Dr Thi-Phuong-Ahn Cao

Dr Grégory Nocton

Dr Louis Ricard

Dr Xavier F Le Goff

Keywords: Iminophosphorane, nickel, one electron-oxidation, high valent metal, phosphasalen

A Tetracoordinated Phosphasalen Nickel(III) Complex

Thi-Phuong-Anh Cao, Grégory Nocton, Louis Ricard, Xavier F. Le Goff, and Audrey Auffrant* ((Dedication----optional))

N,N'-bis(salicylidine)ethylenediamine (salen) derivatives constitute a very successful class of ligand that have found numerous applications in inorganic chemistry and catalysis. [1] Its extraordinary popularity comes from its easy synthesis, allowing a variety of skeleton variations, and its ability to coordinate a great number of different metal ions. Moreover the presence of two phenoxy and imine functions is reminiscent of the coordination environment generated by two histidines and two tyrosines, encountered in several enzymes. Thus, 15 years ago, the one-electron oxidation of a (salen)copper(II) was described, and the resulting persistent phenoxy radical was established as a biomimetic functional model of galactose oxidase. [2] Since then, one-electron oxidized salen metal complexes of first-row transition metals have received considerable attention. [3] Depending on the nature of the redox-active orbitals, oxidation of salen complexes leads either to high valent metal (metal centered oxidation) or metal-ligand radical species (ligand centered oxidation). Therefore, various factors may affect the oxidation locus, as it was nicely illustrated in the case of nickel salen complexes. In non coordinating solvents, one electron oxidized nickel salen complexes are generally highly delocalized ligand radical species with a contribution of the nickel orbitals. [3b, 4] However, in coordinating solvents, or in the presence of exogenous ligands such as pyridine or coordinating anions, the geometry of the complex is modified and so does the locus of the oxidation, allowing the formation of Ni III complexes. [4a,b, 5] Temperature dependant behaviour, viz. valence tautomery, was also described in a few examples. [5b, 6] Moreover, the overall electronic distribution may be finely tuned by the substituents of the phenoxy rings. They influence not only the oxidation locus (which takes place at the most electron rich phenoxide) as evidenced with complexes featuring unsymmetrical salen, [3c, 7] but also the extent of the radical delocalization. [3b, 8] The hybridization of the nitrogen atoms is also of prime importance since it may influence the oxidation locus, the spin state of the metal, and the magnetic interaction within the molecule. [5b, 9] Astonishingly, despite numerous structural variations proposed for salen ligands, the introduction of heteroatoms in its backbone was rarely investigated. [10] As we are interested in iminophosphorane (P=N) based ligands as alternative to conventional imine ones, [11] we developed a new class of ligand featuring iminophosphoranes in place of imines, which we termed phosphasalen (or Psalen). We have recently demonstrated that these ligands are more flexible and electron-donating than their carbon analogue [12] and that they provide efficient and selective initiators for rac-lactide polymerization. [13] In the present work, we wish to report the synthesis and characterization of a phosphasalen nickel(II) complex and its one electron oxidation which leads to a temperature persistent high valent tetracoordinated Ni III complex.

With the objective of studying the oxidation of a [Ni(Psalen)] complex, we chose to use a ligand featuring substituents on the ortho and para positions of the phenoxides in order to increase its stability, and preferred tertbutyl groups for solubility reason. The required phosphasalen ligand 1 was prepared following a procedure we recently published. [13b] Reaction of 1 with [NiBr2(DME)] in THF led, after removal of the potassium salts, to the formation of the nickel(II) complex 2 as a purple solid in 92% yield (Scheme 1). In the 31 P{ 1 H} NMR spectrum the coordination is indicated by a deshielding of the phosphorus nuclei appearing as a singlet at (CD2Cl2) = 36.0 ppm. This complex was characterized by multinuclear NMR spectroscopy, elemental analysis, and X-ray diffraction analysis. As expected for a d 8 complex, 2 adopts a square planar geometry around the metal with a slight deviation from planarity (8.5°, see SI).

Scheme 1. Synthesis of nickel complexes 2 and [2 + ][SbF6 -]

The cyclic voltammogram of 2 exhibits three oxidation waves in CH2Cl2 at +0.16, +0.96, and +1.22 V versus ferrocene/ferrocenium (Fc/Fc + ). The first oxidation is much easier than the one reported for the related [Ni( t Bu-salen)] (E~420 mV, table 1), [5b, 14] in agreement with the better electron donating ability of the phosphasalen. Moreover the difference between the first and second oxidation waves of 2 is large (E 1/2 = 0.8 V) compared to that observed for the salen analogue (E 1/2 = 0.46 V). [5b, 14] In salen nickel complexes, the two first oxidations are known to occur on both phenoxides. Within the Ni(II) salen complexes series, the rather large separation between the two first half-wave potentials observed for [Ni( t Busalen)] was interpreted as a result of the strong electronic communication between the two phenoxides. [5b, 14] Therefore the even larger peak separation observed for 2 may indicate that the first oxidation in this complex does not occur on the ligand but may be metal centered. [Bu4N][BF4]: 0.12 M -1 ; 50 mV s -1 ) and [Ni( t Bu-Salen)]. [5b, 14] Potential values given relative to E1/2(Fc + /Fc). This latter result prompted us to investigate the chemical oxidation of 2. The cation [2 + ] was synthesized by addition of one equivalent of a silver salt to a CH2Cl2 solution of 2. This induced a rapid color change from blue to red-brown. Monitoring the reaction with 31 P{ 1 H} NMR spectroscopy shows the disappearance of the starting material and the formation of a paramagnetic species, whose 1 H NMR signals spread in the chemical shifts values already observed for Ni(III) complexes. [15] Variable temperature experiments confirmed the temperature dependence of those chemical shifts. Different silver salts were used and led to similar results nevertheless only [2 + ][SbF6 -] did crystallize, which encouraged us to focus on this complex. Complex [2 + ] is stable for months in the solid state and days in toluene or dichloromethane solution, at room temperature under an inert atmosphere. In order to get insights in the structural features of 2 and [2 + ], DFT calculations were carried out on both complexes. For the oxidized complex, calculations were carried out with and without the anion giving comparable results (see SI). The geometry optimizations well reproduce the experimental structures, including the non-symmetric contraction of the coordination sphere upon oxidation (see table S6). In the neutral complex 2, the two highest occupied molecular orbitals, which are of similar energy, result from antibonding interactions between the dxz or dyz orbital of Ni and p orbitals of the oxygen atoms (see SI). Upon oxidation this degeneracy is suppressed, as one electron is removed from the orbital featuring an antibonding interaction of the dyz orbital of the Ni and the p orbitals of O(1) and N( 2). This explains why the oxidation causes a stronger contraction on this particular axis. Importantly, this coincides with the highest occupied quasirestricted orbital calculated for [Ni( t Bu-Psalen)] + [2 + ] as well as the Mulliken spin distribution, which is mainly concentrated on the Ni (0.69), O(1) (0.13), and N(2) (0.12) atoms. Therefore, from the DFT calculations, and contrary to its salen analog [Ni( t Bu-salen)] + , [2 + ] may be described as a high valent Ni III complex rather than a ligand radical metal complex. X-Band EPR of [2 + ][SbF6 -] was recorded in a toluene/benzene mixture at different temperatures. At 5 K, [2 + ][SbF6 -] displayed a large g-tensor anisotropy (Figure 3) with g1= 2.29, g2= 2.215 and g3 = 2.061 (giso = 2.19), indicative of a S=1/2 ground state in a rhombic symmetry centered on the nickel ion, i.e. a Ni III metal center. When temperature is increased, the EPR signal tends to decrease but no change in the g strengths have been noted up to 100 K. Above this temperature, the signal is very weak which prevents reasonable simulation and disappears so that [2 + ][SbF6 -] is EPR-silent at room temperature. Large g values account for significant orbital contributions that allow low lying states to mix through spin-orbit coupling. This is rather usual in d 7 complexes, and may also relate to the loss of intensity of the EPR signal at higher temperature. [16,17] In a perfect square planar geometry the g matrix orientation would be as gx = gy >> gz. [18] In the case of [2 + ][SbF6 -], g1 > g2 >> g3 and indeed indicates a predominance of the xy plane (g1 and g2) over the z axis (g3) but with a significant distortion in the xy plane (g1 differs from g2). This observation is in good agreement with the X-ray data that feature two different sets of distances in the xy plane for the N(1)-Ni-O(2) and N(2)-Ni-O(1) axes but, unfortunately, no hyperfine coupling with the 14 N and/or the 31 P atoms in the x and y axis could be observed to further sharpen this analysis. Powder EPR of [2 + ][SbF6 -] is also in agreement with a Ni III complex (see Supporting Information) ruling out an eventual axial solvent coordination in solution. EPR g-tensor calculations were also performed on [2 + ][SbF6 -] and gave a very similar outcome (g1 = 2.34, g2 = 2.21 and g3 =2.09, giso = 2.21; see Supporting Information). As the decrease of EPR intensity signal may have been associated with valence tautomery, [5b] temperature dependant magnetic data were recorded in the solid state for [2 + ][SbF6 -]. The data show a Curie behaviour over the 4-300 K temperature range (χM=C/T) (See SI). The χMT value (C = 0.46 cm 3 .K/mol) is a little higher than expected for a d 7 low spin complex but can be fitted with a giso value of 2.22 which is in very good agreement with the giso obtained from EPR (giso = 2.19). No decrease of the χMT value was observed in a 4-300K temperature range which reinforces that the oxidized phosphasalen nickel features a d 7 high valent Ni III which persists upon increasing the temperature. Visible spectroscopy was also performed on 2 and [2 + ][SbF6 -] at room temperature in CH2Cl2 and toluene. Complex 2 presents two bands at max = 565 nm (17700 cm -1 ) and max = 665 nm (15040 cm - 1 ) (see Figure S4) that account for MLCT charge transfer, typical in nickel(II) square planar complexes. [4a] When one electron is removed and 2 is oxidized to [2 + ][SbF6 -], the purple colour of 2 changes to black and is indicative of high absorption in the visible range. The visible spectrum indeed possesses high absorption coefficients and two broad bands centered in toluene at max = 490 nm (20400 cm -1 ) and max = 990 nm (10100 cm -1 ) with absorption coefficients over 5000 cm -1 M -1 (Figure 4). In order to better understand this particular spectrum, a Time-Dependant DFT analysis was performed using the optimized geometry of [2 + ] described above. This predicts three principal transitions at 11337 cm -1 (I, fosc = 0.066) and 11661 cm -1 (II, fosc = 0.064) and 22409 cm - 1 (III, fosc = 0.094), with energies that agree to some extent with the experimental transitions (viz. 10100 cm -1 and 20400 cm -1 ) (see Figure S10). All of them have a multiconfigurational character that do not allow sharp conclusions, [19] but it appears clearly from this analysis that I and II are essentially composed of LMCT and explain well broad intense transitions and dark coloured complexes. No other significant transition is predicted at lower energy which reinforces the picture of a localized Ni III center versus a delocalized system. [4a] In summary, a nickel(II) complex incorporating an original tertbutyl substituted phosphasalen ligand was synthesized and characterized. Upon oxidation, this complex led to a high valent nickel(III) [START_REF] Grove | For examples of pentacoordinated Ni III complexes see a)D[END_REF] complex as demonstrated by X-ray, EPR, magnetism, and visible spectroscopy as well as DFT calculations. [2 + ][SbF6 -] is therefore one of the rare tetracoordinated Ni III complexes , [15-16, 18a, 21] and to the best of our knowledge the only one featuring a phenoxide ligand. This observation markedly differs from precedents in the salen chemistry where Ni III complexes were observed at low temperature, in presence of additional ligands or anions. It can be rationalized by the better electron donating ability of iminophosphoranes compared to imines, and by the decreased electron density at the phenol rings which are substituted by an electron-demanding phosphorus atom. [12] Preliminary investigations concerning the reactivity of [2 + ][SbF6 -] evidenced a fast oxidative ability that remains yet difficult to control. Our current efforts are devoted to this, considering a fine tuning of the ligand structure. Further studies will also focus on the oxidation of other phosphasalen complexes differing from 2 by the nature of the metal. , 57.42; H, 5.71; N, 2.48. Found: C, 57.50; H, 5.80; N, 2.37.

Complex E1/2 1 , V E1/2 2 , V E1/2 3 , V 2

Figure 1 .

 1 Figure 1. ORTEP view of the cation [2 + ]. The SbF6 anion, solvent molecules and hydrogen atoms have been omitted for clarity. Thermal

Figure 2 .

 2 Figure 2. Spin density distribution in complex [2 + ]

Figure 3 .

 3 Figure 3. X-band EPR spectra of [2 + ][SbF6 -] at different temperatures. [2 + ][SbF6 -] -(4 mM) in toluene:benzene, frequency 9.3690 GHz, P = 0.1mW at 5K, 60 mW at other temperatures. (b) Experimental spectrum (at 5k) and simulated one.

Figure 4 .

 4 Figure 4. UV spectra of [2 + ][SbF6 -] in dichloromethane (black) and in toluene (grey).

Table 1 .

 1 Electrochemical data for 2 (3.0 mmol L -1 in CH2Cl2;

Entry for the Table of Contents (Please choose one layout)

P=N does not equal C=N!

One electron oxidation of a nickel(II) complex incorporating an original phosphasalen ligand ( t Bu-Psalen) leads to a rare tetracoordinated Ni III complex. The oxidation locus is demonstrated by the use of various spectroscopic tools such as X-ray, EPR, magnetism, and visible spectroscopy as well as with DFT calculations.

ABSTRACT

The oxidation of a nickel(II) complex bearing a tetradentate phosphasalen ligand, which differs from salen by the presence of iminophosphoranes (P=N) in place of imines, was easily achieved by addition of a silver salt. The locus of this oxidation was investigated with a combination of different techniques (NMR, EPR, UV spectroscopy, X-ray crystallography, magnetism measurement) as well as DFT calculations. All data are in agreement with a high valent nickel(III) center concentrating the spin density. This markedly differs from precedents in the salen series for which oxidation on the metal was only observed at low temperature or in presence of additional ligands or anions. Therefore, thanks to the good electron donating properties of the phosphasalen ligand, [Ni(Psalen)] + represents a rare example of a tetracoordinated high valent nickel complex in presence of a phenoxide ligand.


F L Constantin

Sensing electromagnetic fields with the AC-Stark effect in two-photon spectroscopy of cold trapped HD +

Keywords: cold trapped ions, hydrogen molecular ions, electrometry, two-photon spectroscopy, ac-Stark shift, dynamic polarizability, polarization ellipse, reference frames ' L

This contribution evaluates the potential for SI-traceable measurements of electromagnetic fields from precision measurements of two-photon rovibrational transitions of cold trapped HD + ions interpreted with accurate theoretical models. Zeeman spectroscopy of a hyperfine component of the (v,L)=(0,0)→(2,2) two-photon transition is exploited for the measurement of a static magnetic field. The absolute sensitivity and accuracy is estimated at the 10 -10 T level in the case of frequency measurements at the quantum projection noise limit. Measurements of the AC-Stark shifts of different Zeeman components of the (v,L)=(0,0)→(2,0) two-photon transition at different orientations of the magnetic field are exploited to measure the polarisation and the intensity of a THz-wave off-resonantly coupled to HD + rotational levels. The sensitivity is estimated at the 10 -7 W/m 2 level. A reference THz-wave with an intensity of 1 W/m 2 can be calibrated in intensity with a fractional accuracy limited at the 10 -2 level by the accuracy of the theoretical calculations and at the 10 -4 level by the experimental errors. In addition, an approach for retrieval of the full polarisation ellipse is demonstrated with a selected THz-wave. The fractional accuracy of the calibration is better than 5% for the amplitudes and better than 10% for the phases of the electric field components of the THz-wave.

INTRODUCTION

Precise sensing of electromagnetic fields has applications in inertial navigation, definition of frequency standards, geophysical exploration and biological imaging. Particularly, the detection of microwave fields by atomic spectroscopy may provide SI-traceable measurements, sub-wavelength resolution, accuracy, precision and long-term reproducible operation [START_REF] Fan | Atom based RF electric field sensing[END_REF] , comparing to the traditional technology with an antenna and a solid-state rectifier. The Rydberg states are highly sensitive to microwave electric fields. Rabi oscillations in Rydberg atom spectroscopy allowed building compact electric field sensors in vapour cells [START_REF] Sedlacek | Microwave electrometry with Rydberg atoms in a vapour cell using bright atomic resonances[END_REF] . The hydrogen molecular ions are simple quantum systems for which transition frequencies [START_REF] Korobov | Fundamental transitions and ionization energies of the hydrogen molecular ions with few ppt uncertainty[END_REF] and systematic shifts [START_REF] Bakalov | Magnetic field effects in the transitions of the HD + molecular ion and precision spectroscopy[END_REF][START_REF] Schiller | Static and dynamic polarizability and the Stark and blackbody-radiation frequency shifts of the molecular hydrogen ions H 2 + , HD + , and D 2 +[END_REF] were calculated accurately. The experiments exploited co-trapping with laser-cooled Be + ions in Paul traps for sympathetical cooling and spatial confinement for HD + . Spectra of HD + were efficiently detected by dissociation and a mass-spectroscopy technique. A relatively small static magnetic field was used to indicate the quantification axis and to split HD + magnetic levels. The rovibrational lines were measured with 10 -9 precision by Doppler-limited spectroscopy [START_REF] Biesheuvel | Probing QED and fundamental constants through laser spectroscopy of vibrational transitions in HD +[END_REF] . Doppler-free rotational spectroscopy in the Lamb-Dicke regime of cold trapped HD + ions provided 10 -10 -level precision for the (v,L)=(0,0)→(0,1) transition at 1.3 THz 7 . Two-photon infrared spectroscopy, addressed in this contribution, may provide resolution and accuracy at the 10 -12 -level or better [START_REF] Constantin | Double-resonance two-photon spectroscopy of hydrogen molecular ions for improved determination of fundamental constants[END_REF] . This contribution evaluates the performances of Zeeman spectroscopy to measure the magnitude of a static magnetic field. A new approach for THz-wave electrometry is proposed, based on measurements of the AC-Stark shifts induced on Zeeman sub-components of a two-photon rovibrational transition of HD + .

TWO-PHOTON SPECTROSCOPY OF HD +

The hydrogen molecular ions, as the simplest molecules, benefit from accurate models for their energy levels. The main contribution to the energy is the rovibrational energy, calculated neglecting the hyperfine structure and in absence of perturbation from external fields. The ab-initio results are expressed as the sum of the nonrelativistic Scrödinger energy with a series expansion of corrections taking into account the relativistic, radiative and nuclear-size related contributions *fl.constantin@univ-lille1.fr (the most accurate calculations were reported in ref. [START_REF] Korobov | Fundamental transitions and ionization energies of the hydrogen molecular ions with few ppt uncertainty[END_REF] ). The next important contribution comes from the hyperfine energy, which was calculated in the nonrelativistic limit at the order α 2 using a Breit-Pauli spin Hamiltonian [START_REF] Bakalov | High-precision calculation of the hyperfine structure of the HD + ion[END_REF] , electron e S r , to yield the total angular momentum . In presence of a small magnetic field, the hyperfine levels are split into magnetic sublevels labelled with the quantum number J z of the projection of on the magnetic field axis. The total energy of a Zeeman level is expressed as:

J r J r ( ) ( ) ( ) ( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + ≈ = - Δ + + = 2 2 0 ; B z J z J v,L,F,S,J, r z J v,L,F,S,J, q B z J z J v,L,F,S,J, t h B ; z J , J , S , F , L , v E B ; z J , J , S , F , L , v E B z J , J , S , F , L , v z E J , S , F , L , v hf E L , v rv E z J , J , S , F , L , v E (1)
In the previous equation, the Zeeman energy sublevels |n,J z > are approximated with a quadratic dependence depending on parameters t, q, and r that have been calculated in ref. [START_REF] Bakalov | Magnetic field effects in the transitions of the HD + molecular ion and precision spectroscopy[END_REF] (h is the Planck constant). Figure 1 presents a selection of Zeeman energy levels that are addressed by two-photon spectroscopy and some off-resonant couplings that are exploited for THz-wave sensing. The proposed experimental setup is sketched in Figure 2. About 10 2 HD + ions are co-trapped with ~10 3 laser-cooled Be + ions to form a Coulomb crystal in a radiofrequency trap. The detection of the rovibrational transitions is performed by REMPD [START_REF] Roth | Ion-neutral chemical reactions between ultracold localized ions and neutral molecules with single-particle resolution[END_REF] . The fluorescence at 313 nm from the laser-cooled Be + ions is used to monitor the loss of trapped HD + ions when the secular motion of HD + ions is excited in the trap. A static magnetic field, controlled with three orthogonal coil pairs, splits the two-photon rovibrational transitions in σ ± (ΔJ z =±2) and π (ΔJ z =0) components. The orientation of the magnetic field defines the quantisation axis, by assuming that the THz magnetic field is negligible as well as the effects (change in populations, mixing of energy levels) of the coupling of the THz electric field to the HD + energy levels. This contribution investigates the components of the two-photon rovibrational transition (v,L)=(0,0)→(2,0) of HD + . Dopplerfree spectroscopy is performed with two counterpropagating lasers tuned around the rovibrational transition at 55.909 THz. The population transferred to the (v,L)=(2,0) level is dissociated with a 175 nm laser. The interaction with blackbody radiation at room temperature recycles continuously population in the ground vibrational level of HD + . The change of the population in the rovibrational levels, upon application of the spectroscopy and dissociation lasers, was calculated with a set of rate equations in order to derive the lineshape of the two-photon resonance [START_REF] Constantin | Double-resonance two-photon spectroscopy of hydrogen molecular ions for improved determination of fundamental constants[END_REF] . The full-width half measured linewidth is 20 Hz, for a two-photon transition rate of 10 s -1 , a dissociation rate of 200 s -1 , and a REMPD time of 10 s.

-3 -2 -1 0 1 2 3 -2 -1 0 1 2 (2,1,1,2,2,3) (2,1,1,2,2,2) (2,1,1,2,2,1) (2,1,1,2,2,0) (2,1,1,2,2,-1) (2,1,1,2,2,-2) (2,1,1,2,2,-3) (2,0,1,2,2,2) (2,0,1,2,2,1) (2,0,1,2,2,0) (2,0,1,2,2,-1) (2,0,1,2,2,-2) (0,0,1,2,2,2) (0,0,1,2,2,1) (0,0,1,2,2,0) (0,0,1,2,2,-1) (0,0,1,2,2,-2) (0,1,1,2,2,3) (0,1,1,2,2,2) (0,1,1,2,2,1) (0,1,1,2,2,0) (0,1,1,2,2,-1) (0,1,1,2,2,-2) (0,1,1,2,2,-3) (v,L,F,S,J,J z ) σ - THz-
The transitions between spin states with maximal J and maximal magnitude of the projection J z =±J (stretched states) of the levels (v,L)=(0,0) and (v,L)=(2,0) are not splitted by the magnetic field. They can be observed with arbitrary orientation of the magnetic field and are suitable for probing THz electric fields coupled off-resonantly to the rotational transition (v,L)=(0,0)→(0,1) at 1.315 THz or (v,L)=(2,0)→(2,1) at 1.197 THz, respectively. The lightshifts, measured as the change of the two-photon frequency upon application of the THz-wave, are exploited to recover the magnitude of the THz electric field vector and its polarisation state. When the spectroscopy laser is referenced to a frequency standard, this approach allows absolute calibration of the amplitudes and the phases of the THz electric field components, using ab-initio calculations of HD + energy levels, their Zeeman shifts and dipole moments of the rotational transitions. The experimental setup and reference frames. HD + ions embedded with laser cooled Be + ions in a Coulomb crystal are probed by two-photon spectroscopy. The magnetic field is generated with three pairs of coils oriented orthogonally that define the Laboratory Coordinate Frame (x,y,z). The orientation of the magnetic field, defined with the Euler angles (α,β), is along the z c axis of the Cartesian Molecular Ion Coordinate frame (x c ,y c ,z c ). Two couterpropagating waves from the IR laser are used for probing the (v,L)=(0,0)→(2,0) and (v,L)=(0,0)→(2,2) transitions of HD + . The 313 nm laser is used for cooling of Be + ions. The 175 nm laser is used for dissociation of the (v,L)=(2,0) level of HD + . The polarisation ellipse of the THzwave is retrieved from the measurements of the lightshifts of Zeeman components of the (v,L)=(0,0)→(2,0) transition.

The fractional frequency uncertainty is estimated with the expression of the Allan variance for the quantum projection noise limit :

( ) τ π 1 τ σ c ion y T N Q × = ( 2 )
where Q, the quality factor of the two-photon transition, is expressed as Q = f 2ph /Δf HWHM in terms of the half-linewidth, determined ultimately by the natural lifetimes of the HD + ion energy levels [START_REF] Amitay | Rotational and vibrational lifetime of isotopically asymmetrized homonuclear diatomic molecular ions[END_REF] . The cycle time T c is associated to a single measurement with N ion ions at two-photon resonance and successive measurements are averaged during an interrogation time τ. Assuming the case of single ion spectroscopy experiment with T c = τ, the frequency uncertainty for (v,L)=(0,0)→(2,0) line is estimated at 2.49 Hz and for (v,L)=(0,0)→(2,2) line at 2.57 Hz.

The measurement of the Zeeman shift δf of a two photon transition |vLFSJ,J z >=|n 1 ,J z1 >→|n 2 ,J z2 > allows to calculate the magnitude of the magnetic field using the relevant values for the theoretical parameters of the Zeeman effect :

{ } { } 2 2 2 1 1 1 n n n n n n th r , q , t , r , q , t U = ( ) ( ) ( ) ( ) ( ) 1 2 2 1 1 2 2 2 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 2 8 n n z n z n n n z n z n z n z n z n z n th q q J r J r q q J r J r δf J t J t J t J t δf , U B - + - - + - + ⋅ - ⋅ ± ⋅ - ⋅ - = r (3)
When two different values are predicted for the magnitude of the magnetic field, the sign in the previous equation is selected such as the value of the magnetic field derived by Zeeman spectroscopy matches the value measured with an accurate magnetometer. The frequency uncertainty and the uncertainties of the parameters , assumed equal to one half of the last significant digit calculated in ref. [START_REF] Bakalov | Magnetic field effects in the transitions of the HD + molecular ion and precision spectroscopy[END_REF] , contribute to the uncertainty of the magnetic field. The error propagation formula is exploited to calculate the total uncertainty of the magnetic field, by assuming that the theoretical parameters are correlated (correlation coefficients ).

ph exp,2 σ kHz 10 5 4 - × = σ paramZ , th ( ) ( ) ( ) 1,2 i 1 = = = = , r , t corr r , q corr q , t corr ni ni ni ni ni ni
The (v,L)=(0,0)→(2,2) transition, with a transition probability by five times higher than that of the (v,L)=(0,0)→(2,0) transition [START_REF] Karr | Energy levels and two-photon transition probabilities in the HD + ion[END_REF] , has sensitive magnetic subcomponents. The measurement of the magnetic field may be performed by Zeeman spectroscopy of the subcomponent |v,L,F,S,J,J z >=|0,0,1,2,2,-2>→|2,2,1,2,4,0>, having the highest Zeeman shift of 719.655 kHz at B=10 -4 T. The magnitude of the magnetic field is determined with an uncertainty <6.7×10 -10 T. The dependence of the relative uncertainty with the magnitude of the magnetic field is plotted in Figure 3. The contribution of the theoretical uncertainties is approximatively four orders of magnitude smaller.

LIGHTSHIFTS IN TWO-PHOTON SPECTROSCOPY OF HD +

Second-order THz electric field response of HD + energy levels

Consider the interaction of a HD + ion with a classical THz electric field expressed as :

( ) . c . c Ê t i L + × ε = ω - e 2 1 t E r ( 4 )
where is the complex amplitude of the THz-wave, is the complex polarization vector and

ε ˆω L E
is the angular frequency. The electric field vector is expressed in terms of the standard components of the polarization vector:

{ 1 0 1 , , q ˆ-= ε } ( ) { } z y x , , q q q , L q L e , e i e ˆÊ Ê r r r m = ε ± = ε ε - = ε ± - = - ∑ 0 1 1 0 1 2 with 1 ( 5 )
The coupling between the HD + energy levels and the THz electric field is expressed in the electric dipole approximation with the interaction Hamiltonian :

( ) ( ) . c . H d E d t i L + × ε ⋅ = ⋅ - = ω - e 2 1 t E V r r r ( 6 )
where is the electric dipole operator in the laboratory frame. When the THz electric field is far from the resonance with another energy level, the ac-Stark shift of an energy level, evaluated with the second-order perturbation theory [START_REF] Le Kien | Dynamical polarizability of atoms in arbitrary light fields: general theory and application to cesium[END_REF] , reads:

d r ( ) ( ) ( ) ( ) ∑ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ γ + γ + ω + - ε ⋅ ε ⋅ + γ + γ - ω - - ε ⋅ ε ⋅ - = δ + + r n r n r n r n r L n i E E n d r r d n i E E n d r r d n Re E E 2 2 4 1 2 h h r r h h r r (7)
in terms of the matrix elements of the dipole operator, the unperturbed energy levels E r , E n , and their decay rates . The tensor formalism allows to express the matrix elements between Zeeman sublevels of the standard components n r ,γ γ

q q d d ε ⋅ = - r
of the electric dipole operator :

( ) ( )( )( )( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + + + + × δ - = + + + - + + - 0 0 0 1 1 1 1 2 1 2 1 2 1 2 1 1 ' L L L J S ' J ' L ' J q J ' J J ' L L ' J J µ ' J , ' J , ' S , ' F , ' L , ' v d J , J , S , F , L , v z z ' S , S L L J S ' J J ' L ' v , vL z q z z (8)
in function of the reduced matrix elements of the dipole moment that have been calculated in the nonrelativistic approximation in ref. [START_REF] Bakalov | Static Stark effect in the molecular ion HD +[END_REF] .

' L ' v , vL µ

Dynamic polarizabilities of HD + energy levels

Equation ( 7) can be expressed as the expectation value of a Stark-shift operator and further expanded in function of the squared magnitudes of the standard components of the THz electric field :

( ) ω Θ ( ) ( ) { } ( ) ( ) 1 1 1 4 1 4 1 0 1 2 2 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⋅ ω + Ε - Η ⋅ + ⋅ ω - Ε - Η ⋅ = ω α ω α - - = ω Θ = δ - - - = - ∑ n d d n n d d n Re E J , J , S , F , L , v J , J , S , F , L , v E E q * n q q n * q q , n q , n , , q q , L q z z L n h h (9) ( ) ω α q , n
by introducing the standard (labeled as σ -, π, σ + for q=-1,0,1) dynamic polarizabilities of the HD + energy levels , which are defined on the second line of eq. ( 9) using the complex operator

Γ + = Η h i H
, defined such as , , and its eigenvalues .

2 n n n i E γ + = Ε h n ' nn E ' n H n δ = 2 n ' nn ' n n γ δ = Γ
In addition, the expression of the AC-Stark shift can be further developed by introducing the reduced scalar, vector and tensor (K=0,1,2) polarizabilities for HD + ion hyperfine levels :

( ) ( ) { } ( ) ( ) ( ) ∑ ∑ = - - = ω α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - - ε ⊗ ε - - = δ 2 1 0 2 1 1 1 4 , , K K vLFSJ z z J J µ , K * K ... K µ µ K L n J µ J J K J Ê E z (10) { } µ , K * ε ⊗ ε in function of the irreducible tensor product components
. The approach to calculate the reduced polarizabilities of the hyperfine HD + levels is indicated in the Appendix.

Numerical calculation of the polarizabilities and their uncertainties

The standard dynamic polarizabilities of the Zeeman subcomponents of the (v,L)=(0,0) and (v,L)=(2,0) states are calculated with eq. ( 9), by summing over all electric dipole couplings to sublevels of the (v,L)=(0,1) and (v,L)=(2,1) states, respectively. The standard dynamic polarizability :

(11) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) ( ) ( ) ; µ ; 1 ; 0 1 0 1 0 1 0 1 0 0 ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ γ γ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = Δ = Δ = = = = = α = = = THz , v L , v , v L , v v , v z z hf hf rv rv THz q , J , J , S , F , L , v f , , , B L , v E B J , S , F , L , v E L , v E J , S , F , L , v E L , v E L , v E f f z
is a function of a set of theoretical parameters: the rovibrational, hyperfine and Zeeman energies, the dipole moment, the natural linewidths of energy levels (multiple values are regrouped within squared brackets), and two experimental parameters: the magnitude of the magnetic field and the polarization and the frequency of the THz-wave.

In order to avoid divergences and to maintain the approximation of far-detuned THz electric field, the contribution of the resonant coupling was neglected on a small frequency domain centered on each resonance (10 Hz for the polarizabilities of the (v,L)=(0,0) states, 1.1 kHz for the polarizabilities of the (v,L)=(2,0) states).

The uncertainty of the dynamic polarizability is estimated conservatively with the root sum of squares of contributions from:

the uncertainty of the theoretical rovibrational energies [START_REF] Moss | Calculations for vibration-rotation levels of HD + , in particular for high N[END_REF] , estimated with the assumed fractional value of , ( )

12 10 - = σ rv r E
-the uncertainty of the theoretical hyperfine energies, assumed at one half of the last significant digit of the values from ref. [START_REF] Bakalov | High-precision calculation of the hyperfine structure of the HD + ion[END_REF] , that is ( )

Hz 500 = σ hf E ,
-the uncertainty of the theoretical Zeeman energies calculated using eq. ( 1) with parameters given in ref. [START_REF] Bakalov | Magnetic field effects in the transitions of the HD + molecular ion and precision spectroscopy[END_REF] , having accuracies of , ( ) ( ) ( )

Hz 5 0. r q t n n n = σ = σ = σ
-the uncertainty of the magnitude of the magnetic field, calculated with error propagation formula applied to eq.

(3), by assuming an experimental frequency uncertainty of , and correlated theoretical parameters from ref. [START_REF] Bakalov | Magnetic field effects in the transitions of the HD + molecular ion and precision spectroscopy[END_REF] , with uncertainties of , the uncertainty of the theoretical dipole moments, assumed at one half of the last significant digit of the values calculated in ref. [START_REF] Bakalov | Static Stark effect in the molecular ion HD +[END_REF] , that is ,

( ) ( ) a.u. 10 3 1 µ µ 4 21 20 01 00 - × = σ = σ . , ,
-the uncertainty of the theoretical radiative linewidths of the energy levels, estimated at and

( ) ( ) ( ) ( ) ( ) ( ) Hz 49 0 1 2 0 2 . , L , v , L , v = γ σ = γ σ = = ( ) ( ) ( ) Hz 10 5 2 7 1 0 - = × = γ σ . , L , v
using values of energy level lifetimes from ref. [START_REF] Amitay | Rotational and vibrational lifetime of isotopically asymmetrized homonuclear diatomic molecular ions[END_REF] , -the uncertainty of the THz-wave frequency, assumed at 10 -12 in relative value.

First, the uncertainty of the Zeeman shift is calculated as the root sum of squares of the contributions of the uncertainties of the theoretical parameters and of the uncertainty of the magnitude of the magnetic field. Next, the uncertainty of the total energy of a Zeeman sublevel is calculated as the root sum of squares of the uncertainties of the rovibrational energy, the hyperfine energy and the Zeeman shift. The uncertainty of an electric dipole coupling term from the expression of the dynamic polarizability is calculated with the root sum of squares of the contributions from the total energy of each Zeeman sublevel, the dipole moment, the energy level linewidth and THz-wave frequency. Finally, the uncertainty of the dynamic polarizability is conservatively estimated as the root sum of squares of individual contributions of couplings between Zeeman sublevels.

At the limit of a static electric field, the value of the polarizability of a stretched state at zero magnetic field is ( )

a.u. 001 392 0 1 2 2 2 1 0 0 . , , , , , , = = ω α
, that agrees with the value calculated in ref. [START_REF] Schiller | Static and dynamic polarizability and the Stark and blackbody-radiation frequency shifts of the molecular hydrogen ions H 2 + , HD + , and D 2 +[END_REF] . The fractional uncertainty is 2.9×10 -4 .

Figure 4.(a) displays the standard dynamic polarizabilities of the stretched state |0,0,1,2,2,2> at a magnetic field B=10 -4 T. The dependence for σ + dynamic polarizability has a resonance corresponding to the coupling to the |0,1,1,2,3,3> level. The σ -dynamic polarizability has two resonances in this frequency range, corresponding to the couplings to the |0,1,1,2,3,1> and 0,1,1,2,2,1> levels, respectively. The π dynamic polarizability has two resonances corresponding to couplings to the |0,1,1,2,3,2> and 0,1,1,2,2,2> levels, respectively. The uncertainties are plotted in Figure 4(b). The increase of the magnitude of the dynamic polarizability, when the THz-wave is tuned near a resonance, is associated with an increase of the uncertainty. |v,L,F,S,J,J z >=|0,0,1,2,2,2> Each dipole-allowed transition between the Zeeman components of the (v,0) and (v,1) states constitutes a sensor for a THz electric field with a frequency tuned near the resonance frequency. The positions of the resonances can be adjusted by changing the magnitude of the magnetic field. Figure 5 indicates the dependence of the resonance frequencies on the magnitude of the magnetic field up to 5 Gauss for a set of hyperfine transitions detuned by 10 MHz to the hyperfine-free rotational frequency. Each component has a dependence with a specific slope. The highest slope has an absolute value of about 7 GHz/T. Transitions between Zeeman levels of (v,L)=(0,0)->(0,1) coupled by the THz-wave polarized:

σ + π σ - Figure 5
. Magnetic field dependence of the resonance frequencies for selected Zeeman components of the rotational transition |v,L>=|0,0>→|0,1>. The frequency detuning is calculated to 1 314 935 827.3 kHz (the hyperfine-free rotational frequency).

Probing THz electric fields with lightshifts: accuracy and sensitivity issues

Let's consider now the detection of a THz-wave with σ + polarization tuned around 1.3 THz from the measurement of the lightshift induced on the frequency of the two-photon transition between the stretches states. At a given value of the magnetic field, the lightshift can be translated to the THz-wave intensity by multiplication with the appropriate factor depending on the THz wave frequency and polarization. The dependence of the lightshift on the detuning of the THzwave is plotted in Figure 6.. For the given parameters of the THz-wave, the lightshift is higher than the uncertainty of the measurement of the two-photon transition for a detuning of a few MHz around the resonances with the relevant rotational transitions, that indicates the bandwidth of the THz-wave detection.

In order to evaluate the performances of precision and sensitivity in THz-wave detection by lightshift measurements, one should asses the uncertainties of the determination of THz-wave intensity. The frequency measurement errors translate to uncertainties of the lightshift and of the magnetic field magnitude. The uncertainties of the theoretical parameters contribute to the uncertainty of the dynamic polarizability. The overall contribution to the uncertainty of the THz-wave intensity is estimated with the error propagation formula:

( ) ( ) { } ( ) [ ] ( ) { } ( ) [ ] 2 2 2 2 2 2 THz th THz th THz F THz f , B , q , U B B f , B , q , U I f c I α Δ σ × ∂ α Δ ∂ + δ σ × = σ ( 1 2 )
in function of the lightshift measurement uncertainty and the uncertainty of the magnetic field magnitude . The prefactor ( )

f δ σ ( ) B σ ( ) 3 0 2 a c c F π - = h
is a combination of fundamental constants (the reduced Planck constant , the speed of light in vacuum , the Bohr radius ). The ab-initio calculated differential dynamical polarizability depends on a set of theoretical constants, the magnetic field, the polarization and the angular frequency of the THz-wave. Therefore, one can make a choice for different parameters, for example the Zeeman subcomponents of the two-photon transition, the h c α Δ 0 a magnitude of the magnetic field in order to minimize the uncertainty at particular intensity, polarization and frequency of the THz-wave.

2> at the quantum projection noise limit. THzwave frequency 1 314 947 502.3 kHz and polarizations σ -, π and σ -+ . (b) Dependence of the fractional uncertainty for THzwave calibration (calculation with combined experimental and theory errors in black squares, calculation for experimental errors only in red circles) with measurements of the lightshifts of the |v,L,F,S,J,J z >=|0,0,1,2,2,2>→|2,0,1,2,2,2> two-photon transition. The magnetic field value is optimized to minimize total uncertainty. THz-wave frequency 1 314 947 502.3 kHz and polarization σ -.

Figure 7.(a) displays the dependence of the sensitivity in function on the magnitude of the magnetic field, for the given THz-wave polarizations and frequency. The lowest THz-wave intensity is estimated with eq. ( 12) by taking into account solely the contribution coming from the frequency uncertainty . The limit is at 1.7×10 -7 W/m 2 , which translates to a sensitivity to THz electric fields of 1.1×10 -5 V/m.

( )

f δ σ
Next, the uncertainty in the calibration of a THz-wave using lightshift measurements is estimated for an optimized value of the magnetic field in the range 0-10 -4 T, that minimizes the total uncertainty estimated with eq. ( 12). Figure 7.(b) displays the dependence of the fractional uncertainty of the THz-wave intensity. For THz-wave intensities higher than 0.1 W/m 2 the optimized value for the magnetic field is negative and therefore fixed at 0. The best fractional uncertainty is at the 10 -3 level. The fractional uncertainty is limited by the accuracy of the dynamic polarizability calculation at the 10 -2 level, for a THz-wave intensity of 1 W/m 2 . The contribution to the fractional uncertainty arising solely from the frequency measurement errors (this dependence is also plotted in Fig. 7.(b)) is more than an order of magnitude smaller than the total fractional uncertainty. The experimental fractional uncertainty is at the 10 -4 level for a THz-wave intensity of 1 W/m 2 .

VECTOR TERAHERTZ ELECTROMETRY

Coordinate frames for THz-wave sensing

The THz electric field vector is decomposed in three orthogonal linearly polarized components along the three axis of a fixed Cartesian laboratory coordinate frame ( )

z y x e , e , e LCF r r r : ( ) ( ) { } . c . c e E z , y , x j t i j j j + × = ∑ = ϕ + ω - e 2 t E r r ( 1 3 )
The THz electric field is defined with real positive amplitudes and phases . One can fix, for simplicity, one phase to zero:

. In this case, the THz electric field vector traces a polarization ellipse (PE) characterized with five parameters (three amplitudes and two relative phases). The magnetic field is generated with three coil pairs in Helmholtz configuration which are oriented orthogonally along e r , such as the direction of is along the direction of the magnetic field that defines the quantization axis. The THz electric field vector is expressed in the using the standard components:

MICF ( ) ( ) ( ) { } . c . c e E , , q t i q q q q + × - = ∑ - = ϕ + ω - - 1 0 1 e 2 1 t E r r ( 1 4 )
with real positive amplitudes and phases having linear or circular polarization defined by

1 0 ± E , E 1 0 ± ϕ ϕ , ( ) 2

E

The relations between electric field amplitudes in different coordinate system, can be calculated with the rotation matrix formalism, and read [START_REF] Koepsell | Measuring the polarisation of electromagnetic fields using Rabi-rate measurements with spatial Resolution : Experiment and theory[END_REF] :

( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) ( ) ( ) (
E E ϕ β α ± ϕ β α ϕ - ϕ β ± ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + + = ϕ - ϕ β α + ϕ α β + ϕ α β + β α + α + β = β α π β α σ β α π ± 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 m ( 1 5 )

Measuring the polarization ellipse of the THz-wave

In the previous equations, the field amplitudes and phases in the laboratory frame ( )

y x z y x , , E , E , E ϕ ϕ
are related to three standard components of the electric field in the molecular ion frame . For a given orientation of the magnetic field, three independent measurements of lightshifts allow to derive all standard components of the THz electric field, by solving a system of three equations with the appropriate differential standard dynamic polarizabilities. These measurements have to be independent, that is, the determinant of the system should be nonzero. That can be done by using different magnetic field magnitudes or probing on different two-photon transitions (for example, the Zeeman σ ± and π components of a hyperfine transition or three different hyperfine transitions). In order to determine all parameters of the THz electric field in the laboratory frame, it is necessary also to change the orientation of the magnetic field. The choice of orientations of the magnetic field is linked to the periodicity of the trigonometric functions in eq. ( 15) and measurements for minimum two different orientations are required. Therefore, in order to determine five parameters of in the laboratory frame, five independent measurements are necessary for the standard components at two different orientations of the magnetic field. For example, one can exploit the measurement of the set of amplitudes

( ) ( ) β α σ β α π ± , , E , E ( ) { 3 2 1 , , k , f , k = δ β α } ( ) t E r ( ) ( ) β α σ β α π ± , , E , E ( ) β α, ( ) ( ) ( ) ( ) ( ) ( ) 2 0 2 0 0 0 0 0 0 0 π σ π σ σ σ π - + - + , , , , , E , E , E , E , E
to derive analytically the parameters of the THz electric field in the laboratory frame [START_REF] Koepsell | Measuring the polarisation of electromagnetic fields using Rabi-rate measurements with spatial Resolution : Experiment and theory[END_REF] .

The experimental uncertainties of the THz electric field measurement arise from lightshift frequency measurements and from the uncertainties in setting the magnitude and the orientation of the magnetic field. Here, stray magnetic fields are neglected as well as nonorthogonality in relative orientation of the coil pairs. The magnitude of the magnetic field generated along , respectively, using separately each pair of coils can be calibrated with Zeeman spectroscopy, as discussed previously. In addition, errors in the calibration of the THz electric field arise from the uncertainties of the standard dynamic polarizabilities. . A set of six theoretical lighshifts are calculated for the two-photon transition between the stretched states |v,L,F,S,J,J z >=|0,0,1,2,2,2>→|2,0,1,2,2,2>, for two orientations of the magnetic field ( ) ( ) ( ) ( )

2 0 0 0 π = β α = β α , , , , ,
, and three different values of the magnitude of the magnetic field B 1 = 10 -6 T, B 2 = 5 × 10 -6 T, B 3 = 10 -5 T. The experimental lightshifts are simulated by adding to the theoretical lightshifts random values with a Gaussian distribution with zero mean and covariance equal to the squared uncertainty of the frequency measurements at the quantum projection noise limit ( ) ( ) ( )

2 2 0 ph exp, , theor , n , exp , n , f f σ ℵ + δ = δ β α β α
. The standard components of the THz electric field are subsequently derived using the differential standard dynamic polarizabilities, estimated as the theoretical values plus a random contribution with a Gaussian distribution with zero mean and variance equal to the squared uncertainty of theoretical calculation

( ) ( ) ( ) ( ) 2 0 B , f THz theor , q , n THz q , n THz theor , q , n , B , f B , f α Δ σ ℵ + α Δ = α Δ
. From the calculated values of the standard components, all parameters of the THz electric field in the laboratory frame are determined analytically. The polarization ellipse of the THz electric field is represented on Figure 8 for 10 simulated sets of lightshifts.

The values and the uncertainties of the parameters of the THz-wave determined from 100 simulations of lightshifts are:

( The accuracy of the calibration is better than 5% for the components of the electric field and better than 10% for the phases. The shift between the initial assumed value and the determined value for each parameter of the THz-wave is consistent with the calculated uncertainty. Precision measurements of lightshifts for different orientations of the magnetic field allow full characterization the THz electric field in the laboratory frame and, therefore, represent feasible alternatives to the Rabi-rate measurements with Rydberg atoms [START_REF] Koepsell | Measuring the polarisation of electromagnetic fields using Rabi-rate measurements with spatial Resolution : Experiment and theory[END_REF][START_REF] Thiele | Self-calibrating vector atomic magnetometry through microwave polarization reconstruction[END_REF] .

( ) ( ) ( ) ( ) ( ) 1 6 )

CONCLUSION

This contribution demonstrates that the measurements of the frequency shifts induced by external fields on narrow twophoton transitions of cold trapped HD + ions are feasible approaches for measuring electromagnetic fields. The Zeeman shift for a selected hyperfine component of a two-photon rovibrational transition is exploited for determination of the magnitude of a static magnetic field with an uncertainty at the 10 -10 T level. The magnitude of the electric field of a THzwave is determined by the lightshift that is induced on a selected Zeeeman sub-component of a two-photon transition. A THz-wave with an intensity of 1 W/m 2 can be calibrated with an accuracy limited at the 10 -2 level by the accuracy of the theoretical calculations and at the 10 -4 level by the experimental errors. In addition, a self-consistent numerical approach for retrieval of the full polarisation ellipse is demonstrated for a selected THz-wave. The calibration accuracy is better than 5% for the electric field components and better than 10% for the phases..

APPENDIX

The reduced polarizability is expressed as : are the reduced matrix elements of the dipole operator. In the case where the detunings to the intermediary energy levels are larger than the hyperfine structure splittings, the reduced polarizability can be further expanded:

( ) ( ) ( ) ( ) ( ) ( ) ∑ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ γ + γ - ω - - - + γ + γ + ω + - × ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + - = ω α + ' J ' n ' J ' n nJ ' J ' n nJ K ' J ' n nJ ' J ' n nJ J K K nJ i E E i E E Re nJ d ' J ' n ' J ' n d nJ J ' J J K K 2 1 2 1 1 1 1 2 1 2 h h h h ( 1 7
( ) ( ) ( ) ( ) ( ) ( ) ω α ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + - = ω α + + + K nL K J S L K nJ L S L J K J J 1 2 1 ( 1 8 )
with the reduced polarizability for a rovibrational level :

( ) ( ) ( ) ( ) ( ) ( ) ∑ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ γ + γ + ω + - - + γ + γ - ω - - × ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + - = ω α + + ' L ' v ' J ' n nJ nJ ' J ' n K ' J ' n nJ nJ ' J ' n L K K vL i E E i E E Re nL d ' L ' n ' L ' n d nL L ' L L K K 2 1 2 1 1 1 1 2 1 1 2 h h h h (19)
and the reduced matrix element of the dipole operator :

( ) ( )( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + - μ = 0 0 0 1 1 2 1 2 1 ' L L ' L L ' L ' n d nL

Figure 1 .

 1 Figure 1. Zeeman components of the hyperfine energy levels of HD + ions addressed in two-photon spectroscopy and relevant couplings for THz-wave sensing.

Figure 2 .

 2 Figure2. The experimental setup and reference frames. HD + ions embedded with laser cooled Be + ions in a Coulomb crystal are probed by two-photon spectroscopy. The magnetic field is generated with three pairs of coils oriented orthogonally that define the Laboratory Coordinate Frame (x,y,z). The orientation of the magnetic field, defined with the Euler angles (α,β), is along the z c axis of the Cartesian Molecular Ion Coordinate frame (x c ,y c ,z c ). Two couterpropagating waves from the IR laser are used for probing the (v,L)=(0,0)→(2,0) and (v,L)=(0,0)→(2,2) transitions of HD + . The 313 nm laser is used for cooling of Be + ions. The 175 nm laser is used for dissociation of the (v,L)=(2,0) level of HD + . The polarisation ellipse of the THzwave is retrieved from the measurements of the lightshifts of Zeeman components of the (v,L)=(0,0)→(2,0) transition.

Figure 3 .

 3 Figure 3. Relative uncertainty of the magnitude of the magnetic field determined by Zeeman spectroscopy of the |v,L,F,S,J,J z >=|0,0,1,2,2,-2>→|2,2,1,2,4,0> transition of HD + . The frequency uncertainty is estimated with the quantum projection noise-limited instability for integration time 1 s, cycle time 1 s.

Figure 4 .

 4 Figure 4. Dependence of the value and the uncertainty of the standard dynamic polarizability of the stretched state |v,L,F,S,J,J z >=|0,0,1,2,2,2> on the frequency detuning to 1 314 945 902.3 kHz (corresponding to the hyperfine resonance |v,L,F,S,J>=|0,0,1,2,2>→|0,1,1,2,3>). Magnetic field 10 -4 T.

Figure 6 .Figure 7 .

 67 Figure 6. Dependence of the ac-Stark shift of the two-photon transition |v,L,F,S,J,J z >=|0,0,1,2,2,2>→|2,0,1,2,2,2> on the frequency detuning to 1 314 945 902.3 kHz (resonance |v,L,F,S,J>=|0,0,1,2,2>→|0,1,1,2,3>), and 1 196 702 588.7 kHz (resonance |v,L,F,S,J>=|2,0,1,2,2>→|2,1,1,2,3>), respectively. Magnetic field 10 -4 T. THz-wave intensity 0.3 W/m 2 , frequency 1 314 947 502 kHz, polarization σ + .

  by varying the currents in the coils. The relative orientation of the two coordinate frames is defined with the Euler angles ( . Each standard component of the THz electric field, denoted with , couples off-resonantly to the π or σ ± Zeeman subcomponents of (v,L)=(v,0)→(v,1) transitions and induces a lightshift in proportion with its squared amplitude.

  Let's consider detection of a THz electric field oscillating at frequency f THz = 1 314 947 502.3 kHz with the following parameters in the laboratory frame ( )

Figure 8 .

 8 Figure 8. Determination of the polarization ellipse of the electric field of a THz-wave at 1 314 945 902.3 kHz from frequency measurements of the lightshifts of the two-photon transition |v,L,F,S,J,J z >=|0,0,1,2,2,2>→|2,0,1,2,2,2>.

  )

		,	z	cos
				E	x	cos	E	y	sin	sin
				E	z	E	x	sin	cos	cos	x
				E	z	E	y	sin	sin	cos	y
				E	y	E	x	sin	sin	cos	x	y
	E	,		E	x				E	y	E	z	E	,
				E	x	E		y	cos	sin	x	y
				E	x	E	z	sin	sin	sin	x
				E	y	E	z	cos	sin	sin	y