K el N DS (-eV ) = -K el N DS (eV )). From the right panel of this figure, where we consider |eV | ≥ ǫ D , we find that there is a small step at ω = ∆ǫ D and a sharp peak at ω = -∆ + ǫ D . The peak is assymetric, and its height is much larger than that of the step. When ω < -∆ + ǫ D , K el N DS changes sign and becomes negative. The voltage bias eV mainly affects the amplitude of the peak and of the step in K el N DS . The left panel of Fig. 8 describes K el N DS when |eV | < ǫ D . The peak height decreases quite fast as a function of eV , and its location is shifted at ω = -∆ + eV . The peak is symmetric for large bias.

K el N DS (ω, eV, ǫ D , η) and K inel N DS (Ω, eV, ǫ D , η) separately. The weight function K el N DS (ω, eV, ǫ D , η) is plotted in Fig.
We turn now to the truly photo-assisted processes, which involve either absorption or emission of energy. The kernel K inel N DS (Ω, eV, ǫ D , η) is plotted in Fig. 9 as a function of frequency Ω, which corresponds to the total energy of two electrons, for ǫ D > 0. In the left panel, eV is negative, and in the center panel, eV is positive but eV < ǫ D , and finally the right panel of Fig. 9 describes eV ≥ ǫ D . We find that when eV < ǫ D , there is a step at Ω = ǫ D + eV . When we increase eV close to ǫ D , the step still dominates K inel N DS but there is a small peak at Ω = -∆ + eV . When eV ≥ ǫ D this (inverted) peak is very sharp. This is explicit in the right panel. The inverted peak, which has a large amplitude, makes it now difficult to observe the step. The (inverted) peak is located at Ω = -∆ + ǫ D . Again, eV mostly affects the amplitude of K inel N DS . For ǫ D < 0 (not shown), the result is similar to that of ǫ D > 0 with opposite eV , but the amplitude of the peak is doubled compared to that of ǫ D > 0 when |eV | ≥ |ǫ D |.

Note that understanding the behavior of the two weight functions as a function of the different parameters (eV and ǫ D ) of the detector is crucial. It allows to control the effect of the device voltage bias eV d on the DC current of the detector, and it is therefore the key for extracting the noise from the measurement of this DC current.

C. Application to a Quantum Point Contact

We now calculate ∆I P AT from Eq. ( 51) with the spectral density of excess noise of a Quantum Point Contact, given by Eq. ( 39). We consider the PAT current as a function of the detector voltage eV for several values of the device voltage eV d , which are shown in Fig. 10. We find that there are two values of eV at which ∆I P AT changes drastically. First, there is a step located at eV = ǫ D and second, a Fano-like peak appears at eV = -ǫ D . The (negative) derivative at eV = -ǫ D seems to diverge. The high of both the peak and the step increase in a monotonous manner as a function of the ratio of the device voltage eV d divided by the dot level ǫ D . When eV d is small, the peak is much higher than the step. Increasing eV d , the peak further increases, but the step height increases faster, starting from the threshold device voltage eV d = ∆ǫ D . In Fig. 10, we find that for ǫ D = 0.4 and eV d = 0.8, the peak height is still higher than the step, but with ǫ D = 0.6, and eV d = 0.6, the step becomes higher than the peak. Here for specificity, we only consider the case where ǫ D > 0, but results for ǫ D < 0 can be obtained in a similar manner, exploiting electron hole symmetry.

In order to further understand the behavior of ∆I P AT , we consider the different contributions of this photoassisted current, which are shown in Fig. 11. Specifically, we plot the elastic current renormalized by the environment, as well as the right and left inelastic currents. We find that the elastic part is symmetric between eV positive and negative. It is almost equal to zero when |eV | < ǫ D . It shows a step at |eV | = ǫ D . This can be understood from the fact that at the threshold eV = ǫ D electrons tunnel from the normal metal lead to the superconductor predominantly by making resonant transitions through the dot. Electrons easily tunnel to the quantum dot, in a sequential manner becoming a Cooper pair in the superconductor. For eV = -ǫ D , the same reasoning can be made for incoming holes, or equivalently for electrons exiting the superconductor: a Cooper pair in the superconductor is split into two electrons, which tunnel to the quantum dot and then to the normal metal lead. Because of the energy conservation condition, it is then necessary to have eV < -ǫ D . the two Cooper pair electrons on the dot: this small feature can only be seen by zooming in the picture. When eV is negative, the absorption of energy from the environment becomes much more favorable. Because of this, the analog of the step corresponding to eV = -ǫ D in the elastic current is smoothed out, and it saturates around eV = -ǫ D . Nevertheless, I inel L also contains contributions where electrons emit energy to the environment. Starting from eV < 0, the process of the Cooper pair tunneling to the normal metal lead and emitting energy to environment has first a small contribution to I inel L but it really becomes noticeable below eV = -ǫ D and eventually saturates for lower bias voltage (not shown). As the voltage of the mesoscopic device is lowered (left to right panels of Fig. 11), two things occur: first, the amplitude of all current contributions decreases, second, the smoothing of the I inel L is reduced because the range of energy available to absorption and emission is reduced.

In brief the sum of the contributions for emission and absorption in I inel L have a tendency to compensate the elastic current at voltages where saturation is reached. ¿From the above considerations, we therefore can interpret the curve of ∆I P AT , and we understand when the detector circuit absorbs or emits energy from/to mesoscopic device: when eV is negative, the PAT current is mainly due to absorption for |eV | < ǫ D , and emission for |eV | > ǫ D .

Note that in all our numerical results, the PAT current is in units of 2e 3 R 2 γ 2 1 γ 2 2 T (1 -T )/π 3 h2 ∆ 3 R K . To check the observability of these predictions we estimate: γ 1 = 0.2∆, γ 2 = 0.02∆, ∆ = 240µeV , R = 0.03R K , T = 0.5 (see also Ref. 10,11,12,23). This implies, e.g. for the PAT current in Fig. 10 when the detector bias is close to ±ǫ D , I P AT ≃ 5 -10pA with the device bias V d = 0.8∆/e = 48µV . This value is acceptable if one compares it with the value of current which has been estimated in Ref. 10. It is also acceptable with present day detection techniques.

IV. CONCLUSION

In conclusion, we have presented a new capacitive coupling scheme to study the high frequency spectral density of noise of a mesoscopic device. As in the initial proposal of Ref. 10, the effect of the noise originating from a mesoscopic device triggers an inelastic DC current in the detector circuit. This inelastic contribution can be thought as a dynamical Coulomb blockade effect where the phase fluctuations at a specific junction in the detector circuit junction are related to voltage fluctuations in same junction. In turn, such voltage fluctuations originate from the current fluctuations in the nearby mesoscopic device, and both are related by a trans-impedance. The novelty is that here, because the junction contains a superconducting element, in the subgap regime, two electrons need to be transferred as the elementary charge tunneling process. In a conventional elastic tunneling situation, the two electrons injected from (ejected in) the normal metal lead need to have exactly opposite energies in order to combine as a Cooper pair in the superconductor. Here, this energy conservation can be violated in a controlled manner because a photon originating from the mesoscopic circuit can be provided to/from the constituent electrons of the Cooper pair in the tunneling process.

We have computed the DC current in the detector circuit for two different situations. In a first step, we considered a single NS junction, and we computed all lowest order inelastic charge transfer processes which can be involved in the measurement of noise: the photo-assisted transfer of single (and pairs of) electrons (with energies within the gap) into quasiparticle(s) above the gap, and photo-assisted Andreev transfer of electrons as a Cooper pair in the subgap regime. It was shown that the latter process dominates when the source drain bias is kept well within the gap. Close but below the gap, the absorption of quasiparticle dominates, and we observe a crossover in the current between the Andreev and quasiparticle contributions. For the above processes, we demonstrated the dependence of the DC current on the voltage bias of the mesoscopic device, chosen here to be a quantum point contact. When this bias eV d is increased, the overall amplitude of the spectral density and its width scale as V d , so that the phase space (the energy range) of electrons which can contribute to the Andreev processes is magnified. We therefore have gained an understanding about how the measurement of the DC current can provide useful information on the noise of the mesoscopic device. Nevertheless, one should point out that with this NS setup, it is difficult to isolate the contribution of the photo-assisted current which involves, respectively, the absorption and the emission of photons from the mesoscopic device. For biases close to the chemical potential of the superconductor, both will typically contribute to the photo-assisted current.

Next, we considered a more complex detector circuit where the normal metal and the superconductor are separated by a quantum dot which can only accommodate a single electron at a given time. There, the dot acts as an additional energy filtering device, with the aim of achieving a selection between photon emission and absorption processes. We decided to restrict ourselves to the photo-assisted Andreev (subgap) regime, assuming that the dot level is well within the gap. By computing the total excess photo-assisted current as well as its different contributions for absorption and emission, and right and left currents, we found that for DC bias voltages comparable to eV = -ǫ D it is possible to make such a distinction. The NDS detection setup could therefore provide more information on the spectral density of noise than the NS setup, but its diagnosis would involve the measurement of smaller currents than the NS setup, because of the presence of two tunnel barriers instead of one.

Note that normal lead -quantum dot -superconductor setups have already been investigated theoretically [START_REF] Fazio | [END_REF] and experimentally recently [START_REF] Gräber | Quantum Information and Decoherence in Nanosystems[END_REF] . In such works, the emphasis was to study how the physics of Andreev reflection affected the Kondo anomaly in the current voltage characteristics. In Ref. 29, the quantum dot consisted of a carbon nanotube making the junction between a normal metal lead and a superconductor. Here, we did not consider the quantum dot in the Kondo regime, and we included interactions on the dot in the Coulomb blockade regime.

A central point of this study is the fact that all contributions to the photo-assisted current, for both the NS and NDS setups, can be cast in the same form:

∆I P AT (eV ) ∝ dΩ |Z(Ω)| 2 Ω 2 S + ex (±Ω)K process (Ω, eV, ...) . (52)
Where K process is a Kernel which depends on the nature (elastic or inelastic) as well as the mechanism (sin-gle quasi-particle or pair tunneling) of the charge transfer process. When dealing with an elastic process, one understand that the environment renormalizes the DC current even when no photon are exchanged between the two circuits. In the case of inelastic tunneling only, the frequency Ω corresponds to the total energy of the two electrons which enter (exit) the superconductor from (to) the normal metal lead. Finally, the sign of the frequency (and thus the bound of the integral in Eq. ( 52), which are left "blank" here) decides whether a given contribution corresponds to the absorption or to the emission of a photon from the mesoscopic circuit.

The present proposal has been tested using a quantum point contact as a noise source, because the spectral density of excess noise is well characterized and because it has a simple form. It would be useful to test the present model to situations where the noise spectrum exhibits cusps or singularities. Cusps are known to occur in the high frequency (close to the gap) noise of normal superconducting junctions. Singularities in the noise are know to occur in chiral Luttinger liquid, tested in the context of the fractional quantum Hall effect 30 . Such singularities or cusps should be easy to recognize in our proposed measurement of the photo-assisted current.

On general grounds, we have proposed a new mechanism which couples a normal metal -superconductor circuit to a mesoscopic device with the goal to understand the noise of the latter. The present setup suggest that it is plausible to extract information about high frequency noise. High frequency noise detection now constitutes an important subfield of nano/mesoscopic physics. New measurement setup schemes which can be placed "on-chip" next to the circuit to be measured are useful for further understanding of high frequency current moments.

VII. APPENDIX C

In this part, we present the denominator products which appear in the tunneling current through the NS junction as a Cooper pair. Such denominators come from the energy denominators of the transition operator T.

(D 0 ← ) -1 = 1 E ′ + ǫ + iη 1 E -ǫ -iη + 1 E + ǫ -iη , (56)
(D el ← ) -1 = 1 E ′ + ǫ + ω + iη 1 E -ǫ -iη + 1 E + ǫ -iη + 1 E ′ + ǫ + iη 1 E -ǫ + ω -iη + 1 E + ǫ + ω -iη , (57)
(D inel ← ) -1 = 1 E ′ -ǫ ′ + iη + 1 E ′ + ǫ + iη × 1 E + ǫ ′ -iη + 1 E + ǫ -iη + 1 E -ǫ -iη + 1 E -ǫ ′ -iη . (58)
Change variables in I inel → , I inel

← as Ω = ǫ + ǫ ′ , δ = ǫ -ǫ ′ .
We define In this Appendix, we present the denominator products which appear in the tunneling current through the NDS junction. D 0 is the original denominator which is not affected by the environment

χ(x, η) ≡ ∞ ∆ dE 1 √ E 2 -∆ 2 1 E -x -iη = π + 2 arcsin( x+iη ∆ ) 2 ∆ 2 -(x + iη) 2 , ( 59
(D 0 ← ) -1 = 1 (E ′ + ǫ + iη)(E ′ + ǫ D + iη)(ǫ + ǫ D + iη)(E + ǫ D -iη) × 1 (-ǫ + ǫ D -iη)(E -ǫ -iη) + 1 (ǫ + ǫ D -iη)(E + ǫ -iη) , (62)
and D el ← is the denominator product affected by the environment

(D el ← ) -1 = 1 (E ′ + ǫ + ω + iη)(E ′ + ǫ D + ω + iη)(ǫ + ǫ D + iη)(E + ǫ D -iη) × 1 (-ǫ + ǫ D -iη)(E -ǫ -iη) + 1 (ǫ + ǫ D -iη)(E + ǫ -iη) + 1 (E ′ + ǫ + iη)(E ′ + ǫ D + iη)(ǫ + ǫ D + iη)(E + ǫ D + ω -iη) × 1 (-ǫ + ǫ D -iη)(E -ǫ + ω -iη) + 1 (ǫ + ǫ D -iη)(E + ǫ + ω -iη) , (63)
where D inel is the denominator product attributed to the inelastic current (affected by environment) and it is defined as

(D inel ← ) -1 = 1 (E ′ + ǫ D -ǫ -ǫ ′ + iη)(E ′ -ǫ ′ + iη) + 1 (E ′ + ǫ D + iη)(E ′ + ǫ + iη) × 1 (ǫ D -ǫ ′ + iη)(ǫ D -ǫ -iη) 1 (E + ǫ D -ǫ -ǫ ′ -iη)(E -ǫ -iη) + 1 (E + ǫ D -iη)(E + ǫ ′ -iη) + 1 (ǫ D -ǫ ′ + iη)(ǫ D -ǫ ′ -iη) 1 (E + ǫ D -ǫ -ǫ ′ -iη)(E -ǫ ′ -iη) + 1 (E + ǫ D -iη)(E + ǫ -iη) . (64)
We define

Π(x 1 , x 2 , η) = ∞ ∆ dE 1 √ E 2 -∆ 2 1 (E -x 1 -iη)(E -x 2 -iη) . (65)
If

x 1 = x 2 then Π(x 1 , x 2 , η) = 1 2(x 1 -x 2 ) π + 2 arcsin( x1+iη ∆ ) ∆ 2 -(x 1 + iη) 2 - π + 2 arcsin( x2+iη ∆ ) ∆ 2 -(x 2 + iη) 2 , else Π(x 1 , x 2 , η) = (x 1 + iη)(π + 2 arcsin( x1+iη ∆ )) 2(∆ 2 -(x 1 + iη) 2 ) 3/2 + 1 ∆ 2 -(x 1 + iη) 2 .
Then we define

Ψ 0 ← (ǫ, ǫ D , η) = ∞ ∆ dE ∞ ∆ dE ′ 1 √ E 2 -∆ 2 E ′ 2 -∆ 2 1 D 0 ← = Π(-ǫ, -ǫ D , -η) Π(ǫ, -ǫ D , η) (ǫ + ǫ D + iη)(-ǫ + ǫ D -iη) + Π(-ǫ, -ǫ D , η) (ǫ + ǫ D + iη)(ǫ + ǫ D -iη) , (66)
Ψ el ← (ǫ, ǫ D , ω, η) = ∞ ∆ dE ∞ ∆ dE ′ 1 √ E 2 -∆ 2 E ′ 2 -∆ 2 1 D el ← = Π(-ǫ -ω, -ǫ D -ω, -η)Π(ǫ, -ǫ D , η) + Π(ǫ -ω, -ǫ D -ω, η)Π(-ǫ, -ǫ D , -η) (ǫ + ǫ D + iη)(-ǫ + ǫ D -iη) + Π(-ǫ -ω, -ǫ D -ω, -η)Π(-ǫ, -ǫ D , η) + Π(-ǫ -ω, -ǫ D -ω, η)Π(-ǫ, -ǫ D , -η) (ǫ + ǫ D + iη)(ǫ + ǫ D -iη) , (67)
Ψ inel ← (ǫ, ǫ ′ , ǫ D , η) = ∞ ∆ dE ∞ ∆ dE ′ 1 √ E 2 -∆ 2 E ′ 2 -∆ 2
Since D 0 → (ǫ) = D 0 ← (-ǫ), D el → (ǫ) = D el ← (-ǫ), so Ψ 0 → (ǫ) = Ψ 0 ← (-ǫ), Ψ el → (ǫ) = Ψ el ← (-ǫ). But in fact, if we change the name of variable ǫ → ǫ ′ then change variable ǫ ′ = -ǫ, we will obtain the same formula for both cases eV > 0 and eV < 0. Since ǫ, ǫ ′ are independently equivalent, so that it is evidently Ψ inel → (ǫ, ǫ ′ , ǫ D , η) = Ψ inel ← (ǫ, ǫ ′ , ǫ D , η). Hereafter, we neglect the ← or → index in these functions.

If eV > 0, the elastic current contributions in I → exist but the elastic current contributions in I ← vanish (in contrast to the case of eV < 0).

We change variables in inelastic contributions as 1 S.R. Eric Yang, Solid State Commun. 81, 375 (1992).

Ω = ǫ + ǫ ′ , δ = ǫ -ǫ ′ ,
2 Y.M. Blanter and M. Büttiker, Phys. Rep. 336, 1 (2000).

FIG. 2 :

 2 FIG. 2: Electronic tunneling in a NS junction: a) Quasiparticle electron tunneling. b) Andreev reflection. c) Photoassisted electron tunneling as a quasiparticle in the superconductor. d) Photo-assisted Andreev reflection.

  FIG.5: ∆IP AT plotted as a function of the DC bias voltage, for a mesoscopic device voltage bias eV d : 0.3 (continuous line), 0.5 (dashed line) and 0.8 (dotted line). The left/center/right panels depict single quasiparticle tunneling/two quasiparticle tunneling/Andreev reflection. ∆IP AT is in units of C2e, with T = 0.6 (see text).

FIG. 6 :

 6 FIG.6: Andreev reflection contribution to ∆IP AT plotted as a function of the DC bias voltage, for a mesoscopic device voltage bias eV d : 0.3 (continuous line), 0.5 (dashed line) and 0.8 (dotted line). Same units as in Fig.5. The inside panel is a zoom of the same contribution at small eV , displaying linear behavior.

FIG. 7 :

 7 FIG.7: Crossover between Andreev reflection (crossed line) and two quasiparticle tunneling current (uncrossed line) close to the gap, for a mesoscopic device voltage bias eV d : 0.3 (continuous line), 0.5 (dashed line) and 0.8 (dotted line). Same units as in Fig.5.

  FIG. 8: The weight function K el NDS as a function of frequency ω. Right panel, |eV | ≥ ǫD; left panel |eV | < ǫD.

  2χ(-ǫ, -η) + χ(-ǫω, -η)] [χ(ǫ, η) + χ(-ǫ, η)] + χ(-ǫ, -η) [χ(ǫω, η) + χ(-ǫω, η)]} ,

  ǫ + ǫ ′ǫ D , ǫ ′ , -η) + Π(-ǫ D , -ǫ, -η)) × Π(ǫ + ǫ ′ǫ D , ǫ, η) + Π(-ǫ D , -ǫ ′ , η) (ǫ Dǫ ′ + iη)(ǫ Dǫiη) + Π(ǫ + ǫ ′ǫ D , ǫ ′ , η) + Π(-ǫ D , -ǫ, η) (ǫ Dǫ ′ + iη)(ǫ Dǫ ′iη) .

  (ω, eV, ǫ D , η) = eV -eV dǫΨ el.tot (ǫ, ǫ D , η) , .tot (ǫ, ǫ D , ω, η) = 2Ψ 0 (ǫ, ǫ D , η) + Ψ el (ǫ, ǫ D , ω, η).

Acknowledgments

We thank Richard Deblock for valuable discussions.

V. APPENDIX A

In this Appendix, we define Ψ 0← , K el 2e← (ω, eV, η), and K inel 2e← (Ω, eV, η) in the

VI. APPENDIX B

In this Appendix, we recall the definition of Keldysh Green's functions [START_REF] Ketterson | Superconductivity[END_REF] . First, we define the anomalous Green's function, describing the pairing of electrons with opposite spin in the superconductor:

If both t and t ′ are in the upper branch, and t > t ′ or both t and t ′ are in the lower branch, and

. These Green's functions enter the description of the Andreev process.

If we consider the single quasiparticle tunneling in the superconductor, we use the conventional definition of the Green's function as for normal metals.

Secondly, we define the Green's function of the one level QD: G Dσ (tt ′ ) ≡ T K c Dσ (t)c + Dσ (t ′ ) . Simplifications occur because the quantum dot has a singly occupied level with energy ǫ D , the first electron is transferred to the lead before the second hops on the quantum dot so that in our work, we only consider the QD Green's function where both time quantities t and t ′ are in the upper or lower branch, and the Green's function values only when t > t ′ if t, t ′ in the upper branch

. The Green's function in the normal metal lead reads:

) . In our work, we consider the cases where two electrons tunneling from/to normal metal lead, so that we only consider normal metal Green's function where t and t ′ are in the different branches. For the case electrons tunneling from the superconductor to the normal metal, we use the greater Green's function G > Lσ (k, tt ′ ) = e -i(ǫ k -µL)(t-t ′ ) with ǫ k > µ L . For the case electrons tunneling from the normal metal to the superconductor, we use the lesser Green's function G < Lσ (k, tt ′ ) =c † kσ (t ′ )c kσ (t) = -e -i(ǫ k -µL)(t-t ′ ) with ǫ k ≤ µ L .