S Bernon

T Vanderbruggen

R Kohlhaas

A Bertoldi

A Landragin

P Bouyer

Heterodyne non-demolition measurements on cold atomic samples: towards the preparation of non-classical states for atom interferometry

Keywords: 07.05.Fb, 37.10.Gh, 42.50.Dv, 07.60.Ly, 32.60.+i

We report on a novel experiment to generate non-classical atomic states via quantum non-demolition (QND) measurements on cold atomic samples prepared in a high finesse ring cavity. The heterodyne technique developed for the QND detection exhibits an optical shot-noise limited behavior for local oscillator optical power of a few hundred µW, and a detection bandwidth of several GHz. This detection tool is used in single pass to follow non destructively the internal state evolution of an atomic sample when subjected to Rabi oscillations or a spin-echo interferometric sequence.

Introduction

Quantum metrology [START_REF] Lee | A quantum Rosetta stone for interferometry[END_REF] is the field of quantum information that studies the ultimate amount of information obtainable when measuring a given observable of a system. Quantum state engineering techniques are thus developed to maximize the sensitivity of measurement devices.

In quantum optics, after the first observation of squeezed states [START_REF] Slusher | Observation of squeezed states generated by four-wave mixing in an optical cavity[END_REF], squeezing enhancing techniques have been strongly investigated, leading to very highly squeezed states [START_REF] Vahlbruch | Observation of squeezed light with 10-dB quantum-noise reduction[END_REF] and NOON states [START_REF] Afek | High-NOON states by mixing quantum and classical light[END_REF]. The benchtest to study the practical application of quantum technology is gravitational wave detection: the instruments used nowadays are reaching classical limit due to the optical radiation pressure on the mirrors. Hence, further sensitivity improvements require quantum engineering techniques. Recently, squeezing enhancement has been proven in a gravitational wave detector prototype [START_REF] Goda | A quantum-enhanced prototype gravitationalwave detector[END_REF].

State-of-the-art atomic detectors such as clocks [START_REF] Santarelli | Quantum projection noise in an atomic fountain: a high stability cesium frequency standard[END_REF], inertial sensors [START_REF] Gauguet | Characterization and limits of a cold-atom Sagnac interferometer[END_REF], and magnetometers [START_REF] Wasilewski | Quantum noise limited and entanglement-assisted magnetometry[END_REF] have also reached the standard quantum limit given by the atomic shot-noise. To go further, the sensitivity can be enhanced either increasing the number of atoms, or reducing the effect of quantum noise. In this context, atomic spin-squeezed states [START_REF] Kitagawa | Squeezed spin states[END_REF] have been recently achieved [START_REF] Schleier-Smith | States of an ensemble of two-level atoms with reduced quantum uncertainty[END_REF][START_REF] Louchet-Chauvet | Entanglement-assisted atomic clock beyond the projection noise limit[END_REF][START_REF] Esteve | Squeezing and entanglement in a bose-einstein condensate[END_REF][START_REF] Gross | Nonlinear atom interferometer surpasses classical precision limit[END_REF][START_REF] Riedel | Atom-chip-based generation of entanglement for quantum metrology[END_REF], and they allowed sensitivity enhancement in atomic clocks [START_REF] Louchet-Chauvet | Entanglement-assisted atomic clock beyond the projection noise limit[END_REF][START_REF] Wineland | Spin squeezing and reduced quantum noise in spectroscopy[END_REF].

In this paper we present an apparatus designed to generate cold atomic samples in an optical cavity and to perform heterodyne non-demolition measurement in the perspective of spin-squeezed states preparation. At first an overview of the experimental apparatus and a characterization of the high-finesse optical cavity are presented. The second part is dedicated to the heterodyne nondestructive measurement of an atomic population. After a presentation of the experimental set-up, we demonstrate the non-demolition measurement of Rabi oscillations and characterize the influence of the probe on the oscillations. We finally show the real-time measurement of the internal atomic state evolution in a Ramsey interferometer.

The experimental apparatus

The apparatus is composed of two chambers: in the first a 2D MOT is operated, and used as a source of cold atoms; the second one is the science chamber where the 3D MOT and the experiments are performed in the center of a crossed optical cavity (figure 1).

2D and 3D MOTs

The 2D MOT vacuum chamber is a titanium machined piece with glue sealed windows, which are antireflection coated at 780 nm on both sides. The science chamber is made of non magnetic stainless steel (AISI type 316LN) and titanium; its viewports are antireflection coated at 780 nm. To operate the resonator enhanced optical trap, the viewports on the horizontal plane are additionally antireflection coated at 1560 nm.

The pressure in the 2D MOT chamber is below 7×10 -8 mbar, whereas in the science chamber it is below 10 -9 mbar. The differential vacuum operation relies on a reduced conductivity between the two chambers.

The 2D MOT produces a bright source of pre-cooled atoms used to load the 3D MOT. The rubidium background pressure is obtained from the vapor pressure of a 1 g rubidium sample.

The loading rate of the 3D MOT by the 2D MOT is about 5 × 10 8 at.s -1 at room temperature. The MOT lifetime is ∼ 20 s. The center of the 3D MOT is placed 3 mm above the 2D MOT jet to avoid direct collisions with thermal atoms, and is centered on the cavity crossing region where the atoms can be optically trapped in the resonator. The 3D MOT coils are mounted in vacuum and are mechanically tightened using ceramic supports to the titanium plate holding the The MOT beams are generated by external cavity diode lasers [START_REF] Baillard | Interference-filter-stabilized external-cavity diode lasers[END_REF] amplified to 1 W by tapered amplifiers. Fiber coupled splitters (Schäfter+Kirchhoff and OzOptics) are adopted to obtain the independent MOT beams in a compact and reliable design.

The repump light frequency is fixed on the |F = 1 → |F = 2 atomic transition while the cooling and probing light are frequency locked to the repump laser. They can be continuously frequency shifted over a range of 400 MHz.

The high finesse cavity

The optical cavity, that is resonant for both 1560 nm and 780 nm, is of dual interest in this experiment. In the first instance, it allows to produce a deep optical potential to trap atoms using a low power laser source by exploiting the build-up effect of the resonator at 1560 nm. On the other hand, it enhances the light-atom coupling thus improving the non-demolition measurement efficiency realized at 780 nm.

Mechanical design

The cavity, shown in figure 2, is made up of four mirrors placed at the corners of a square with a diagonal of 90 mm. All the mechanical components of the cavity are realized in titanium to maintain a low magnetic environment for the trapped atoms. The size of the cavity results from a compromise to maximize the optical access while minimizing the cavity mode to ensure a tight trapping. The square geometry enables a 90°crossing angle which avoids an interference pattern in the crossing region once the polarization is set to be in the cavity plane. The mirrors are all identical and are plane-concave with a radius of curvature of R = 100 mm. They have a diameter of 1/2 inch. All the parts are tightly mounted on a 15 mm thick titanium plate, which gives its mechanical stability to the set-up. Two mirror mounts are completely fixed. One mount is actuated under vacuum by piezoelectric actuators (Newfocus Picomotors) which allow for a coarse alignment of the cavity. The last mount is a piezo-actuated three-axis nano-positioning system (Madcitylabs M3Z) with maximal angular displacement of 2 mrad and translation of 50 µm. It is used to finely adjust the cavity crossing angle and dynamically control the cavity length.

Spectral properties

The four mirrors of the cavity are highly reflecting at both 1560 nm and 780 nm. The free spectral range (FSR) of the cavity is 976.2 MHz and the cavity FWHM linewidth is γ = 546 kHz at 1560 nm. The FSR is measured by simultaneously injecting the carrier and a sideband created by an electro-optic modulator (EOM) in two adjacent longitudinal modes. γ is obtained from the Lorentzian profile of the transmission when the sideband is scanned across the resonance. The finesse is then F 1560 = FSR/γ = 1788. The corresponding mirror amplitude reflectivity obtained from the formula F = πr 2 /(1 -r 4 ), is r = 99.956% at 1560 nm. At 780 nm, the reflectivity measured by the manufacturer is r = 99.99923%, i.e. an expected finesse F 780 = 102000.

Geometrical characterization

To estimate the mode parameters, the resonator has been analyzed with the ABCD matrix formalism [START_REF] Kogelnik | Laser beams and resonators[END_REF]. In more detail, the cavity is converted into an equivalent sequence of lenses and self consistency is postulated. In the adopted geometry, an astigmatism arises from the nonzero angle of incidence θ on the curved mirrors. The effective radii of curvature are then R = R cos θ in the plane defined by the cavity (horizontal plane) and R ⊥ = R/ cos θ in the orthogonal plane. The red dashed lines represent equation (1) for the two orthogonal directions when only the off axis incidence on the mirrors is taken into account. For the horizontal direction, the black dashed-dotted line includes also the corrections on the effective radius of curvature. In inset, the cavity transmission signal when the cavity length is scanned for a horizontal (black, TEM m0 ) and a vertical (blue, TEM 0n ) misalignment of the injection beam.

We measured the geometrical properties of the cavity by analyzing the transverse mode spectrum at 1560 nm. Since the cavity is astigmatic, the transverse modes have Hermite-Gauss profiles. The horizontal mode inter-spacing is 78.9 MHz and the vertical one is 164.6 MHz. The mode eigenfrequencies are related to the Gouy phase [START_REF] Gouy | Sur une propriété nouvelle des ondes lumineuses[END_REF] acquired by the photons while traversing the beam waist and are thus linked to the cavity geometry. The absolute mode frequencies are given by [START_REF] Siegman | Lasers[END_REF]:

ν p,m,n = c 2πL 2pπ + m + 1 2 Arg A m + B m q m + n + 1 2 Arg A n + B n q n , (1)
where p is the longitudinal mode order, m and n are the horizontal (TEM m0 ) and vertical (TEM 0n ) transverse mode numbers, A m and B m factors refer to the ABCD matrix element in the m direction, and q m is the complex radius of curvature of the beam. The parameters A, B and q contain the geometrical information of the cavity (length, angle and radius of curvature of the mirrors). Figure 3 compares the experimental measurement of the transverse mode inter-spacing and its a priori calculation in red dashed line. The discrepancy of the horizontal splitting is corrected by a phenomenological factor α that, for example, accounts for aberrations. This factor is applied on the horizontal effective radius of curvature R = α R cos θ, and is adjusted on the slope of the data giving α = 1.020(5) (black curve in figure 3). In the adopted quasi-concentric configuration, the parallel direction is the closest one to the instability regime. A 2% deviation of α induces a noticeable frequency shift as it can be seen in figure 3. Including this correction for the astigmatism, we infer a horizontal waist ω = 93.1 µm to be compared with 98(1) µm found using a tomographic measurement [START_REF] Brantut | Light-shift tomography in an optical-dipole trap for neutral atoms[END_REF][START_REF] Bertoldi | In situ characterization of an optical cavity using atomic light shift[END_REF] and reported in section 2.2.5. The vertical waist is calculated to be ω ⊥ = 129.8 µm. The corresponding Rayleigh ranges are z r = 17.46 mm and z ⊥ r = 33.9 mm.

Frequency lock of the 1560 nm laser to the cavity

The optical cavity is pumped with the radiation produced by a single longitudinal mode distributed feedback erbium doped fiber laser (DF-EDFL) near 1560 nm (Koheras laser from NKT Photonics), amplified with a 5 W erbium doped fibered amplifier (EDFA, from Keopsys). The typical laser linewidth is of the order of a few kHz, its output power is 100 mW single mode and is linearly polarized. The radiation is coupled to the resonator through a beam expander and a tilted doublet, whose angle allows us to mode-match the astigmatism. The laser is locked to a transversal cavity mode using the Pound-Drever-Hall technique [START_REF] Drever | Laser phase and frequency stabilization using an optical resonator[END_REF]. A low frequency feedback is applied on the piezoelectric element controlling the laser cavity length. The fast frequency correction is realized with an acousto-optic modulator in double-pass configuration with a bandwidth of 250 kHz. More details about the implemented lock scheme and its noise performance can be found in [START_REF] Bertoldi | In situ characterization of an optical cavity using atomic light shift[END_REF].

Light-shift tomography of the cavity optical potential

The atoms released from the MOT, operated in the crossing region of the two cavity arms, are used to characterize in-situ the resonator geometry. The light at 1560 nm that pumps the cavity is close to the rubidium transitions 5P 3/2 -4D 3/2,5/2 at 1529 nm. This causes a strong differential light shift on the two levels of the D 2 transition. More precisely, at 1560 nm the scalar polarizability of the 5P 3/2 level is 47.7 times that of the 5S 1/2 level. As a consequence, by changing the detuning of the probe beam with regard to the D 2 transition it is possible to bring into resonance atoms placed at different positions in the optical potential [START_REF] Brantut | Light-shift tomography in an optical-dipole trap for neutral atoms[END_REF]. This method provides the isopotential lines of the crossed dipole trap, as shown in figure 4, where the 1560 nm laser is locked to the fundamental transversal mode (TEM 00 ). In each shot, the probe detuned by δ is absorbed by the atoms with potential energy U (r) = -0 cRe(α)I(r)/2 = hδ/(47.7 -1) in the ground level; for this atomic class the probe is shifted into resonance because of the local differential light shift. The optical density was projected on an axis parallel to one cavity arm (lower row in figure 4), and the profile was fitted to obtain the distance between the couples of isopotential lines. Plotting the position of these isolines versus the probe detuning provides the cavity mode profile, as shown in figure 5. A fit with the first Hermite-Gauss mode gives a waist ω = 98(1) µm. In [START_REF] Bertoldi | In situ characterization of an optical cavity using atomic light shift[END_REF] a similar analysis was performed also for the TEM 10 and TEM 20 modes, obtaining consistent results.

Scattering and losses

The differential light shift presented in section 2.2.5 is proportional to the intracavity intensity and to the atomic polarizability (α = 6.83 10 -39 Jm 2 V -2 [START_REF] Arora | Magic wavelengths for the np-ns transitions in alkali-metal atoms[END_REF][START_REF] Safronova | Frequency-dependent polarizabilities of alkalimetal atoms from ultraviolet through infrared spectral regions[END_REF]). In a four mirror geometry, the intracavity power P intra is related to the power at one output P out by P intra = 2(1 + x)FP out /π, where x = κ 2 L /t 2 represents the loss of the mirrors κ L in units of the mirror transmission t. Combining the measurement of the differential light shift caused by the intracavity power with the output power in one arm, we infer scattering losses on the mirrors that are 1.5 times the mirror transmission.

Considering these extra-losses, the maximum coupling efficiency achievable is 36%. The maximum coupling obtained experimentally is 35(2)%. The backward scattering on the cavity mirrors is measured to be 1.9 in intensity compared to the forward direction. This light, that propagates in the reverse direction, interferes with the forward propagating light and causes a 4.38% modulation depth of the optical potential. The standing wave formed by the retroreflection is phase related to the mirror positions and is then sensitive to thermal drift and acoustic noise affecting the mirrors.

Trapping in a folded cavity

The optical resonator is used to enhance the 1560 nm input intensity to obtain a high intracavity power suitable to realize a far off resonance optical dipole trap for neutral atoms. More precisely, with an incident power of 5 W an intracavity power of 200 W is reached. Moreover, at the cavity crossing, the potential presents high trapping frequencies in all spatial directions. At full power, a trap depth of U 0 = 1.4 mK is obtained and the trapping frequencies are ω x /2π = ω y /2π = 1.2 kHz and ω z /2π = 1.6 kHz.

The dipole trap is loaded from a MOT in a similar way as described in [START_REF] Clément | All-optical runaway evaporation to Bose-Einstein condensation[END_REF]. For the loading, the depth of the FORT starts at ∼ 100 µK. A compressed MOT phase is operated by detuning the MOT beams to 5 Γ from the atomic resonance for 50 ms. The cloud is compressed and the atomic density increases in the FORT region. The cooling beams are then further detuned to 40 Γ for 100 ms. The high detuning has been chosen to red-detune the light with respect to the atomic transition even in the presence of the strong light-shift. Moreover, in the FORT region the repump light is out of resonance and the atoms are accumulated in the F=1 hyperfine state, realizing in this way a dark MOT induced by the atomic light-shift. Finally, the cooling radiation is turned off and the power in the FORT is ramped up to full power in 10 ms.

At the end of the loading sequence about 20 million atoms are loaded at the crossing region of the dipole trap. After thermalization, the sample temperature is 230 µK, leading to a temperature to trap depth ratio k B T /U 0 ∼ 6. The trapping time is limited to 6.6 s by one body losses.

Heterodyne non-demolition measurement 4.1 Collective measurement

An ideal quantum non-demolition measurement can be seen as the textbook quantum projective measurement realized on an operator that commutes with the hamiltonian. In the case of multi-particles measurement, the result of the measurement collectively constrains the sample and not individual atoms. Correlations between atoms have been created in what is called the measurement back-action. The back-action collapses the wavefunction and can be used to prepare spin-squeezed states [START_REF] Vanderbruggen | Spin-squeezing and Dicke-state preparation by heterodyne measurement[END_REF].

In the following, we consider an ensemble of N at two levels atoms {|a i , |b i }. We introduce the collective-spin operators:

J x = 1 2 i (|b i a i | + |a i b i |) , (2)
J y = i 2 i (|a i b i | -|b i a i |) , (3)
J z = 1 2 i (|b i b i | -|a i a i |) , (4)
which obey the commutation relations J i , J j = i ijk J k . The observable of the measurement is

J z = ∆ N /2
, which is one half of the population difference between the two internal atomic states. For a coherent spin state (CSS) polarized along J x , one has ( J x , J y , J z ) = (N at /2, 0, 0), and (∆ J 2 x , ∆ J 2 y , ∆ J 2 z ) = (0, N at , N at ) /4. A non-demolition measurement based on optical techniques requires to reach a high signal-tonoise detection together with a low light level to limit the decoherence induced by spontaneous emission. Spontaneous emission can be reduced using off-resonance probing, which is a dispersive measurement [START_REF] Kuzmich | Atomic quantum non-demolition measurements and squeezing[END_REF]. In that case, the off-resonance modification of the refractive index is measured by the phase shift induced on the probe optical field. Several techniques can be implemented to measure this phase shift, including Mach-Zehnder interferometry [START_REF] Louchet-Chauvet | Entanglement-assisted atomic clock beyond the projection noise limit[END_REF], mapping phase fluctuations into intensity fluctuations using a cavity tuned on the side of its resonance [START_REF] Schleier-Smith | States of an ensemble of two-level atoms with reduced quantum uncertainty[END_REF], or comparing the probe dephasing to a far from resonance local oscillator [START_REF] Teper | Backaction noise produced via cavityaided nondemolition measurement of an atomic clock state[END_REF][START_REF] Vanderbruggen | Spin-squeezing and Dicke-state preparation by heterodyne measurement[END_REF]. The latter technique, called heterodyne detection, is adopted in the following.

The detection scheme considered is based on frequency modulation spectroscopy [START_REF] Drever | Laser phase and frequency stabilization using an optical resonator[END_REF][START_REF] Black | An introduction to Pound Drever Hall laser frequency stabilization[END_REF][START_REF] Bjorklund | Frequency Modulation Spectroscopy[END_REF]. A laser beam is phase modulated to produce frequency sidebands; one sideband is placed close to an atomic transition and experiences a phase shift φ at passing through the atomic sample. The detection of the beat note at the modulation frequency then allows us to estimate the atomic population of the probed state.

This scheme highly benefits from the almost noninteracting carrier used as a local oscillator. For a laser, limited by its quantum fluctuations (section 4.4), the signal-to-noise ratio (SNR) of the detection is

SNR ∝ √ N s N c ( √ N s ) 2 + ( √ N c ) 2 + ( √ N e ) 2 sin φ at , (5)
where N s represents the total number of photon detected in the sideband and N c the photon number in the carrier. The contribution of technical noises in units of photon number is N e . We underline that the two first terms of the denominator in equation ( 5) are the quantum phase noise of the laser expressed in terms of photon number. For a high power in the local oscillator, the shot noise overcomes the detection noise N c N e , and equation ( 5) simplifies to SNR ≈ √ N s sin φ at , which corresponds to a sideband shot noise limited detection, independently of the sideband power. The probe sensitivity to the fluctuations of the optical path is considered in the next section.

A low sensitivity to common mode noise

In homodyne detections, such as Mach-Zehnder interferometry, the sensitivity to path length fluctuations is determined by the optical wavelength λ. In a heterodyne detection scheme for which frequency components are spatially overlapped, the relevant length is the modulation wavelength λ mod = Ω/(2πc). For a modulation in the microwave range, λ mod is of the order of a few centimeters, whereas λ is of the order of a micrometer. As a consequence, the sensitivity to length fluctuations is greatly reduced.

In the following, we focus on the sensitivity of a phase modulated detection with respect to optical path length fluctuations, and consider the effect of residual amplitude modulation (RAM) [START_REF] Gehrtz | Quantum-limited laser frequencymodulation spectroscopy[END_REF]. An optical field E, phase modulated at Ω, can be written as:

E = E 0 ∞ n=-∞ J n (β) cos (ω c t + nΩt) , (6)
where J n are Bessel functions of the first kind and β is the modulation depth. For a small modulation depth (β 1) the expansion can be limited to the first orders (J 0 (β), J 1 (β), J -1 (β)) ≈ (1, β, -β). Let φ 0 , φ 1 , and φ -1 be the different atomic phase shift experienced by each frequency component of the optical field. The photocurrent i det at the output of the photodetector is

i det = ηP opt 1 + β cos Ωt - ΩL c ∆Φ + + β sin Ωt - ΩL c ∆Φ -, (7)
where L is the distance between the modulator and the detector, P opt the optical power in the local oscillator, η the detection sensitivity of the photodiode, and

∆Φ + = cos (φ 1 -φ 0 ) -cos (φ 0 -φ -1 ) , (8)
∆Φ -= sin (φ 1 -φ 0 ) -sin (φ 0 -φ -1 ) . (9)
After the demodulation of i det by sin(Ωt + Φ dem ) with Φ dem = ΩL/c, and for length noise fluctuations δL of the optical path integrated in the detection bandwidth ∆f , the dispersive signal obtained is S = S 0 + δS L , where:

S 0 = ηβP opt ∆Φ -, (10)
δS L = 2πηβP opt δL λ mod ∆Φ + . ( 11
)
δS L is the component of the noise that is due to fluctuations of the optical path length. Since the atomic contributions to the phase shift are small φ -1 , φ 0 , φ 1 1, we have at second order:

∆Φ + = 1 2 (φ 1 -φ -1 ) (2φ 0 -φ 1 -φ -1 ) , (12)
∆Φ -= φ 1 + φ -1 -2φ 0 . ( 13
)
Considering the case depicted in figure 7, we have φ 1 = φ at 1 and φ -1 = φ 0 = 0. From equations ( 11), ( 12) and ( 13), the detection noise δS L is ηβP opt φ 2 at δL/λ mod . Hence, for a pure phase modulation, a small atomic phase shift φ at makes the signal even less sensitive to length fluctuations than δL/λ mod .

A major noise source in this heterodyne scheme arises from the residual amplitude modulation due to phase modulation imperfections. The residual amplitude modulation unbalances the field amplitude in the sidebands and the optical electric field can be written as E = E 0 e iωct 1 + β(1 + )e iΩt -β(1 -)e -iΩt . The signal after demodulation is:

S = ηβP opt (∆Φ -+ ∆Φ -,AM ) + 2π δL λ mod (∆Φ + + ∆Φ +,AM ) , (14)
where ∆Φ +,AM and ∆Φ -,AM are the atomic phase contributions for a pure amplitude modulation:

∆Φ +,AM = 2 + 1 2 (φ 1 -φ 0 ) 2 + (φ -1 -φ 0 ) 2 , (15)
∆Φ -,AM = φ 1 -φ -1 . (16)
In the case of figure 7, the overall noise contribution of optical path length fluctuations is:

δS L = ηβP opt (φ 2 at + ) δL λ mod . ( 17
)
As a consequence, in a phase modulated detection, the length fluctuations are rejected by a factor (φ 2 at + )λ/λ mod with respect to a Mach-Zehnder interferometer (equation ( 17)). For our experimental parameters, = 10 -2 , λ mod =10 cm, λ = 1 µm, and for φ at < 100 mrad, it represents 7 orders of magnitude of rejection on δS L . In addition, δS L /δS shot scales as P opt . As a result, if the photodetector is shot noise limited for a low optical power in the local oscillator (section 4.4), there is no need of a dynamical control of the optical path length.

Optical set-up

The optical set-up is presented in figure 8. The detection beam is generated by an extended cavity diode laser, frequency locked to the reference laser [START_REF] Appel | Versatile digital GHz phase lock for external cavity diode lasers[END_REF]. The detuning of the carrier, and thus of the sideband from the atomic resonance, is electronically controlled by a Phase Locked Loop (PLL). An acousto-optic modulator (AOM) is used as a switch for the detection beam and generates pulses with 300 ns rise time. A polarization maintaining fibered electro-optic modulator (EOM) produces frequency sidebands that probe the atomic sample. The beating signal of the carrier with the sidebands is detected on a fast photodiode (see section. 4.4), and the demodulated signal is acquired with a digital oscilloscope. The internal atomic state is manipulated with a microwave signal at about 6.834 GHz generated by a home-made frequency chain. This chain relies on a low noise quartz oscillator (Wenzel Associates, mod. Blue Top Ultra Low Noise Oscillator) which has -113dBc/Hz phase noise at 1 Hz. After a suitable amplification, the microwave is shined on the atoms with an antenna placed at about 20 cm from the sample.

The optical detector

The beat-note signal is detected on a fast GaAs PIN photodiode with integrated transimpedance amplifier designed for high-speed optical communications at 850 nm (Finisar, model. HFD3180-203). The electronic scheme for the photodiode and the amplifier is presented in figure 9. The integration of the photodiode and amplifier in the same package greatly reduces parasitic capacitances and allows a high detection bandwidth of a few GHz together with a high transimpedance gain of the order of one kΩ. The detector is mounted on a home-made high frequency printed circuit board. The output of the amplifier is connected to a 50 Ω matched coplanar waveguide and AC-coupled to a SMA connector.

The noise at the output of the photodiode has been measured with a spectrum analyzer. The output of the transimpedance amplifier was amplified by 36 dB to overcome the noise level of the spectrum analyzer. The noise power spectral density of the illuminated detector was measured at 1 GHz and 2 GHz, as reported in figure 10.

The measurement shows a very good linearity of the noise power spectral density versus the incident power, which means that both the laser and the detector are shot-noise limited at the adopted frequencies. The figure of merit κ considered is the electronic noise equivalent light shot-noise, that is the optical power required to generate the same noise as the detection electronics [START_REF] Windpassinger | Ultra-low noise differential AC-coupled photodetector for sensitive pulse detection applications[END_REF]. We measured κ = 165 µW at 1 GHz, and κ = 469 µW at 2 GHz. Moreover, the slope provides a direct measurement of the photodiode gain G PD = V out /P opt = gR F η where η ∼ 0.5 A/W is the sensitivity of the photodiode (see figure 9 for notations). At the output of the transimpedance amplifier, voltage fluctuations induced by the light shot noise are v n = gR F 2eηP opt ∆f , where e is the charge of the electron and ∆f the detection bandwidth. Hence the noise power spectral density on a resistance load R L of 50 Ω is:

PSD n = v 2 n R L ∆f = 2e G 2 PD R L η P opt , (18)
From the slopes of figure 10, one gets G PD = 1466 V/W at 1 GHz and G PD = 880 V/W at 2 GHz, values in good agreement with the nominal value of 1250 V/W.

5 Non-demolition measurement of an evolving atomic state.

We now switch to the application of our heterodyne scheme to the measurement of a cold atomic cloud in free fall during the measurement. The spontaneous emission rate induced by the probe is first measured. Then, a real-time measurement of atoms undergoing Rabi oscillations is realized. Finally, the detection tool is used to follow continuously the evolution of an atomic state during an interferometric sequence.

Effective scattering rate

In experiments that concern measurement induced squeezing, the probe induced spontaneous emission is among the limiting factors to reach highly entangled states [START_REF] De Echaniz | Conditions for spin squeezing in a cold 87 Rb ensemble[END_REF]. To quantify the destructivity induced by the probe in our set-up, the atoms prepared in the |F = 1 hyperfine state are released from the dipole trap and continuously probed. The cloud contains 4.7(3) million atoms in an initial rms size of σ at = 41 µm, and expands with a temperature of 55(5) µK. The optical frequency detunings with respect to the atomic transitions are presented in figure 11(a). The modulation frequency is fixed to 2.808 GHz. The detuning (δ) of the probing sideband with respect to the atomic transition is changed by moving the absolute frequency of the carrier. As a consequence, the quadrature set by the demodulation phase is independent of the detuning δ.

Spontaneous emission induced by the probe optically pumps the atoms to |F = 2 where they become transparent for the probe. The measured exponential decay time τ of the signal (inset of figure 11(b)) expressed as a decay rate γ = 1/(πτ ) is then closely linked to the spontaneous emission rate of the probe.

The predicted decay rate γ, shown in figure 11(b) as a gray zone, is calculated from the beam waist (245 µm), the carrier power (P c = 120 µW), and the power in each sideband (P s = 76 nW) of the linearly polarized probe, and is expressed as:

γ = γ s + γ c + γ exp = b 2 ΓI s /(2I sat,2 ) 1 + 4 δ Γ 2 + I s /I sat,2 + 2 i=0 b i ΓI c /(2I sat,i ) 1 + 4 ∆ i Γ 2 + I c /I sat,i + γ exp , (19)
where I sat,i is the saturation intensity for the π transition from |F = 1 to |F = i , and the Zeeman sub-levels are considered equally populated. We have

I

Rabi oscillations

Microwave driven Rabi oscillation is a well understood physical phenomenon and a typical demonstration of coherent manipulation. It is therefore very convenient to precisely characterize how a non-destructive probe affects the oscillating system [START_REF] Windpassinger | Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements[END_REF][START_REF] Chaudhury | Continuous nondemolition measurement of the Cs clock transition pseudospin[END_REF][START_REF] Smith | Continuous Weak Measurement and Nonlinear Dynamics in a Cold Spin Ensemble[END_REF].

The atomic system is released from the MOT and optically pumped in the hyperfine ground state |F = 1, m F = 0 . At t = 0, a microwave driven Rabi oscillation is induced between |F = 1, m F = 0 and |F = 2, m F = 0 levels. A magnetic bias field of 0.5 G is applied along the vertical axis so that other Zeeman transitions are off-resonance. The cloud contains ∼ 10 7 atoms and expands with a temperature of 80 µK. The probe used in this experiment has a waist of w 0 = 800 µm and delivers pulses of 1.25 µs with a repetition rate of 100 kHz or 50 kHz. The carrier power is 70 µW and the sideband power is 90 nW. As described in figure 12(a), one sideband is blue detuned by δ from the |F = 2 → |F = 3 transition whereas the carrier is 2.5 GHz away from the resonance. Hence, the configuration is probing the population in the |F = 2 state.

Figure 12 shows two examples of real time measurements of the atomic evolution obtained in a single shot. Figure 12(b) was taken with a probe detuning of 7.9 Γ and shows a trace of Rabi oscillation undergoing decoherence. Figure 12(c), which was taken with a detuning of 0.8 Γ, presents an up-lift of the average oscillation. This effect is qualitatively explained by taking into account the probe induced spontaneous emission which transfers atoms from |F = 2, m F = 0 to |F = 2, m F = 0 . There, the atoms still contribute to the detected signal while being offresonance for the microwave excitation.

Figure 13(a) presents the difference of Rabi oscillation frequency δΩ R for two different repetition rates of 100 kHz and 50 kHz as a function of the detuning (δΩ R = Ω R,100kHz -Ω R,50kHz ,). The Rabi frequency is obtained by fitting the first 800 µs of the oscillation where the expansion has little effect. It clearly shows that the probe light modifies the Rabi frequency around its unperturbed value of (Ω R ∼ 6.6 kHz). For the experimental parameters given above, the carrier induces a light shift ∆E c /h ∼ 2 kHz for a 100 kHz repetition rate whereas the sideband lightshift is below 380 Hz for the smallest detuning considered. The difference in Rabi frequency calculated from the carrier light shift is δΩ R = 227 Hz, in reasonable agreement with the 197 Hz average of figure 13(a). In addition, from the fitting procedure, we extract that a SNR=1 is obtained for a destructivity of 2.6 × 10 -6 scattering event per atoms. The exponential damping of the Rabi oscillations is described by its damping rate β which is the sum of three independent terms:

β = β spont + β shift + β else . (20)
β spont represents the damping of the oscillation by the spontaneous emission process of both the carrier and the sideband, β shift results from the light shift inhomogeneity induced by the Gaussian light profile [START_REF] Windpassinger | Inhomogeneous light shift effects on atomic quantum state evolution in non-destructive measurements[END_REF], which is essentially limited to the carrier effect, and β else represents the probe independent decoherence effects such as the microwave inhomogeneity and the expansion of the cloud expansion. The behavior of β versus the probe detuning is presented in figure 13(b). The spatial inhomogeneity of the carrier light-shift contributes to damp the oscillation in a characteristic time that is proportional to the inverse of the Rabi frequency dispersion

∆Ω R = Ω 2 R -Ω R 2
, where the averages are taken over the cloud profile. It results in a damping rate β shift = α∆E 2 c / 2h 2 Ω R where α characterizes the inhomogeneity over the cloud size, and depends on the atomic and probe sizes. The theoretical curves plotted in figure 13(b) are given for β else = 90 Hz and α = 0.162.

The sum of the three damping coefficients reproduces well the observed behavior. Close to resonance, the damping of the oscillation is dominated by spontaneous emission events. Tuning the probe further from resonance greatly reduces the effect of spontaneous emission from the sideband, and the damping becomes limited by the inhomogeneous light shift induced by the carrier.

Interferometric phase in a spin-echo sequence

We present here the non-demolition measurement of the atomic state during an interferometric sequence. We are particularly interested in the evolution of the atomic pseudo-spin in the equatorial plane of the Bloch representation, since it is directly associated to the evolution of the interferometric phase.

The atoms are initially prepared in the |F = 1, m F = 0 state and probed in the configuration of figure 12(a). The interferometer is a π 2 -π -π 2 spin-echo sequence. The whole sequence was run in less than 500 µs to avoid loss of signal due to the atomic cloud expansion and free fall. The π pulse has a duration of 74.5 µs. The intermediate π pulse maps the interferometric phase on the measured observable J z + N /2, which is the projection of the atomic spin vector on the z-axis in the Bloch representation of figure 14(c). To modify the interferometric phase, the frequency of the microwave driving field is scanned with respect to the ground state hyperfine transition.

Figure 14(a) shows the traces of the atomic state interferometer acquired in real time for different microwave detunings. The result is well understood in terms of rotations of the pseudospin vector on the Bloch sphere (figure 14(b)-14(c)). The amplitudes of the signal oscillation during the intermediate π pulse were extracted from the traces and compare well with the predicted ones (figure 14(d)). The characterization shows that the superposition state of an interferometer can be measured without necessarily blurring the fringes. This indicates that the measurement was realized in the weak measurement regime, where the atomic state is not strongly projected. Moreover, in a usual interferometric sequence, the atoms are lost after the destructive measurement. With such a non-demolition measurement at the end of the sequence, the atoms can be reused so as to increase the duty cycle of atomic interferometers [START_REF] Lodewyck | Nondestructive measurement of the transition probability in a Sr optical lattice clock[END_REF].

Cavity-enhanced measurements

As shown in [START_REF] Vanderbruggen | Spin-squeezing and Dicke-state preparation by heterodyne measurement[END_REF][START_REF] Oblak | Quantum-noise-limited interferometric measurement of atomic noise: Towards spin squeezing on the cs clock transition[END_REF] the optimal level of squeezing ξ 2 = 1/(1 + κ 2 ) is related to the signal to noise ratio κ of the measurement:

κ 2 = φ 2 N at N s 2 ∝ ρ 0 η, ( 21
)
where φ is the single atom induced phase shift, ρ 0 is the resonant optical density, and η is the single atom spontaneous emission probability for a pulse of N s photons. The coherence loss due to spontaneous emission cannot be avoided since η enters the equation of κ. Nevertheless, using a cavity to strengthen the atom-light interaction increases the coupling φ by the cavity finesse F. This results in a SNR enhancement proportional to √ F [START_REF] Lye | Nondestructive dynamic detectors for Bose-Einstein condensates[END_REF] for a given spontaneous emission rate (η ∝ N s ).

To prevent for the light-shift effect of the carrier, the measurement can be realized in reflection [START_REF] Teper | Backaction noise produced via cavityaided nondemolition measurement of an atomic clock state[END_REF]. In the scheme presented in figure 15, to measure the population difference, the detection realized with the two sidebands is shot noise limited because of the strong local oscillator. In addition, the effective interaction between atoms and photons can take advantage of the trapping and probing that exploits the same cavity mode.

Conclusions

We have presented an apparatus designed for the production of spin-squeezed cold atomic samples in a dual-frequency high-finesse cavity. We have shown that, thanks to the intra-cavity power enhancement, the chosen configuration is suitable for all-optical trapping of atoms using standard optical telecommunication technologies.

We described a new non-demolition heterodyne detection scheme. This scheme is limited by the optical shot noise of the probe and presents a high rejection of optical path length fluctuations. We demonstrated the method by performing nondestructive measurements of the coherent evolution of internal atomic states. From the nondestructive measurement of Rabi oscillations, we have shown that the decoherence induced by the probe can be limited by the inhomogeneous light shift from the carrier and not by the spontaneous emission induced by the sideband. We also presented a real-time measurement of the atomic state evolution in a Ramsey interferometer with low decoherence.

Our experiment is particularly suited for metrology with a high squeezing level of large particles numbers. Because of the power build-up at 1560 nm, we will achieve large degenerate samples with low intensity and heterodyne detection can be enhanced by using the cavity at 780 nm. Moreover, using the cavity enhanced atom-light coupling combined with the intrinsic atomic and optical mode overlap should allow us to reach the ultimate sensitivity of the heterodyne measurement device while avoiding inhomogeneous light-shift from the carrier.

Figure 1 :

 1 Figure 1: Overview of the vacuum system showing the compact 2D MOT on the right and the science chamber with the high finesse cavity on the left.

Figure 2 :

 2 Figure 2: The folded optical cavity. (a) Schematic of the cavity geometry and (b) the inner vacuum set-up with the suspended cavity and the coils generating the MOT magnetic field.

Figure 3 :

 3 Figure 3: Frequency splitting of the transverse mode relatively to the closest fundamental mode. The red dashed lines represent equation (1) for the two orthogonal directions when only the off axis incidence on the mirrors is taken into account. For the horizontal direction, the black dashed-dotted line includes also the corrections on the effective radius of curvature. In inset, the cavity transmission signal when the cavity length is scanned for a horizontal (black, TEM m0 ) and a vertical (blue, TEM 0n ) misalignment of the injection beam.

Figure 4 :

 4 Figure 4: (Top) Tomographic images of the optical potential setting the probe frequency to different detuning with respect to the D 2 line. (Bottom) Integral optical density obtained by projecting the upper images on the 45°dashed line crossing one arm of the cavity.

Figure 5 :

 5 Figure5: Optical potential depth when the 1560 nm laser is locked to the fundamental transversal mode of the optical resonator. The measurement was obtained by projecting the signal relative to one arm of the cavity. The result of the fit with the first Hermite-Gauss mode is shown with a solid line.

5 Figure 6 :

 56 Figure 6: Number of atoms in the trap versus trapping time. The solid line is an exponential fit for one body losses. The inset shows an image of the trapped atomic sample after 500 ms.

Figure 7 :

 7 Figure7: Schematic view of a population sensitive heterodyne measurement. The circles are instantaneous phase representations of the optical frequency components in the frame rotating at the carrier frequency. The positive sideband (+, in red) starts in phase with carrier and turns with angular frequency Ω while the negative sideband (-, in blue) starts in opposite phase and turns with angular frequency -Ω. The positive sideband is close to the atomic transition ω 0 , it accumulates an additional phase φ at that is extracted after demodulation.

Figure 8 :Figure 9 :

 89 Figure8: Set-up of the single-pass non-destructive measurement. The probe light is generated by an extended cavity diode laser and offset locked to the atomic transition via a phase locked loop (PLL) and a proportional integrator feedback loop. The absolute frequency of the light is controlled by changing the reference frequency of the PLL. The atoms are probed with the optical pulses generated by the AOM. The sidebands for the heterodyne detection are inserted by a phase modulator.

Figure 10 :

 10 Figure 10: Noise power spectral density versus the incident optical power for two different detection frequencies, 1 GHz (circles) and 2 GHz (triangles). The inset presents the set-up used for the noise measurement.

Figure 11 :

 11 Figure 11: Destructivity as a function of probe detuning. (a) Frequency position of the optical triplet with respect to the D 2 atomic line. (b) Probe induced scattering rate. The gray shaded area, with red dashed borders represents the theoretical decay rate. The only parameter is the expansion rate of the cloud (γ expansion ). In inset is shown the demodulated atomic signal at δ = 4.81Γ, with the relative exponential fit.

  sat,0 = 16.67 W/m 2 , I sat,1 = 26.7 W/m 2 , and I sat,2 = 61.23 W/m 2 . The branching probabilities b i to spontaneously scatter from |F = i to |F = 2 are b 0 = 0, b 1 = 1/5, and b 2 = 1/2. I s = 2P s /(πw 2 ) is the sideband intensity while I c is carrier intensity, ∆ i = 2.808 + ∆ 2i GHz is the carrier detuning from the transition and ∆ 2i is the angular frequency difference between |F = 2 and |F = i . γ exp = 120 Hz is a constant to take into account the signal loss due to the combination of fall and expansion of the cloud; it is the only adjusted parameter in figure 11(b). A simulation of the loss of signal with the expansion gives a decay rate of 159 Hz, showing good qualitative agreement with the experimental value. The gray zone in figure 11(b) represents the expected region for the decay rate versus the probe detuning when uncertainties in the experimental parameters are considered. It shows a good agreement with the experimental data. Although the probe is very weak compared to the carrier, it dominates the spontaneous emission rate for δ < 5Γ in this specific experimental realization (figure 11(b)).

Figure 12 :

 12 Figure 12: Non-demolition measurement of the atomic population in |F = 2 when Rabi oscillations are driven in the atomic sample. The probing sequence is realized with a 1.25 µs long pulse repeated every 10 µs. Each graph is realized in a single experimental cycle. The detuning of the probe to the transition is set to 7.9Γ (a) and 0.8Γ (b). The red curve is a fit to the data and the dashed black line helps as a guide to the eyes.

Figure 13 :

 13 Figure 13: Dependence of the Rabi oscillation parameters from the probe detuning. (a) Difference of the Rabi frequency for data sets having repetition rates of 100 kHz and 50 kHz. (b) Damping rate of the oscillations. In red open diamond are presented data with a repetition rate of 100 kHz, whereas blue squares represent data at 50 kHz. The solid lines are the expected decay rates taking into account spontaneous emission, inhomogeneous light-shift, and expansion. The dashed lines are the decay rates without the spontaneous emission contribution.

Figure 14 :

 14 Figure 14: Spin-echo sequence. (a) Non-demolition measurement of the number of atoms in |F = 2 during the interferometric sequence. Each curve is the average of three repetitions. The measurement uses 1.25 µs long detection pulses with a repetition rate of 100 kHz. The different curves are obtained by scanning the microwave frequency. The microwave detuning from the atomic transition is 0 Hz (red curve), 1 kHz (blue curve), 1.2 kHz (green curve) and 1.8 kHz (black curve). The colors in (b) and (c) refer to the same microwave detuning. (b) Simulation of the expected signal during the evolution of the atomic sample. (c) Trajectories of the atomic state on the Bloch sphere. (d) Amplitude of the oscillation of the signal during the intermediate π pulse. The red dots are experimental data and the black dashed line is the expected amplitude.

Figure 15 :

 15 Figure 15: Scheme of the Pound-Drever-Hall like measurement method. The carrier is locked at the center of two cavity resonances to suppress cavity noise. Each sideband probes one of the hyperfine states.

Acknowledgment

We thank David Holleville, Thierry Botter and Remi Geiger for their participation in the early stage of the experiment. This work was supported by IFRAF, DGA, European Union (with STREP program FINAQS), and ESF (EUROQUASAR program). A. B. acknowledges support from EU under an IEF Grant. S.B. acknowledges the Fulbright foundation for support.


L Lajaunie
email: luc.lajaunie@cnrs-imn.fr

M L David

J F Barbot

Physical properties of Co/n-Ge Schottky contacts

Keywords: numbers: 73, 30, +y, 73, 40, Ns

To investigate the role of the interface state on the physical properties of Schottky contacts, Co/n-Ge Schottky diodes that have undergone various cleaning methods (HF etching and in-situ thermal cleaning) were studied by Transmission Electron Microscopy (TEM), Deep-level Transient Spectroscopy (DLTS) and by a detailed analysis of the temperature dependence of the diodes characteristics. It is shown that Schottky barrier height characteristics are sensitive to the nature of the interface. The strongest Fermi level pinning and the highest spatial inhomogeneities are observed for intimate metal/semiconductor contacts. The presence of a thin oxide interlayer, even of Ge native oxide, allows the Fermi level to be released towards the conduction band and leads to more homogeneous contacts. Finally our results suggest that a pure GeO 2 oxide interlayer should present a better depinning efficiency than the native Ge oxide.

Introduction

Owing to its high charge carrier mobility and its good compatibility with high-k materials, [1,2] germanium is a potential candidate to replace silicon as channel material in sub-22 nm Complementary Metal Oxide Semiconductor (CMOS) technologies. However Ge n-MOSFETs (Metal Oxide Semiconductor Field Effect Transistors) present lesser performance than silicon n-MOSFET mostly because of the fast diffusion of ntype dopants in germanium which prevents the formation of shallow junctions. [3] To overcome such issue Schottky Source/Drain MOSFETs are promising alternatives. [4] However whereas such devices require low Schottky barriers (SBHs) to compete with conventional MOSFETs, [5] most of the direct contacts metal/Ge yield to high SBHs due to a strong Fermi level-pinning effect. [6] SBHs larger than the germanium band gap have even been reported suggesting the formation of an inversion layer at the interface metal/Ge [7] consecutively of the small germanium band gap and of the charge neutrality level lying close to the valence band. [8] Promising studies have shown that by introducing a thin oxide layer at the metal/Ge interface, the Fermi level (FL) can be depinned, [4,9] i.e the FL of the metal is released toward the conduction band of germanium, thus yielding a lower SBH. For instance, by inserting a thin layer (2nm) of Al 2 O 3 at the metal/semiconductor (MS) interface, SBHs of Co/n-Ge diodes has been found to decrease from 0.62 to 0.35 eV. [9] Moreover it has been shown that the depinning efficiency of the oxide is not only dependent on the nature of the metal, [9] but also on the thickness and on the chemical nature of the oxide [4]. The underlying mechanisms leading to the FL pinning are still subject to debate. According to the metal induced gap states (MIGS) theory, the FL pinning is due to the wave function of the metal penetrating into the semiconductor and inducing additional states in the band gap. [6] However the role of the dangling bonds at the MS interface has been pointed out as well. [10] In the case of germanium, the different experimental observations tend to strengthened one or the other theory. [4,[11][12][13] However, in most of the case, possible SBHs non-homogeneities which can alter the electrical properties of real MS contacts [14][15][16][17] are not taken into account.

Surface preparation of the substrate prior to metal deposition could be a possible source of SBH inhomogeneities. In the studies concerning the characterization of SBHs on germanium, HF etching is the most often used cleaning method. [9,18] As the roughness of the germanium surface increases by a factor two with this cleaning method [19] it could enhance the spatial inhomogeneities at MS interfaces. Moreover the metal impurities at the germanium surface are not removed with such a cleaning in contrary to what is observed after an HCl cleaning for instance. [20] This may also influence the SBH characterization. Another way to clean the germanium surface is to realize the thermal decomposition of the native oxide layer GeO X as its desorption occurs at 430 • C under high vacuum. [21] An annealing in UHV at 360 • C during 15 minutes has even been reported to be sufficient to obtain an oxide-free germanium surface [22]. Besides the modifications that can be induced by different surface cleaning prior to MS contact formation, implantation induced defects created during the metal deposition also modify the properties of MS contacts. [23,24] In this work, electrical and microstructural characterizations of Co/n-Ge contacts have been conducted to understand the influence of the interface state on the electrical properties of MS contacts. In particular a detailed analysis of the temperature dependence of the diodes characteristics has been carried out to take into account the inhomogeneities of the SBHs. This paper is divided into six parts. In the first part, the theoretical background related to SBH inhomogeneities analysis is recalled. In the second part, the experimental details are given. In parts three to five, the microstructural and electrical characterization of the samples are presented. Finally the results are discussed in the framework of the Fermi level-pinning concept.

Background

Under the assumption of thermionic current being the major forward current distribution, the diode current of a homogeneous MS interface can be determined as: [START_REF] Rhoderick | Metal-semiconductor contacts[END_REF]

I = I S (e q(V -IR S ) nkT -1) (1)
where V is the applied voltage, q the elementary charge, n the ideality factor, k the Boltzmann's constant, T the temperature and R S the series resistance. The saturation current, I S , is given by:

I S = AA * T 2 e -qφ IV B kT (2)
where A is the Schottky contact area, A * the Richardson's constant and φ IV B the SBH derived from the I-V characteristics. The image force lowering has been neglected due to the low doping of the germanium. The SBH can also be estimated from C-V measurements using the following relation:

qφ CV B = q(V bi + kT q + kT q ln N C N D ) (3)
where N C is the effective density of states in the conduction band, V bi the built-in potential and N D the free carrier concentration.

Most of the interpretations of electrical data from real Schottky contacts implicitly assume the uniformity of the SBH at the MS interface. However, several discrepancies with the thermionic emission theory such as differences between φ IV B and φ CV B , variation of the ideality factor with the temperature as well as the non-linearity of the Richardson's plot are often noted in the literature. [15,16,[START_REF] Kim | [END_REF][27][28] Fluctuations of the built-in potential ascribed to spatial inhomogeneities along the MS interface alter the electrical behavior of real Schottky contacts. These fluctuations are not taken into account in the thermionic (TE) model but are well reproduced by the model of Werner and Güttler [14] (the WG's model in the following) by assuming a Gaussian distribution of the SBH of mean value φ B and standard deviation σ. The authors have shown that the SBH determined from C-V measurements is not dependent on the standard deviation, i.e. that φ CV B (T ) = φ B (T ). Eqs (1) and (2) giving the flow of current in the TE model are similar than in the WG's model, but with an apparent SBH given by:

φ IV B (T ) = φ B (T ) - qσ 2 (T ) 2kT (4)
Both, φ B and σ 2 are temperature dependent according to: [14] φ B (T ) = φ 0 + α φ T (5)

σ 2 (T ) = σ 2 0 + α σ T (6)
where φ 0 and σ 0 are the mean barrier height and the standard deviation extrapolated at 0 K; α φ and α σ are their temperature coefficients, respectively. φ 0 and α φ are determined by plotting φ CV B as a function of the temperature. The standard deviation at 0 K and its temperature coefficient are obtained by combining equations ( 4) and ( 6):

φ CV B (T ) -φ IV B (T ) = qσ 2 0 2kT + qα σ 2k (7)
The conventional Richardson's plot is then modified as follows to take into account the SBH inhomogeneities:

ln I S T 2 - q 2 σ 2 (T ) 2k 2 T 2 = ln (AA * e -qα φ k ) - qφ 0 kT (8)
The Richardson's constant, A * , and the mean SBHs at 0 K, φ 0 , are extracted from the modified Richardson plot, Equation (8). In the following we have labeled this last value φ RICH 0 for the distinction with the value deduced from the C-V-T measurements (Φ 0 (T ), Equation ( 5)).

In the WG's model, the temperature dependence of the ideality factor is of importance as it could modify the extracted parameters of the Gaussian distribution. This dependence is related to the deformation of the Gaussian distribution under the applied bias: [14] 1

n -1 = -ρ 2 + ρ 3 2kT (9)
where ρ 2 and ρ 3 quantify the voltage deformation of the mean barrier height and the standard deviation, respectively. As n reflects the deformation of the Gaussian distribution under the applied bias, the SBH extracted from I-V characteristics is already corrected from the voltage deformations. However, any voltage dependencies of the mean barrier height will impair the comparison between φ CV B (T ) and φ IV B (T ) which is needed to deduce the standard deviation of the Schottky contacts. From this point it is assumed in the WG's model that the mean SBH shows under reverse bias the same dependence than under forward bias. Then, the correct zero-bias barrier, φ B0V , extracted from C-V characteristics is given by:

φ B0V = V bi (1 -ρ 2 ) + kT q + (E C -E F ) q (10)
The free carrier concentration has then to be corrected accordingly:

N D0V = N D (1 -ρ 2 ) (11)
It must be pointed out that this later assumption has never been experimentally confirmed even if some of these corrections have already been used in the literature. [29] 3. Experiments N-type (001) germanium wafers (ρ ∼ 20 Ω.cm) were used in this study. After different surface cleaning (pre-treatment) that will be detailed in the following, cobalt films of 22 nm thick were deposited at room temperature by using an electron beam deposition system under high vacuum (10 -8 torr). Thermal pre-treatments were performed in-situ in the deposition chamber using four lamps of 2000 W located in the vicinity of the sample holder. For each run, two samples were mounted in the deposition chamber. One of the samples was dedicated to the structural characterization and the film was deposited on all the sample surface. For the other one, the Co film was evaporated through a circular shadow mask of 2 mm in diameter to allow the electrical study of the Co-based Schottky contacts. No possible effects related to the perimeter of the diodes have been highlighted by a variable-area diode preliminary study.

Two different surface pre-treatments were studied: HF etching and thermal cleaning. For HF etching, the sample labeled PHF was dipped successively in acetone, ethanol solutions, HF (4%) during 30 seconds, and then rinsed three times in deionised water. Finally it was dried under N 2 air before being loaded in the deposition chamber. The characteristic time of passivation of a Ge surface being of the order of 15 min, [30] a great care was taken to load the samples in the deposition chamber as fast as possible knowing that the usual time to load the samples in the deposition chamber is about 5 min. For the thermal cleaning: two temperatures were chosen: 400 • C and 700 • C (samples labelled P400 and P700). The duration and the ramp were set at 40 min and 20 min, respectively. Note that no chemical treatment was performed on the surface of the samples before any thermal pre-treatment. Finally a reference sample, labelled REF, which was not submitted to any pre-treatment, was also studied.

TEM experiments were carried out to characterize the nature of the metal/semiconductor interface and the microstructure of the Co thin films. TEM samples were prepared in the cross-section geometry; they were mechanically thinned using a tripod polisher down to 10 µm, and then ion milled in a GATAN-PIPS apparatus at low energy (2.5 keV Ar) and low incidence (±8 • ) to minimize irradiation damage. They were studied by High Resolution Transmission Electron Microscopy (HRTEM) using a JEOL 3010 microscope (300 kV, LaB 6 , point to point resolution=0.19 nm). The electrical characterizations (C-V, I-V, DLTS) of the diodes were performed using a BioRad DL8000 (bridge capacitance at 1 MHz) apparatus with a He cryostat allowing a temperature variation from 40 to 350 K with a temperature sensitivity of 0.1 K. Moreover, optical microscopy images were used to accurately determine the area of each diode. Figure 1 displays cross-sectional HRTEM micrographs of the four studied samples after cobalt deposition. At the top of the samples, a continuous layer of polycrystalline cobalt, 22-25 nm thick, with nanometer sized grains is observed. According to selected area electron diffraction patterns analyses (not shown here), for any pre-treatments the cobalt thin films exhibits the hexagonal closed-pack structure as expected [START_REF] Gerl | Physique des matériaux[END_REF]. The cleaning procedure has thus no effect on the microstructure of the cobalt film.

Microstructural characterization

Between the cobalt layer and the germanium substrate, a thin amorphous-like layer (2 nm thick) attributed to germanium oxide is observed in both the reference sample (REF) and the sample pre-treated at 400 • C (P400). No evidence of this amorphous layer is seen on samples pre-treated at 700 • C (P700) nor by HF (PHF). These observations show that a threshold temperature T S , 400 • C < T S < 700 • C, exists, above which the native oxide can be removed under annealing in high vacuum. This temperature is, however, strongly dependent on the atmosphere; the reduction of GeO X under wet N 2 ambient [START_REF] Zou | [END_REF] was observed at 550 • C whereas the desorption in ultra high vacuum of the native germanium oxide was observed by heating the substrate at 360 • C during 15 minutes [22]. The temperature is thus not the only parameter which takes part in the removal of the oxide, and the other parameters, as the pressure inside the deposition chamber, must be taken into account. Finally, although not revealed by HRTEM, the chemical composition of the oxide interlayer could have been modified by the annealing at 400 • C [21], leading to different SBHs. This will be confirmed by electrical characterization.

From these HRTEM observations, it is clear that the MS interface is sensitive to the cleaning history of the samples; therefore differences in the Schottky barriers properties are thus expected. DLTS measurements were performed on the four samples in order to reveal a possible contamination which would have been able to occur during the preparation of the surface and/or during the deposition. In Figure 2, the DLTS spectra of the four samples are shown. Several peaks are observed, the energy of the corresponding deep levels with respect to the bottom of the conduction band edge and their apparent capture cross section are reported in Table 1. A hole trap, H 1 , is observed for the P700 and PHF samples. The observation of minority carrier traps in Ge Schottky barriers has already been reported for Co and Cr n-Ge Schottky barriers. [33,34] This is attributed to the creation an inversion layer near the surface. [34,35] As it will be detailed in the following, the presence of an inversion layer is also highlighted by the SBHs characteristics of these two diodes.

Deep levels characterization

Most of the signatures could be assigned to metallic impurities. For instance, from their signatures and by comparing with DLTS-spectra measured on samples where copper was intentionally introduced, E 7 and H 1 could be assigned to the two acceptor levels of copper Cu , respectively. Moreover, E 2 and E 3 could respectively be assigned to the two acceptor levels of gold [36] and E 6 and E 4 to nickel [37] and titanium in substitutionnal position. [23] The nature of the other traps E 1 , E 5 , E 8 is still unclear. Irradiation-induced defects, created by some energetic particle originating from the region of the filament as observed by Auret et al. [24,38] can however be ruled out since the signature of these deep levels does not correspond to any of the electron-irradiation induced traps reported in the literature. [39] Table 1. Signatures of the different DLTS lines on the as-deposited Co/n-Ge diodes. The capture cross section of the hole trap H 1 should be considered with caution since it has been calculated from majority carrier (electron) characteristics.* All the energies are given with respect to the conduction band edge except for the hole trap H 1 , where it is given with respect to the valence band edge. The concentrations are relative to the samples which are not in brackets.

Sample

Deep Levels The total defect concentration is below 5 × 10 12 cm -3 , so 5% of the doping concentration in the region probed (3.5-10 µm from the surface). Due to the low defect concentration, it has been difficult to obtain reliable defect profiles. However, one can estimate from the high diffusion coefficients of the metals detected in these samples [START_REF] Claeys | Germanium-based technologies: from materials to devices[END_REF] that the defect concentration closer to the surface is not going to reach a level where it can affect the I-V forward characteristics neither by generation-recombination nor by compensation. For Nickel which is known to be the most efficient lifetime killer in germanium, a concentration larger by several orders of magnitude would be necessary to influence the I-V forward characteristics. [START_REF] Simoen | [END_REF] The concentration of defects is not significant. Indeed, it varies for diodes having undergone the same procedure, suggesting a random contamination during either the pre-treatement or the deposition. Copper is a common contaminant in germanium [33,34,[START_REF] Simoen | [END_REF] but it is still unclear whether the Cucontamination is coming from the wafer itself or from the co-sputtering of copper during the metal deposition. [34] On the other hand, E 1 , E 2 , E 3 and E 4 are only observed in the pre-annealed samples P400 and P700, with a higher concentration in P700 than in P400. This clearly shows that a high temperature pre-treatment introduces metallic impurities, the higher the temperature, the higher the concentration of impurities. This contamination could be due to impurities located in the deposition chamber, that desorb and then diffuse in the germanium substrate during the pre-annealing.

E C -E T (eV) σ(cm 2 ) N T (cm -3 ) Possible identity P700, (P400) E 1 - - - ? P700,(P400) E 2 0.07 3 × 10 -15
In the following, due to the low concentration of deep-levels, the generationrecombination current has been neglected and the current flow has been considered as predominantly thermionic.

Schottky barrier characterization

Electrical characterization at 160 K

To minimize the thermal generation of intrinsic carriers (n i ∼ 10 13 at/cm 3 at 300 K), the four diodes were studied at a given temperature of 160 K. Figure 3 shows typical semi-logarithmic I-V plots of the four diodes measured at this temperature. As seen, for each pre-treatment, a good quality rectifying Schottky barrier is observed with several orders of magnitude between the forward and the reverse currents. However, depending on the pre-treatment, different behaviours are observed. For instance, the REF and P400 diodes exhibit much higher forward currents than the two other diodes. The P700 diode presents the lowest reverse current, lower than the limit current of our measurement set-up. The forward currents were fitted with the TE model (plain lines on Figure 3) which was assumed to be the dominant transport mechanism for such low doping concentration. As seen, a good agreement between experimental and theoretical data is observed. The parameters, namely the saturation current I S , the Schottky barrier height φ IV B , the ideality factor n and the series resistance R S were extracted from the TE fits for each diode. The as-derived series resistance and ideality factors were also confirmed by using the plots of experimental current over conductance (I/G) as a function of I (not shown here). [42] From the C-V measurements at 160 K, the doping concentration was found to be (8.6 ± 0.5) × 10 13 cm -3 in good agreement with the manufacturer specification (9.0 × 10 13 cm -3 ). The main parameters extracted at 160K are summarized in Table 2. All the diodes present ideality factors greater than unity. The SBHs both extracted from C-V and I-V measurements are found to be strongly dependent on the pretreatment procedure. Furthermore, for all the diodes, the SBH derived from C-V measurements is much higher than that derived from I-V measurements (qφ CV B > qφ IV B ).

6.2. Electrical characterization in the temperature range 80 -270 K To get further insights on the carrier transport mechanisms through the Co/n-Ge contacts, I-V and C-V characteristics were recorded in the temperature range 80 -270 K every 10 K for each samples. Figure 4 shows the forward I-V characteristics of the two diodes REF and PHF which present similar behaviors with the P400 and P700 diodes, respectively. An increase of the current with the temperature is observed in good agreement with the thermionic model. It is noteworthy that from a temperature of 220 K, a small bump on the PHF and P700 diodes I-V characteristics is observed, pointing out clearly the presence of a second transport mechanism. To get ride of this mechanism, the study will be limited to the low SBH (low temperature side).

The saturation currents were extracted from the I-V curves and used to plot the conventional Richardson plots shown in Figure 5. As seen, the expected linear behavior is only observed for the REF diode. However, the SBH and the Richardson constant derived from this plot are abnormally low: 0.02 eV and 2×10 -7 A.cm -2 K -2 respectively. This later value is several order of magnitude lower than the theoretical value of A * for n-(001) Ge (143 A.cm -2 K -2 in reference [START_REF] Sze | Physics of semiconductor devices[END_REF]). The values of n determined from the fit procedure are plotted in Figure 6. As seen, the ideality factor is strongly dependent on temperature on the low temperature side (below 200 K) except for the REF diode where it stays constant on all the temperature range studied.

These observations conjugated with the differences between φ CV B and φ IV B at 160 K are good indicators of SBH inhomogeneities. To successfully apply the WG's model, a temperature range of application has been determined by considering that i.) the current flow must be governed by a single thermionic mechanism and ii.) φ CV B must be larger than or equal to φ IV B (see Equation ( 7)). From these two points, the temperature range has been restricted below 180 K. Figure 7 and Figure 8 show the SBH extracted from C-V-T measurements and φ CV B (T ) -φ IV B (T ) as a function of temperature, respectively. As seen, these plots are straight lines and qφ CV B higher than qφ IV B is observed in all the temperature range. The derived parameters from the linear fits, namely the mean barrier height at 0 K φ 0 , its temperature coefficient α φ , the standard deviation at 0K σ 0 , and its temperature coefficient, qα σ ,were extracted and are reported in Table 4. The modified Richardson plot according to Equation ( 8) is plotted in Figure 9. As seen, the expected linear behavior is recovered after the WG's model corrections. The Richardson's constant and the mean barrier height at 0 K (φ RICH 0 ) were derived as explained in previous section and are reported in Table 4. The Co/n-Ge diodes prepared by using different cleaning procedures show significantly different SBH properties. Furthermore, for all the diodes, the values of A * are close to the theoretical value. In addition, the mean SBHs at 0 K determined using the modified Richardson plots, φ RICH 0 , are in good agreement with those extracted from the C-V-T characteristics, φ 0 . These two observations confirm that the experimental data are well described by a Gaussian distribution of SBH. In this temperature range (80-170 K), in good agreement with the WG's model, a plot of 1/(n-1) versus 1/T yields to a straight line as shown in Figure 10 for all the diodes. From these plots, the voltage coefficients ρ 2 and ρ 3 were extracted and reported in Table 3. As seen, they are strongly dependent on the cleaning history. Table 3 lists also the experimental free carrier concentration as well as the corrected free carrier concentration N D0V at 160 K. Not only this correction cannot be applied to the P400 diode (as ρ 2 > 1) but it leads also to an increase of the discrepancy of the free carrier concentration without any physical reason. Obviously, the mean SBH of samples φ B does not show under reverse bias the same dependence than under forward bias; therefore no further correction are needed on the SBH extracted from C-V characteristics. 4, all the surface pre-treatments lead to both an increase of the mean SBH at 0 K, φ 0 , and of its temperature coefficient, α φ . The PHF and P700 diodes, both being intimate contacts, present similar values of φ 0 . The passivation of the dangling bonds by HF pretreatment has thus no or few effects on the FL pinning.

Analysis of barrier inhomogeneities using a Gaussian distribution model

The presence of a thin oxide interlayer (REF and P400 samples) leads to the decrease of Φ 0 . However, the P400 sample exhibits a 80 % higher mean SBH at 0 K than the REF sample for equal oxide thickness. This clearly points out the different chemical nature of the oxide between the P400 and the REF sample. According to the experiments of Prabhakaran et al., [21] the oxide interlayer of the REF sample is made of GeO 2 and other suboxides while it mainly consists of suboxides for the P400 sample. The oxide thicknesses of both REF and P400 are similar, this shows that the depinning efficiency is higher for GeO 2 than for other suboxides. Finally it is worth noticing that the PHF and P700 samples exhibit a higher mean SBH than the germanium band gap in the temperature range studied (see Figure 7), as already reported in previous experimental and theoretical studies on n-type germanium. This is attributed to the FL pinning and to the formation of an inversion layer [7,8,[START_REF] Yao | [END_REF] thus explaining the observation of a minority carrier trap for these two diodes. The temperature dependence of the mean SBH (α φ ) can also be roughly explained in the framework of the Fermi level pinning concept. [15,45] According to the band to which the FL is pinned, the temperature coefficient varies from 0 to dE g /dT , where E g is the indirect band gap of germanium. In the temperature range 100-300 K, the value of dE g /dT for germanium is close to (-0.4)-(-0.5) meV/K. [46,47] The temperature coefficient of the REF diode thus indicates a pinning between the conduction band and the middle of the band gap. However, for high SBHs, qα φ is four times higher than the expected value and therefore the FL pinning is not sufficient to understand the variation of SBH with the temperature.

The standard deviation at 0 K is a measure of the barrier inhomogeneity: the higher the value σ 0 , the higher the SBH inhomogeneity. The highest values of σ 0 are found for the PHF and P700 diodes, indicating thus, that the intimate contact of Co on Ge enhances the inhomogeneity of the SBH distribution. This result is of importance as it points out the need to take into account the SBH inhomogeneities even if no interfacial layer is observed at the MS interface. Inhomogeneous SBH can be induced by the roughness of the MS interface [14]. It is well known that a HF pre-treatment yields an increase of the roughness of the Ge substrate [19,48] thus explaining the higher σ 0 observed for the P HF diode than for the REF and P400 diodes. Even if no study on the impact of the Ge roughness by thermal pre-treatments has been found in the literature, similar statement can be derived for the P 700 diodes according to the value of σ 0 . Finally, it should be noted that, contrary to what was observed for φ 0 , the standard deviation is less affected by the chemical nature of the oxide interlayer.

The temperature coefficient, α σ , is strongly modified by the nature of the pretreatment as well. The temperature coefficients are found to be negative for the REF diode, and superior to 0 for the PHF, P400 and the P700 diodes, indicating thus that the inhomogeneity increases with the temperature. According to Zhu et al., [15] the physical meaning of the negative value of α σ can be understood by the pinch-off model proposed by Tung. [49] In this model, the variation of SBH between low and high SBH areas decreases with the temperature. The barrier heights appear more homogeneous than they are. Therefore, the dependence with the temperature of the standard deviation could not be understood by the pinch-off model for the P400 and the P700 diodes. Such behavior, according to us, has never been reported in the literature.

Conclusion

In summary, by conducting together microstructural and electrical characterizations of Co/n-Ge diodes that have undergone various cleaning methods, we have linked the SBH properties to the interface state. For any pre-treatments, the electrical behaviors of the Schottky contacts could not be explained by the thermionic model only, pointing out the necessity to think in term of barrier inhomogeneity. The electrical measurements can be satisfactorily explained by assuming a Gaussian distribution of SBH as already reported for Ni/n-Ge Schottky contacts. [50] Whatever the cleaning diode history the intimate MS contact (HF etching or thermal pre-treatment at 700 • C) leads to the strongest Fermi level-pinning and to the highest spatial inhomogeneities without any relevant differences on SBH characteristics. It should be noted that during the thermal pre-treaments metal-lic impurities coming from the deposition chamber were introduced in the Ge substrate however in too low concentrations to affect the I-V forward characteristics. The insertion of a thin oxide interlayer (reference sample and pre-treatment at 400 • C) yielded to a depinning of the FL and to a more homogeneous contacts. While the nature of the oxide interlayer did not show significant differences on the SBH inhomogeneities it greatly affects the FL depinning. Ours results suggest that a pure GeO 2 oxide interlayer should present a better depinning efficiency than the native Ge oxide. Even for basic Schottky contacts on n-Ge, the electrical behaviors of the diodes are strongly modulated by the spatial inhomogeneities. Our work highlights the necessity of taking into account these inhomogenities in the interpretation of electrical characteristics. In particular the values deduced from the classic Richardson plots can be strongly inaccurate.

Figure 1 .

 1 Figure 1. TEM bright-field micrographs of Co thin films on Ge substrate deposited by electron beam evaporation. a) REF b) P 400 c) P 700 d) P HF . The white arrows highlight the oxide interlayer.

Figure 2 .

 2 Figure 2. DLTS scan normalized by the capacitance for the four as-deposited samples. The amplitude of the encircled area have been divided by four for convenience.

Figure 3 .

 3 Figure 3. Reverse and forward I-V currents at 160 K for the four Co/Ge contacts. Fitting curves using the thermionic model (TE) are in plain lines.

Figure 4 .

 4 Figure 4. Forward I-V characteristics at different temperatures for the two diodes REF and PHF. The arrow highlights the presence of a second transport above 220 K.

1 )Figure 5 .

 15 Figure 5. Conventional Richardson's plots of Co/Ge contacts having undergone various pre-treatments.

Figure 6 .

 6 Figure 6. Ideality factor, n, extracted from forward I-V characteristics as a function of the temperature for the four Co/Ge contacts.

Figure 7 .

 7 Figure 7. Variation of φ CV B with the temperature and linear fits from Equation (5).

Figure 8 .

 8 Figure 8. Variation of φ CV B (T ) -φ IV B (T ) with 1T and fitting curves from Equation (7) for the as-deposited samples.

Figure 9 .

 9 Figure 9. Modified Richardson's plot and linear fits from Equation(8). To minimize the uncertainty on the determination of both σ 0 and α σ , the term σ 2 (T ) 2k 2 T 2 has been determined by using the experimental φ CV B (T ) -φ IV B (T ) data.

Figure 10 .

 10 Figure10. n -1 -1 vs T -1 for the as-deposited samples and the linear fits from Equation (9).

Table 2 .

 2 SBHs extracted from C-V measurements φ CV B , from I-V measurements φ IV

	B

Table 3 .

 3 Voltage coefficients, ρ 2 and ρ 3 deduced from Figure10. Free carrier concentrations extracted from C-V characteristics at 160 K without (N D ) and after corrections of the voltage deformation (N D0V ).

	Sample ρ 2	ρ 3 (meV ) N D (cm -3 ) N D0V (cm -3 )
	REF	0.67	0	8.3 × 10 13	2.7 × 10 13
	PHF	0.29	-7	9.1 × 10 13	6.5 × 10 13
	P400	1.30	24	8.1 × 10 13	X
	P700	-0.05 -1	9.6 × 10 13	1.1 × 10 14

Table 4 .

 4 Mean barrier height at 0 K φ 0 , and its temperature coefficient α φ , Standard deviation at 0 K σ 0 and its temperature coefficient α σ . Richardson's constant, A * , the mean barrier height at 0 K derived from the modified Richardson plot, φ RICH

	0	.

Acknowledgments

We would like to thank M. Marteau (Pprime) for the thin film depositions and E. Simoen (imec) for the fruitful discussions.