Yaovi Holade
email: yaovi.holade@enscm.fr

Qing Wang

Hazar Guesmi

Sophie Tingry

David Cornu

Shelley D Minteer
email: minteer@chem.utah.

Unveiling the Pitfalls of Comparing ORR Kinetic Data for Pd-based Electrocatalysts Without the Experimental Conditions of the Current-Potential Curves

The road to solid-polymer-electrolyte fuel cells [START_REF] Jiao | Designing the next generation of protonexchange membrane fuel cells[END_REF] relies on the sluggish electrocatalytic reduction of oxygen (ORR) [START_REF] Debe | Electrocatalyst approaches and challenges for automotive fuel cells[END_REF] that bridges the fields of metal-air batteries and alkaline fuel cells. Among many electrocatalysts being explored each year, only Pd can replace Pt with akin ORR activity in alkaline media [O 2 + 2H 2 O + 4e -→ 4HO -, E eq ~1. 18 V RHE (RHE: reversible hydrogen electrode)], [START_REF] Yang | Combinatorial Studies of Palladium-Based Oxygen Reduction Electrocatalysts for Alkaline Fuel Cells[END_REF][START_REF] Lima | Catalytic Activity-d-Band Center Correlation for the O2 Reduction Reaction on Platinum in Alkaline Solutions[END_REF] thus the option to reduce the economic stress on Pt alone. As any electrocatalytic reaction, ORR's main metrics [START_REF] Debe | Electrocatalyst approaches and challenges for automotive fuel cells[END_REF][START_REF] Kocha | Best Practices and Testing Protocols for Benchmarking ORR Activities of Fuel Cell Electrocatalysts Using Rotating Disk Electrode[END_REF][START_REF] Li | Some thoughts about reporting the electrocatalytic performance of nanomaterials[END_REF][START_REF] Dix | Critical Practices in Rigorously Assessing the Inherent Activity of Nanoparticle Electrocatalysts[END_REF][START_REF] Mayrhofer | Measurement of oxygen reduction activities via the rotating disc electrode method: From Pt model surfaces to carbon-supported high surface area catalysts[END_REF] are the kinetic current density (j k , A mg -1 metal or A cm -2 ECSA , ECSA = electrochemically active surface area), the Tafel slope (b, mV dec -1 , potential required to achieve 10-fold increase, its interpretation is more complex [START_REF] Li | Some thoughts about reporting the electrocatalytic performance of nanomaterials[END_REF] ), the half-wave potential (E 1/2 , potential for a half value of the limiting current), the onset potential (E onset , potential at which ORR starts) and the exchange current density (j 0 ). The higher j k (E 1/2 , E onset and j 0 ) is, the more active the electrocatalyst is. Although some testing protocols or "best practices" are reported, [START_REF] Kocha | Best Practices and Testing Protocols for Benchmarking ORR Activities of Fuel Cell Electrocatalysts Using Rotating Disk Electrode[END_REF][START_REF] Dix | Critical Practices in Rigorously Assessing the Inherent Activity of Nanoparticle Electrocatalysts[END_REF][START_REF] Mayrhofer | Measurement of oxygen reduction activities via the rotating disc electrode method: From Pt model surfaces to carbon-supported high surface area catalysts[END_REF][START_REF] Shinozaki | Oxygen Reduction Reaction Measurements on Platinum Electrocatalysts Utilizing Rotating Disk Electrode Technique: I. Impact of Impurities, Measurement Protocols and Applied Corrections[END_REF] a deep analysis of the literature points out an impossibility for a fair comparison of ORR kinetic parameters due to a lack of full disclosure of experimental conditions. The latter is, however, essential to "cross the valley of death" [START_REF] Siegmund | Crossing the Valley of Death: From Fundamental to Applied Research in Electrolysis[END_REF] in scientific research from fundamental to applied fields.

In 0.1 HClO 4 , Shinozaki et al. [START_REF] Shinozaki | Oxygen Reduction Reaction Measurements on Platinum Electrocatalysts Utilizing Rotating Disk Electrode Technique: I. Impact of Impurities, Measurement Protocols and Applied Corrections[END_REF] used the method of linear sweep voltammetry (LSV) to reveal that j k,anodic of the anodic sweep (from lower to higher potential) is considerably higher than j k,cathodic of the cathodic sweep (from higher to lower potential): at 0.9 V RHE , j k,anodic /j k,cathodic = 5.3 (bulk Pt) and 2.4 (nano Pt). Our recent data show a huge discrepancy, j k,anodic /j k,cathodic of an order of magnitude for supported PdFe electrocatalysts in 0.1 M KOH. [START_REF] Cassani | Fe-Modified Pd as an Effective Multifunctional Electrocatalyst for Catalytic Oxygen Reduction and Glycerol Oxidation Reactions in Alkaline Media[END_REF] The summarized random sample of publications in alkaline media (period of 2015-2021) in Table S1 clearly shows that the majority of the reports do not state the scan direction and/or do not use the same protocol, 3,12- 15 which means a conflicting comparison of the activity where a claimed outstanding performance or superiority over the others could not be. Viewpoint aims to answer these questions and draw the attention of researchers to experimental situations that can lead to an erroneous evaluation/comparison of ORR kinetic parameters for Pdbased electrocatalysts in alkaline media. We have performed a set of experiments at the forefront of electrochemical, electrocatalytic and computational sciences to: (i) interrogate current practices by using commercial Pd/C, (ii) and encourage researchers to adopt transparency and reproducible methodologies in fundamental and applied research in energy conversion systems. As the kinetic activity of ORR is size-/shape-dependent, [START_REF] Shao | Electrocatalysis on Platinum Nanoparticles: Particle Size Effect on Oxygen Reduction Reaction Activity[END_REF] we firstly used transmission electron microscopy (TEM) to precisely characterize the commercial electrocatalyst of carbon black supported Pd (Pd/C, 20 wt.% Pd loading, 3-4 nm crystallites size, Premetek Co., USA). Figure 2a shows the high-resolution TEM image of an individual Pd particle of 3.2 ± 0.1 nm that is consistent with the supplier's data (Figures S2-S6 To better understand the electrocatalysis of ORR using RRDE [disk = 0.196 cm 2 glassy carbon disk coated with 50 µg Pd cm -2 , ring = 0.072 cm [START_REF] Debe | Electrocatalyst approaches and challenges for automotive fuel cells[END_REF] Pt bulk], cyclic voltammetry (CV) experiments were recorded in the absence and presence of O 2 . Figure 1b shows that if two different studies did not use the same scan directions, they will report different behaviors and performance for j k ; however, it is the same electrocatalyst. Figure 2b shows that in the outgassed electrolyte (lower potential limit fixed at 0.3 V RHE to avoid hydride formation for lower values [START_REF] Tateishi | Hydrogen electrode reaction on electrodes of glassy carbon-supported ultrafine Pd particles in alkaline media[END_REF][START_REF] Hu | Voltammetric investigation of palladium oxides-I: Their formation/reduction in NaOH[END_REF] ), Pd reacts with OH -to produce Pd(OH) x (from 0.52 V RHE ) that further evolves into PdO (from 0.69 V RHE ), and the latter is reduced back to metallic Pd with a peak at 0.67 V RHE (the monolayer charge of 424 μC cm -2 for 1.45 V RHE upper potential limit is used to calculate ECSA). S2] where the rate is limited by the mass transport of O 2 from the bulk electrolyte to active sites.

Third, the mixed kinetic-diffusion controlled region (characterized by E 1/2 ) where the Koutecky-Levich model [with two implicit criteria, the existence of an electron transfer process that is the rate-determining step (rds) and a first-order reaction with respect to O 2 ] 20 enables to extract the kinetic data from the potential region corresponding to 0.1×j diff < j < 0.8×j diff . 8 E onset should be considered for j < 0.01×j diff . The more positive E onset (high) is, the more active the electrocatalyst is because the cell voltage is U = E (cathode) -E (anode) , so a high E (cathode) is sought for ORR.

We note that these control experiments in the absence and presence of O 2 enable us to clearly Fermi level, the stronger the interactions with adsorbates. This correlation is confirmed by the calculated binding energies of O 2 on the three selected systems (see Table S3), -0.99 eV on pristine Pd(111), -1.01 eV on hydroxylated Pd(111) and -0.12 eV on the oxidized Pd(111 confirm the experimentally distinguished three regions of palladium surface engaged in the ORR (metallic vs hydroxylated vs oxidized), which means that the full experimental conditions related to the electrode potential scanning should be always provided for a fair comparison.

Impact of the scan direction (potential sweep): anodic vs cathodic. The preceding results have

shown that for a fixed potential range, e.g. 0.3-1 V RHE , the starting point immediately changes the surface state. This consequently affects the nature of the "electrocatalyst-O 2 " interaction and thus the electrocatalytic activity, but we currently lack data on the extent of the discrepancy. So, to provide representative results based on the current research practices, we have used two concentrations of the electrolyte, which are 0.1 and 1 M KOH. Data of Table S2 show that for 4e -ORR at 1600 rpm, j diff (mA cm -2 ) = -5.7 and -3.6 for 0.1 and 1 M, respectively. While 0.1 M is routinely used, 1 M is closer to the realistic OH -concentration in practical situations. [START_REF] Yang | High-Loading Composition-Tolerant Co-Mn Spinel Oxides with Performance beyond 1 W/cm2 in Alkaline Polymer Electrolyte Fuel Cells[END_REF] In 1 M KOH, j diff is low because of the reduced solubility and diffusion coefficient of O 2 in bulk electrolyte. [START_REF] Yang | Combinatorial Studies of Palladium-Based Oxygen Reduction Electrocatalysts for Alkaline Fuel Cells[END_REF][START_REF] Davis | The solubility and diffusion coefficient of oxygen in potassium hydroxide solutions[END_REF] Figure 3a shows that, regardless of the electrolyte concentration, the LSV curves (1600 rpm, 5 mV s -1 ) shift to higher potentials in the kinetic region (E ≥ 0.8 V RHE ). This is supported by a drastic change of the kinetic parameters when switching from the cathodic to anodic direction (Figures 3b-e, S8-S9, Table S2). For E 1/2 (Figure 3b), ΔE 1/2 , anodic/cathodic = 42-51 mV. For j k (Figures 3c-d), j k,anodic /j k,cathodic = 10.0 (0.1 M KOH) and 7.7 (1 M KOH) at 0.90 V RHE , while at 0.85 V RHE , j k,anodic /j k,cathodic = 12.2 (0.1 M KOH) and 2.8 (1 M KOH). There is a slight decrease of the Tafel slope of b anodic/cathodic = 5-10 mV dec -1 (Figure 3d). Figures 3f andS9a show that the scan direction impacts the mechanism of ORR with a high amount of HO 2 -from the incomplete ORR (O 2 + H 2 O + 2e -→ HO 2 -+ OH -) when considering the cathodic scan. This could already be seen in Figure 3a with the current recorded at the Pt ring for its oxidation back into O 2 . Even though, herein, the overall transferred number of electrons per O 2 molecule is closed to 4 (Figure S9b), for fuel cells applications, 2 hydrogen peroxide species are not desired because their accumulation throughout the operation will diminish the performance and the lifetime by decreasing the faradaic efficiency and the durability of the solid polymer membranes. Based on the previous electrochemistry and DFT data, the origin of this discrepancy is the surface state of the electrocatalyst, depending on where LSV starts. For the cathodic direction, the sweep begins in a region where the electrocatalyst surface is covered by oxidized species and subsequently, the reduction of these species and O 2 proceeds concomitantly. This can be seen in Figures 2a and3a with a small deviation of the current around 0.7 V RHE leading to a high plateau.

For the anodic direction, the scan starts from metallic surface and subsequently builds up as the potential increases from reducing to oxidizing. [START_REF] Shinozaki | Oxygen Reduction Reaction Measurements on Platinum Electrocatalysts Utilizing Rotating Disk Electrode Technique: I. Impact of Impurities, Measurement Protocols and Applied Corrections[END_REF] The result is the change of the electrocatalyst-O 2 interface caused by several sources such as the nature of actives sites [Pd, Pd(OH) x , PdO]. The latter means a change in the types of electronic interaction and charge transfer resistance as well as the available number of actives sites and ECSA. These results demonstrate that when researchers do not specify the direction in which the LSV curves were obtained, the comparison of performance cannot be fair. Or, when reporting ORR data, it is not relevant to compare data obtained by one direction with another direction (disclosed or not) because we have seen here in alkaline media (elsewhere in acidic media [START_REF] Shinozaki | Oxygen Reduction Reaction Measurements on Platinum Electrocatalysts Utilizing Rotating Disk Electrode Technique: I. Impact of Impurities, Measurement Protocols and Applied Corrections[END_REF] ) that we have 2 to 12 times difference in activity. One can, unfortunately, notice that, for manuscripts on ORR, the tables summarizing the comparison with previous data by other researchers never mention the used scan direction. That said, when not disclosed, the probability that it would be the same scan direction is 50%, which is not trivial.

For everybody, the baseline of comparison must be the same, which is unfortunately not the case as summarized in Table S1 for a random sample of publications in alkaline media (2015-2021).

Impact of the scan rate. Having shown that the scan direction during LSV affects dramatically the magnitude of the measured kinetic metrics of ORR because of the dependence on the oxidation state of palladium surface, we next interrogated the possible impact of the scan rate by using 50, 20, 10 and 5 mV s -1 . This is very important because, for a fixed potential range, the electrocatalyst will spend more time in each of the above three regions at low scan rates. While 5 mV s -1 should be closed to the pseudo steady-state, 50 mV s -1 appears to be too fast (10 times).

This means that not all the oxidized species of the palladium will be reduced at higher scan rates as confirmed by LSV curves in Figures S10a andS11a (the undesired HO 2 -amount is incredibly high for the cathodic direction), making it difficult in extracting reliable kinetic data for scan rates higher than 20 mV s -1 . We note that Shinozaki et al. [START_REF] Shinozaki | Oxygen Reduction Reaction Measurements on Platinum Electrocatalysts Utilizing Rotating Disk Electrode Technique: I. Impact of Impurities, Measurement Protocols and Applied Corrections[END_REF] have observed the reverse trend in 0.1

M HClO 4 at Pt where the kinetic data systematically decrease upon the increase of the scan rate.

Here, while the extracted kinetic data in both electrolytes clearly highlight a big difference between the anodic and cathodic directions (Figures 4, S10 and S11), there is no monotonous evolution of the kinetics. This is possible because of the nature of the electrolyte, the active sites and the catalytic metal. Overall, 10 mV s -1 appears to be the best compromise for the tested conditions. At 10 mV s -1 , j k,anodic /j k,cathodic = 14.0 (0.1 M KOH) and 5.5 (1 M KOH) at 0.90 V RHE while ΔE 1/2 , anodic/cathodic = 43 mV and b anodic/cathodic = 5-5 mV dec -1 . All this confirms that the literature comparison should be made under the same conditions with respect to the scan rate. Impact of the voltammetry method for: CV vs LSV. Having shown that the direction in which the potentials were scanned and the speed at which this scanning was done have a major impact on the magnitude of the kinetic parameters, we finally sought to know if the polarization method also has a substantial effect. It is worth of mentioning that practical fuel cells devices are tested using pseudo steady-state conditions (potentiostatic or galvanostatic) and even ORR was investigated by the potentiostatic polarization of stepwise potentials. 9 The outcomes are shown in Figures 5a-f and S12 (data are gathered in Table S1). There is still a big difference between the anodic and cathodic directions where the polarization curves are positively shifted towards high potentials when the electrode scanning starts at the lower potential limit and increases. For j diff (mA cm -2 ) = -5.6 (LSV) and -4.8 (CV) in 0.1 M KOH, the difference of 14% does not impact significantly the other quantitative metrics of the E 1/2 (Figure 5b), j k (Figure 5c-d) and Tafel slope (Figure 5e). The big discrepancy is the amount of undesired HO 2 -(Figures 5f andS12) that increases with the augmentation of the electrolyte concentration.

While this part again emphasizes the need to indicate the direction of the scan, the observed difference might result from the discrepancy between the two techniques where LSV allows to get close to "steady-state" conditions. [START_REF] Li | Some thoughts about reporting the electrocatalytic performance of nanomaterials[END_REF][START_REF] Dix | Critical Practices in Rigorously Assessing the Inherent Activity of Nanoparticle Electrocatalysts[END_REF][START_REF] Shinozaki | Oxygen Reduction Reaction Measurements on Platinum Electrocatalysts Utilizing Rotating Disk Electrode Technique: I. Impact of Impurities, Measurement Protocols and Applied Corrections[END_REF] Actually, half of a CV may not be merely associated with an LSV, which explains why LSV is very popular for the study of ORR, HER, OER, etc.

As a general discussion, the purpose of the present work is not to impose a method for choosing the scan direction but to reveal its impact and how to avoid the pitfalls of a conflicting comparison of ORR kinetic data at Pd-based electrocatalysts. There are two schools of thoughts, one in favor of the cathodic direction (decreasing potentials) and the other in favor of the anodic direction (increasing potentials). The defenders of the cathodic direction argue on the functioning of the fuel cell device where the voltage [U = E (+) -E (-) = E (cathode) -E (anode) ] is maximum when each of the half-cells has its highest value, mathematically speaking, high for E (cathode) and low for E (anode) . So, we should work by gradually increasing the overall cell voltage, thus decreasing E (cathode) , that of ORR. However, in practice, it may turn out that higher currents are required as we start operating a fuel cell, this is the argument of those in favor of the anodic direction. In the middle, one can rationally think that it would be better to do both directions during ORR. Then, we either report data for both directions, as is already done for the kinetic current density (j k , A mg -1 metal and A cm -2 ECSA ), or we represent only the average value, which is in fact already done during fuel cell tests. For metallic electrocatalysts (Pd herein and Pt elsewhere [START_REF] Shinozaki | Oxygen Reduction Reaction Measurements on Platinum Electrocatalysts Utilizing Rotating Disk Electrode Technique: I. Impact of Impurities, Measurement Protocols and Applied Corrections[END_REF] ), one thing is sure, the scan direction and the rate at which the potentials are run in that direction should be clearly indicated in the reports. The cathodic direction can start at an onset potential slightly positive from OCP, so, essential to disclose. We note that the activity discrepancy is expected for materials with surface reactions within the electrochemical window of ORR, that is, 0.5-1 V RHE .

We can see from the Figure 2b that the kinetics region of ORR is situated in the potential window where the surface of palladium is oxidized. Given that the surface reaction is affected by the reactivity of the electrocatalyt with the water molecules or electrolyte, the electrochemical response corrolarely depends on the surface state. Therefore, OCP of O 2 -saturated electrolyte is a mixed electrode potential between ORR and redox processes involving the surface species (M, M(OH) x , MO x ). It means that the reduction thermodynamics and kinetics of ORR will, also, depend on those of the oxidized species M(OH) x and/or MO x . So, OCP should not be used as a guarantee of performance. Finally, there is an appeal for other types of commonly used materials, for instance, the PGM-free electrocatalysts that present or not a surface oxidationreduction profiles in the potentials range of interest for ORR, that is, 0.5-1 V RHE .

To conclude, our results show that a possibly unintentional non-disclosure of all experimental conditions is very problematic when comparing ORR kinetic data of Pd-based electrocatalysts in alkaline electrolytes. Specifically, the used scan direction to record the polarization curves is generally ignored while the experimental data show that the anodic scanning (low to high

Figure 1

 1 Figure 1 clearly shows that if two persons do use the same scan direction during LSV to record

Figure 1 .

 1 Figure 1. ORR kinetic activity extracted from the anodic scanning [from 0.35 V RHE to OCP

  for extended data). The interplanar space of 0.23 nm implies the predominance of the Pd(111) crystalline plane. Based on this finding, theoretical and computational investigations on Pd model semi-infinite surface were considered to gain insights into the interaction of the O 2 molecule with the electrocatalyst surface.

Figure 2 .

 2 Figure 2. (a) TEM of Pd/C. (b) Steady-state CV recorded at RRDE in the presence and presence of O 2 (1 M KOH, 1600 rpm, 10 mV s -1 ): anodic direction vs cathodic direction. (c)

  distinguish three domains: (A) metal-based surface, (B) hydroxyl-based surface and (C) oxidebased surface. Similar to Pt(111) that is the model electrocatalytic surface for ORR, and based on the above characterization, we next performed first principle Density Functional Theory (DFT) calculations to reveal how the nature of the Pd electrocatalyst surface (thus the potential range) modifies its interaction with O 2 molecule and strongly impacts the measured kinetic activity.

Figure

  Figure2cshows the computed local density of states (LDOS) of the 4d-bands of Pd atoms in

Figure 3 .

 3 Figure 3. Scan direction effect. (a) LSV (1600 rpm, 5 mV s -1 ) of ORR at an RRDE: disk

Figure 4 .

 4 Figure 4. Effect of the scan rate from LSV (in mV s -1 ) at 1600 rpm. (a) Tafel plots in 0.1 M

Figure 5 .

 5 Figure 5. Effect of the voltammetry method. (a) LSV and CV polarization curves (1600 rpm, 10 mV s -1 ) of ORR at an RRDE step up: disk (bottom) and ring (top). (b) Half-wave

  potentials) leads to ultrafast electrocatalytic kinetics with a positive shift of the half-wave potential and the dramatic increase of the kinetic current density 4 to 15 times. The exact origin of this discrepancy is the surface state of the metallic electrocatalyst. The electrocatalysis research community needs to be careful to the likely over-/under-estimation of the ORR's figures of merit (kinetic current density, Tafel slope, half-wave potential and onset potential) before claiming any outstanding or superior performance whereas this could not be the case. The present work contributes towards the best practices and encourages every researcher to strike transparency and reproducible methodologies in electrochemical energy conversion systems. ASSOCIATED CONTENT Supporting Information. The following files are available free of charge Experimental details, summary of ORR data from relevant Pd-based electrocatalysts at RDE/RRDE in alkaline media for the last years, (HR)TEM images, and extended Figures/Tables.

  While a plethora of Pd-based electrocatalysts are synthesized every year for ORR in alkaline media, research questions of fundamental importance are still unanswered: (i) what should be the best way to evaluate their performance and compare them to those already existing? (ii) to what extent the comparison is biased if, to record the current-potential polarization curves of ORR using a rotating disk/ring electrode (RDE/RRDE), the scan direction (anodic vs cathodic), the scan rate (5 to 50 mV s -1 ) and the scan nature (LSV vs CV) are not clearly made explicit? This

  [START_REF] Hu | Voltammetric investigation of palladium oxides-I: Their formation/reduction in NaOH[END_REF][START_REF] Grdeń | Electrochemical behaviour of palladium electrode: Oxidation, electrodissolution and ionic adsorption[END_REF] Under O 2 with an OCP of ca. 1.03 V RHE , the forward (from 0.35 V RHE to OCP) and backward (from OCP to 0.35 V RHE ) scans give 3 domains of the current density of the disk (bottom) and ring (top). RRDE has

a collection efficiency of 24.5 ± 0.5% (Figure

S7

) and will enable quantifying the incomplete hydroperoxide anion HO 2 -in alkaline media (pK a (H 2 O 2 /HO 2 -) = 11.75) and the overall number of exchanged electrons using Eqs. S1-S11. First, the activation region at high potential (low overpotential, characterized by E onset ) where the reaction rate is limited by the electrons transfer from the electrocatalyst surface to O 2 . Second, the diffusion-limiting region at low potential [high overpotential, characterized by a limiting current density j diff (Levich's relationship), Table

  The stronger adsorption energies of O 2 together with the elongated O-O band distances on metallic and hydroxylated Pd(111) suggest that the barriers for O 2 dissociation are much lower on these surfaces than on oxidized one. Moreover, these results

). The computed geometric parameters of O 2 suggest that O 2 does not strongly interact with the oxidized Pd(111) because the O-O bond distance is 1.33, 1.27 and 1.23 Å [2.04-2.06, 2.13, 3.66 Å for the distance of O 2 from the surface] for metallic Pd(111), hydroxylated Pd(111) and oxidized Pd(111), respectively.

ACKNOWLEDGMENT Y.H. thanks European Institute of Membranes of Montpellier (COGENFC, PAT-Energy-Axis-2018) and LabEx CheMISyst (ANR-10-LABX-05-01) for startup funds support. This work was financially supported by the Army Research Office MURI (W911NF-14-1-0263). This work was granted access to the France HPC resources of [CCRT/CINES/IDRIS] under the allocation 2020 [x2020087369] made by GENCI (Grand Equipement National de Calcul Intensif). The authors acknowledge Dr. Erwan Oliviero (MEA platform, Université de Montpellier) for TEM analysis.

Notes

The authors declare no competing financial interest.


To prevent biofouling (micro-and macrofouling), seawater is treated with biocide compounds:

-1,3-propylene diamine (e.g., Hatacide, Hatasperse, Drewsperse, …) -(alkyl amino)-3 aminopropane (e.g., MEXEL-432®) Organic matter (NOM + organic pollutants)

K HBrO = 15 -167 M -1 s -1 K HClO = 0,7 -5 M -1 s -1
The total organohalogen content in seawater can only be determined by the EOX protocole

  dissolved gaseous chlorine, sodium hypochlorite or electrochlorination -mode : continuous at low doses or intermittent at high doses -dose : 0.5 to 1.5 mg/L (low dose) to 10 mg/L (high dose) -discharge rates : 7,000 m 3 /h to 30,000 m 3 /h (6 to 9 months/year) Total Residual Oxidant Example of A TRO Analyzer (Hach 1735)™ Based on a colorimetric reaction of chlorine (or bromine) and its inorganic derivatives with DPD (N,N-diethyl-p-phenylenediamine) NH 2 Cl + 3 I -+ H 2 O + H + → NH 4 OH + Cl -+ I 3 -(2) NHCl 2 + 3 I -+ H 2 O + 2H + → NH 4 OH + 2 Cl -only free chlorine and inorganic forms chlorinated compounds = Cl 2 , HClO/ClO -, NHCl 2 , NH 2 Cl, NCl 3 and CNClIn seawater (and brackish waters), chlorine reacts rapidly with Br -(~ 65 mg/L) as follows:

•

  Extraction of organohalogen compounds by MTBE (in presence of Na 2 SO 4 ) Combustion/pyrolysis at 900 °C Micro-coulometric analysis of HX Field campaign, Gulf of Fos, France, 2017/10/products identified at the outlets of coastal industrial plants Trihalomethanes (THM): CHBr 3 (95-98%) >> CHBr 2 Cl > CHBrCl 2 Haloacetonitriles (HAN) : Dibromoacetonitrile CHBr 2 CN + H 2 O → CHBr 2 CONH 2 (dibromoacetamide), CHBr 2 CONH 2 + H 2 O → CHBr 2 CO 2 H + NH 3 Haloacetic acids: Dibromoacetic acid Obviously, many other compounds should be formed according reactivity of HBrO/BrO -At each sampling point, two samples were collected (at the surface and 7-meter depth) using 6-L Niskin bottle -1-L bottle for TOC, TN, Br -analysis -1-L bottle for analysis of halogenated byproducts: EOX, THMs, HAAs, HPs, HANs, HALs -Temperature, pH, and salinity were determined on-site during water sampling using a CTD-typemulti-parameter probe (MS5) -Free and total chlorine were determined on-site with DPD spectrometry • Evaluation of toxicity of bromoform, tribromoacetic acid and tribromophenol • Test based on: mortality, abnormal development rate, stunted growth • 48 hours of exposure of urchins from NPS Picture of 48 hours urchin larvae after exposure to anthropogenic compounds Eco-toxicity: Dose-response methodology From Karine Lebaron Ph'D works For the lowest observed effect concentration (LOEC)  20% of pluteus  « Synergetic » effect of pollutants Eco-toxicity: cocktail effects From Karine Lebaron Ph'D works Conclusions and perspectives -Only 15% of CBP identified in marine environments (according EOX measurement) -Nature and level of CBP affected by many biotic and abiotic parameters -CHBr 3 the most found CBP in coastal industrial plants -Dibromoacetonitrile, dibromoacetic acid, bromal hydrate intermediates of bromoform -Bromophenols : accumulate in the food web, in sediments (potential health impact?) -Long-term impact of presence of cocktail CBP on the environment? -And on human health? -Mithridatization or irreversible impacts (genomic studies)?

Acknowledgements

Thanks to Tarek, Djamel, Carine, Laurent, Karine, Sylvain, Maxime, Julien, … Thanks to the Pittcon Organizing committee Thanks to : the French National Research Agency (ANR) The Regional Council Provence-Alpes Côte d'Azur The Eco-citizen Institute for knowledge of pollution Thank you for your attention!

• Air Sampling:


domain: spi phys

text:
Jeanne Bernard
email: jeanne.bernard@onera.fr

Yannick Bidel

Malo Cadoret

Clément Salducci

Nassim Zahzam

Sylvain Schwartz

Alexis Bonnin

Cédric Blanchard

Alexandre Bresson

Atom interferometry using σ +σ -Raman transitions between | F = 1 , m F = 1 〉 and | F = 2 , m F = ± 1 〉

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.

I. INTRODUCTION

During the last two decades, light-pulse atom interferometers (LPAIs) [1] which exploit the wave-like interference of atoms, have become unique instruments for precision measurements of inertial forces, with applications in both applied and fundamental science. For example, atom interferometric techniques have been employed in measurements of gravitational [2,3] and fine structure constants [4,5], test of the equivalence principle [6], searches for dark sector particles [7][8][9], and even proposed for use in gravitational wave detection [10,11]. They have also enabled the realization of high performance accelerometers [12][13][14][15], gyroscopes [16][17][18][19][20][21][22][23] and gravimeters [24][25][26][27][28] demonstrating great promise for fielded inertial sensors based on atom interferometry [29][30][31]. In addition, they can also be utilized for probing the field gradient of an external field such as gravity [32][33][34] or magnetic fields [35,36].

In a LPAI, sequences of laser pulses are used to split, deflect and recombine matter-waves to create atom interference. In inertial sensors, these sequences of light pulses commonly use counterpropagating two-photon Raman transitions with large one-photon detuning [1] between hyperfine ground states of alkali atoms (e.g |F = 1 and |F = 2 for 87 Rb). They form the basic atom-optics elements by finely controlling the external degree of freedom of the atoms through the generation of coherent superposition of momentum states. In a counterpropagating configuration, the transfer between the two internal ground states is always accompanied with a change of ±hk eff of the momentum state, where k eff is the effective wave vector.

In order to achieve high precision measurements, the two counterpropagating Raman lasers are usually obtained thanks to a retroreflected geometry where a single laser beam with two laser frequencies is retroreflected off a mirror. This geometry allows to mitigate parasitic effects induced by wave front distortions which are critical to achieve good accuracy and long term stability [28,37]. It also reduces interferometer phase noise as most vibration effects are common to the two laser fields. In addition, in order to avoid systematic errors induced by first order Zeeman effect, the atoms are commonly manipulated in the magnetically-insensitive m F = 0 sublevels in the interferometer.

In this work, we report on the experimental realization of a Raman transition-based LPAI between magneticallysensitive internal states in a Mach-Zehnder type geometry. Using the supplementary internal degree of freedom of atoms manipulated in sensitive magnetic sublevels, we realize a sensor which simultaneously measures inertial and magnetic accelerations. Our work focuses on the specific case of 87 Rb. Using a σ + -σ -polarized light arrangement, we manipulate the atoms in the interferometer between the two magnetically-sensitive ground states 5S 1/2 |F = 1, m F = ∓1 → |F = 2, m F = ±1 , also used as atomic clock transition for magnetically trapped 87 Rb [38], taking benefit of their similar first-order Zeeman shift. Using this technique we perform the measurement of the horizontal component of acceleration, in a close-to-zero velocity regime, using a single-diffraction Raman scheme [15], without need for alternative techniques to lift the degeneracy of the double diffraction process [39]. We demonstrate a short-term sensitivity of 25 × 10 -5 m.s -2 .Hz -1/2 for absolute acceleration measurement. We then discuss some specifics of our technique in comparison with usual magnetically-insensitive Raman-based atom interferometers, such as spontaneous emission rate, additional sensitivity to magnetic field inhomogeneity and light-shifts. Finally, in light of the advantages of this technique, we propose atom interferometer designs which could be of interest in metrology, as well as for improving the performances of cold-atom inertial sensors in operational field conditions [13,[29][30][31].

II. METHOD A. Principle and advantages of the method

We implement our method using a horizontal Mach-Zehnder type LPAI based on counterpropagative stimulated two-photon Raman transitions between the |F = 1, m F = ∓1 and |F = 2, m F = ±1 hyperfine levels of the 5S 1/2 ground state of 87 Rb. These two states are coupled via an intermediate state, using lasers of frequencies ω 1(2) detuned from the 5P 3/2 , F = 1(2) state by the one-photon detuning ∆ 1(2) (see Fig. 1(a)). A static bias magnetic field of magnitude B aligned with the Raman lasers is applied to define a quantization axis for the atoms. This field shifts the |F, m F ground state magnetic sublevels by ∆E = µ B g F m F B as a first approximation, where F is the atomic total angular momentum, m F = 0, ±1, ..., ±F are its projections on the quantization axis, µ B is the Bohr magneton, and g F is the Landé factor, equal to -(+)1/2 for the F = 1(2) states respectively. The two counterpropagating Raman beams are generated using a retroreflective setup. Contrary to many LPAI experiments using a lin⊥lin polarization configuration and a large one-photon detuning allowing to exclusively drive counterpropagative Raman transitions between the magnetic-insensitive m F = 0 sublevels, we implement here a σ + -σ -configuration (see Fig. 1(b)): the Raman beams have a σ + polarization in one direction, and a σ -polarization in the retroreflected direction. The quantum state at the input of the interferometer is prepared to be in one single Zeeman sublevel |F = 1, m F (m F = +1 or m F = -1). Thus, according to the electric dipole transition selection rules, only one counterpropagating transition is possible. Indeed the σ + -σ - Raman laser configuration only allows ∆m F = ±2 transitions. Consequently, the two-photon Raman transition couples the magnetically-sensitive hyperfine states |F = 1, m F = ±1 ↔ |F = 2, m F = ∓1 with an effective Rabi frequency [START_REF] Steck | Rubidium 87 D Line Data[END_REF]:

Ω eff = Γ 2 √ I 1 I 2 4I sat 1 4 √ 3 1 ∆ 1 - 1 ∆ 2 (1)
where Γ = 2π × 6.07 MHz is the natural line width, I sat = cπhΓ/3λ 3 = 1.67 mW.cm -2 is the saturation intensity (with c the speed of light, h the Planck's constant and λ = 780 nm), I 1(2) are the Raman laser intensities and ∆ 1(2) are the one-photon detunings with respect to the hyperfine levels 5P 3/2 , F = 1(2) (see Fig. LPAIs usually manipulate atoms in the magneticallyinsensitive m F = 0 sublevels. For zero-velocity atoms the use of retroreflected Raman beams leads naturally to a double-diffraction scheme: two stimulated Raman transitions with opposite momentum transfer ±h k eff are simultaneously resonant [39]. Our scheme has the advantage of having only one counterpropagating Raman transition allowed despite the retroreflection. In addition, we can very easily implement the k eff -reversal technique [24] to eliminate some systematics by alternatively preparing the atoms in the |F = 1, m F = ∓1 states. Moreover the Raman beams have the same polarization as the magnetooptical trap beams, enabling a more compact and simple sensor.

B. Experimental apparatus and lasers

The experiment was carried out in the LPAI setup described in [15,[START_REF] Perrin | [END_REF]. The usual steps of atom interferometry (preparation, interferometry and population detection) were performed with the laser system described in [42]. On the one hand an Erbium distributed feedback fiber laser (DFB-FL) at 1.5 µm, locked to a rubidium transition through a saturated absorption setup [43], is used to cool and detect the atoms. On the other hand a DFB laser diode at 1.5 µm, frequency controlled by a beat-note with the fiber laser, provides the LPAI laser source. The two Raman frequencies are generated using a fibered phase modulator [44]. Both lasers are finally combined at 1.5 µm through an electro-optical modulator acting like a continuous optical switch between each laser, before seeding a 5 W Erbium-doped fiber amplifier (EDFA). The output of the EDFA is sent to a second harmonic generation bench. The complete laser setup and optical bench description can be found in [15].

C. State preparation and Raman spectroscopy

We investigate our method by selecting the atomic input state and implementing Raman spectroscopy. A cold 87 Rb atom sample is produced in a three-dimensional magneto-optical trap (MOT) loaded from a background vapor in 840 ms. An optical molasses cools the atoms down to 2 µK in 8 ms. After turning off the cooling beams, the atoms are in free fall. Then a horizontal bias magnetic field B ∼ 400 mG is switched on and a microwave π-pulse is applied, followed by a blow-away beam, allowing to select the atoms in the |F = 1, m F = -(+)1 state. Raman spectroscopy is performed using a Raman pulse of duration τ = 10 µs. Finally an internal state-selective vertical light-induced fluorescence detection is used to measure the proportion of atoms in each hyperfine state |F = 1 and |F = 2 . The cycling time of the experiment is T cycle = 1 s.

Figure 2(a) displays the measured transition probability as a function of the Raman frequency difference (ω 1 -ω 2 )/2π. The atoms being prepared in the |F = 1, m F = -1 state, the electric dipole transition selection rules state that only two transitions are possible with our Raman beam polarization configuration: a copropagating transition |F = 1, m F = -1 → |F = 2, m F = -1 (almost insensitive to Doppler effect and therefore narrower) and a counterpropagating transition |F = 1, m F = -1 → |F = 2, m F = 1 (see Fig. 1(a)). A third transition can be observed in Fig. 2(a) due to spontaneous emission: a fraction of the atoms are transferred by spontaneous emission to the m F = 0, 1 magnetic sublevels of F = 1 and undergoes one of the two degenerate transitions

|F = 1, m F = 1 → |F = 2, m F = 1 , |F = 1, m F = 0 → |F = 2, m F = 2 .
Spontaneous emission will be further discussed in Section IV A.

One can notice not only that the σ + -σ -transition frequency is independent of the magnetic field magnitude B at first order (see Fig. 2(c)), but also that the ±k eff transition frequencies (corresponding to |F = 1, m F = ∓1 ↔ |F = 2, m F = ±1 ) are the same (see Fig. 2(b)), which is very useful when implementing the k eff -reversal technique (see Section III A).

In conclusion, once the Raman frequency is properly tuned, it is only the state preparation in the magnetic sublevel m F = ∓1 that defines which transition ±k eff will be addressed during the interferometer. This means that reversing the effective wave vector is different from LPAIs using the |F = 1, m F = 0 -|F = 2, m F = 0 transition, where the Raman frequency needs to be changed in order to reverse k eff .

III. ATOM INTERFEROMETRY WITH σ

+ -σ - TRANSITIONS
Our Mach-Zehnder type LPAI in a horizontal configuration consists of a π/2 -π -π/2 Raman pulse sequence, with each pulse separated by a time T . Due to free fall of atoms across the laser beams of waist 5.5 mm (1/e 2 radius), the interrogation time is limited to 2T = 33 ms. At the output of the interferometer the phase shift is the sum of two terms: ∆φ = ∆φ propagation + ∆φ laser where ∆φ propagation is the difference in the action computed along the classical trajectory of each interferometer arm, and ∆φ laser is the phase difference imprinted on the atoms by the Raman lasers at different locations [45]. The complete calculation of ∆φ propagation is done in Section IV B and shows that ∆φ propagation = k eff • a B T 2 + smaller terms (see Eq.( 12)) with a B an acceleration due to a magnetic force. This magnetic acceleration depends on the transition ±k eff and is ex-

pressed as a B = -µ B m g F m F ∇B = -h m α m ∇B,
where m is the 87 Rb atomic mass, h is the reduced Planck constant, α m = ±2π × 0.70 MHz/G is the Zeeman shift of |F = 1, m F = ∓1 respectively [START_REF] Steck | Rubidium 87 D Line Data[END_REF], and ∇B is the gradient of the magnetic field magnitude B. In the infinitely short, resonant-pulse limit, the second phase term of ∆φ is given by ∆φ laser = k eff • a inertial T 2 where a inertial is the acceleration of the atoms due to gravito-inertial effects. This interferometer geometry thus exhibits at its output an atomic phase shift sensitive to the combined acceleration of the atoms due to gravito-inertial effects and a force due to a magnetic field gradient:

FIG. 2.
Raman resonance spectra obtained by scanning the Raman frequency (ω1 -ω2)/2π across the resonances.

(a) Raman spectrum with the atoms prepared in the |F = 1, mF = -1 sublevel.

The magnetic field amplitude is B = 411 mG. Three resonance peaks are observed and correspond to four transitions:

|F = 1, mF = -1 → |F = 2, mF = -1 , |F = 1, mF = -1 → |F = 2,
= k eff • a T 2 = k eff • a inertial - h m α m ∇B T 2 (2)
At the end of the interferometric sequence we measure the proportion of atoms in each output port of the interferometer |F = 1 and |F = 2 by fluorescence. The normalized population in the state |F = 2 at the LPAI exit is a sinusoidal function of the interferometer phase shift:

P = P m - C 2 cos ∆φ = P m - C 2 cos k eff • a T 2 (3)
where P m is the fringe offset and C is the fringe contrast. In the following we neglect the smaller terms from ∆φ propagation (see Eq.( 12)) which will be studied in Section IV B.

The force responsible of the magnetic acceleration depends on the magnetic field inhomogeneities. To evaluate this force in our setup, we proceed as follows: just as in a LPAI gravimeter we apply a radio frequency chirp β (Hz/s) to the effective Raman frequency to scan the interference fringes. The sinusoidal dependence of the probability (see Eq.( 3)) leads to an ambiguity in the acceleration measurement. We solve this issue by measuring interference fringes for different interrogation times T . Reversing the sign of the effective wave vector (i.e. preparing the atoms in |F = 1, m F = ∓1 alternatively), the magnetic acceleration changes sign and the phase shifts are respectively:

∆φ ± = ± k eff • ( a inertial + a B± ) -2πβ T 2 = ± k eff • a inertial + k eff • a B+ -2πβ T 2 (4)
where

a B± = a B (F = 1, m F = ∓1) = a B (F = 2, m F = ±1) a B+ = -a B-
This means that whatever the pulse separation T , the phase shift is zero when the chirp reaches β 0± = k eff 2π • (±a inertial + a B+ ). From this we easily extract the magnetic acceleration a B+ = 2π 2k eff • (β 0+ + β 0-). Its numerical value is -7.79 × 10 -3 m.s -2 , i.e. several interfringes i = λ/2T 2 = 1.43 × 10 -3 m.s -2 (T = 16.5 ms). The corresponding magnetic field gradient is ∂ x B = -24.2 mG.cm -1 and is therefore responsible for a non negligible bias on the inertial acceleration measurement. The k eff -reversal technique (i.e. preparing the atoms in the |F = 1, m F = ∓1 states alternatively) is essential to eliminate such a bias.

A. Correlation fringes

When scanning the interference fringes by varying the frequency chirp β, the fringes are washed out because of vibration noise (typically as soon as T ≥ 6 ms). To recover the interference fringes, we perform a correlationbased technique [46] which combines both measurements of the LPAI output signal P and of the classical accelerometer fixed to the Raman mirror. Figure 3 shows the typical fringe pattern obtained by plotting the transition probability P at the LPAI output versus the acceleration measured by the classical accelerometer. The fringe contrast obtained from a sinusoidal least-squares fit of the data is C = 13 %, which is the best contrast that we obtained when adjusting the Raman laser intensity at a fixed Raman pulse duration of τ = 10 µs. We demonstrated in [15] a horizontal hybrid accelerometer with a contrast of 40 % on the same experimental setup. In Section IV we investigate the loss of contrast associated to the σ + -σ -technique.

B. Accelerometer sensitivity

We analyze the sensitivity and stability of the horizontal atom accelerometer by hybridizing the classical and the atomic sensors [29,46]. The sign of the effective Raman wave vector k eff is reversed every measurement cycle, i.e. the atoms are alternatively prepared in the |F = 1, m F = ∓1 states. We calculate the inertial acceleration by computing the half sum of the phase shifts measured on each correlation fringe pattern ±k eff . The fringe ambiguity is removed by assuming that the mag-netic acceleration has the same value as calculated for smaller interrogation times, i.e. a B = -7.79 × 10 -3 m.s -2 . Figure 4 displays the Allan standard deviation (ADEV) of the hybridized atom interferometer signal. We achieve a short-term sensitivity of 25 × 10 -5 m.s 2 / √ Hz, which is not as good as the state of the art for horizontal configurations [14,15]. In Section IV we discuss several arguments to explain this sensitivity. The ADEV of the atomic sensor scales as τ -1/2 and reaches 3.8 × 10 -6 m.s 2 after 3300 s integration time. No conclusion can be drawn regarding the long-term stability of the atom accelerometer because of angular drifts of the Raman mirror. An auxiliary tilt sensor could be used to monitor the angle between the Raman beam and the horizontal direction during the measurements.

IV. SPECIFICS OF THE METHOD A. Spontaneous emission

As shown in Eq.( 1) the transition probability amplitudes for each excited state |F = 1 and |F = 2 interfere destructively. In order to address the counterpropagating transition, the laser detuning is set between the |F = 1 and |F = 2 levels and therefore induces spontaneous emission and coherence loss. Assuming that atoms which undergo spontaneous emission do not interfere anymore, the contrast at the atom interferometer output is reduced, and so is the sensor sensitivity. We experimentally tuned the Raman laser frequency to minimize the spontaneous emission rate. The detuning ∆ 2 from the excited state |F = 2 is adjusted via a beatnote between the fiber laser and the Raman laser diode. The probability of transfer by spontaneous emission in |F = 2 is estimated by measuring the transfer probability during a 10 µs out-of-resonance Raman pulse. Figure 5 shows the experimental result (red dots) as a function of the detuning ∆ 2 , along with the theoretical prediction of the probability of transfer by spontaneous emission in the |F = 2 level during a 10 µs pulse. In order to match the experimental result with the theoretical prediction, we introduce an empirical multiplicative factor κ = 2.5 for the laser intensity (see Appendix A for detailed calculations). Considering the dipole matrix elements we can derive an estimation of the total spontaneous emission probability during the whole interferometer both from data and theory (blue dots and black line in Fig. 5). Here again we assume κ = 2.5 in the theoretical calculations. The need of this empirical factor κ is not understood. It could come from experimental imperfections (non perfect π/2 Raman pulses) and from our theoretical treatment of spontaneous emission that is too simple (the resolution of Bloch optics equations could be used in a more elaborated model [47]).

The optimal detuning is given by the curve minimum: ∆ 2 = -100 MHz both theoretically and experimentally. We can conclude from Figure 5 that for ∆ 2 = -100 MHz, 75 % of the atoms undergo spontaneous emission during the interferometer duration. Such a loss of atoms reduces the contrast by a factor 4 and could explain our relatively low contrast value of 13% (see Figure 3). As a matter of fact, we demonstrated in [15] a horizontal hybrid accelerometer on the same experimental setup with a contrast of 40%. Since 40% × 1/4 = 10%, the visiblity loss is probably due to spontaneous emission. Nevertheless, we investigate in the next subsection the phase shift sensitivity to magnetic field and show how it also affects the LPAI contrast and bias.

Setting the detuning close to resonance still has an advantage: the transition requires low Raman intensity (∼ I sat ) compared to the magnetically insensitive Raman transition (∼ 10 I sat ) for which the detuning is set far from resonance. However spontaneous emission can be drastically reduced if the Raman transition is performed on the D 1 line instead of the D 2 line of 87 Rb, since the hyperfine levels are further apart. Theoretical calculations of spontaneous emission on the D 1 line (without taking into account the factor κ)show that only 10 % of the atoms undergo spontaneous emission during the interferometer sequence, which is comparable to LPAIs using the |F = 1, m F = 0 -|F = 2, m F = 0 transition with a commonly used one-photon detuning from |F = 3 of ∼ -1 GHz.

B. Sensitivity to magnetic field

As shown in Eq.( 2) the LPAI is sensitive to both inertial acceleration and magnetic forces from field inhomogeneities. Using the k eff -reversal technique one can extract each contribution by computing either the half sum or the half difference of the phase shifts (see Section III). This is only valid under the assumption of a constant magnetic field gradient from shot-to-shot. We perform in this Section the detailed calculation of the phase shift due to magnetic field by taking into account spatial inhomogeneities of the magnetic field up to order two. From this study we estimate the bias and the loss of contrast induced by the magnetic field.

We have considered so far a weak magnetic field and a linear relationship between magnetic energy levels and magnetic field, with the same shift for |F = 1, m F = ∓1 and |F = 2, m F = ±1 . For the ground state manifold of the D transition, the exact calculation of the energy levels is given by the Breit-Rabi formula [48]. In the case of 87 Rb in the |F = 1, m F = -1 ≡ |1, -1 and |F = 2, m F = 1 ≡ |2, 1 levels, the energies are given by:

E |1,-1 = h (α m -∆α) × B ≡ hω a E |2,+1 = h (α m + ∆α) × B ≡ hω b (5)
with α m = g J -g I 4h µ B and ∆α = g I h µ B . Here we do not take into account the hyperfine splitting, as it is a constant which cancels out in the phase shift calculation.

a, b stand for the two hyperfine states |1, -1 , |2, +1 of the LPAI. g I is the nuclear g-factor, g J is the Landé factor and µ B is the Bohr magneton. Under the assumption of spins following the magnetic field adiabatically during free fall, B represents the magnitude of the magnetic field. We define the average energy shift α m = 2π × 0.70 MHz/G (see Eq.( 2)) and the differential energy shift ∆α = 2π × -1.4 kHz/G coming from the Breit-Rabi formula. Similarly, for the -k eff transition between |1, +1 and |2, -1 , the energy levels are described by Eq.( 5), only (α m , ∆α) are replaced with (-α m , -∆α).

We use the Feynman path integral approach to compute the magnetic phase shift between the two arms (u, d) of the interferometer. Using a perturbative calculation for the effect of the magnetic field [45], the phase ϕ (i) accumulated along each arm i = u, d of the LPAI is given by the classical action S cl along the unperturbated classical path divided by h:

ϕ (i) = S (i) cl /h = 2T 0 L[ r(t)]/h dt (6)
The phase difference at the output of the LPAI is then:

∆φ path = ϕ (u) -ϕ (d) = (S (u) cl -S (d) cl )/h = 1/h 2T 0 L[ r (u) (t)] dt - 2T 0 L[ r (d) (t)] dt = T 0 ω b r (u) (t) -ω a r (d) (t) dt + 2T T ω a r (u) (t) -ω b r (d) (t) dt (7
) Taking into account the Breit-Rabi correction (see Eq.5), the phase difference at the output of the interferometer can be split into two terms:

∆φ path = α m 2T 0 B r (u) (t) -B r (d) (t) dt + ∆α T 0 B r (u) (t) + B r (d) (t) - 2T T B r (u) (t) + B r (d) (t) dt (8)
The first term (proportional to α m ) arises from the magnetic field variation between the upper and the lower arms of the interferometer, whereas the second term (proportionnal to ∆α) is accounting for the variation of the mean field B between [0-T ] and [T -2T ].

The phase shift calculation is performed on the unperturbed trajectories r (u) (t), r (d) (t) whose expressions are:

     r (u) (t) = r 0 + ( v 0 + v eff )t -v eff (t -T )H(t -T ) + 1 2 gt 2 r (d) (t) = r 0 + v 0 t + v eff (t -T )H(t -T ) + 1 2 gt 2 (9
) where v eff = h k eff /m, ( r 0 , v 0 ) are the position and velocity vectors at the first Raman pulse and H(t) is the Heaviside function.

The magnetic field magnitude B is supposed to be time-independent and can therefore be expressed through its Taylor expansion in space:

B( r) = B 0 + B 1 • r + 1 2 r • B2 • r = B 0 + ∂ x Bx + ∂ y By + ∂ z Bz + 1 2 (∂ 2 x Bx 2 + ∂ 2 y By 2 + ∂ 2 z Bz 2 + 2∂ x ∂ y Bxy + 2∂ x ∂ z Bxz + 2∂ y ∂ z Byz) (10)
Using Eq.(9,10) the phase shift calculation of Eq.( 8) leads to two terms proportional to α m and ∆α respectively:

∆φ path = ∆φ 1 + ∆φ 2 (11)
where

                                 ∆φ 1 = α m T 2 ∂ x Bv eff + ∂ 2 x Bv eff [x 0 + (v 0x + v eff /2)T ] + ∂ x ∂ y Bv eff [y 0 + v 0y T ] + ∂ x ∂ z Bv eff z 0 + v 0z T + 7/12gT 2 ∆φ 2 = -∆αT 2 2∂ x B(v 0x + v eff /2) + 2∂ y Bv 0y + 2∂ z B(v 0z + gT ) (12)
The first term of ∆φ 1 is the magnetic acceleration of Eq.( 2) since

α m ∂ x Bv eff T 2 = α m ∂ x B h m k eff T 2 = k eff a B T 2 .
The other terms of ∆φ 1 are due to magnetic field curvatures whereas the terms of ∆φ 2 come from the differential shift ∆α and the magnetic field gradient. From Eq.( 12) one can deduce the bias and loss of contrast induced by the σ + -σ -method.

The k eff -reversal technique enables to eliminate any systematic effect whose sign does not change when reversing k eff . The bias is then due to the remaining phase terms. In our case, since (α m , ∆α) change sign when reversing k eff , we deduce from Eq.( 12) that the bias induced by magnetic effects is:

1 2 α m ∂ 2 x Bv 2 eff T 3 -2∆α∂ z BgT 3 (13)
The first term is an inertial phase due to the atom recoil and the magnetic field curvature: it arises as soon as the magnetic field gradient is different for the upper and lower arm of the interferometer. It was estimated theoretically by computing the magnetic field produced by the horizontal field coils (∂ 2

x B ∼ 5 G.m -2 ) and leads to a bias of 2 × 10 -6 m.s -2 (we recall T = 16.5 ms). The second term is an energy dependent phase term which comes from time variation of the differential Zeeman energy shift induced by the magnetic field gradient ∂ z B seen by the atoms during free fall. In order to estimate it we measured the magnetic field vertical gradient by Zeeman spectroscopy and found ∂ z B = 1.2 G.m -1 . The resulting bias is 2 × 10 -4 m.s -2 . In comparison, ref. [49] demonstrates a high-performance magnetic shield for long baseline atom interferometry with inhomogeneities below 3 × 10 -7 G/cm: the associated bias would be 3 × 10 -9 m.s -2 , at the cost of a more complex magnetic field shield. From Eq.( 12) one can easily notice that this bias can be eliminated through an atomic fountain design with a properly set vertical velocity at the first Raman pulse v 0z = -gT . One can also set the quantization field to its magic value [38] corresponding to the magnetic field at which the derivative of the energy difference h(ω b -ω a ) is null. But this configuration does not cancel out the bias perfectly because it requires to change the magnetic field sign when changing the sign of k eff , since the magic field for the ± k eff transitions are respectively ±3.2 G [38].

Regarding the contrast reduction induced by magnetic effects, it is due to the position and velocity dependent terms in Eq.( 12). Indeed any phase shift sensitivity to position (or velocity) results in each atom (or velocity class) providing its own fringe pattern. Since the atom detection protocol averages over these patterns, the fringe contrast is reduced. In our case the interrogation time T = 16.5 ms is short enough to neglect the loss of contrast arising from velocity dependent phase shifts in Eq.( 12) (see Fig. 6(a)). The contrast reduction due to averaging over position on the other hand is much more significant. As an example we analyse the phase shift α m ∂ 2

x Bv eff x 0 T 2 : it reduces contrast by a factor e -1 2 (αm∂ 2

x Bv eff T 2 σx) 2 [50], where σ x = 1 mm is the typical size of the atomic cloud. Fig. 6(b) illustrates this loss of contrast as a function of the magnetic field curvature ∂ 2

x B: the contrast is typically reduced by 50 % in the presence of ∂ 2

x B ∼ 50 G.m -2 . As comparison Fig. 6(a) shows how the velocity dependent phase shift α m ∂ 2

x Bv eff v 0x T 3 does not affect contrast. Since the magnetic field along the Raman beams cannot be measured precisely enough on our experimental setup, we estimated theoretically the curvature due to our magnetic field coils and found ∂ 2

x B th ∼ 5 G.m -2 . From this we can state that inhomogeneities of the magnetic field created by the coils do not affect the fringe contrast. However the presence of another magnetic field source creating a non negligible field curvature responsible of a contrast reduction is to be considered. Loss of contrast can also be interpreted as in [50]: position and velocity dependent phase terms in Eq.( 12) are responsible for the opening of the interferometer in momentum and position respectively. We introduce the notation Γ = 2 h m α m ∂ 2 x B since the magnetic field curvature is the exact analog of a gravity gradient in a vertical LPAI. The forces associated with Γ tend to open up the trajectories of the atoms and lead to an open interferometer with nonvanishing relative position and momentum displacements at the output of the LPAI. As demonstrated in [50] a position dependent phase shift results in a momentum displacement δP at the output of the LPAI, and a velocity dependent phase shift results in a position displacement δX. Both displacements are given by the following expressions to first order in ΓT 2 [50]:

δX = ΓT 2 hk eff m T (14) δP = ΓT 2 hk eff (15)
In our case the change of position associated with Eq.( 14) is very small compared to the coherence length which is estimated by the thermal De Broglie wavelength: δX ∼ 10 -8 m l c = λ DB = h/ √ 2πmk B θ ∼ 10 -7 m for 87 Rb atoms at θ = 2 µK. On the other hand the momentum displacement given by Eq.( 15) is not as negligible as the position displacement. Even though [51] suggests a protocol to mitigate the loss of contrast due to gravity gradients Γ through a suitable adjustment of the laser wavelength at the second Raman pulse, this technique cannot be applied here. Indeed the σ + -σ -method requires the detuning ∆ to stay between the |F = 1 and |F = 2 sublevels to avoid spontaneous emission (see Section IV A), which means that the mitigation technique proposed by [50] would inevitably reduce the fringe contrast.

C. Light shifts

In most cases one can eliminate the differential onephoton light shift by adjusting the intensity ratio between the two Raman lasers. In our σ + -σ -Raman transition scheme, there is no intensity ratio that cancels out the differential one-photon light shift (see Appendix B for complete calculation of light shifts), which means that an intensity variation between the first and the last pulses of the LPAI leads to a residual parasitic phase shift. For an intensity variation of 10 % between the first and the last pulses (due to the position of the atoms in the horizontal Raman beam with gaussian intensity profile), the corresponding bias due to the one-photon light shift is -1.5 × 10 -5 m.s -2 . However, assuming the same intensity variation for the +k eff and the -k eff interferometers, this light shift is nearly rejected through the k eff -reversal technique, since δ diff LS1 (+k eff ) δ diff LS1 (-k eff ). More generally it is essential to note that the one-photon light shift doesn't represent a limit to our σ + -σ -technique, since one can find an intensity ratio canceling it out on the D 1 line of 87 Rb.

In our setup, the two-photon light shift arises from the off-resonant copropagating Raman transitions detuned by ± 2µ B g F h B from the considered ±k eff counterpropagating transition. Its effect decreases with the magnetic field. When B = 400 mG, it is calculated to be negligible compared to the one-photon light shift and it cancels out as well when the k eff -reversal technique is applied.

V. APPLICATIONS

In this section we propose some possible applications of our technique in both metrology and for development of new designs of atom interferometers dedicated to field applications.

A.

h/mX measurement Atom interferometers allow to determine the finestructure constant α based on measuring the recoil velocity v r = hk/m X of an atom X of mass m X absorbing a photon of momentum hk, where h = h/2π, and k is the photon wave number. With an accurate measurement of k, h/m X can be measured and α can be determined allowing to test the standard model and beyond [4,5].

Here, using our interferometric design we propose to measure h/m X by combining the interferometric measurement of the magnetic acceleration a B and an independent measurement of the magnetic field gradient ∂ x B thanks to a micro-wave spectroscopy. In Eq.( 2), using the k eff -reversal technique, one can isolate the magnetic acceleration, leading to:

h m X = 2π a B α m ∂ x B (16)
Considering state-of-the-art atom accelerometers at their best level of accuracy (∼ 10 -8 m.s -2 ) [28], combined with a micro-wave Ramsey interferometer with a freeevolution time T =10 ms and a signal-to-noise ratio SNR = 10 3 , one could measure magnetic fields at the level of ∼ 100 nG. Thus, under an acceleration a B = 1 m.s -2 one could obtain a relative uncertainty at the level of 10 -8 on h/m X measurement. Although this performance is less than the already achieved accuracy [4,5], this independent technique which is not based on the atom-recoil measurement could be interesting for metrology.

B. Force-balanced atom accelerometer

The supplementary internal degree of freedom provided by the magnetically-sensitive states used in the interferometer allows to exploit the sensitivity of the atoms to magnetic field gradients and transfer a magnetic acceleration onto them. The magnetic acceleration of a Rb atom in a m F = ±1 sublevel due to a magnetic field gradient ∂ z B is a B ∼ 32.1 × ∂ z B m.s -2 . For example, applying a magnetic field gradient of magnitude ∂ z B ∼ 30 G.cm -1 could compensate for gravity acceleration on Earth. Thus, we propose to use our interferometric scheme to create a force-balanced atom accelerometer where the inertial acceleration undergone by the atoms, and measured by an auxiliary classical accelerometer, could be compensated by applying a magnetic acceleration. The basic principle of the technique is depicted in Fig. (7). The atom accelerometer is hybridized with a conventional classical accelerometer in order to track the bias of the classical accelerometer such as in [30]. Our coil configuration consists of two vertical coils above and below the atoms. A pair of Helmholtz coils generates a bias field B q along the z-axis and defines the quantization axis. This quantization axis is aligned with the direction of the Raman lasers in order to enhance the Raman transition between the magnetically-sensitive states |F = 1, m F = ∓1 and |F = 2, m F = ± 1 . A second pair of coils with counterpropagating currents (anti-Helmholtz configuration) in the same housing, is set to create the magnetic field gradient along the same direction. The current output of the classical accelerometer, proportional to the inertial acceleration, is then used as an input signal to counter-balance the inertial force undergone by the atoms. In pratice, depending on the electric current fed into the anti-Helmholtz coils, one can adjust the magnetic acceleration to a B = γi where i is the electric current and γ a scale factor which can be precisely determined through calibration of the magnetic acceleration in a lab-based environment. Thus, it is for example possible to levitate the atoms against gravitational acceleration and therefore extend the evolution time in earthbound laboratories. Additionally, using three pairs of Helmholtz coils in 3-orthogonal directions, this scheme could be further extended to compensate for any inertial acceleration in the 3 dimensions, where 3 classical accelerometers are fixed to 3 atom accelerometers to form a 3-axis hybrid accelerometer. With this scheme, one could simultaneously apply a magnetic force on the atoms in the 3 dimensions. This could benefit onboard atom accelerometers submitted to spurious accelerations which limit the dynamic range because the atoms drop out from the laser beams and the detection zone.

VI. CONCLUSION

We reported on the experimental demonstration of a horizontal cold-atom interferometer using counterpropagating Raman transitions between |F = 1, m F = ∓1 and |F = 2, m F = ±1 of 87 Rb. Using the same σ + -σ - polarized light arrangement as the MOT, we generated Raman coupling between the two states of the interferometer and showed that this technique allows to perform single-diffraction Raman process in a close-tozero velocity regime without the need for alternative techniques [15]. We demonstrated that this technique presents both advantages and disadvantages compared to the |F = 1, m F = 0 -|F = 2, m F = 0 transition usually used in atom interferometry (see TABLE I).

Absolute horizontal acceleration measurement with a short-term sensitivity of 25 × 10 -5 m.s -2 .Hz -1/2 was achieved. In our setup, limitations of the sensitivity arise from spontaneous emission, leading to a reduction of the interferometer contrast. The accuracy of the atom accelerometer is mainly limited by the bias caused by the magnetic field gradient at the level of ∼ 2 × 10 -4 m.s -2 .

Although the short-term sensitivity is bigger by almost one order of magnitude in comparison with state-of-theart horizontal atom accelerometers [14,15], it could be improved by changing the Raman excitation scheme to the D 1 line, thus reducing spontaneous emission. Additionally, one could reduce this acceleration bias at the level of ∼ 3 × 10 -9 m.s -2 (with T = 16.5 ms) considering a high-performance magnetic shielding leading to a magnetic-field inhomogeneity of 3 × 10 -7 G.cm -1 [49]. Moreover, we showed that acceleration bias could be suppressed by performing the atom interferometer using a fountain geometry. Finally, we believe that using the supplementary internal degree of freedom provided by atoms manipulated in magnetically sensitive levels provides interesting features such as levitation schemes for inertial applications requiring compact setups. We start by calculating the effective two-photon Rabi frequency Ω eff describing the Raman coupling between the hyperfine states |F = 1, m F = -1 and |F = 2, m F = 1 of the interferometer. We compute the single-photon scattering rate R F sc from atoms starting in |F = 1 or |F = 2 . From these the probability of transfer by spontaneous emission for an entire interferometer can be calculated.

We describe the intensity I n of each EOM sideband n at the output of the phase modulator as I n = I • J n (β) 2 where I is the total laser intensity, J n is the Bessel function of the first kind of order n, and β is the modulation index of the EOM. Here we only take into account two sidebands (n = 0 and n = 1), the others being detuned enough to be neglected. I sat stands for the saturation intensity and Γ is the natural line width. We calculate the effective two-photon Rabi frequency:

Ω eff = Γ 2 2 I I sat J 0 (β)J 1 (β) F M F ,+ 1,-1 • M F ,- 2,1 ∆ F (A1) where M F ,± F,m F = F, m F | er |F , m F ± 1 J = 1/2||er||J = 3/2
are the rubidium D 2 dipole matrix elements for σ ± transitions, expressed as multiples of J = 1/2||er||J = 3/2 , as given in [START_REF] Steck | Rubidium 87 D Line Data[END_REF].

∆ F = ω L -ω F 2
is the detuning of the carrier (of frequency ω L ) relative to the transition

F = 2 → F (of energy hω F 2 ).
Assuming a π/2 Raman pulse of duration τ with Rabi frequency Ω eff • τ = π/2, we can deduce from Eq.(A1) the laser intensity I(∆ F ) as a function of the detuning ∆ F .

The rate of spontaneous emission for atoms starting in

|F = 1, m F = -1 or |F = 2, m F = 1 is: R F sc = Γ I I sat n={0,1},F ,σ=± M F ,σ F,m F 2 1 + 4 ∆ n F,F 2 Γ 2 + I I sat J n (β) 2 (A2) where ∆ n F,F = ω L + nω hfs -ω F
F is the detuning of the sideband n (of frequency ω L + nω hfs ) relative to the transition F → F (of energy hω F F ). The ground state hyperfine splitting ω hfs is also the EOM driving frequency.

From these, the probability of spontaneous emission for an entire interferometer can be calculated:

P SE = 1 -exp -R 1 sc • 2τ -R 2 sc • 2τ (A3)
where we consider that the atoms spend as much time (2τ ) in state |F = 1 and in state |F = 2 , the total Raman interaction duration being 4τ .

We measure experimentally the number of atoms that are transferred from |F = 1 to |F = 2 when the Raman detuning is off the two-photon resonance. In order to compare these measurements with our theoretical model, we calculate the scattering rate R 1→2 sc for atoms initially in the state |F = 1, m F = -1 , undergoing a single-photon transition to an excited state |F , m F , and transferred to |F = 2 by spontaneous emission. This corresponds to the scattering rate R 1 sc presented in Eq.(A2) with the difference that one needs to take into account the spontaneous emission rates Γ F →2 of each deexcitation |F , m F → |F = 2, m F . Therefore we have:

R 1→2 sc = I I sat n={0,1},F ,σ=± Γ F →2 × (M F ,σ 1,-1 ) 2 1 + 4 (∆ n 1,F ) 2 Γ 2 + I I sat J n (β) 2 (A4)
where the spontaneous emission rates Γ F →2 are the following:

               Γ 0→2 = 0 Γ 1→2 = Γ 6 Γ 2→2 = Γ 2 Γ 3→2 = 0 (A5)
As presented in the associated article (see Figure 5), we introduce an empirical parameter κ in the theoretical formulas to account for the difference between the effective Raman intensity and the theoretical prediction. Thus we write the total loss of atoms (i.e. in the |F = 1 and |F = 2 levels) by spontaneous emission during the whole interferometer as :

P SE = 1 -exp -κ R 1 sc • 2τ + R 2 sc • 2τ (A6)
Likewise, we calculate the probability of transfer by spontaneous emission in |F = 2 during a pulse of duration τ as follows :

P SE F=2 = 1 -exp -κR 1→2 sc • τ (A7)
Appendix B: One-photon and two-photon light shifts

In a Mach-Zehnder type atom interferometer, uncompensated differential light shifts δ diff LS from the Raman lasers result in an additional phase contribution given by [52]:

∆φ LS = -arctan tan Ω eff,1 τ 2 δ diff LS,1 Ω eff,1
+ arctan tan Ω eff,3 τ 2

δ diff LS,3 Ω eff,3 (B1)
where the effective Rabi frequencies and the differential light shifts for each pulse i are given by Ω eff,i and δ diff LS,i respectively. τ is the duration of the first and last pulses.

M F ,σ F,m F 2 ∆ n F,F J n (β) 2 (B2)
The differential one-photon light shift is then:

δ diff LS1 = δ 2,1 LS1 -δ 1,-1 LS1 (B3)
There is no intensity ratio that cancels out the differential light shift, contrary to LPAIs using the |F = 1, m F = 0 -|F = 2, m F = 0 transition. This means that a difference of intensity between the first and the third pulses leads to an additional phase shift [52]. However this light shift is nearly rejected through the k-reversal technique, since δ diff LS1 (+k eff ) δ diff LS1 (-k eff ). We recall that reversing the sign of k eff means preparing the atoms alternatively in the |F = 1, m F = ∓1 states.

In our setup, the two-photon light shift arises from the off-resonant copropagating Raman transitions detuned by ±∆ B = ±2 × µ

             δ 1,-1 LS2 = Ω -1 eff,co 2 4∆ B δ 2,1 LS2 = Ω +1 eff,co 2 -4∆ B (B4)
where ∆ B = 2× µ B g F h B is the first order Zeeman splitting between the magnetic sublevels m F = 0 and m F = 2. The effective Rabi frequency of the copropagating transitions are:

             Ω -1 eff,co = Γ 2 2 I I sat J 0 (β)J 1 (β) F ,σ=± M F ,σ 1,-1 • M F ,σ 2,-1 ∆ F Ω +1 eff,co = Γ 2 2 I I sat J 0 (β)J 1 (β) F ,σ=± M F ,σ 1,1 • M F ,σ 2,1
∆ F (B5) Finally, the differential two-photon light shift is:

δ diff LS2 = δ 2,1 LS2 -δ 1,-1 LS2 (B6)
Just as for the one-photon light shift, the differential two-photon light shift is almost completely rejected through the k-reversal technique, because δ diff LS2 (+k eff ) δ diff LS2 (-k eff ).

FIG. 1 .

 1 FIG. 1. (a) Scheme of the σ + -σ -Raman transitions between Zeeman sublevels of the two hyperfine ground states of 87 Rb in the presence of a magnetic field. Solid lines represent the σ + -σ -polarized beams performing the |F = 1, mF = -1 ↔ |F = 2, mF = 1 transition. Dashed lines represent the σ --σ + polarized beams performing the |F = 1, mF = 1 ↔ |F = 2, mF = -1 transition. (b) Schematic setup of two-photon Raman transitions in the commonly used retroreflected geometry. A two-level atom is interacting with two pairs of counterpropagating light fields with σ + -σ -polarizations. This polarization arrangement allows for only one pair of couterpropagating light fields to drive the Raman coupling leading to a single diffraction Raman process despite zero Doppler shift.

  FIG. 2.Raman resonance spectra obtained by scanning the Raman frequency (ω1 -ω2)/2π across the resonances.(a) Raman spectrum with the atoms prepared in the |F = 1, mF = -1 sublevel.The magnetic field amplitude is B = 411 mG. Three resonance peaks are observed and correspond to four transitions:|F = 1, mF = -1 → |F = 2, mF = -1 , |F = 1, mF = -1 → |F = 2,mF = 1 and two degenerate transitions |F = 1, mF = 1 → |F = 2, mF = 1 and |F = 1, mF = 0 → |F = 2, mF = 2 . (b) Raman spectrum with the atoms prepared in the |F = 1, mF = +1 sublevel. The magnetic field amplitude is B = 411 mG. (c) The atoms being prepared in the |F = 1, mF = -1 sublevel, the magnetic field amplitude is tuned from 200 mG to 400 mG. The transition |F = 1, mF = -1 → |F = 2, mF = 1 is insensitive at first order to a magnetic field amplitude variation.

  ∆φ

FIG. 3 .

 3 FIG.3. Horizontal atom interferometer fringe pattern. The total interferometer time is 2T = 33 ms and the Raman pulse duration is τ = 10 µs. The solid line is a sinusoidal leastsquares fit using Eq.(3) and considering error-free independent variables. The estimated fringe contrast is C ∼ 13 %. The fringes are shifted from zero because of the magnetic acceleration and the bias of the classical accelerometer.

FIG. 4 .

 4 FIG. 4. Allan standard deviation (ADEV) of the hybridized horizontal atom accelerometer (blue line). The dashed line illustrates the τ -1/2 scaling.

FIG. 5 .

 5 FIG. 5. Probability of spontaneous emission (SE) as a function of the detuning ∆2 from the excited state |F = 2 . Red dots represent the experimental data of probability of transfer by spontaneous emission in the |F = 2 level during the first Raman pulse of duration 10 µs. The data was obtained from the detected background on Raman spectra plotted for different detunings ∆2 at constant effective Rabi frequency. The dashed line represents the theoretical probability of transfer by spontaneous emission (adjusted with a parameter κ = 2.5) in the |F = 2 level during a Raman pulse of 10 µs. Blue squares represent the total probability of spontaneous emission in the |F = 1 and |F = 2 levels during the whole interferometer. It was derived from the experimental data in red dots. The black line is the theoretical total probability of spontaneous emission during the whole interferometer (corrected by a factor κ = 2.5).

FIG. 6 .

 6 FIG. 6. Theoretical contrast calculated as a function of the magnetic field curvature. Parameters: interrogation time T = 16.5 ms, gas temperature θ = 2 µK, typical atomic cloud size σx = 1 mm, velocity σv = kBθ/m = 1.4 cm.s -1 . (a) Contrast loss due to the velocity dependent phase shift αm∂ 2x Bv eff v0xT 3 . (b) Contrast loss due to the position dependent phase shift αm∂ 2x Bv eff x0T 2 .

FIG. 7 .

 7 FIG. 7. Scheme of the force-balanced accelerometer. AHC: Anti-Helmholtz Coil ; HC: Helmholtz Coil.

Γ 2 4 I

 4 The one-photon Raman light shift is imprinted onto both hyperfine states |F = 1, m F = -1 and |F = 2, m F = 1 by out-of-resonance Raman lasers. Each light shift δ F,m F LS1 has the following expression: δ F,m F LS1 = I sat n={0,1},F ,σ=±

  B g F h B from the considered ±k eff counterpropagating transition. The level|F = 1, m F = -1 is perturbed by the coupling Ω -1 eff,co of the copropagating transition |F = 1, m F = -1 ↔ |F = 2, m F = -1 . Likewise, the level |F = 2, m F = 1 is perturbed by the coupling Ω +1 eff,co of the copropagating transition |F = 1, m F = 1 ↔ |F = 2, m F = 1 .The corresponding light shifts are:

TABLE I .

 I Summarized advantages and disadvantages of a Mach-Zehnder cold-atom interferometer using counterpropagative Raman transitions between the state |F = 1, mF = ∓1 and |F = 2, mF = ±1 of 87 Rb.

	Advantages	Disadvantages
	Same polarization arrange-	Spontaneous emission
	ment MOT/Raman	(reduced on the D1 line of
		87 Rb)
	Single counterpropagating	Remaining copropagating
	transition in retroreflec-	Raman transition
	tive geometry despite zero
	Doppler shift
	Supplementary internal de-	Require precise control and
	gree of freedom provided by	mapping of the magnetic
	adressing mF = 0 Zeeman	field
	sublevels
	(force-balanced
	accelerometer)
	Appendix A: Probability of transfer by spontaneous
	emission

ACKNOWLEDGMENTS M. C. acknowledges funding from ONERA through research project CHAMOIS (Centrale Hybride AtoMique de l'Onera et Inertie Strapdown).