J M Bizau

C Blancard

M Coreno

D Cubaynes

C Dehon

N El

F Folkmann

M F Gharaibeh

A Giuliani

J Lemaire

A R Milosavljevi

C Nicolas

R Thissen

Photoionization study of Kr + and Xe + ions with the combined use of a merged-beam set-up and an ion trap

Keywords: numbers: 32, 80, Fb, 32, 80, Zb

Photoionization study of Kr+ and Xe+ ions with the combined use of a merged-beam set-up and an ion trap

Introduction:

Atomic physics provides unique tools for plasma diagnostics. In particular, based on photoionization studies on ionic species one can deduce essential spectroscopic data for the determination of important plasma parameters such as abundance and temperature. In addition, dipole selection rules allow further simplification of the data by reducing the number of allowed transitions. To be useful, quantities like the energies and oscillator strengths of discrete lines or direct photoionization cross sections must be measured in absolute value with high accuracy for a large number of ionic species and over a wide range of photon energy. Photoionization is also one of the main processes contributing to plasma opacity and radiation transfer inside plasmas.

Krypton and xenon ions are of particular importance in tokamak plasmas. Injection of high-Z gases, essentially Kr and Xe, has been proposed as a technique to mitigate disruption [1,2], i.e. the uncontrolled and sudden loss of tokamak plasma current and energy. Disruption can produce severe damage on the vessel wall. The situation becomes more delicate for large machines such as ITER (International Thermonuclear Experimental Reactor), where incident loading energy may reach several GW/m 2 [3,[START_REF]ITER Physics Basis[END_REF]. After injection of the gases, highly charged Kr and Xe ions are dominant in the core of the plasma, and low-charged and singly charged ions are abundant near the edge. Experimentally, photoionization data on ionic targets have been obtained using mainly two techniques: DLPP (Dual Laser Produced Plasma) and merged-beam in synchrotron radiation facilities. While the former technique measures photoabsorption spectra [5], the latter provides absolute single and multiple photoionization cross sections [6,7]. Up to now, only the K shell photoionization of the Kr + ion has been reported [8], and all studies on the Xe + ion have focused on the region of 4d subshell excitation and ionization using the merged beam technique [9][10][11][12][13]. To best of our knowledge, no studies using these techniques have been published on the Kr + ion or on the Xe + ion in another energy range and, in particular, close to the thresholds. On the top of that no theoretical results on the photoionization processes are available for these ions outside the region of 4d excitation in the Xe + ion. One difficulty, suffered both by the DLPP and merged-beam techniques, relates to the composition of the ionic target. Very often, it is made of a mixture of ions in the ground and metastable states, and most of the time in unknown relative proportion. Attempts have been achieved to determine experimentally the population of excited states inside the target. In DLPP experiments, this can be done, to a certain extent, by taking advantage of the time-and space-resolved capabilities of the technique which allows probing with the ionizing radiation regions of different temperatures in the plasma target, and enhancing the contribution of ions in the ground state or the metastable states [14]. In merged-beam experiments, the populations can be determined by measuring the attenuation of the ion beam passing through a gas cell, taking advantage of the different magnitude of the electron capture cross section for the ions in the ground state and the metastable states [15][16][17][18]. Another method, applied in the case of the Sc 2+ ion, consisted of comparing, via the detailed balance principle, the intensity of the resonant photoionization cross section of Sc 2+ with the previously measured dielectronic recombination cross section of Sc 3+ [19,20]. This procedure was possible only because one channel strongly dominates the radiative decay of the recombination resonances in Sc 2+ . Except for these few examples, in all other studies the relative populations were estimated by normalization to some theoretical results, or by assuming a statistical population when the mean electron kinetic energy in the plasma used to produce the ions is well above the energy needed to excite metastable states. In 2008, we proposed an alternative method to avoid the problem introduced by the production of metastable states in the target: to use a trap to store the ions before irradiation during a time long enough to allow the excited states to radiatively decay to the ground state and, after irradiation in situ, to analyze the charge of the ions. The feasibility of the concept was demonstrated using a Fourier Transform Ion Cyclotron Resonance (FT-ICR) trap at the Gas Phase beam line of ELETTRA [21]. The photoionization spectrum of Xe + ion in the pure 5p 5 2 P 3/2 ground state was measured in the 20-25 eV photon energy range. Before this experiment, only a few attempts to couple an ion trap with a beam of XUV light had been performed. Kravis et al [22,23] have obtained the charge distribution of the photoions formed by broadband irradiation with synchrotron light of Ar 2+ ions stored in a Penning trap. More recently, Epp et al [24] have used the ultra-brilliant light from the Free Electron Laser (FEL) in Hamburg to demonstrate the feasibility of performing laser spectroscopy on Fe 23+ ions produced in an Electron Beam Ion Trap (EBIT). Recently, the same EBIT was used at BESSY II to record photoionization spectra of some multiply-charged ions [25]. Hirsch et al [26] have also used a linear trap at BESSY II to measure photoionization spectra of mass selected clusters. In this paper, we present our results on photoionization of Kr + and Xe + ions in the energy range covering 15 eV above the first ionization threshold, using both merged-beam and FT-ICR devices. With the merged beam set-up, the absolute photoionization cross sections for a mixture of ions in the np 5 2 P 3/2 ground level (n = 4 for Kr and n = 5 for Xe) and np 5 2 P 1/2 metastable level have been determined. With the trap, the photoionization cross section for the ions in the pure np 5 2 P 3/2 ground level has been recorded in relative value. Combination of the two techniques allows the data obtained with the trap to be placed on an absolute scale. MCDF calculations have been performed to help in the interpretation of the data. In the following, after the description of the two devices and the experimental procedures, we will present and discuss the results. To conclude, future developments of the ion trap technique for photoionization studies on ionic species will be commented upon.

Experimental procedures

Merged-beam experiment

Measurements were performed using the ASTRID merged-beam set-up at the University of Aarhus in Denmark. The characteristics of the set-up and the experimental procedure used to extract absolute photoionization cross sections from the measurements have already been described in detail [27]. In short, the photons are supplied by synchrotron radiation from the 30period undulator beam line at the ASTRID storage ring. Synchrotron radiation is monochromatized by a Miyake type monochromator [28] composed of one 1200 lines/mm plane grating covering the 15-180 eV photon energy range. In order to avoid higher-order radiation from being diffracted by the grating, a Mg filter is inserted at the exit of the monochromator. The photon flux is measured by an Al 2 O 3 photodiode. Photodiode efficiency and photon energy are determined in real time using an ionization chamber of the double-collector type originally developed by Samson [START_REF] Samson | Techniques of Vacuum Ultra Violet Spectroscopy[END_REF]. For energy calibration, np s → 3 excitation lines in argon and np s → 2 excitation lines in neon were used [START_REF] Yu | NIST Atomic Spectra Database (ver. 4.0.0)[END_REF]. Accuracy on the photon energy scale is estimated to be 20 meV. Kr + and Xe + target ions are produced using a hot-filament plasma ion source of the Nielsen type [START_REF] Almen | [END_REF]. The ions are extracted from the ion source and accelerated using a 2 kV extraction voltage, mass separated by a dipole magnet, collimated and merged electrostatically with the photon beam. The interaction between photons and ions takes place in a ~50 cm long cylinder, biased to +600 V in order to obtain a well defined interaction length by energy marking of the photoions. To measure the profiles of the two beams and their mutual overlap, three sets of xy scanners are placed along this cylinder. Typical currents of 20 nA of target ions were obtained in the interaction region. The signal of photoions and the primary ion beam are separated by a second dipole magnet. The current of primary ion beam is measured in a Faraday cup. To reduce the background of scattered ions, the beam of photoions is further separated by an electrostatic analyser before the ions are detected by a particle detector similar to the one described by Rinn et al [32]. A chopper is inserted at the exit of the monochromator to measure the background signal on the detector produced by ions scattered by slits, ionization resulting from collisions between ions from the target beam and residual gas molecules, and highly-excited metastable ions that are emitted by the ion source and that decay in the interaction region by autoionization. Typical signal/noise ratio measured on the detector was 30 Hz/10 Hz at photon energy 10 eV above threshold.

The absolute photoionization cross sections σ are calculated from measurements of the current I of the target-ion beam, the photon beam intensity (=J/eη, from the photodiode current J with efficiency η), the ion and photon beam profiles, the velocity v and charge stage q of the target ions, the photoion counting rate S, the efficiency of the particle detector Ω, and the length of the interaction region L as follows:

∆ ∆ Ω = L z yF x dz IJ vq Se 0 2 ) ( η σ (1)
∆x∆yF(z) is an effective beam area (z is the propagation axis of the two beams), and F is a two dimensional form factor defined by where i xy = i(x,y)∆x∆y and j xy = j(x,y) ∆x∆y are the ion and photon currents, respectively, recorded during the beam-profile measurements and ∆x and ∆y the step sizes involved in these scans, typically 0.2 mm. Accuracy of the cross-section data was determined by statistical fluctuations and the systematic contributions resulting from the measurement of the different parameters in equation (1). Above 25 eV photon energy, the latter is estimated to be 15% and is dominated by the uncertainty in the determination of the absolute photon flux, in particular due to the possible contribution of highorder radiation, and the determination of the detector efficiency [27]. Below 25 eV, the magnitude of photoionization cross sections might be underestimated and the systematic uncertainties cannot be estimated, since second order radiation is not filtered out by the Mg filter and contributes in an unknown proportion to the measured photon flux.

FT-ICR ion trap

In the present experiment, we have used the FT-ICR ion trap already used at ELETTRA [21]. The concept of coupling a FT-ICR to a light source has been already applied with great success to perform spectroscopy of selectively prepared ions by Infra-Red Multi-Photon Dissociation (IRMPD) on Infra-Red FEL [33,34]. The MICRA (Mobile Ion Cyclotron Resonance Analyzer) spectrometer, the same used at the CLIO Infra-Red FEL in Orsay, France and the Gas Phase beam line at ELETTRA, Italy, has been installed on branch B of the DESIRS beam line of the storage ring SOLEIL, Saint Aubin in France. The synchrotron radiation, produced in a 15-period electromagnetic undulator, is monochromatized by a normal incidence monochromator delivering photons in the 5-40 eV energy range. A new 400 lines/mm low dispersion grating and 700 µm exit slit width were used to obtain high flux, resulting in a bandpass of 100 meV. Optics cut-off produced by the normal incidence of the light on the grating provides spectral purity above 21 eV photon energy. The FT-ICR traps are based on a strong magnetic field, trapping the ions in two dimensions, and an electrostatic field creating a potential well in the third dimension. The particularity of FT-ICR traps is that the ions can be excited in-situ and create an image current on electrodes, which presents a harmonic behavior directly related to the mass of the trapped ions. Through a Fourier transform of the recorded signal, it is therefore possible to measure the ions mass to charge ratio m/z, and their relative abundance. MICRA is a very simple and robust movable spectrometer, based on a 1.24 T permanent magnet, and has been specifically designed to allow a light beam to traverse the whole set-up [35]. The magnetic field allows for mass selection and detection in the m/z range from 4 to 500 u/q, with an ultimate resolving power of 73,000 at the mass of Xe + ion. The ICR cell is a cube with 2 cm sides. The two walls of the cell perpendicular to the photons propagation axis, used to excite the ions cloud, are replaced by an open structure to provide optical access without interference from surfaces. We estimate the number of trapped ions at a few hundred thousand and the volume of the ion cloud in the range of 0,05 cm 3 leading to an ion density in the middle of the trap to be of the order of 10 6 cm -3 . The nominal pressure in the cell is in the low 10 -9 mbar. Figure 1 illustrates a typical time sequence produced for data acquisition at a given photon energy. The target ions are first produced by ionization of rare gas atoms in the gas pulse by 25 eV kinetic energy electrons. In this example, after a delay of ∆t =300 ms, the ions are irradiated for 1100 ms by the monochromatized synchrotron radiation, and the mass spectrum of ions in the trap is then recorded. Chopping of the synchrotron radiation beam is performed using a pneumatic activated shutter. The mass spectrum measured at one photon energy is then obtained from the Fourier transform of the signal resulting from the average of 60 to 100 repetitive cycles, similar to the one presented on figure 1. It is important to note that, in the energy range considered here, neither the energy of the photons or the electrons alone is sufficient to produce doublycharged ions from double ionization of the atoms (the thresholds are 38.36 eV and 33.10 eV for Kr and Xe atoms, respectively [START_REF] Yu | NIST Atomic Spectra Database (ver. 4.0.0)[END_REF]). We have checked that no doubly-charged ions are produced when the photon pulse and/or the electron pulse are missing in the cycle. One useful characteristic of the FT-ICR ion trap is its ability to precisely select the ions stored inside the trap by increasing the kinetic energy of the unwanted ions until they hit the wall of the ICR cell. After irradiation the parent singly-charged ions were ejected to improve doubly-charged ion detection. The best signal to noise ratio on the mass spectrum was obtained by performing the Fourier transform on the very beginning of the recorded signal, typically from 1.5 ms from the beginning to 3 ms. To obtain the photoionization spectra, the height of the doubly-charged signal in the mass spectrum was followed as a function of photon energy and corrected for the variation of the photon flux recorded using a gold mesh inserted at the exit of the photon beam line. For krypton, we used the height of the 84 Kr isotope line, and for xenon the mean of 129 Xe, 131 Xe and 132 Xe lines heights. The photon energy of the spectra was calibrated on the energy measured at ASTRID.

Results and discussion

Merged-beam experiment

The variation of the photoionization cross sections determined with the merged-beam set-up for Kr + and Xe + ions are shown on the upper-and lower-panel of figure 2, respectively. These spectra were obtained with a fixed width of the monochromator slit, resulting in a bandpass (BP) continuously increasing from 50 meV at threshold to 100 meV at 15 eV above threshold for the Kr + ion spectrum, and from 30 meV to 100 meV for the Xe + ion spectrum. The relative statistical uncertainties on the data at 30 eV photon energy are 6% and 16% for Kr + and Xe + ions, respectively. The cross section presented here for the Xe + ion is in good agreement with the one we have measured previously with lower resolution and statistics, using the same merged-beam set-up but a different ion source [21]. We note the close similarity of the Kr + and Xe + spectra in shape and intensity. Resonant structures are observed above the threshold and are followed by a smoothly decreasing continuum. Such similarity was already observed in the case of Br and I atoms, which are isoelectronic of Kr + and Xe + ions, respectively [36,37]. The integrated oscillator strengths of the spectra, calculated using [38]

- × = υ υ σ dh h f ) ( 10 11 . 9 3 , (3)
are also of the same order, 1.40 and 1.49 for Kr + and Xe + ion, respectively . Enlargements of the resonance regions, obtained with an improved resolution compared to the spectra of figure 2, are shown on figure 3 for Kr + ion (BP = 20 to 40 meV) and figure 4 for Xe + ion (BP = 20 to 30 meV). The relative statistical uncertainties on these measurements are 7% and 4% for Kr + and Xe + ions, respectively. Kr + and Xe + ions are halogen-like with one hole in the p outersubshell, giving an electronic configuration np 5 2 P 3/2 (n = 4 for Kr and n = 5 for Xe) in their ground level. The long vertical lines above the spectra indicate the position of the various np 4 3 P 2,1,0 and 1 D 2 ionization thresholds (the 1 S 0 threshold is at higher energy than covered by the figures) [START_REF] Yu | NIST Atomic Spectra Database (ver. 4.0.0)[END_REF].

The first excited level np 5 2 P 1/2 is metastable, with lifetimes of 340 ms and 49 ms for Kr + and Xe + ions, respectively [39]. For both ions, the strong lines observed below the first ionization threshold of the ions in the ground level testify to the contribution of ions in the 2 P 1/2 metastable level. These ions are produced inside the ion source and have enough time to travel to the interaction region, the time of flight needed for the transfer being a few µs. The np 4 ionization thresholds for the ions in the metastable level are also indicated on figures 3 and 4 [START_REF] Yu | NIST Atomic Spectra Database (ver. 4.0.0)[END_REF].

The lines observed on the spectra can be attributed to excitation of the valence shell electron by electric-dipole allowed processes, giving rise to the transitions md ms np np , 4 5

→

. The np 4 ms,md excited states decay via autoionization to the ground level of the doubly-charged ion. Rydberg states of ms and md character converging to each of the four upper np 4 levels are expected to be observed in photoionization spectra, giving rise to rich structures of intricate Rydberg series. The situation is even worse once we consider the important contribution of the ions in the metastable level, doubling the number of expected lines. For halogen atoms, only the very sophisticated eigenchannel R-matrix approach has been able to satisfactorily reproduce the experimental spectra, in particular the strongly asymmetrical profile of some Rydberg series resulting from an interference effect between the direct and the resonant photoionization processes [40]. Considering the moderate energy resolution chosen in this work to compensate for the low density of target ions, we have not attempted a detailed identification of the observed structures, which are most of the time composed of several unresolved lines. An analysis can be achieved using the formula:

2 2 ) ( δ - - = ∞ n RZ E E n (4)
where R, ∞ E , Z and δ are the Rydberg constant, the converging limit, the electric charge of the core ion and the quantum defect, respectively. The ionization thresholds ∞ E were taken from [START_REF] Yu | NIST Atomic Spectra Database (ver. 4.0.0)[END_REF] and we assumed Z = 2. The measured energy E n , the quantum defect and the assignment resulting from this analysis are given in tables 1 and 2 for Kr + and Xe + ions, respectively. For both ions, the dominant structures are associated with Rydberg series converging to the 1 D 2 limit, with principal quantum number m going up to 13. The positions of the lines of these series are shown as the short vertical lines on figures 3 and 4, both for ions initially in the ground and the metastable terms. The relative intensities of the two series indicates a relative population of the two terms close to the statistical weight. For Kr + ion, series converging to the 1 S 0 can also be identified and the quantum defect is almost constant along the series. The analysis is more difficult for Xe + ion, only the series converging to the 1 D 2 limit being clearly identified but with a less constant quantum number along the series than for Kr + ion. We have performed multiconfiguration Dirac-Fock (MCDF) calculations using the code developed by Bruneau [41] to calculate the direct photoionization cross section. We considered all of the levels with an np 5 configuration in the initial state and np 4 in the final ionic state. For each initial state, the photoionization cross sections were calculated with a regular interval of the photon energy of 0.1 eV close to thresholds and 0.5 eV above. The result of these calculations for the Kr + ion is shown as the solid line on the top panel of figure 2. In the case of the Xe + ion (lower panel), the solid line is the result of our previously published calculations performed using the same code [21]. Both theoretical cross sections have been reconstructed assuming a statistical population of the ions in the 2 P 3/2 ground level and 2 P 1/2 metastable level. Let us note that, save at the opening of the various ionization thresholds, the magnitude of the reconstructed cross section is quite insensitive to the relative abundance of ions in the two levels. Our calculations show that the photoionization cross section is almost the same for the ions in the ground and the metastable levels, as can be expected from a monoconfigurational approach, demonstrating the weak effect of electron correlations for this process. For both ions, the calculations reproduce correctly the variation of the cross sections with photon energy, but with a magnitude 15 to 30% systematically lower than the measurements. On the upper panel of figure 2, clear window resonances are observed at photon energy of 31.38 eV and 31.69 eV. They can be interpreted as the n = 5 term of the Kr + 4s4p 5 ( 3 P 2,1 ) np Rydberg series which have a series limit at 38.73 eV and 39.16 eV [START_REF] Samson | Techniques of Vacuum Ultra Violet Spectroscopy[END_REF]. Kr + 4s4p 5 ( 3 P) 6p member could be associated with the weak structure at 35.0 eV. Kikas et al [42] propose 31.62 eV and 34.86 eV for the n = 5 and n = 6 members of the 4s 4p 5 ( 3 P) np 2 S series, respectively.

FT-ICR ion trap

The spectra obtained for Kr + ion using the FT-ICR ion trap at different delay times ∆t between the production of the ions and their irradiation by synchrotron radiation are displayed on the two lower panels of figure 5. They are compared with the spectrum recorded using the merged-beam set-up shown on the upper panel. It is the same spectrum than the one on figure 3, but convolved with a Gaussian profile to achieve the same apparent excitation bandpass (100 meV) and thus to make comparison between the spectra recorded with the two techniques easier. The spectra obtained with the ion trap are placed on an absolute scale assuming that the integrated area under the FT-ICR ion trap spectrum is identical to the merged-beam spectrum. The one on the middle panel was obtained by irradiating the Kr + ions just after their production inside the trap (∆ t=0). Despite a larger statistical scatter, it displays the same structures as those seen on the merged-beam spectrum, demonstrating that the ions produced in the trap are both in the ground 2 P 3/2 and in the metastable 2 P 1/2 levels, and with abundances close to those observed in the merged-beam set-up.

On the lower panel, the delay between production and irradiation was increased to ∆t =520 ms, a longer time than the natural lifetime of the metastable level (350 ms) allowing most of the ions in this level to radiatively decay to the ground level. The intensity of the structures below the first ionization threshold at 24.36 eV has been strongly reduced, demonstrating that the ions irradiated inside the trap now are mostly in the ground level. The main loss of trapped ions often comes from charge exchange with the water molecules in the residual gas inside the trap. As the ionization potential of xenon atom is lower than water, charge exchange with water is inhibited and the trapping time of Xe + ions can be extended without major losses. Figure 6 shows the spectrum recorded at ELETTRA with the same trap with a delay of more than one second [21], corresponding to twenty times the natural lifetime of the 2 P 1/2 meta- stable level. It is compared to the spectrum of figure 4 convolved with a Gaussian profile to achieve the same overall resolution. As was the case previously, the ion trap spectrum was placed on an absolute scale by equating the integrated areas of the two spectra. Once again, the structures below the first ionization threshold (20.98 eV) are strongly reduced in intensity, thus demonstrating that most of the irradiated ions are in their ground level. We observe no effect of the trapping magnetic field of 1.24 T on the position of the lines. The MCDF code has been used to quantify the magnetic field effect on both Xe + 5p 5 and Xe 2+ 5p 4 levels. Each ionic charge state has been separately considered. For each M J value, the energies of the sublevels have been calculated by diagonalization of the Hamiltonian matrix including contributions of the magnetic field interaction evaluated on the basis of unperturbed wave functions. The shift produced by a field of 1.24 T has a maximum amplitude of 0.15 meV for Xe + 5p 5 and 0.21 meV for Xe 2+ 5p 4 . Such values are well below the resolution of our experiments.

Conclusion

We have measured the photoionization cross sections of halogen-like Kr + and Xe + ions in an energy range extending up to 15 eV above the threshold. Two different set-ups were used, a merged-beam set-up at ASTRID and a FT-ICR ion trap at SOLEIL. The merged-beam set-up has allowed the identification of the main resonant structures in the spectra, as well as the determination of the absolute cross sections for the direct photoionization process. Comparison of the photoionization spectra obtained using the trap and the merged-beam experiment has allowed extraction of the pure ground state ionization cross section on an absolute scale. MCDF calculations suitably reproduce the variation of the direct photoionization cross section. This work demonstrates that the implementation of an ion trap experiment at a third generation synchrotron radiation beam line is a valuable tool that allows the measurement of the photoionization cross section of ionic species in their ground state if the lifetimes of metastable states are significantly shorter than the ion storage time. Photoionization experiments on stored ions are still difficult owing to (i) the very low density of target ions and (ii) the limitation in trapping time resulting from charge exchange and reactions with the residual gas. Nevertheless, there is no doubt that such experiments will be developed in the future, as demonstrated by the recent results obtained on multiply-charged ions [25] and ionic clusters [26]. Concerning the coupling of a permanent magnet FT-ICR trap with synchrotron radiation, improvements can be obtained by increasing the number of stored ionic species and by achieving a better overlap between the ion cloud and the photon beam. Such a device is under development with a cylindrical permanent magnet producing in its centre a magnetic field coaxial with the axis of the cylinder. The cell will also be larger than in the MICRA instrument. In this way ions can be produced externally to the magnetic field region and then be transferred inside the trap. Since all high pressure events are in a separate chamber the analyzer cell is operated with improved vacuum. This is useful for investigations over a wide wavelength range. Radio-Frequency traps are also very promising instruments for the study of mass selected ions as demonstrated recently at SOLEIL [START_REF] Giuliani | Private Communication[END_REF], but they operate with a helium buffer gas which restricts their use to photon energies smaller than 24.5 eV, the helium ionisation threshold. The high resolution of the FT-ICR mass spectrometer (MS) and its MS/MS capability make it a powerful tool for the study of photoionization processes on molecular ions. Up to now, only two experiments have been published, both on the CO + ion using merged-beam set-ups [START_REF] Andersen | [END_REF]45]. Such studies with merged-beam set-ups have not been further developed, partly due to the strong background which can be observed in the photoionization spectra. We will use the MICRA trap to study photoionization processes on small molecular ions, in particular of interest in planetology.

Figure 1 :

 1 photons detection

Figure 2 :

 2 Figure 2: Variation as a function of the photon energy of photoionization cross sections measured with the merged-beam set-up for the Kr + ion (upper panel) and the Xe + ion (lower panel). The solid curves give the results of our MCDF calculations for the direct photoionization process.

Figure 3 :

 3 Figure3: Enlargement of the photoionization cross section for Kr + ion close to thresholds recorded with an improved bandpass (mean value of 30 meV). The long vertical lines above the spectrum give the position of the np4 (n = 4) ionization thresholds for the ions in the np 5 2 P 3/2 ground level and np 5 2 P 1/2 metastable level[START_REF] Yu | NIST Atomic Spectra Database (ver. 4.0.0)[END_REF], and the short ones the position of the np 4 1 D 2 md Rydberg series (see text).

Figure 4 :

 4 Figure 4: Same legend as figure 3 for Xe + ion (n=5) and a mean BP = 25 meV.

Figure 5 :

 5 Figure 5: Comparison of the spectra obtained for the Kr + ion using the merged-beam set-up (upper panel) and the FT-ICR ion trap at two different delays between the production of the ions inside the trap and their irradiation by synchrotron radiation: ∆t=0 (middle panel) and ∆t=520ms (lower panel).

Figure 6 :

 6 Figure 6: Comparison of the spectra obtained for the Xe + ion using the merged-beam set-up (upper panel) and the FT-ICR ion trap (lower panel).

Table 1 .

 1 Energy, quantum defect and assigment of the Rydberg series obtained using equation (4) for the Kr + ion.

	Energy (eV)	δ		Assignment
	23.712	0.02	2 P 1/2	3 P 1 10d
	23.845	0.29	2 P 1/2	1 S 0 4d
	23.902	0.18	2 P 1/2	1 D 2 6d
	24.334	0.20	2 P 1/2	1 D 2 7d
	24.501	0.29	2 P 3/2	1 S 0 4d
	24.562	0.19	2 P 3/2	1 D 2 6d
	24.613	0.21	2 P 1/2	1 D 2 8d
	24.811	0.17	2 P 1/2	1 D 2 9d
	24.989	0.23	2 P 3/2	1 D 2 7d
			2 P 1/2	1 D 2 10d
	25.047	0.15	2 P 1/2	1 D 2 11d
	25.119	0.20	2 P 1/2	1 D 2 12d
	25.18	0.15	2 P 1/2	1 D 2 13d
	25.280	0.20	2 P 3/2	1 D 2 8d
			2 P 1/2	1 S 0 5d
	25.475	0.19	2 P 3/2	1 D 2 9d
	25.605	0.23	2 P 3/2	1 D 2 10d
	25.710	0.19	2 P 3/2	1 D 2 11d
	25.785	0.19	2 P 3/2	1 D 2 12d
	25.842	0.23	2 P 3/2	1 D 2 13d
	25.91	0.38	2 P 3/2	1 S 0 5d
	26.10	0.33	2 P 1/2	1 S 0 6d
	26.58	0.31	2 P 1/2	1 S 0 7d
	26.74	0.38	2 P 3/2	1 S 0 6d
	27.23	0.35	2 P 3/2	1 S 0 7d
	27.53	0.38	2 P 3/2	1 S 0 8d
	31.38		2 P 3/2 4s4p 5 3 P 2 5p

Table 2 .

 2 Same as table 1 for the Xe + ion

	Energy (eV)							δ						Assignment
	20.195							0.16		2 P 1/2		1 D 2 6d
	20.594							0.25		2 P 1/2		1 D 2 7d
	20.876							0.28		2 P 1/2		1 D 2 8d
	21.062							0.34		2 P 1/2		1 D 2 9d
	21.210							0.30		2 P 1/2		1 D 2 10d
	21.425							0.29		2 P 3/2		1 D 2 6d
	21.886							0.29		2 P 3/2		1 D 2 7d
	22.177							0.30		2 P 3/2		1 D 2 8d
	22.375							0.31		2 P 3/2		1 D 2 9d
	22.510							0.35		2 P 3/2		1 D 2 10d
	22.610							0.41		2 P 3/2		1 D 2 11d
	22.683							0.51		2 P 3/2		1 D 2 12d
	22.747							0.49		2 P 3/2		1 D 2 13d
	At higher photon energy, the smoothly decreasing cross sections are due to the direct
	photoionization process
	ns	2	np	5	2	P 3	/	2	1 ,	/	2	+	h ν	→	ns	2	np	4	3	P 2	, 1 ,	0	1	D 2	1	S	0	+	e	-	(5)

Acknowledgements

The authors would like to thank the support of the European Community -Research Infrastructure Action under FP6 Structuring the European Research programme (IA-SFS, contract number RII3-CT-2004-506008). They are also grateful to the ISA staff of the Aarhus University and to the staff of SOLEIL and, in particular, of the DESIRS beam line for their helpful assistance. One of us (ARM) acknowledges the support by the Ministry of Science and Technological Development of Republic of Serbia under Project No. 141011.


Ilija Stefanovi
email: ilija.stefanovic@rub.de

Nader Sadeghi

Jörg Winter

The influence of C 2 H 2 and dust formation on the time dependence of metastable argon density in pulsed plasmas

Diode laser absorption at 772.38 nm is used to measure the time resolved density of Ar*( 3 P 2 ) metastable atoms in a capacitively coupled radio-frequency (RF) discharge running in argon/acetylene mixture at 0.1 mbar. The RF power is pulsed at 100 Hz and the density of Ar*( 3 P 2 ) atoms in the 5 ms ON time and in the afterglow are recorded. Different plasma conditions, namely: 1) pure argon, 2) argon + 7% acetylene before powder formation, 3) argon + 7% acetylene after dust particles have been formed and 4) argon with dust particles remained in the plasma volume but without acetylene are studied. The measured steady-state Ar*( 3 P 2 ) density in the middle of the reactor is always about 10 times larger in dusty argon plasma than in pure argon discharge. This is mainly a consequence of the enhancement of electron temperature after dust formation. Both steady state densities and decay times in the afterglow indicate that the degree of dissociation of C 2 H 2 in the plasma volume can be as high as 99%. It is shown that in our plasma conditions on the surface of dust particles the loss of Ar*( 3 P 2 ) atoms is negligible compared to their loss by diffusion to the electrodes.

Introduction

Basic properties and application of dusty plasmas (plasmas containing nano-to micro-sized particles) have been discussed by many authors in the literature [START_REF]See a comprehensive study of dusty plasma basics and application[END_REF]. Especially, the change in plasma properties by immersing or growing dust particles in plasma was addressed. Experimental results showed that if the dust particle density was sufficiently large, the important electron loss on these particles leads to a significant decrease of electron density. This forces the plasma to "respond" by increasing the electron production rate, obtained by an enhancement of the electron temperature [START_REF] Boufendi | [END_REF]3,4]. Important changes on electrons temperature and density, as well as on optical emission properties have been reported [5,6,7,8,9]. Also, the dusty plasma models predicted differences in behaviour when comparing dusty and dust free plasmas, like change in phase shift between voltage and current (α-γ´transition) [10] or different power deposition and electron energy distribution [11][12][13]. Argon metastable atoms (Ar*) have been used to investigate the kinetics of fast electrons in numerous discharge configurations [14][15][16][17]. These atoms also served as a probe to monitor the kinetic energy of heavy particles in the plasma, deduced from the Doppler profile of an absorption line [18][19][20]. Recently, the density of Ne* in neon with dust particles immersed [21] and of Ar* in argon/acetylene with plasma-polymerized dusts has been analyzed [22]. However, the kinetics of Ar* metastables in the afterglow of a pulsed complex plasma have not yet been investigated. Pulsing the radiofrequency (RF) discharge permits to control the generation of dust particles in reactive plasmas [23]. It was found [24] that by varying the pulsed frequency and duty cycle the dust particle nucleation could be increased or suppressed. So, it is of prime importance to understand the basic properties of dusty plasma afterglow, in particular the behaviour of Ar* metastables who posses about 11.5 eV of internal energy and can initiate chemical reactions in the afterglow or induce electron detachment from negatively charged dust particles. To measure the Ar* density and its time behaviour, we used the Laser Absorption Spectroscopy (LAS) on argon 3 P 2 metastable state. The details of the experimental technique are given in the following paragraph and results relative to the time dependence of metastable density during the RF pulse and in the afterglow for different plasma conditions are analyzed and discussed in the last paragraph. With a fixed RF power, we have considered four different situations: 1-pure argon plasma (Ar); 2argon/acetylene plasma before dust formation (Ar/C 2 H 2 ); 3-argon/acetylene plasma after dust particles have been formed and grown to reach a steady-state (Ar/C 2 H 2 /dust); 4-after step 3, the plasma is maintained running but the acetylene flow is stopped. This way, dust particles remain inside "pure" argon plasma (Ar/dust).

Experiment

The detail of plasma chamber used in this work to generate the plasma has been described elsewhere [25]. The capacitively coupled (CCP) symmetrically driven radio-frequency (RF) 13.56 MHz discharge is produced by two parallel-plane 30 cm diameter electrodes located inside a 50 cm diameter 30 cm high stainless steel cylinder chamber. The electrode gap, which can be varied externally, was fixed to 7.0 cm in these experiments. The discharge was driven in cw or pulse modes, with variable pulsing frequency between 100 Hz and 1 kHz. Most results reported in this paper have been obtained at 100 Hz. Pure argon (8 sccm) or with 0.5 sccm C 2 H 2 mixture was supplied to the discharge chamber and a throttle valve in the exit port kept constant the gas pressure at 0.1 mbar. The discharge input power ranged between 10 W and 80 W and for each plasma condition the match box was adjusted to keep the reflected power to less than a few percent of the forward power. The carbonaceous nanoparticles (dust) have been formed in the process of plasma polymerization of acetylene monomer [25,26]. To follow the time variation of Ar*( 3 P 2 ) metastable atoms, we have employed the laser absorption spectroscopy (LAS) technique on 772.38 nm argon line (1s 5 -2p 7 transition in Paschen notation [27]), described in [28]. The light source was an external cavity single mode diode laser, ECDL (Sacher Lasertechnik), whose width (<10 MHz) was much smaller than the typical bandwidth of the absorption line (≅0.8 GHz). The laser frequency could be finely and accurately scanned by changing the voltage on the piezoelectric element that moved one of the cavity mirrors. The beam from the ECDL was split in three parts with beam splitters. A first beam was guided to a 20 cm long confocal Fabry-Perot interferometer (375 MHz free spectral range) and detected with a photodiode PD1, to perform precise laser frequency calibration. The 2 nd beam crossed a low pressure low current argon discharge tube and then detected with a photodiode PD2. It served for the absolute scale frequency calibration of the laser [28]. The last beam, attenuated down to <10 µW, by a set of neutral density filters to avoid optical saturation [29], was launched into the discharge chamber, passed parallel to the electrodes, and was detected on the opposite side of the chamber with the photodiode PD3 (figure 1). In steady-state studies, all three signals from PDs were simultaneously recorded and averaged in different channels of a digital oscilloscope and then treated by a computer. In time resolved studies, the signal from PD2 is used to set the laser frequency at the centre of the absorption line, where the maximum absorption is reached, and the signal from PD3 is averaged on the digital oscilloscope, triggered synchronously with the RF discharge, and hence recorded in the computer.

Results and discussion

Gas temperature

Laser absorption spectroscopy is based on Beer-Lambert's law, which in low pressure and low density plasmas, when the Doppler Broadening controls the line-shape, writes [29]:

- - = 2 D 2 0 D eff ul 2 0 0 ) ( ) ( ) 2 ln( 4 exp / ) 2 ln( 2 8 ) ) ( ) ( ln( δν ν ν δν π π λ N l A g g v I v I l u (1)
where I 0 ( )/I( ) are the detected intensity at frequency without and with absorbing atoms, A ul =5.18 10 6 s -1 is the emission Einstein's coefficient of the 0 =772.38 nm argon line [27], g u =3 and g l =5 are the statistical weights of the upper and lower levels of the line, respectively, N is the densities of absorbing atoms, l eff =30 cm is the effective absorption length (diameter of the electrodes) and D is the Doppler width of the line given by [29]:

M T M RT c Hz / 10 16 . 7 ) / ( 2 ln 2 ) / 2 ( ) ( g 0 7 g 0 D ν ν δν - ⋅ = = (2)
When the laser frequency is set at the line centre 0 , the density of Ar*( 3 P 2 ) atoms can be deduced from the measured ratio of I 0 /I, if the Doppler width, which depends on T g , is known and assuming homogeneous density in radial direction:

For T g measurements, the plasma is run in continuous regime and the laser frequency is scanned at about 25 Hz around the line centre. Figure 2 shows as an example, the absorption line profile and the F-P transmission signal, after the time scale of the oscilloscope has been converted to GHz by means of F-P peaks, separated by 0.375 GHz. In pure argon, T g values deduced from the fit are 293 K and 295 K at 15 W and 50 W of RF power, respectively. Considering the ±10 K estimated uncertainty on these values, we conclude that in pure argon plasma the gas remains at room temperature. When the laser beam was moved close to the electrode (inside the sheath or in the negative glow) the measured T g corresponded also to the room temperature. In argon with the presence of dust (4: Ar/dust) we have measured 333 K and 365 K at 15 W and 50 W of RF power, respectively. These values are 10% and 20% higher than the room temperature. But close to the electrodes, T g was found to be 306 K and 320 K at 15 W and 50 W, respectively. We believe that, this small increase of T g in the plasma bulk after dust particles are formed is due to the increase of the plasma electric field by an order of magnitude, as was reported in [9]. This induces an enhancement of the kinetic energy of ions in the bulk of the discharge. Part of this energy is transferred to argon atoms in charge exchange and elastic collisions. But near the electrodes, the gas remains in equilibrium with electrodes which are at room temperature.

Time dependent Ar* density

Figure 3 shows the time variation of Ar* ( 3 P 2 ) metastable density at the middle of the electrodes for different gas composition and at 20 W and 80 W applied RF powers. The RF modulation frequency was around 100 Hz, with 50% duty cycle. Reported densities are deduced from (3), assuming T g = 300 K. They therefore could be underestimated at maximum by 10% in dusty plasmas. We observe that in all cases the steady-state density is reached at about 1 ms after the plasma ignition (at t = 0). We will first discuss the variation of these steady-state density with plasma parameter and hence analyze changes induced by these parameters on the lifetime of Ar* atoms in the afterglow.

Steady-state density.

Under our experimental conditions, the main production mechanism of metastable atoms is the direct electron-impact excitation from the ground state argon atom. The steady state density results therefore from an equilibrium between production and loss frequencies of Ar* atoms [30]:

τ ] Ar .[ ). ( *] Ar [ e e e n T k = (4)
where k e is the electron temperature (T e ) dependent excitation rate, n e the electron density, [Ar] argon density and the lifetime of metastable atoms, which can have some n e and T e dependence [28,30]. In pure argon, the steady-state Ar* density is about 2 10 16 m -3 at 20 W and only slightly increases with RF power at 80 W. As n e should increase linearly with power, we can conclude that the enhancement of excitation frequency is almost compensated by a decay of the lifetime of Ar* atoms by electron impact de-excitation of these atoms [30].

With acetylene added to the feed gas (Ar/C 2 H 2 graphs in figure 3), Ar* density falls to 1 10 15 m -3 at 20 W. This decrease by a factor of 20 can be attributed to the efficient quenching of Ar* atoms by C 2 H 2 , for which the measured quenching rate coefficient is k q =5.6 10 -16 m 3 s -1 [31]. However, considering the respective flow rates 8 sccm and 0.5 sccm for Ar and C 2 H 2 and the 0.1 mbar total pressure, the estimated density of C 2 H 2 would be 1.5 10 20 m -3 , which should result in a quenching frequency of Ar* by C 2 H 2 of ν q 8 10 4 s -1 . As it will be discussed in the next section, the corresponding 12 µs lifetime is more than 2 orders of magnitude smaller than the measured lifetime of Ar* in pure argon. Hence considering that adding C 2 H 2 can not introduce an important enhancement of n e and T e , according to (4) the effective decrease in Ar* density should be even larger. We believe that due to the high degree of dissociation of C 2 H 2 by the discharge, its density in the plasma bulk is much lower than the estimated value from relative flow rates and pressure. At 80 W, the Ar* density is reduced by only a factor of 4 when C 2 H 2 is added to argon. In fact, the degree of dissociation of acetylene increases with RF power. After dust particles are formed in Ar/acetylene discharge (Ar/C 2 H 2 /dust graphs in figure 3), Ar* density increases significantly and even overpasses its value in pure argon. This important enhancement has several origins. First, in the presence of dust particles the plasma becomes more resistive and the RF power dissipated in the bulk increases [9,12]. Second, the excitation zone which was near the electrodes in the wave riding α mode, becomes more homogeneous with dusts [11,13].

So increasing the Ar* production term in the halfway from electrodes, where data shown in figure 3 have been taken. Finally, with higher T e in dusty plasma, the degree of dissociation of C 2 H 2 becomes larger, hence inducing an enhancement of the lifetime of Ar* atoms. Close to the electrode (data not shown), the rise of Ar* density with dust is much less spectacular than in the plasma centre.

Stopping the C 2 H 2 flow leaves dust particles in the plasma and induces further rise of Ar* density (Ar/dust graphs in figure 3). In fact, Ar* is no more quenched by C 2 H 2 and its lifetime increases. As expected, this rise is more pronounced in 20W plasma than in 80W plasma because C 2 H 2 was less dissociated in the former case. Measured densities in argon with dust are almost an order of magnitude larger than in pure argon. The origin of this large difference relays on the enhancement of T e combined to the spatially homogeneous excitation of Ar* atoms in γ' regime, associated to the presence of dust, whereas in pure argon the excitation zone is localized near the electrodes, α regime [11].

Lifetime in the afterglow.

After the RF plasma is switched off, production of Ar* atoms stops and their density decays almost exponentially by diffusion to the electrodes and by reaction in the plasma volume with other particles. In pure argon, using the diffusion coefficient of Ar* atoms in Ar D 0 .[Ar]=1.8 10 20 m -1 s -1 [28] and considering our experimental conditions, the loss frequency of Ar* by diffusion to the electrodes is ν d =150 s -1 . At 0.1 mbar, loss frequencies by two and three body collisions with argon are 5 s -1 and 1 s - 1 , respectively [28], hence negligible. The lifetime deduced from decay curves of Fig. 3 With C 2 H 2 stopped and dust particles remained in the discharge (Ar/dust), the lifetime reaches almost its value in pure argon, remaining however always slightly lower. This lower value could either be due to the lost of Ar* atoms on dust particles or to quenching by small amount of carbonated molecules desorbed from reactor walls and electrodes. To rule out the first possibility, we can estimate the related loss frequency, ν D , assuming a density N D =1 10 12 m -3 of dust particles of 30 nm radius. These are the characteristic values measured by light scattering in our reactor and working conditions [32] and would correspond to a surface of about S D =1 10 -14 m 2 for each particle. A rough calculation, assuming that the number of atoms of mean velocity w impinging on a surface s is 1/4×N×<w>s, would provide:

1 D D * Ar D 1 ) 4 1 ( - ≅ > < = s S N w ν (5)
where <w Ar* >=400 m/s is the mean velocity of Ar* atoms impinging on dust particles. In fact, this loss frequency on dust is much smaller than the loss by diffusion, ν d =150 s -1 and is hence negligible. Do et al [21] have observed the quenching of Ne* metastable atoms in the sheath boundary, where dust particles were concentrated. However, in that work, if the density of dust particles were comparable to ours, the 10 µm size of their SiO 2 particles corresponds to more than 4 orders of magnitude larger surface, highlighting the influence of dust particles on quenching of Ne* atoms. An important point that we would like to point out is the faster decay of Ar* metastable atoms in the few 100s of µs following the end of the discharge in 80 W pure argon (Ar graph in Fig. 3b). Almost absent in the 20 W pure argon plasma, this early fast decay is also missing in the 80 W Ar/dust plasma. We attribute this decay to the quenching of Ar* atoms by plasma electrons, which in fact transfer Ar* metastables to the Ar* ( 3 P 1 ) resonant state (for more detail see refs. 26 and 28):

Ar* ( 3 P 2 ) + e Ar* ( 3 P 1 ) + e (6) This reaction is endoergic by only 75 meV and can be very efficient in the early afterglow. For T e above 0.2 eV, the reported rate coefficient of k t =2 10 -13 m 3 .s -1 [28,30] would correspond to a decay frequency of ν t =1000 s -1 for n e = 5 10 15 m -3 . In reality, due to the radiation trapping in large size plasmas, the density of atoms in the 3 P 1 resonance state remains high and the reverse process of (6) brings back to the Ar*( 3 P 2 ) metastable state a part of the transferred atoms. As a consequence, the overall rate coefficient of electron impact quenching of Ar*( 3 P 2 ) atoms is reduced to a few times 10 -14 m 3 s -1 (see [30] for discussion), resulting in a decay frequency of a few hundreds s -1 , comparable to the decay frequency of Ar* by diffusion. In the pure argon 80 W plasma the above mentioned n e is reached [33] but in 20 W plasma n e is about 4 times smaller and an order of magnitude smaller in the presence of dust particles [32][33][34]. Consequently, the effective depopulation of Ar*( 3 P 2 ) state by electrons will be negligible in these later cases.

Conclusion

In this work, we have shown that Ar*( 3 P 2 ) density can be an excellent probe of dust formation in argon/acetylene plasmas. Due to the enhancement of the electron temperature and the change of discharge from α regime to γ' regime after dust formation in the plasma, the steady-state population of Ar*( 3 P 2 ) metastable state becomes an order of magnitude larger in dusty argon plasma than in pure argon discharge. Measured decay times of metastable atoms in the afterglow indicate that C 2 H 2 is highly dissociated in the plasma volume, its degree of dissociation reaching 99% at 80 W argon/acetylene discharges`after powder formation. Our results also show the depopulation in the early afterglow of metastable state by electron impact transfer to the resonant Ar*( 3 P 1 ) state. But under our experimental conditions, the quenching of metastable atoms on the surface of dust particles can be neglected.

Figure1.

  Figure1. Experimental set-up, top view. ECDL -External Cavity Diode Laser, F-P -Fabry-Perot etalon, RC -reference cell (low pressure low current Ar discharge), BS -Beam Splitters, PD1, 2, 3 -Photodiodes.

Figure 2 .

 2 Figure 2. Fabry-Perot transmission signal with 0.375 GHz peak separation (lower part) and absorption line profile at two different discharge powers of pure argon plasma: 15 W (triangles) and 50 W (solid circles). Gaussian fits to experimental points (solid lines) provides the gas temperature (T g ) and metastable density (N m ): N m = 1.22×10 16 m -3 , T g = 293 K at 15 W and N m = 1.65×10 16 m -3 , T g = 295 K, 50 W.

Figure 3 .

 3 Figure 3. Ar* ( 3 P 2 ) metastable density versus time for (a) 20 W and (b) 80 W input powers. Different lines/symbols are for different gas mixtures: solid line -pure Ar; open squares -Ar/C 2 H 2 ; solid circles -Ar/C 2 H 2 /dust; crosses -Ar/dust. The vertical dot-dashed line indicates the OFF time of the RF power.

  is about 3.4 ms, indicating the presence of a small amount of impurity (probably C 2 H 2 trace left in the reactor chamber). The very fast decay of Ar* density when acetylene is added to the feed gas results from its quenching by C 2 H 2 . However, the measured lifetimes, about 200 and 500 µs at 20 W and 80 W, respectively, are much larger than the expected 12 µs with the amount of C 2 H 2 introduced into the chamber. This proves that in the plasma volume, the dissociation rate of C 2 H 2 is about 95% at 20 W and approaches 98% at 80 W. After the apparition of dust particles, the lifetime increases up to about 1.2 ms at 80 W and it is again larger than at 20 W. This large lifetime indicates that in the bulk plasma C 2 H 2 is dissociated to about 99%. The enhancement of the electron temperature should cause the observed high degree of dissociation of acetylene.

Acknowledgments

This work was supported by the DFG through Project No. WI 1700/3-1, Graduierten Kolleg 1051 and Research Department "Plasma with Complex Interactions", Ruhr Universität Bochum.


Olivier Pinel

Julien Fade

Daniel Braun

Pu Jian

Nicolas Treps

Claude Fabre

Ultimate sensitivity of precision measurements with Gaussian quantum light : a multi-modal approach

Keywords: numbers: 03.65.Ta, 42.50.Ex, 42.50.Lc, 42.50.St

Multimode Gaussian quantum light, which includes multimode squeezed and multipartite quadrature entangled light, is a very general and powerful quantum resource with promising applications in quantum information processing and metrology. In this paper, we determine the ultimate sensitivity in the estimation of any parameter when the information about this parameter is encoded in such light, irrespective of the information extraction protocol used in the estimation and of the measured observable. In addition we show that an appropriate homodyne detection scheme allows us to reach this ultimate sensitivity. We show that, for a given set of available quantum resources, the most economical way to maximize the sensitivity is to put the most squeezed state available in a well-defined light mode. This implies that it is not possible to take advantage of the existence of squeezed fluctuations in other modes, nor of quantum correlations and entanglement between different modes.

Optical techniques are widely used in many areas of science and technology to make accurate measurements and diagnostics, from microscopy, spectrography, chemical analysis, to gravitational wave detection and ranging. There are many reasons for this: light allows us to extract information in a remote and non destructive way, it carries information in a massively parallel way, and perhaps more importantly, optical measurements can reach very high precision and sensitivity levels.

It is therefore important to know what is the ultimate limit of sensitivity that can be possibly achieved in the estimation of a parameter θ that is encoded by one way or another in a light beam, given some constraints such as a fixed mean photon number N . This limit is imposed by the unavoidable quantum fluctuations of light and depends on the quantum state of light which conveys the information about θ. When the light is in a coherent state, this limit is called 'standard quantum limit' and scales as 1/N 1/2 .

Many studies have been devoted to finding ways to enhance the sensitivity of parameter estimation beyond the standard quantum limit using quantum resources. It has been shown that enhanced sensitivity can be achieved by using squeezed light [1] or entangled light [2]. This has been first experimentally demonstrated for measurements in which the information about the parameter θ is carried by the total intensity [3] or by the phase [4] of a light beam. Later situations were considered where the parameter θ does not change the total intensity of the light but modifies the details of the repartition of light in the transverse plane [5] (for example to estimate a very small lateral displacement of a beam [6]). As the energy of the squeezed state increases with the squeezing factor, the ultimate limit with squeezed state for a fixed total energy scales as 1/N 3/4 .

If one uses instead entangled states such as NOON states [7] one reaches the so-called Heisenberg-limit (HL) which scales as 1/N . However, in the present state of technology real measurement schemes using these states do not lead to very high sensitivities, because of the small values of N experimentally reachable (so far, the highest achievable NOON state has N ∼ 100 [8]), and decoherence tends to rapidly destroy these states, therefore limiting the performance of the measurement to a 1/N 1/2 scaling for large N [9][10][11]. In [12] a scheme was proposed that reaches the HL without the use of an entangled state. This paper tackles the problem of optimized parameter estimation in a more practical way. Light is considered as a probe to measure a parameter of a physical system (see Fig. 1). As in that case all quantum limits scale as some inverse power on N , it is very important to consider states with very high N values. It turns out that, so far, only multimode Gaussian states are the available non classical states of light with very high mean photon number. They include quantum resources like multimode squeezing and multipartite entanglement that are widely used in quantum optics and quantum information processing. These states are already generated experimentally with impressive amounts of squeezing [13] and entanglement [14] shared by many modes [15]. When they include a coherent state in one of the modes, the mean photon number N can be easily as large as 10 16 [16].

The originality of the present approach is its multimodal character. A multimode quantum state is defined not only by the value of the coefficients of its decomposition on the multimode Fock state basis |n 1 , n 2 , ..., n ℓ but also on the spatio-temporal shape of the different modes on which these Fock states are defined. This leaves us two kinds of degrees of freedom on which to act: as we will see below, the ultimate sensitivity is obtained not only by choosing the best possible Gaussian quantum state, but also by putting this state in an optimized mode basis.

Expression of the Quantum Cramer Rao bound for pure states -Our aim is to measure the smallest possible variation of a parameter θ around a given value that we take to be 0 by an appropriate change of origin. The quantum state which contains the information about this parameter is described by a density matrix ρθ . The error in the estimation of θ based on Q repeated measurements of an observable  on this state is given by [17]

δθ = δA 2 est 1/2 θ √ Q ∂ ∂θ A est θ , (1)
where A est is an unbiased estimator of θ that depends on the results of the measurements of Â. By optimizing over all estimators A est and all measurements, Braunstein and Caves [17] showed that the best achievable sensitivity for measuring a small variation of θ is bounded by the socalled quantum Cramér-Rao (QCR) bound

δθ ≥ δθ min ≡ 2 Q s(ρ θ , ρθ+dθ ) dθ -1 , (2)
where s(ρ θ , ρθ+dθ ) is the Bures distance between ρθ and ρθ+dθ , which, in the case of pure states |ψ 1 and |ψ

2 is equal to 2(1 -| ψ 1 |ψ 2 |).
Let us now consider a pure quantum state of light |ψ θ spanning over M different spatial or temporal modes {v i (r, t)} (i = 1, ..., M ). For mixed states with parameter independent mixing probabilities, the sensitivity can at most be as good as for the pure states from which it is mixed [18]. We call âi the annihilation operator in the mode v i , and introduce the quadrature operators xi = âi + â † i and pi = i(â † i -âi ). We define the column vectors x = (x 1 , . . . , xM ) ⊤ , p = (p 1 , . . . , pM ) ⊤ , and X = (x, p) ⊤ .

The overlap between the states |ψ θ and |ψ θ+dθ reads:

| ψ θ |ψ θ+dθ | 2 = (4π) M W θ (X)W θ+dθ (X) d 2M X, (3)
W θ being the Wigner function of |ψ θ :

W θ (x, p) = 1 (2π) M e iξ.p x-ξ |ψ θ ψ θ | x+ξ d M ξ (4)
At second order in dθ, the overlap is equal to

| ψ θ |ψ θ+dθ | 2 ≃ 1 - dθ 2 2 (4π) M (W ′ θ (X)) 2 d 2M X .
(5) The first order vanishes because the states are pure. Throughout this letter, for any function depending on the parameter θ, we use the convention

f ′ θ ≡ ∂f ∂θ θ=0
, regardless of what other explicit variables f might depend on.

This leads to the QCR bound for pure states

δθ min = 2Q (4π) M (W ′ θ (X)) 2 d 2M X -1/2 . ( 6
)
This intermediate result is very interesting as it gives a simple expression of the QCR bound for any pure quantum state. In the remainder of this paper, we will apply this formula to Gaussian states. QCR bound for pure Gaussian states -For a Gaussian state |ψ θ , the Wigner function takes the form

W θ (X) = 1 (2π) M exp - 1 2 (X -X θ ) ⊤ Γ -1 θ (X -X θ ) (7)
where X θ is the column vector of the expectation values of the quadratures for the different modes, and Γ θ the symmetrized covariance matrix: Γ θ,[i,j] = 1 2 ( X i X j θ + X j X i θ ). As we treat the problem in all its generality, both possibly depend on θ. One finds from ( 6)

δθ min = Q -1/2   X ′⊤ θ Γ -1 θ X ′ θ + tr Γ ′ θ Γ -1 θ 2 4   -1/2 . ( 8
)
The expression in the outermost bracket of Eq. ( 8) corresponds to the quantum Fisher information I Fisher for a pure Gaussian state. It consists of two terms which represent the information about θ that can be extracted respectively from the mean field and from the noise. In the limit of very large values of N , more precisely when the quantum field fluctuations are so small compared to N that one can treat them to first order, the second term turns out to be negligible compared to the first, and we will neglect it from now on. This approximation is a consequence of the practical approach we consider in this paper and corresponds to realistic experimental implementations. Let us stress that such a linearization procedure has been widely used in the literature to determine the Gaussian quantum state which is produced by nonlinear effects such as parametric down-conversion or four-wave mixing.

Let us now use our freedom of choice of the mode basis in which to describe the quantum state: we will see that I Fisher can be expressed in more physical terms if one introduces a mode basis { v i (r, t)} specific to our problem. We first define the normalized mean photon field mode as:

u θ (r, t) = a θ (r, t) a θ (9)
where â(r, t) = i âi v i (r, t) is the local annihilation operator, a θ (r, t) = ψ θ | â(r, t) |ψ θ the mean photon field, and a θ its norm:

a θ = |a θ (r, t)| 2 d 2 rdt 1/2 , ( 10
)
where the spatial integration is made over a surface perpendicular to the light beam propagation, and the time integration over the detection time. In the limit of a narrow-band field, the mean photon field mode u θ is proportional to the mean value of the electric field in the θ-dependent quantum state.

We can now define the detection mode by

v 1 (r, t) = a ′ θ (r, t) a ′ θ . ( 11
)
One then completes the basis starting with mode v 1 by other orthonormal modes v n>1 . The modes v n do not depend on θ since the derivative in (11) has been taken at the value θ = 0

The expression of the Fisher information in the { v i (r, t)} mode basis is very simple as it involves only one matrix element of Γ -1 θ :

I Fisher = 4Γ -1 θ=0,[1,1] a ′ θ 2 ( 12
)
where Γ -1 θ=0, [1,1] is the first left, top element of the matrix Γ -1 θ in the basis { v i (r, t)} taken at the value θ = 0 of the parameter.

In particular, the Fisher information for a single surement involving a coherent state (Γ θ = 1), that we will call I 0 , is found to be

I 0 = 4 a ′ θ 2 = N θ 4 u ′ θ 2 + N ′ θ N θ 2 , ( 13
)
where N θ = a θ (r, t) 2 is a quantity that tends to the mean photon number N in the high N limit where fluctuations can be linearized. We obtain finally the following expression of the QCR bound for parameter estimation using quantum Gaussian states

δθ min = QN θ 4 u ′ θ 2 + N ′ θ N θ 2 Γ -1 θ,[1,1] -1/2 . ( 14
)
It depends on 3 factors: the first one is as usual the mean total number of photons measured QN θ . The second one is related to the variation as a function of θ of the displacement of the mean field mode and the mean photon number. The more the light properties are affected by the variation of θ, the better the sensitivity one can expect for its estimation. While the general argument is obvious, the explicit formula ( 14) is not. The last factor is the influence on the measurement of the quantum fluctuations of the state, which is remarkably contained in a single element of the inverse covariance matrix in our specific mode basis.

Optimized multimode Gaussian state for parameter estimation -Let us now discuss under which conditions nonclassical multimode Gaussian states can be put to best use in the estimation of θ. We will take the point of view of an experimentalist who wants to use the minimum possible amount of quantum resources that allow him to reach the QCR bound. He will start from the simplest way known to date to generate multimode quantum Gaussian states [19], which consists in linearly mixing several single mode squeezed beams produced by independent "squeezers", such as degenerate parametric amplifiers. We will call σ 2 min the smallest quadrature noise among all these squeezed modes. σ -2 min is the largest eigenvalue of the inverse covariance matrix in the initial basis of the independent squeezed modes. With the help of linear couplers i.e. of a θ dependent unitary transformation of the mode basis, the multimode squeezing can be transformed into a multimode entangled/squeezed Gaussian state in a mode basis the spatio-temporal shape of which can also be tailored at will [20]. One can show that, under such unitary transformations, the diagonal matrix elements of the inverse of the covariance matrix are bound by the spectral radius of Γ -1 θ=0 , which is equal to 1/σ 2 min . Equality is reached only if the detection mode 1 is an eigenmode of the covariance matrix with the eigenvalue σ 2 min , and thus when the most squeezed state is put in the detection mode, with no quantum correlations with any other mode. The QCR bound corresponding to the quantum resources that we have just described is thus

δθ min = σ min √ QN θ 4 u ′ θ 2 + N ′ θ N θ 2 -1/2 . ( 15
)
We have shown here an important result: the only way to saturate the Cramér-Rao bound in the configuration that we have just described is to put the most squeezed state available into the detection mode and not to have correlations with the other modes. The presence of other squeezed modes, or of any kind of entanglement, will not help to improve the sensitivity: one cannot take advantage of squeezed fluctuations or quantum correlations coming from different modes to improve the estimation of a single parameter [21]. We therefore advise experimentalists to produce a single vacuum squeezed state, to put it in the detection mode, and to mix it with a coherent state of high mean photon number N in the mean photon field mode u θ (r, t) . Doing that, they will be sure that nobody else will make a more sensitive estimation of the variation of θ around 0 for a given shape u θ (r, t) of the mean field.

A possible experimental implementation that reaches the QCR bound -The determination of the Quantum Cramér-Rao bound is very general and does not tell us which kind of detection, and which kind of measurement strategy is to be used in order to reach it. We show in this paragraph that a homodyne detection scheme in which the local oscillator is precisely taken in the detection mode allows us to reach the QCR bound.

If one uses an intense local oscillator in mode v 1 , the balanced homodyne detection operator, for a null relative phase between the local oscillator and the measured beam, is given by D = x1 √ N LO , where N LO is the mean photon number of the local oscillator and x1 the real quadrature operator of the mode v 1 . A balanced detection set-up therefore allows us to measure the projection of a multimode field on the oscillator mode, even in presence of many other modes.

For a small variation of the parameter θ around 0 the mean value of the homodyne signal is given by: (20) which is indeed equal to the QCR bound (15) for a single measurement.

In conclusion, we have derived the expression of the ultimate limit for parameter estimation using pure Gaussian multimode states. We have shown that this limit can be reached with the help of a balanced homodyne detection scheme. We have also shown that multimode squeezing and multipartite entanglement are of no help, and that it is very important to shape in the best way the mode in which to put the non-classical Gaussian state in order to reach the ultimate limit in the most economical way. These results are good news for the experimentalists because single mode highly squeezed Gaussian states can be readily generated experimentally and because a simple homodyne detection scheme, easily achievable in a laboratory, is sufficient for reaching the best possible sensitivity.

FIG. 1 .

 1 photodetection and information processing

We acknowledge the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under the FET-Open grant agreement HIDEAS, number FP7-ICT-221906, and of the ANR project QUALITIME.


Marco Abbarchi

Vincenzo Ardizzone

Thimotée Lecomte

Aristide Lemaitre

Isabelle Sagnes

Pascale Senellart

Jacqueline Bloch

Philippe Roussignol

Jérôme Tignon

Timothee Lecomte

Jerome Tignon

One

Keywords: numbers: 78, 47, 1p, 71, 36, 1c

come

I. I: INTRODUCTION

Parametric scattering processes (PSPs) and optical parametric oscillation (OPO) have been intensively studied for their numerous applications in non-linear optics. [START_REF] Shen | The Principles of Nonlinear Optics[END_REF] Parametric amplification, generation of new laser frequencies, twin beams, and photon entanglement are important features of PSPs and OPO. [START_REF] Shen | The Principles of Nonlinear Optics[END_REF][START_REF] Armstrong | [END_REF][3] However, poor conversion efficiency, large cavity extension and driving laser power are typical drawbacks of conventional PSPs and OPO. The increasing need of efficient and miniaturized light sources for scalable quantum computation and quantum information protocols [START_REF] Bouwmeester | The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, QUantum Computation[END_REF] stimulated the recent realization of OPO in semiconductor microcavities (MCs) [START_REF] Kavokin | Microcavities[END_REF] toward the realization of electrically-driven quantum emitters in solid state. [START_REF] Fürst | [END_REF] Original proposals for the exploitation of parametric phenomena in planar MCs (2D-MCs) were based on the strong lightmatter coupling (SC), which rules the shape of upper and lower polariton branches (UP and LP respectively) dispersions in energy-momentum space. [START_REF] Kavokin | Microcavities[END_REF]7,8 The excitation at the inflection point of the S-shaped LP branch allows the efficient realization of PSPs with energy and momentum conservation (phase matching conditions, PMCs). 7 It is now well established that very large χ 3 polaritonic non-linearities in 2D-MCs can be used in order to achieve low-threshold OPO between non degenerate signal and idler beams. Nevertheless, this approach suffers from intrinsic drawbacks: (i) the required strong light-matter coupling is usually achieved only at low temperature, (ii) the pump laser must be adjusted to a specific angle, (iii) signal and idler beams are intrinsically strongly unbalanced, thus precluding realistic applications and possible study of quantum correlations.

More recently interbranch parametric scattering in vertical multi-2D-MCs allowed for relevant improvements in the exploitation of 2D-MCs-based OPO: 9,10 relaxing the requirement of SC, multi-2D-MCs permit the generation of signal and idler beams at higher operating temperatures in the weak coupling regime (WC). 10 In this case the twin beams are degenerate in momentum and split in energy (vertical process), preventing an easy spatial separation of the generated signal and idler beams. In addition, with this method, the PMCs impose an equal energy spacing in energy between the three photonic bands and still the extraction efficiency of the idler beam is poor preventing the generation of balanced beams.

In this contribution we show a different approach for the generation of OPO in wire-shaped one-dimensional microcavities (1D-MCs). 11 The multiplicity of the 1D-MC photonic bands 12 and their fine structures 13 allow for a versatile engineering of different scattering channels and of twin beams polarization properties. The choice of horizontal scattering processes (signal and idler degenerate in energy and split in momentum space) relaxes the requirements of SC 10,11 and does not require an equal spacing of the photonic/polaritonic modes splitting, necessary for achieving vertical OPO. 10,13 Most importantly, the twin beams generated in horizontal OPO have intrinsically balanced intensity.

We performed polarization resolved measurements both under non-resonant and resonant excitation, at normal incidence. Signal and idler linear polarization may be opposite to that of the excitation laser demonstrating the generation of Rayleigh-free photon beams via polarization inversion mechanism. The onset of a threshold in the power dependence of signal and idler intensities obtained in horizontal processes demonstrates the realization of balanced OPO both at 10 K in SC as well as 100 K in the WC.

The investigated sample is grown in a molecular beam epitaxy reactor and is based on a λ/2 2D-MC composed of Ga0.05Al0.95As. 14 The cavity is sandwiched between top and bottom Bragg mirrors (respectively 26 and 30 pairs of Ga0.05Al0.95As/Ga0.80Al0.20As layers). The active medium is constituted of three stacks of four GaAs quantum wells (width 7 nm), placed at the antinodes of the cavity mode (one group at the cavity center and the other two at the first antinodes in each Bragg mirror). The layers thickness is wedged, allowing for a fine tuning of the relative energy between the MC photonic mode energy at -→ k = 0 (E2D) and the excitonic resonance (EX). Hereafter E2D -EX is referred as detuning δ. The 1D-MCs are fabricated by processing the planar 2D-MC by reactive ion etching (the long axis of the wire is set orthogonal to the wedge thus δ is kept constant along the 1D-MC). Different set of 1D-MCs having width L ⊥ from 3 µm to 7 µm and same length L = 1000 µm are etched on the same sample. A spacing of 6 µm between two adjacent 1D-MCs allows the investigation of individual emitters.

The CW excitation is provided by a tunable Ti:Sapphire laser focused in a long and sharp spot (∼100 µm by ∼4 µm in full width at half maximum) while its polarization is set perpendicular to the wire axis (direction ⊥). The sample is cooled down to cryogenic temperatures in a cold-finger, liquid-helium cryostat allowing for a fine tuning between 5 K and 300 K.

The far field radiation spectrum of single 1D-MCs is investigated with a method based on the Fourier space imaging (see Fig. 1 a)): the image of the back focal-plane of a micro-objective lens (NA = 0.4, focal length f = 16 mm) is focused on the exit slit of a 50 cm monochromator and detected by a silicon based CCD camera (the spectral resolution is about 100µeV).

In our experiment the 1D-MC long axis (direction ) is set at ∼45 degrees with respect to the entrance slit of the spectrometer (see Fig. 1 b)) making possible the simultaneous detection of the band dispersion of even and odd modes which in turn present an intensity maximum or a minimum at -→ k = 0. All the spectra are discriminated in polarization in order to detect fine structure which represents a key feature of our experiment.

Due to the discontinuity in the refractive index in the direction ⊥, the electric field is also laterally confined. [START_REF] Kavokin | Microcavities[END_REF] This discontinuity changes the photonic mode of the original 2D-MC in a set of modes indexed by j = 0, 1, 2, . . . with energy at -→ k = 0 given by

E j 1D = [E 2D ] 2 + [( c/n C )(π/L ⊥ )(j + 1)] 2
where n C is the refractive index of the cavity material (for this reason we refer to the detuning δ as the energy distance between E X -E j=0 1D ). 16,17 power (P Exc ≃ 2 mW) at 10 K and 100 K, respectively (note that the scale of the horizontal axis (θ Y ) takes into account the relative angle between the 1D-MC axis and the spectrometer entrance slit). The detected polarization in the left (right) panel is (⊥) to the 1D-MC axis. At 10 K the SC is easily achieved by choosing a moderate detunig (δ ≃ -5 meV). At 100 K and large detuninig (δ ≃ -35 meV) the band dispersions are well approximated by hyperbolas (see Fig. 1 d)) showing that the observed cavity modes are prevalently photon-like and the system is in WC. However, when reducing δ, we observe strong reduction in the intensity of the cavity modes masking all the far field features and preventing an efficient PSP (not shown). In fact at 100 K the investigation is limited to δ 20 meV.

Fig. 1 e) and f) shows the two polarized spectra at θ Y ≃ 5 degrees (with this choice we can display the emission of modes with odd j and eventually remove the unwanted laser stray light entering in the spectrometer). As already discussed in Reference 13, polarization fine structures are present: each mode is split in two sub-branches (with linear polarization and ⊥ at lower and higher energy, respectively). The splitting (∆ j = E j 1D (⊥) -E j 1D ( )) was ascribed to mechanical stress due to the different thermal expansion coefficients of the copper cold finger of the cryostat (where the sample was glued at room temperature) and that of the GaAs 1D-MC sample. 13 At 10 K and SC the measured value of ∆ j are quite large and increase with j (∆ 0 = 0.86 meV, ∆ 1 = 1.02 meV and ∆ 2 = 1.23 meV). At 100 K and WC the polarization splitting is strongly reduced as evidenced in Fig. 1 e): while ∆ 1 is still observable, ∆ 0 is negligible (∆ 0 = 0.1 meV, ∆ 1 = 0.26 meV, ∆ 2 = 0.57 meV). In previous reports the large value of ∆ j allowed both to investigate the polarization dynamics of polaritons modes and enabled the realization of vertical OPO via linear polarization inversion mechanism. 13 Based on the same idea we can obtain balanced twin beams in horizontal OPO as discussed later.

Let's now discuss the resonant excitation regime and the generation of OPO. Fig. 2 a) shows a colour-intensity map of a 1D-MC dispersion for polarization and describes the scheme used for achieving PSPs at 10 K. The value of ∆ 0 , much larger then the line broadening associated to the modes 0 ⊥ and 0 (see Fig. 1 d)), permits the horizontal PSPs (0 ⊥ , 0 ⊥ ) → (0 , 0 ) by resonantly exciting the 1D-MC at the minimum of the 0 ⊥ band at -→ k = 0 (this process is schematically depicted on top of Fig 2 a)). The inset in Fig 2 a) shows the corresponding measured process resolved in energy and angle while Fig 2 c) shows the full far field, where the two elongated and bright spots (S and I) are parallel to the laser spot (L) and perpendicular to the wire axis.

At 100 K the small value of ∆ 0 (see Fig. 1 d)) does not allow for an easy separation of S and I from the laser spot whenever exciting on the j = 0 mode. The small spread in angle of S and I associated to the process (0 ⊥ , 0 ⊥ ) → (0 , 0 ), would prevent a clear imaging. As depicted on top of the measured dispersions at 100 K (see Fig. 2 c)) we can exploit the larger splitting ∆ 1 and resonantly stimulate the process (1 ⊥ , 1 ⊥ ) → (0, 0)) by pumping at the minimum of 1 ⊥ at -→ k = 0 (in this case both final states in the modes 0 ⊥ and 0 are possible 13 ). The inset in Fig. 2 b) and Fig. 2 d) show the scattering process resolved in energy-angle space and the corresponding full far field, respectively.

In order to clarify the origin of the measured scattering we perform polarization resolved measurements under resonant excitation which are summarized in Fig. 3. In the top panel is shown the case of PSP at 10 K where we observe the bright S and I emission at θ Y = ±7 degrees only for polarization. In particular this demonstrates a complete polarization inversion being the ratio of the intensity in the two modes ⊥ / smaller than 0.5%. In fact, among all the possible PSPs the observed (0 ⊥ , 0 ⊥ ) → (0 , 0 ) is the only allowed by the PMCs. 13 At 100 K the picture is quite different: as shown in the bottom panel of Fig. 3 relevant scattering is present for both polarizations ⊥ and at θ Y = ±10 degrees (note that the two bands 0 ⊥ and 0 are largely overlapped both in energy and momentum). This is consistent with the fact that both (1 ⊥ , 1 ⊥ ) → (0 , 0 )) and (1 ⊥ , 1 ⊥ ) → (0 ⊥ , 0 ⊥ ) PSPs are allowed by the PMCs. 13 The relatively large intensity of S and I for polarization demonstrates an efficient polarization inversion mechanism populating the mode 0 which intensity is at least the 50% of that in 0 ⊥ .

The separation in angle between S and I in the process (0 ⊥ , 0 ⊥ ) → (0 , 0 ) at 10 K is smaller than that in (1 ⊥ , 1 ⊥ ) → (0, 0) at 100 K. This is consistent with the smaller value of ∆ 0 with respect to the energy separation E j=1

1D -E j=0 1D which in turn rules the spread in angle of the final states of PSPs at 10 K and 100 K respectively.

Further information about the nature of the observed S and I in the 0 mode can be gathered by power dependence measurements performed in the same resonant conditions previously described. From the full far field images (see Fig. 2 c) and d)) we extract the total intensity of S and I as a function of P Exc . The results of this analysis at 10 K (100 K) are displayed in left (right) panel of Fig. 4. Both the cases of SC and WC present two distinct regimes: at low P Exc the detected intensity in the 0 mode is dominated by laser scattering and incoherent relaxation (barely linear with P Exc ) while above the OPO threshold value (P T h ) it rapidly increases. By taking into account the dimension of the laser spot on the sample surface we estimate the measured P T h at 10 K and 100 K in ∼6 KW/cm 2 and ∼14 KW/cm 2 , respectively. More generally the measured values of P T h range between ∼1 KW/cm 2 for δ ≃ -5 meV up to ∼60 KW/cm 2 for δ ≃ -20 meV. This finally demonstrates the generation of OPO of signal (S) and idler (I) beams with equal intensities, as well as relevant spatial separation, in SC and WC.

In conclusion we have shown the realization of OPO in 1D-MCs based on horizontal PSPs. The modes fan of 1D-MCs joined with large linear polarization splitting ∆ j permits an efficient engineering of the parametric scattering channels and allows the generation of OPO both in the SC and in the WC at low and high temperature. The twin beams generated in horizontal processes are intrinsically balanced in intensity and well separated in momentum space. Thanks to the polarization inversion mechanism we can manipulate the polarization of S and I which can be cross-polarized respect to the excitation laser. In the present method, differently from vertical OPO, different interband and sub-band splitting is a tuning tool for setting proper S and I angular spread. Note that the limit of 100 K for achieving OPO is related to the particular sample in study which was optimized for low temperature investigation. Higher operating temperatures could be in principle reached with appropriate modes energies E j 1D , detunings δ and by tailoring the polarization splitting ∆ j .

FIG. 1 .

 1 Photon energy (eV)

FIG. 2 .FIG. 3 .FIG. 4 .

 234 Photon energy (eV)