M C Simon

M Schwarz

S W Epp

C Beilmann

L Schmitt

Z Harman

M Baumann

B L Schmitt

T M Baumann

P H Mokler

S Bernitt

R Ginzel

S G Higgins

C H Keitel

R Klawitter

K Kubiček

V Mäckel

J Ullrich

J R Crespo López-Urrutia

Photoionization of N3 + and Ar8 + in an electron beam ion trap by synchrotron radiation

Keywords: numbers: 32.80.Fb, 95.30.Dr, 37.10.Ty

  L'archive ouverte pluridisciplinaire

Introduction

The interaction of light and matter through photoionization (PI) of atoms and ions is a truly fundamental process and many efforts have led to a large body of experimental and theoretical work for atoms and also for ions in low charge states. Comprehensive databases have been setup up, such as the Opacity Project (TOP) [1], the Iron Project [2], and the FERRUM project [3], which have collected the calculations of international collaborations. Unfortunately, for highly charged ions (HCIs) a multitude of sometimes contradictory predictions have only a scarce foundation of accurate laboratory data until now. The main reason for this status quo is the fact that preparation of HCI targets of sufficient density and stability still presents a serious experimental challenge. Our present work introduces a novel experimental method for precise PI measurements using HCIs having high ionization potentials and charge states which were beyond reach until now. Moreover, we performed new, extended fully relativistic calculations in the large-scale multiconfiguration Dirac-Fock (MCDF) frame, which compare well with the present experimental data.

In astrophysics, PI of HCIs has been recently observed in high resolution X-ray spectra of quasars, which display absorption lines due to the presence of HCIs in the line of sight [4]. Such data have revealed the hitherto only presumed existence of a tenuous warm-hot intergalactic medium, the so-called WHIM [5], comprising a large fraction of baryonic matter in large-scale filaments at temperatures, which are consistent with the now detected abundance of such highly ionized species. First estimates of temperature (0.3 to 5 million degrees), density (10 to 100 times the mean baryon density of the universe), and mass of the medium have been obtained through the analysis of the O 5+ spectrum [6,7,8,9,10,11], and quantitative studies have involved O 6+ and O 7+ ions. Further abundant HCIs such as C, N, Ne, Ar, and Fe in various charge states are expected to contribute to the detected photoabsorption [12,13]. Other photoionized astrophysical plasmas such as those found in active galactic nuclei (AGNs) and accretion disks had been already known. At present, the identification of absorption lines due to HCIs in astrophysical spectra is based nearly exclusively on theoretical predictions, which have large uncertainties. In general, it can be said that in astrophysical codes [23] the quantitative modeling of PI is still insufficiently taken into account and a more sophisticated theoretical treatment is needed in future modeling.

Beyond the astrophysical relevance, investigations of HCIs provide stringent tests of atomic theory, since relativistic effects, quantum electrodynamic contributions, and nuclear size effects grow with high powers of the charge state in those ions. They also offer the possibility of performing systematic studies on electron-electron correlation along isoelectronic and isonuclear sequences, a tool capable of providing deep insights into many-body quantum mechanics. Since direct PI measurements are largely missing for HCIs the study of the time-reversed process to PI, namely photorecombination, using ion storage ring [14,[START_REF] Müller | AIP Conf. Proc[END_REF][START_REF] Badnell | [END_REF]17] and electron beam ion trap (EBIT) [18,19,20,21,22] techniques has given access to valuable data for benchmarking theory through the principle of detailed balance. However, these methods only gave a partial view of PI, and direct measurements are of great importance. The application of synchrotron radiation sources to atomic and condensed matter physics has brought the development of high spectral resolution instrumentation (up to E/∆E ≈ 200,000) which can in principle overcome the current resolution limitations of ion storage ring and EBIT electron-ion photorecombination experiments. In this context, the PI of low charge ions already constitutes a technique of remarkable accuracy [24]. These points clearly emphasize the need for PI investigations on HCIs to interpret astrophysical spectra and to guide advanced atomic structure calculations.

Present status

The broad interest in data on PI of ions has lead to the development of sophisticated methods utilizing merged beams (MB), transient plasmas (e.g. dual laser produced plasma -DLP), and ion traps. Unfortunately, none of the established experimental techniques is capable of accessing truly high charge states, or only with very limited resolution. In fact, only few measurements exist in which the ionization potential (E IP ) of the investigated ion is higher than 150 eV (see figure 1). (colour online) Representative overview of ions in charge states q ≥ 2 for which PI experiments have been performed [25,26]. The ordinate indicates the charge state q and the abscissa the atomic number Z. Ions investigated by MB (orange circles), DLP (blue triangles) and Penning traps (green squares) [27,28] are compared to those studied in the course of this work with an EBIT (red stars) [29,30], in order to illustrate the remarkable extension of the accessible charge states. The gray area marks the charge states for each element having an ionization potential below 150 eV, black lines indicate closed shell isoelectronic sequences.

In the last decade, the highest accuracy in PI was achieved by the MB method in measurements of mainly singly and multiply charged ions with charge states q ≤ 3 for lighter elements, e. g., [31], q ≤ 6 for Ba [32] and Fe [33], q ≤ 7 for Xe [34], and preliminary results with q ≤ 9 have been presented for Ce [START_REF] Habibi | th ICPEAC[END_REF], see figure 1.

In those experiments, a beam of ions in charge state q with a typical area density below 10 7 cm -2 is exposed to synchrotron radiation. The ionized species, i. e., photoions having the charge state (q + 1), is separated downstream and detected (cf. review by Kjeldsen [START_REF] Kjeldsen | [END_REF]). However, for higher charge states, the achievable target ion area density drops severely in ion beams, bringing the photoion signal below the noise level. Higher HCI densities are reached in discharge or laser plasmas (cf. review by West [37] and [38,39,40,41,42]). Pulsed radiation sources used as backlighters allow photoabsorption measurements, but separating photoions is impractical. The transient conditions of both plasma and backlighter, as well as of the plasma density gradients, and the simultaneous presence of different ionic species cause large experimental uncertainties.

Only two PI measurements using trapped ions have been reported, both suffering severe density limitations. Ar 2+ ions stored in a Penning trap were exposed to broadband synchrotron radiation producing K-shell vacancies [27]. Photoions up to Ar 8+ resulted from the subsequent electron shake-off. Recently, Xe + was photoionized in a Penning trap [28], and the Fourier transform-ion cyclotron resonance method was then applied to extract the photoion signal. Using trapped ions allows for the suppression of undesirable contributions from target ions in metastable states (up to 50% in MB experiments). However, low target area densities of only 10 6 cm -2 imply that these techniques will neither lead to significant improvements in sensitivity, nor enable HCI studies.

Here we demonstrate a promising new approach towards PI investigations with ions and HCIs, where high photon flux from a synchrotron irradiates the ions confined in an EBIT. Data on N 3+ and Ar 8+ ions show the versatility of this method. All HCIs of astrophysical interest, such as, O 4+ to O 6+ or Fe 6+ to Fe 16+ , as well as those relevant to hot fusion plasmas [43,44,45] are now accessible. The idea had been discussed already in 1984 by Church et al. [46], and an apparatus was built by Kravis and coworkers [47,48,49], but to our knowledge no results on PI of HCIs have been reported so far.

The Be-like N 3+ ion was chosen for demonstration due to its astrophysical abundance, to test predictions on threshold and resonance energies and their (relative) strengths commonly used for plasma modeling. For argon the situation is quite special: In spite of being of astrophysical relevance, motivating numerous theoretical studies, only few PI experiments have been conducted on this element so far. Ar 8+ is a Ne-like system, which is very suitable for this type of experiment, as it can be prepared at relatively high densities. Furthermore, PI investigations of neutral atoms are especially simple on noble gases, hence Ar 8+ is located on an isoelectronic [50,51,52,53] and isonuclear sequence [27,54,55,56] reaching down to the neutral system.

Experimental arrangements

As indicated by dotted boxes in figure 2, the setup consist of three main components: the EBIT produces the target ions, the synchrotron provides the required photons, and an ion extraction beamline is used for photoion detection. The experimental setup with these components as well as the different applied measuring modes are described in the following.

Electron beam ion trap

An EBIT generates a dense target of HCIs, well controlled and understood in terms of charge states, their time evolution and spatial distribution [57,58,59] governed by the electron beam energy (E e kin ) and current (I e ). The electron beam is accelerated by the potential difference between the cathode (U cath ) and the trap (U trap ), which consists of an assembly of ring shaped electrodes, called drift tubes (DTi, i = 1, 2, 3). A strong magnetic field, usually produced by superconducting coils, guides and compresses the electron beam. Atoms or molecules injected into the trap center are ionized through electron impact ionization (EII). Positively charged particles are radially trapped by the negative space charge of the electron beam, which is directly proportional to I e and inversely to E e kin . Potentials applied to the DTs control the axial confinement, forming a cylindrically-shaped trapping volume of ≈ 50 mm length and tens to hundreds of micrometers diameter (ion cloud). The trapped ions are further ionized to higher charge states until the ionization potential lies above the electron beam energy. Even higher charge states will only be produced in traces through two step processes involving long-lived metastable intermediate states. The control over the charge states is of crucial importance for a PI experiment, where the ion cloud is brought to overlap with a photon beam of energy hν. By choosing an electron beam energy just sufficient to produce target ions in the charge state q, the detection of ions with charge (q + 1) can be linked to PI:

E e kin ≈ e • (U trap -U cath ) ≤ E q IP ≤ hν (1)
In principle, a higher electron beam energy beyond the ionization potential could lead to a more efficient population of the target state, but EII by current densities of typically J e ≈ 10 21 electrons s -1 cm -2 would completely dominate PI due to the photon flux of Φ γ ≈ 10 16 photons s -1 cm -2 and reduce the signal to noise ratio for the photoion:

J e • σ EII Φ γ • σ P I (2)
A rough estimate on the target ion area density can be derived by assuming that the total charge of the ion cloud amounts approximately to the total charge of the electron beam within the trap volume

I e l m e 2 e ∆U total electron charge ≈ e Z qmax(Z) q=1 q • N q,Z total ion charge , (3)
where l stands for the length of the ion cloud, m e for the electron mass, e for the elementary charge, ∆U for the electron beam accelerating potential (≈ U trap -U cath ), and N q,Z for the number of ions of a certain element Z in charge state q . By assuming a uniform charge state distribution (N 1 = N 2 = . . . N qmax ), a simple expression replaces the inner sum in (3) and an estimate on the involved ion area density ρ can be obtained:

qmax q=1 q • N q = N qmax q max (q max + 1) 2 ≈ N qmax q 2 max 2 , (4) ρ
(Z, q max ) = N qmax A ≈ f 4 d 2 IC π I e l e q 2 max 2 m e e ∆U , (5)
where ρ is the target ion area density, d IC is a measure for the ion cloud diameter (typically 250 µm) and f is a charge fraction factor accounting for charge contributions by other elements, degree of electron charge compensation and the simplistic assumption of a uniform charge state distribution. An estimate within an order of magnitude should be possible even if f introduces a certain degree of freedom. Typical values for ρ are 10 9 -10 10 cm -2 . These estimates are also supported by our recent quantitative analysis of VUV spectroscopic EBIT data on Fe ions in [60] and [61]. At this point it should be emphasized once more, that this ion area density represents an increase of three to four orders of magnitude, when compared to an ion beam typically utilized in MB experiments. Therefore it is advantageous to directly expose the trapped ion cloud of an EBIT to synchrotron radiation, instead of using extracted HCIs, even if such a trapped target is an ensemble of ions in different charge states. Laser produced plasmas, while reaching much higher densities, survive for only nanoseconds to microseconds, whereas the EBIT ion cloud can be sustained for periods of time from fractions of a second to hours.

Photon source

The work was performed at the BESSY II electron storage ring in Berlin with the transportable FLASH-EBIT, developed at the Max-Planck-Institut für Kernphysik (Heidelberg) for experiments at modern light sources. Using this device, resonant laser spectroscopy was recently extended into the soft X-ray region at the Free electron LASer in Hamburg (FLASH) [62]. That work on resonant excitation of the 2 S 1/2 -2 P 1/2 transition in Li-like Fe 23+ at 49 eV suggested also potential applications to PI studies, since EBITs can produce nearly any ionic system of interest. During two campaigns at BESSY II, the FLASH-EBIT was installed at two undulator beamlines covering different energy regions. N 3+ , having an ionization threshold energy of E N 3+ IP ≈ 77 eV, was investigated at the beamline U125/2-SGM [63], which is designed for photon energies in the range from 40 eV to 180 eV, and equipped with a monochromator having a resolving power E/∆E greater than 50,000. Ar 8+ , with a significantly higher ionization potential of E Ar8+ IP ≈ 422 eV, was studied at the beamline U49/2-PGM1 [64,65,66], supplying photons at energies from 85 eV to 1600 eV. Both beamlines achieve maximal fluxes of 10 13 photons s -1 (at 100 mA ring current and at a bandwidth of 0.1%). The ion cloud was brought to overlap with the photon beam with the aid of a scintillation crystal, which can be moved in and out of the trap. The fluorescence light induced by the photon beam or the electron beam (as a proxy for the position of the ion cloud) were imaged on a CCD detector. More details on this system can be found in [29].

Ion extraction

With an EBIT, detection of photoelectrons or photoabsorption are currently not options for obvious technical reasons (strong electron beam, weak total photon absorption, trap geometry). Hence, the observation of the PI process is bound to the photoion signal. To analyze the trap contents, ions extracted from the trap through the collector were detected (see figure 2). The extracted ion beam was bent off the photon beam axis after passing through the collector by an electrostatic 90 degree deflector. Subsequently, the ions were separated according to their charge-to-mass ratio (q/m) by means of a Wien velocity filter. The electric and magnetic crossed fields of the Wien filter were chosen such that target ions and photoions impinge upon disparate regions of a position sensitive detector (PSD) at highest spatial resolution. High sensitivity can be achieved with this setup, as the PSD (consisting of a micro channel plate and a delay line anode) detects individual ions with an efficiency close to 100% and low background noise.

The ion extraction can be performed in a continuous or pulsed mode depending on the ionization potential of the ion under study. In the continuous mode the potentials of the DTs (see top left in figure 2) are chosen to form a shallow trap, allowing ions which overcome the trap barrier to escape (evaporate). Starting from a deep trap with no evaporative losses and increasing the potential on the central electrode DT2 until the ions are pushed over the trap barrier leads to the pulsed mode (compare to figure 3). In both cases DT3 is kept at a slightly lower potential than DT1 to guarantee that practically all ions leave the trap towards the collector. The trap potential U trap (and the ion charge state q) determine the kinetic energy of the ions.

Typically, low ion currents in the continuous extraction mode are suitable for counting on the PSD, while the pulsed mode requires a careful slow depletion of the trap in order not to saturate the detector. The sensitivity depends on the ion extraction efficiency. In general, ion transport losses are higher at low ion beam energy. Hence, a minimal acceleration voltage is required, setting a lower bound for the trap potential of a few hundred volts. For target ions in a low ionization stage this contradicts the demand expressed by (1). In this case, the target ion preparation (using a low trap potential) has to be separated in time from the photoion detection, consequently performed by a pulsed extraction (at a high trap potential). Data on N 3+ have been obtained in such a cyclic measurement procedure, which is described in detail below. We achieved efficient ion extraction with a trap potential of only U trap = 300 V. Therefore, Ar 8+ with E Ar8+ IP ≈ 422 eV could be measured in the preferable continuous mode. EBIT based measurements on ions with a low ionization potential are especially challenging. Hence, the cyclic (or pulsed) measurement procedure for N 3+ (E N 3+ IP ≈ 77 eV) shall be treated first.

Cyclic measurement mode for low ionization potentials: N 3+ ions

We applied a cyclic measurement mode with a total duration of 6 s and four different phases, as depicted in figure 3. During phase (i), N 3+ ions were generated abundantly, with the cathode biased to only U cath = -50 V and DT2 set to ground potential U DT 2 = 0 V to stay well below the ionization threshold to produce N 4+ ions. The electron beam current under these conditions was ≈ 2 mA. Inserting these values into (5) yields a N 3+ area density of 2•10 10 cm -2 , where a charge fraction factor of f = 0.3 was assumed. After about 3.5 s, the electron beam was turned off, while the photon beam was always on. Without the strong negative space charge of the electron beam the radial confinement only relies on the magnetic field. During this so-called EBIT magnetic trapping mode [67], which is basically equivalent to a Penning trap, the ion cloud diameter increases and the ion density is reduced by approximately one order of magnitude. In phase (ii), lasting about 0.5 s, the confining DTs (DT1 and DT3) were ramped up, while the central DT2 was kept at ground potential to eliminate evaporative axial ion losses. During phase (iii), the DT2 potential was slowly raised over about 1 s to empty the trap gradually (pulsed extraction), in order to prevent an overload of the ion-counting PSD. Finally, in phase (iv) during about 1 s all remaining ions were expelled, and the DT potentials were reset. Then, the electron beam was switched on again, turning back to phase (i) to start a new cycle with an incremented photon beam energy. Further objectives that also demand a cyclic measuring procedure consisting of various phases are for instance the suppression of target ions in metastable states [28] or the variation of the interaction time in order to determine absolute cross sections [30]. In fact, comparing our N 3+ data with results from MB experiments [68] one can see a reduced contribution from metastable states, which will be discussed in more detail below. The demonstration of an absolute cross section measurement performed on Fe 14+ is described elsewhere [30].

Continuous measurement mode for high ionization potentials: Ar 8+ ions

In the continuous measurement all the EBIT parameters were kept constant while the photon energy is scanned. Here, a higher target ion area density is maintained by the electron beam throughout the measurement. Moreover, continuous ion extraction enhances evaporative cooling of the trapped ions [59], thus increasing the achievable resolution by reducing Doppler broadening, and can also have further positive effects on the ion target density by concentrating the trapping volume [57]. Furthermore, unavoidable depopulation of the photoionized species through electron capture or charge exchange is less critical in the continuous mode. Finally, statistical considerations make continuous extraction more favorable than a pulsed one. Cyclic procedures induce more fluctuations in number of target ions, beam transport etc., than steady-state conditions, and thus require longer acquisition times to achieve good statistical quality. Nonetheless, since the reproducibility of the target preparation was good, by normalizing the photoion to the target ion signal a large suppression of those fluctuations could be achieved for both measuring modes.

Photon energy calibration

Photoabsorption by N 2 -molecules in a conventional gas cell at a photon energy of ≈ 400 eV was used to calibrate the Ar 8+ data. The uncertainty of the N 2 calibration is roughly 100 meV, and constitutes the main error source in our results for the resonance energies.

Data on N 3+ were calibrated by injecting an atomic He beam into the trap center with the electron beam switched off. The DTs were set to positive voltages similar to phase (iv) in figure 3. Photoions were extracted and autoionizing resonances were recorded and identified [50]. This technique is based on photoion detection, which is more sensitive than conventional photoabsorption, and only uses the existing setup as shown in figure 2. We achieved a calibration accuracy of 8 meV, the limiting factor being the chosen monochromator slit width. More details on the so-called photon beam ion source (PhoBIS) calibration mode can be found elsewhere [29,69]. Generally, He-like systems are very appropriate for calibration procedures, as precise theory is available and, in contrast to even more precisely calculated H-like systems, PI resonances can be observed. An additional advantage of He-like systems is that EBITs have the best performance when producing closed shells ions, and could therefore help establishing new calibration standards at high photon energies.

Theory

Simulated PI spectra have been calculated by various models, see e.g. the works [1, 2, 3, 70] and references therein. We have calculated for the systems studied here the resonance energies, widths and strengths within a fully relativistic large-scale MCDF method similar to [71,72] and have compared the results with our experimental data on N 3+ and Ar 8+ ions.

The cross section for a given resonant photoionization channel, including quantum interference with the direct photoionization process, can be written as a function of the photon energy ω as

σ PI i→d→f ( ω) = S i→d→f 2 Γ d π 1 -1/q 2 F + 2 /q F 1 + 2 , (6)
with the resonance strength S i→d→f and the unitless energy given by

S i→d→f = 2π 2 c 2 4 ( ω) 2 A re i→d A a d→f Γ d , = 2( ω + E i -E d )/Γ d , (7)
and q F is the Fano asymmetry parameter [73]. Here, i is the initial state (Be-or Ne-like) of the ion. The index d denotes quantities related to the autoionizing state formed by photoabsorption. This intermediate state then decays via autoionization to the final state f . A re i→d stands for the one-photon radiative excitation rate (transition probability per unit time) and A a d→f is the autoionization (Auger) rate of the excited resonant state. Γ d denotes the total natural width of the resonant state, given as the sum of the radiative and autoionization widths:

Γ d = (A r d + A a d )
, where A r d and A a d are the corresponding decay rates summed over all possible decay channels. A shift of the transition energy (E res = E d -E i ), caused by the coupling to the electronic continuum [73], is included in our definition of E d . Interference with direct PI may change the line profile and the position of the maximal cross section but not the resonance energy. Our calculated resonance energies and strengths can be compared to experimental data by fitting the function in Eq. ( 6) to the measured spectra. When interference with the direct PI channel can be neglected, i.e. q F → ∞, the energy dependence of the cross section approaches a symmetric Lorentzian line profile:

σ PI,symm. i→d→f ( ω) = S i→d→f Γ d /(2π) ( ω + E i -E d ) 2 + Γ 2 d 4 . (8)
We calculate the bound state functions involved in the transition rates and the corresponding level energies E i and E d in the framework of the MCDF method [74,75] as in [71]. In this scheme, the many-electron atomic state function (ASF) is given as a linear superposition of configuration state functions (CSFs) sharing common total angular momentum (J), magnetic (M ) and parity (P ) quantum numbers:

|ASF; P JM = nc k=1 c k |CSF k ; P JM . (9)
The CSFs are constructed as jj-coupled N -particle Slater determinants of one-electron wave functions. In Eq. ( 9), a CSF is specified by the orbital occupation and the coupling of subshell angular momenta. The number of CSFs is denoted by n c . As an example, in the calculation of the bound states involved in the PI of Ne-like Ar we take into account relevant configurations which can be constructed from the dominant one by single and double orbital substitutions, resulting typically in hundreds of CSFs. Such an extensive calculation is necessary as relativistic correlation contributions amount to as much as 6-8 eV, while the experimental accuracy is on the 0.1 eV-level.

The contribution of the Breit interaction amounts to 3.3-3.4 eV for the levels involved.

The single-electron one-loop quantum electrodynamic (QED) radiative corrections of typically 2.6 eV have also been included, and QED screening corrections were approximated to be around 0.2 eV by a rescaling of the single-electron values. Mass shift effects are on the order of 0.2 eV. The continuum electron wave functions entering the Auger rates A a d→f are calculated numerically by solving the radial Dirac equation with the screening potential induced by the bound (frozen) electrons, with a method similar to that applied in Ref. [76]. In general, within our model we expect typically an accuracy of 0.3 eV for the resonance energy and 20% for the cross section. Within these uncertainties QED and other subtle effects contribute already noticeably to the resonance positions.

Results and discussion

Photoionization of N 3+

The top of figure 4 shows an overview spectrum for PI of the 1s 2 2s 2 1 S 0 ground state (GS) and the 1s order to populate the 2p ns or 2p nd series of N 3+ . However, MS and GS resonances appear with similar heights in our measurement, indicating that only a small part of N 3+ target ions is in the metastable 3 P 0 state. Comparing the heights of the two ionization edges supports this interpretation. The labels in the plot indicate the doubly excited states 2p nl. Counts of N 4+ photoions for each trap dump were normalized to the corresponding counts of N 3+ ions and to the photon flux, which was monitored by measuring the photocurrent emitted from a highly transparent gold grid behind the monochromator slit. Fano profiles [73] with Fano parameters around q F = -2.3 ± 0.5 were fit when the resolution and statistical significance were sufficient (the three 2pnd resonances); otherwise Gaussian profiles were used (q F → ∞). For the 2p ns series the Gaussian profiles are inverted, indicating very small Fano parameters (q F ≈ 0). For q F = ∞ or q F = 0 the resonance energy coincides with the maximal cross section.

Predictions taken from the Nahar Ohio State University Radiative Atomic Database (NORAD) [77,[START_REF] Nahar | NORAD URL[END_REF] are plotted on the bottom of figure 4 after convolution with a Gaussian profile (FWHM = 75 meV) to account for experimental broadening. Generally, good agreement with the experimental results in relative strengths as well as line profiles is found when adding tabulated values of the GS and MS cross sections with same weights. However, this contradicts the experimental finding, that the majority of the target ions are in the GS. Furthermore, an overall energy shift of ≈ 150 meV towards lower photon energies is clearly visible. Predicted resonance energies of our MCDF calculations are plotted as vertical orange lines with the corresponding diamond giving the cross sections on resonance as they would appear after convolution with an experimental broadening of 75 meV (FWHM). In contrast to NORAD values, the weights for MS and GS have not been chosen equal here.

Only 25% of the target ions are assumed to be in the excited 3 P 0 state, which is in better agreement with our experimental results. In order to account for direct PI and improve comparability to NORAD, resonances excited from MS and GS N 3+ have been increased by 1 and 2 Mb, respectively. For the resonance energies good agreement is found especially for the GS excitation. For MS excitation, the resonance positions all shift to slightly higher energies, indicating a corresponding stronger binding of the MS state. In figure 5 the region around 69.9 eV reveals the doublet structure of the MS 2p 5p resonances. The data was obtained in only one hour of acquisition time, proving the high sensitivity of our method. Due to time constraints, a resolution of 35 meV was chosen. A high accuracy measurement by Müller et al. [24] achieved a remarkable resolution of 3.3 meV (FWHM) on the same features, in narrow scans each encompassing one of the lines. For the whole range 68 to 88 eV data with a 40 meV resolution have been preliminarily reported by the same group [24,[START_REF] Müller | 25 th ICPEAC[END_REF]. The absolute precision of our first demonstration measurement is within a factor of 3 of those results.

Table 1 lists our results and compares them to those of MB experiments by Bizau et al. [68] and, for the 2p 5p 3 P 0 resonance by Müller et al. [24], as well as with predictions partially not shown in figure 4 (TOP: R-matrix calculation from TOPbase [1,[START_REF] Tully | [END_REF], compilation of NIST [81], online calculations from CATS: Cowan Atomic Structure Code [START_REF] Cowan | Theory of Atomic Structure and Spectra[END_REF][START_REF]Los Alamos Atomic Physics[END_REF]). An excellent agreement between the experimental values of Müller on the 10 meV level can be stated. Our data also agree well (within their larger error bars) with the results of Bizau. In one case, their former identification of the GS 1s 2 2p 5s resonance seems questionable. The thresholds for direct PI of N 3+ in its GS and MS (mean value) were also determined. Compared to MB experiments, where GS and MS species are nearly equally populated, the MS fraction is considerably suppressed by our cyclic measurement technique. The ratio of the excitation rate of the 3 P 0,1,2 by the electron beam to the spontaneous decay rate, combined with the collisional quenching rate, determines the strength of the observed resonances excited in phase (i). During phases (ii) and (iii) the spontaneous decay of the short-lived MS 3 P 1,2 to the GS (below 1 ms lifetime) reduces their contribution.

R-matrix Calculations collected in TOP neglect relativistic effects and use only a very reduced number of levels in the N 3+ case. These values should not be used any more, as more accurate and elaborate calculations are available, most notably NORAD [77,[START_REF] Nahar | NORAD URL[END_REF]. The R-matrix calculations shown by Müller [24,[START_REF] Müller | 25 th ICPEAC[END_REF] (values not given here) for the MS 2p 5p 3 P 0 , using a larger set of levels and employing relativistic corrections, achieve the closest agreement with their and our measured data. Resonance energies compiled in the NIST database [81], with recommended values based in part on experiment, seem reliable, but no PI cross sections are given there. Our fully relativistic MCDF calculations yield resonance energies in good agreement with the experiment, even though they are expected to show their real merits especially for heavier systems at higher charge states and photon energies.

Photoionization of Ar 8+

Data on Ar 8+ were acquired in the continuous measurement mode. An electron beam with an energy of 400 eV (U trap = 300 V, U cath = -100 V) and a current of 8 mA was steadily maintained. Ions with closed shells like Ar 8+ , can be prepared in very high abundance, as the electron beam energy can be adjusted well above the ionization potential of the last open shell electron (E Ar7+ IP ≈ 143 eV) without having enough Table 1. Photoionization edges (E) and resonance energies (2p nl) fitted with Fano (F) and Gaussian (G) profiles from this work (Experiment) compared with merged-beam (MB) results from [68] (B), [24] (M -upper two lines), and calculations from this work (MCDF), TOP [1,[START_REF] Tully | [END_REF], NIST data [81], as well as Cowan Atomic Structure Code [START_REF] Cowan | Theory of Atomic Structure and Spectra[END_REF][START_REF]Los Alamos Atomic Physics[END_REF]. Values are given in eV. ≈ 422 eV). This makes it possible to achieve a charge state distribution having its maximum at the highest charge state, i. e. Ar 8+ . Therefore, estimating the area density based on (5) with a charge fraction factor f =1 seems realistic and yields 1.3•10 10 ions cm -2 . An overview spectrum for Ar 8+ , including the threshold and near-threshold resonances, is shown in figure 6. The resonances belong to the 2s (2p 6 ) np Rydberg series (labels in the plot omit the full 2p subshell). Counts of Ar 9+ ions below the threshold energy are assigned to EII of Ar 8+ in excited metastable states. The experimental photon energies show calibration uncertainties of approximately 100 meV. On the bottom of figure 6 three calculations are shown for comparison. Values from TOP have been convoluted with a Gaussian profile of FWHM ≈ 300 meV. Like in the plot for N 3+ our MCDF calculations are indicated by orange drop lines with diamonds giving the cross sections (here a broadening of 800 meV was applied). Liang recently presented resonance energies without giving cross sections indicated as blue vertical markers.

A high resolution measurement on the first resonance, taken with a two hours acquisition time and a step size of 5 meV, is depicted in figure 7. The width of the exit slit for the photon beam was set to 15 µm, corresponding to a nominal resolution of E/∆E = 15,000 (bandwidth: < 0.01%). The asymmetric shape of the resonance is clearly visible and shows good agreement with TOP predictions (full blue line) after a shift of 145 meV and convolution with a Gaussian profile (FWHM = 88 meV). In spite of the threshold region good agreement can be stated. Deviations for resonance energies are on the order of ≈ 300 meV while at the threshold a mismatch of a few eV appears.

Additionally, a fit of a Fano profile convoluted with an experimental broadening was applied (dashed red line), which yields a resolution of E/∆E ≈ 5,000 for this high precision measurement. An asymmetry parameter of q F = -6±1 and a Fano width Γ = 30±5 meV have been obtained by the fitting procedure.

In table 2 experimental data are compared to our MCDF calculations and to predictions found in TOP [1], at NIST [81], as well as to online CATS calculations [START_REF] Cowan | Theory of Atomic Structure and Spectra[END_REF][START_REF]Los Alamos Atomic Physics[END_REF], and to recent predictions by Liang et al. [START_REF] Liang | [END_REF]. The values from TOP show ( 1 0 3 c o u n t s ) ) np Rydberg series only exist for the first three representatives of this isoelectronic sequence (atomic Ne, Na + , Mg 2+ ) and were published by Kennedy et al. [52]. The first autoionizing resonance is 2s 3p for all three systems and dominates the PI spectra. Fano widths, asymmetry parameters (q F ), and series limits have been extracted for this resonance and can be compared to our values obtained by a fit to the first autoionizing resonance (2s 5p) of Ar 8+ (see figure 7). The most extensive experimental study on the Ne-like sequence is given by Chakraborty et al. [51] and reaches up to Si 4+ . Special attention is drawn to the threshold region, where anomalous behavior (extremely low direct PI cross sections) was predicted for Si 4+ . For this charge state the 2s 3p resonance lies very close to the 2s 2 2p 6 -2s 2 2p 5 ionization limit. The dual laser plasma (DLP) technique equipped with a photoelectric detection system was applied, hence photoabsorption was recorded rather than PI. Calculations predict the 2s 3p resonance to lie below the threshold, but the experiment showed an asymmetric Fano profile, indicating interference with direct PI. This was explained as a technical feature of the DLP method, where plasma microfields lower the ionization limit and lead to forced autoionization. Recently Bizau et al. [53] performed MB experiments on Si 4+ in the range from 110-184 eV at two MB facilities and clearly showed that the 2s 3p resonance does not contribute to PI of that ion. However, the accuracy was not high enough to test the predicted anomalous behavior. Furthermore, results for the threshold region of different theoretical methods are compared and disagree significantly, as the given threshold energies vary by several eV. The situation for Ar 8+ is similar. Here, the 2s 4p resonance energy lies close to the ionization threshold. Predictions for E IP scatter by ≈ 3 eV (see figure 2). Inconclusive theoretical predictions, for example in the threshold region of Si 4+ and Ar 8+ , corroborate the need for decisive and benchmarking experiments to solve such open questions and guide theory.

Summary and outlook

Results of the photoionization of N 3+ and Ar 8+ have been obtained with a new experimental method suitable for arbitrary charge states. The ions stored in an EBIT were directly exposed to synchrotron radiation. A high target area density of approximately 10 10 cm -2 , four orders of magnitude higher than in conventional ion beam targets, was achieved. This technique increases tremendously the number of ionic species accessible to PI experiments. In this work two approaches have been demonstrated, useful for both low and high ionization threshold energies respectively. Firstly, a measurement consisting of several phases separating target ion production and photoion detection in time was applied for N 3+ . This cyclic procedure also enabled efficient depopulation of metastable states. With an ionization potential of only 77 eV, N 3+ is close to the lower limit of applicability of this technique. Results in this energy range, with a typical accuracy of 0.05 eV, were compared to those from the established merged beam method, which show excellent agreement. Secondly, Ar 8+ , a case of high ionization potential (422 eV), was measured in a continuous mode giving unprecedented access to ions in high charge states and with high ionization potential. This will enable investigations of long isoelectronic and isonuclear sequences and give interesting insight into electron-electron correlations.

The measured PI values are compared to our extended full relativistic MCDF calculations. The theoretical results compare nicely with the measurements, i.e. within 0.3 eV for the resonance positions and about 20% for the cross section predictions. Highly charged ions at higher Z values will in particular benefit from these sophisticated calculations.

As strong XUV and X-ray absorption lines due to PI of HCIs have recently been observed in astrophysical spectra, and identification of these structures still had to rely almost exclusively on calculations, the present investigations constitute a first step towards better benchmarking of those models. These results together with those of [29,30,62] form a strong case for the installation of EBITs at synchrotrons and free-electron lasers, since astrophysics, plasma physics and atomic structure theory can all profit from the detailed studies now becoming possible.

Figure 1 .

 1 Figure1. (colour online) Representative overview of ions in charge states q ≥ 2 for which PI experiments have been performed[25,26]. The ordinate indicates the charge state q and the abscissa the atomic number Z. Ions investigated by MB (orange circles), DLP (blue triangles) and Penning traps (green squares)[27,28] are compared to those studied in the course of this work with an EBIT (red stars)[29,30], in order to illustrate the remarkable extension of the accessible charge states. The gray area marks the charge states for each element having an ionization potential below 150 eV, black lines indicate closed shell isoelectronic sequences.

  photon beam

Figure 2 .

 2 Figure 2. (colour online) Experimental setup for PI measurements using synchrotron radiation (coming from the right side) and an EBIT (bottom left box) equipped with an ion extraction (right box). Ions are produced and trapped by a compressed electron beam, forming a cloud with area density ≈ 10 10 cm -2 , which is axially overlapped with a photon beam. The electron kinetic energy E e kin = e • (U trap -U cath ) must stay below the photon energy hν. The left top graph shows the potential settings in the EBIT. Target ions and photoions are extracted with a kinetic energy of E ion kin = q • Utrap (where Utrap is characterized by the drift tube potentials U DT i ), bent towards a velocity filter separating them by charge-to-mass ratio (q/m), and counted with a position sensitive detector (PSD).

Figure 3 .

 3 Figure 3. (colour online) Measurement cycle (6 s long) for PI of N 3+ ions (green circles). The electron beam energy stays below the EII-production threshold (77 eV) for N 4+ (green triangles). Target ion production and photoion extraction are performed sequentially in four phases: (i) production of N 3+ target ions with the electron beam on, and turning off the electron beam at the end, (ii) fixing the ion cloud by raising the confining potentials, (iii) slow ion extraction by reducing the trap depth, and (iv) emptying the trap totally before resetting. Timing of DT potentials are shown in upper panel (DT1: green dotted line, DT2: brown full line, DT3: blue dashed line); axial trap configurations are shown below. Cycles are triggered by the monochromator control software.

  s s s e c t i o n ( M b ) P h o t o n e n e r g y ( e V )

Figure 4 .

 4 Figure 4. (colour online) Top: overview spectrum for PI of N 3+ ions. Counts of N 4+ ions are plotted vs. photon energy (green histogram). Structures are due to autoionizing intermediate states (labels) excited from the ground state (GS) and metastable states (MS) of N 3+ . Data are normalized to N 3+ counts and photon flux. Peaks are fitted with Fano or Gaussian profiles (red line). The step size is 100 meV in the range of MS structures and 40 meV in the range of GS structures. Bottom: comparison with available theory by NORAD (blue line) and our MCDF calculations (orange diamonds with drop lines) show good agreement.

Figure 5 .

 5 Figure 5. (colour online) Photoionization of N 3+ ions in metastable states (green dots). The 2p 5p MS resonance doublet with a 5 meV step size and resolution ≈ 35 meV (FWHM). The exit slit of the beamline U125/2-SGM was set to 250 µm. Error bars only account for statistical (counting) uncertainties. Resonance energies are extracted by Gaussian fits (red curve) and indicated by dashed drop lines.

  t o n e n e r g y ( e V ) T h e o r y C r o s s s e c t i o n ( M b )

Figure 7 .

 7 Figure 7. (colour online) Detailed scan over the 2s 5p resonance with a step size of 5 meV. Counts of Ar 9+ are plotted vs. photon energy (green dots), error bars only include counting statistical uncertainties. TOP calculations (full blue line) are shifted by 145 meV. The fit curve (dashed red line) is a convolution of a Fano profile (q F = -6±1 and a Fano width Γ=30±5 meV) with an experimental broadening of 88 meV.

  2 2s 2p 3 P 0 metastable states (MS) of Be-like N 3+ ions in the range from 68 to 84 eV. Data for GS and MS were taken in two individual runs with 40 meV and 100 meV step size, respectively. The observed structures correspond to doubly excited autoionizing intermediate states. Cross sections starting from MS 2s 2p are larger, as a single E1 transition 2s -nl leads to a doubly excited state, while otherwise two 2s electrons have to be excited through a forbidden transition from the GS 2s 2 in

	P h o t o i o n y i e l d N ( c o u n t s ) 4 +	6 9 . 7 5 0 1 0 0 1 5 0 2 0 0 2 5 0 N	3 +	6 9 . 8 3 D 1	6 9 . 9 2 p 5 p	7 0 . 0 3 P 1	7 0 . 1 E x p . F i t
				P h o t o n e n e r g y ( e V )

  Figure 6. (colour online) Top: overview PI spectrum on Ar 8+ in the range from 415 to 495 eV, obtained by counting Ar 9+ photoions (green histogram) in a continuous extraction mode. Counts of Ar 9+ below the threshold are interpreted as EII of Ar 8+ target ions in metastable states. Labeled vertical lines indicate experimental threshold energy and resonance energies of the 2s np Rydberg series.Absolute photon energy calibration with an accuracy of 100 meV is based on photoabsorption by N 2 -molecules in a gas cell. Bottom: comparison to our MCDF calculations (orange diamonds with drop lines) as well as available theoretical predictions by TOP[1] (blue line) and Liang et al.[START_REF] Liang | [END_REF] (blue vertical markers).

Table 2 .

 2 Photoionization edges (E) and resonance energies (2snp) fitted with Fano (F) and Gaussian (G) profiles from this work (Experiment) compared with calculations from this work (MCDF), TOP[1,[START_REF] Tully | [END_REF], NIST data[81], Cowan Atomic Structure Code (CATS)[START_REF] Cowan | Theory of Atomic Structure and Spectra[END_REF][START_REF]Los Alamos Atomic Physics[END_REF], as well as relative values by Liang et al.[START_REF] Liang | [END_REF]. Values are given in eV. Fano profile at 417.3 eV, reflecting the theoretical uncertainty, whether this resonance contributes to PI cross sections or not. NIST values were found for the 2p and 2s ionization edge and deviate nearly by 2 eV. CATS resonance energies are online available and calculated rapidly on demand, explaining larger deviations. A shift to higher values of about 2 eV appears for all resonances, therefore the very good agreement of the ionization edge is probably a coincidence. Recent calculations by Liang[START_REF] Liang | [END_REF] using Breit-Pauli Hamiltonian within the R-matrix theory show deviations from the data too. There, only resonance energies relative to the threshold are given. For comparison we shifted the energy scale such that resonance energies best reproduce the experimental findings. Applying this shift, the threshold is mispredicted by 2.02 eV. The relative agreement within the Rydberg series between our experiment and Liang is on the 130 meV level.Experimental data of other Ne-like systems for comparison of PI resonances of the 2s (2p6

	Transition		Experiment	MCDF	TOP	NIST	CATS Liang
			this work	this work		2p-∞
		E 422.20(12) 422.14(30) 423.953 422.54 422.15 420.197
	2s-5p	F 447.71(10) 447.54(30) 447.56	-	449.90 447.71
	2s-6p	G 463.62(12) 463.35(30) 463.34	-	465.76 463.61
	2s-7p	G 472.80(15) 472.63(30) 472.61	-	475.06 472.92
	2s-8p	G 478.82(15) 478.53(30) 478.51	-	480.98 478.85
	2s-9p	G 482.76(18) 482.58(30) 482.50	-	484.98 482.86
	2s-10p	G 485.94(30) 485.33(30) 485.32	-	487.81 485.69
	2s-∞	-	497.80(40) -	-	497.44 499.47 497.39
	overall good agreement, but the resonance energies given there are generally too
	low (≈ 250 meV). The ionization threshold energy, however, is given at 423.95 eV,
	which is 1.75 eV too high. Moreover, TOP gives values for the 2s 4p resonance with a

Acknowledgments

We thank the staff of the BESSY II electron storage ring for excellent support, and especially G. Reichardt, O. Schwarzkopf, R. Follath, and R. Püttner. K. Bechberger, T. Busch, N. Müller, T. Schiffmann, and S. Vogel of MPIK deserve our special gratitude for their help. B. L. S. was supported by a fellowship from the German-American Fulbright Commission, The work of Z. H. was supported by Helmholtz Alliance HA216/EMMI. The measuring campaigns were funded partly by the Max Planck Advanced Study Group at the Center for Free-Electron Laser Science.


R Kollengode Easwaran

L Longchambon

P.-E Pottie

V Lorent

H Perrin

M Garraway

B M Garraway

RF spectroscopy in a resonant RF-dressed trap

Keywords: numbers: 39, 25, +k, 32, 80, Pj

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Since the initial proposal of Zobay and Garraway [1], adiabatic potentials for ultracold atoms, resulting from the radiofrequency (RF) dressing of atoms in a inhomogeneous magnetic field, have been demonstrated [2] and are used by many groups to propose and realize unconventional traps [3][4][5][6][START_REF] Extavour | Proceedings of the International Conference on Atomic Physics, Atomic Physics[END_REF][START_REF] Jo | [END_REF][9], in particular in the framework of atom chips [6][START_REF] Extavour | Proceedings of the International Conference on Atomic Physics, Atomic Physics[END_REF][START_REF] Jo | [END_REF]. They offer the great advantage of a simple tailoring of the trap potential, allowing the realisation of a double well [6] or a ring trap [4,9] with dynamically adjustable parameters. The potential seen by the dressed atoms is due to the spatial variation of the RF coupling strength or the RF detuning, or both. These RF-dressed traps are usually loaded directly from a conventional magnetic trap, by ramping up the RF amplitude or sweeping the RF frequency.

Depending on the conditions, the adiabatic potential may result essentially from a light shift induced by a variation of the RF coupling, as was first demonstrated in a pioneering experiment at NIST with microwave radiation [10]. In this case, denoted here as the 'off-resonant configuration', the detuning of the RF field to the magnetic Zeeman splitting is negative everywhere and the atoms are expelled from the most resonant central region, leading for example to a double well structure well suited for matter wave interferometry [11]. On the other hand, the trapping potential may be due essentially to the variation of the detuning, the atoms being then attracted towards the regions of zero detuning [1]. In this 'resonant configuration', the atoms sit at points where the RF field is resonant with the magnetic level spacing, which makes them more sensitive to fluctuations of the RF frequency, and heating due to frequency fluctuations was indeed observed [2,12,13]. This heating may be avoided if a low noise RF source is used [13], leading to traps compatible with Bose-Einstein condensates. However, the transfer of the atoms from the initial magnetic trap to the RF-dressed trap is not usually adiabatic, which results in an initial heating in the loading phase and the destruction of the condensate [2,12].

This limitation motivated a theoretical study of evaporative cooling inside the dressed trap, by means of a second RF field [14]. In the case of a non-resonant RF trap, this scheme was used to prepare independent condensates in a double well [11]. In this paper, we present results of spectroscopy in the dressed trap, obtained by recording the atom losses induced by a very weak second RF field [15,16]. The shape of the resonance is directly linked to the energy distribution of the atoms in the dressed trap. We discuss the position of the expected lines and the strength of the second RF coupling. Finally, we discuss the choice of the second field frequency for evaporative cooling of the atoms dressed by the main RF field.

Spectroscopy of the RF-dressed trap

The spectroscopy of an RF-dressed trap is performed in the following way: A weak probing field can induce transitions between dressed states, which results in a loss of atoms from the adiabatic potential. The losses are monitored as a function of the frequency of the probing field, as has been investigated in detail in the offresonant configuration [15]. In this section, the position and strengths of the expected resonances are discussed.

Effect of the dressing field

In the following, we denote as (x, y, z) the axes of the laboratory frame, z being the vertical axis. In the experiment, the adiabatic potential is created by a strong RF field of frequency ω 1 and Rabi frequency Ω 1 , linearly polarised along y, which dresses the atoms placed in an inhomogeneous magnetic field B(r). The direction of this static magnetic field at position r is denoted as Z. The two important parameters which determine the resulting adiabatic potentials are the RF detuning δ(r) = ω 1 -ω L (r) from the local Larmor frequency and the local effective RF coupling Ω eff 1 (r). The latter depends on the angle θ(r) between the RF polarisation y and the static magnetic field direction Z, being zero for collinear fields: Ω eff 1 (r) = Ω 1 sin θ(r). Hence, the static magnetic field geometry is the skeleton of the adiabatic potentials.

The static magnetic field B(r) is that of a compact Ioffe-Pritchard trap [17] characterised by the bias field B 0 = 1.8 G at the centre, the radial gradient b = 223 G•cm -1 and the longitudinal curvature b = 250 G•cm -2 along the trap axis x. We define as ω 0 = |g F |µ B B 0 / the Larmor frequency at the trap centre and as α = |g F |µ B b / the magnetic gradient in units of frequency, where g F is the Landé factor, σ(g F ) = g F /|g F | its sign and µ B is the Bohr magneton. The typical length-scales in this magnetic trap are L = B 0 /b = 850 µm and R = B 0 /b = ω 0 /α = 81 µm. The Larmor frequency at position r is then approximately given by ω L (r) = ω 0 1 + x 2 /L 2 + (y 2 + z 2 )/R 2 in the vicinity of the magnetic trap centre. On the other hand, the local effective coupling to the RF field is approximately

Ω eff 1 (r) = Ω 1 1 -α 2 y 2 ωL(r) 2 .
As a result of the near resonant dressing field, the energy levels of the atom plus RF field system are grouped into manifolds separated by an energy ω 1 . Inside a manifold, the energy spacing of the 2F + 1 levels is denoted Ω(r). The adiabatic potential V mF 1 (r) = σ(g F ) m F Ω(r) then varies in space through the spatial variation of both the detuning δ(r) and the effective RF coupling Ω eff 1 (r) [3]. Given the static magnetic field geometry, the adiabatic potential for a given F, m F adiabatic state reads:

V mF 1 (r) = σ(g F ) m F Ω(r) = σ(g F ) m F δ(r) 2 + Ω 2 1 1 - α 2 y 2 ω L (r) 2 , (1)
where

δ(r) = ω 1 -ω L (r). (2)
This expression corresponds to the eigenenergies at point r of the Hamiltonian H 1 = -δ(r)F Z + Ω eff 1 (r)F X , obtained after applying the rotating wave approximation (RWA) to the RF-atom coupling, which is valid when Ω 1 ω 1 and |δ(r)| ω 1 . Here, X is direction orthogonal to the static field direction Z, in the y -Z plane. The total potential energy is V mF (r) = V mF 1 (r)+Mgz, where g is the gravitational acceleration and M the atomic mass.

The potential minimum for the extreme adiabatic state m F = σ(g F ) F is located on the isomagnetic surface S 0 defined by δ(r) = 0 or equivalently ω L (r) = ω 1 [START_REF]Strictly speaking, the minimum is shifted away from the isomagnetic surface S 0 by gravity along z and by[END_REF]. Schematically, this surface is an elongated ellipsoid, cylindrically symmetric around x with a radius r 0 = ω 2 1 -ω 2 0 /α and a long axis x 0 = r 0 L/R. On the surface S 0 , ω L (r) = ω 1 and the RF coupling only depends on the coordinate y:

Ω eff 1 (y) = Ω 1 1 -α 2 y 2 ω 2 1
. The total potential energy on S 0 then reads F Ω eff 1 (y)+Mgz. It varies through the RF shift between F Ω 1 1 -

α 2 r 2 0 ω 2 1 = F ω0
ω1 Ω 1 at y = ±r 0 on the y axis and F Ω 1 in the y = 0 plane. On the other hand, the gravitational shift also varies as (z) + Mgz)/h for the F = 2 ground state of 87 Rb in our magnetic field configuration, in units of frequency of the dressed states m F = 2, ... -2, plotted along the vertical direction. The dressing frequency is set to ω 1 = 2π × 8 MHz with a Rabi frequency of Ω 1 = 2π × 50 kHz, resulting in a dressed trap centred at position z = -506 µm. The two extreme dressed levels are emphasised. The origin z = 0 corresponds to the static magnetic field minimum, and gravity is taken into account. The probing field ω 2 induces transitions between dressed states at a position z such that Ω(z) = |ω 2 -nω 1 |, n ∈ N. These positions correspond to two points z ± belonging to two surfaces S ± , defined respectively by δ

(r) = ± q (nω 1 -ω 2 ) 2 -Ω eff 1 (r) 2
, on both sides of the trapping surface S 0 defined by δ(r) = 0. a function of z between -Mgr 0 at the bottom (z = -r 0 ) and 0 at the equator (z = 0). Depending on the relative strength of the gravitational shift Mgr 0 with respect to the RF shift F Ω 1 (1 -ω0 ω1 ), there are either two minima at the equator at z = 0 and y = ±r 0 [6] or a single minimum at the bottom of this surface z = -r 0 [START_REF] Morizot | [END_REF]. With the parameters of our experiment, we are always in the second situation [2]. At the minimum, where the atomic density is largest, the RF coupling reaches its maximum value Ω 1 . Away from the minimum, it decreases down to a value depending on the energy and on the RF frequency through the ellipsoid radius r 0 . At a given energy above the potential minimum, the coupling strength reduction is more pronounced for smaller RF frequencies ω 1 . For example, at the operational RF frequency of 8 MHz the reduction is less than 20% at energies below 10 µK. By contrast, at 3 MHz, the reduction remains below 7% at an energy corresponding to 1 µK, and reaches 50% for 10 µK. Nevertheless, the effective coupling Ω eff 1 never vanishes over S 0 . If necessary, the coupling inhomogeneity could be reduced by increasing the bias field B 0 .

In the following, we concentrate on the case of rubidium 87 in the state F = 2. A plot of the adiabatic potentials around z = -r 0 as a function of the vertical coordinate z in the laboratory frame is given in figure 1 for a typical RF frequency ω 1 = 2π×8 MHz and for the parameters of our Ioffe-Pritchard magnetic trap. The resulting trap in the dressed state m F = F = 2 is very anisotropic, with oscillation frequencies near the trap centre of the order of 1 kHz along z and of 5 to 20 Hz in the horizontal directions. In this way, the atom cloud has the shape of a crêpe [2].

Coupling to the probing field: resonance frequencies

A second RF field of frequency ω 2 and Rabi frequency Ω 2 is applied to the dressed atoms. The polarisation of this second field is also linear along y, and again its orientation relative to the static magnetic field slightly varies across the dressed trap. As a result, both polarisations X and Z are present and can couple to the atomic spin. In the following, we neglect the position dependence of Ω 2 and concentrate in each case on the polarisation component having a significant coupling to the atomic spin. Ω 2 is typically a factor 100 times smaller than the dressing Rabi frequency Ω 1 . This second field allows a coupling between dressed states m F and m F with |m F -m F | = 1: see the arrows in figure 1. This coupling occurs either within the same manifold or between different manifolds. The expected resonant frequencies for this coupling at position r are thus ω 2 = nω 1 ± Ω(r), n being an integer [15].

In the 'resonant configuration', the points r in space where the probing field is resonant, i.e. where the condition Ω(r) = |ω 2 -nω 1 | is satisfied, consist of two surfaces S + and S -on both sides of the trapping surface S 0 (see figure 2a). The detuning of the local Larmor frequency with respect to ω 1 has an opposite sign for these two probing surfaces, δ(r 2 for the surface S ± respectively. The shape of the resonant surfaces S ± strongly depends on the value of the probing frequency ω 2 around nω 1 . First, let's remark that the resonance can be reached at point r for probing frequencies ω 2 close to nω 1 only if |ω 2 -nω 1 | > Ω eff 1 (r). Three cases can then be distinguished. First, for frequencies ω 2 satisfying |ω 2 -nω 1 | > Ω 1 , the resonant surfaces are very close to isomagnetic ellipsoids, see figure 2a. The atoms need a minimum non zero energy to reach these surfaces, and the point where the necessary energy is the lowest lies at the bottom of the external surface S -. Second, for |ω 2 -nω 1 | = Ω 1 , the two resonant surfaces merge at the trapping surface S 0 all over the vertical plane y = 0, in particular at the trap bottom, and all atoms may reach this point and be lost, see figure 2b. Third, for |ω 2 -nω 1 | < Ω 1 , resonant surfaces still exist due to the spatial dependence of Ω eff 1 , but each of them is split into two curved surfaces centred around the y axis, see figure 2c and2d. Again a minimum energy is necessary to reach the lowest resonant point. Finally, for |ω 2 -nω 1 | equal to Ω 1 ω 0 /ω 1 , the resonant surfaces collapse to two points at the equator and for lower values of |ω 2 -nω 1 | no atoms are lost.

) = ± (nω 1 -ω 2 ) 2 -Ω eff 1 (r)
In summary, the energy in the dressed trap necessary for an atom to escape through a point resonant with the probing field depends on the probe detuning to nω 1 . Figure 3 gives, as a function of |ω 2 -nω 1 |, the minimum energy E min (ω 2 ) with respect to the trap bottom for an atom to escape, for a dressing frequency of ω 1 /2π = 3 MHz and a Rabi frequency of Ω 1 /2π = 75 kHz. The three regimes discussed above are clearly visible. The final shape of the recorded spectrum can be inferred from this minimum energy. If, over the duration of the experiment, the RF weak probe couples atoms efficiently out of the dressed trap when a resonant surface is reached, all the atoms with an external energy E larger than E min (ω 2 ) should be lost. Hence, the expected fraction of remaining atoms in the trap f rem (ω 2 ) is a function of E min (ω 2 ), namely

f rem = E min (ω2) 0 p(E) dE (3)
where p(E) is the energy distribution in the dressed trap, and the zero energy level for E is taken at the trap bottom. In particular, we expect f rem (nω 1 ± Ω 1 ) = 0 and an asymmetric shape of the spectrum around this value.

The exact shape of p(E) is related to the exact energy potential. Its expression is not analytical, as the trap is far from being harmonic. However, we propose as a simple ansatz an exponential expression for p of the form p(E) = (k B T ) -1 exp(-E/k B T ) and compare the simple prediction f rem (ω 2 ) = 1exp[-E min (ω 2 )/k B T ] to the experimental data, see section 3.2.

Coupling to the probing field: coupling strengths

In the previous section, the two resonant surfaces S ± at which the transitions ω 2 = nω 1 ± Ω(r) can occur have been presented. Let us point out that the strength of these different transitions is not the same, and depends on position in the dressed trap. This is true even if we assume a homogeneous Rabi frequency Ω 2 , which we do in the following for simplicity. For example for n = 1, the coupling strength at position r is Ω2

1 ∓ δ(r) Ω(r)
for the transitions at frequency ω 2 = ω 1 ± Ω(r), respectively [14]. The coupling strength can be interpreted in terms of the number of photons n from the dressing field necessary for the transition to occur. If we consider detunings δ(r) which are large compared to the dressing amplitude Ω 1 (such that Ω(r) |δ(r)| = |ω 1 -ω L (r)|), then the atoms are far from the trapping surface S 0 . In this case, the two RF couplings become independent and the probing field essentially couples undressed states together. As a result, the coupling is of order Ω 2 for the single photon process at frequency ω 2 ω L (r), and drops to a vanishing value Ω 2 Ω eff 1 (r) 2 4δ 2 for the three photon process at frequency ω 2 2ω 1 -ω L (r). Hence, transitions to an untrapped state occur essentially through one of the two surfaces resonant with the probing field: the inner surface S + , closer to the magnetic field minimum, for the ω 2 = ω 1 -Ω(r) transition, and the outer surface S -, below the trapped atoms, for the ω 2 = ω 1 + Ω(r) transition.

Let us now focus on the low frequency transition n = 0, at ω 2 = Ω(r). In contrast to the previous case, this transition presents (as we shall see below) a symmetric coupling Ω eff 1 (r)Ω 2 /ω 2 at the two resonant surfaces S ± , between which the trapped atoms lie. This has some importance in the implementation of evaporative cooling in the dressed trap.

As this transition was not discussed in Ref. [14], we will briefly present its origin. First, we remark that the RF probe field should be π-polarised, meaning that the RF field component parallel to the static magnetic field direction Z is responsible for the transition. This can be understood by considering the transition induced by the probe away from the trapping surface S 0 , where the two RF fields decouple and the resonant condition is simply ω 1 ± ω 2 = ω L (r): the probe and the circularly polarized dressing field ω 1 induce a two-photon transition, that will change m F by 1 unit if the probe field is π-polarized. The Hamiltonian describing the spin evolution at position r in the presence of the two RF fields then reads

H(t) = ω L (r)F Z + 2Ω eff 1 (r)F X cos ω 1 t + 2Ω 2 F Z cos ω 2 t ( 4
)
where Z is the direction of quantization imposed by the static magnetic field and X is the effective polarisation of the dressing RF field ω 1 of Rabi frequency Ω eff 1 (r). To deal with this double frequency RF coupling, we apply a rotation transformation to the Hamiltonian in the spirit of Ref. [14], at an angle θ(t) = (ω 1 t + 2Ω2 ω2 sin ω 2 t) around the Z axis. The idea of this transformation is that the term in θF Z is chosen to cancel the new term 2Ω 2 F Z cos ω 2 t. After the rotation, product terms of the form cos ω 1 t cos ω 1 t + 2Ω2 ω2 sin ω 2 t appear. Since Ω 2 ω 2 , these terms can be developed as products of sine and cosine functions. Finally, application of the rotating wave approximation allows the elimination of fast rotating terms (rotating at about 2ω 1 ). The Hamiltonian in the rotated frame finally simplifies to

H (t) -δ(r)F Z + Ω eff 1 (r)F X - 2Ω eff 1 (r)Ω 2 ω 2 sin ω 2 tF Y . ( 5
)
This consists of the Hamiltonian H 1 = -δ(r)F Z + Ω eff 1 (r)F X for the state dressed by the main RF field, to which a transverse RF coupling of frequency ω 2 and effective amplitude

Ω eff 1 (r)Ω2 ω2
is added. This coupling allows transitions at a frequency ω 2 = Ω(r). Again, the condition is fulfilled at the two resonant surfaces S ± , but now with the same coupling amplitude at two points symmetric with respect to the surface S 0 , in contrast to the n = 1 case.

Experimental results

In this section, experimental results of RF spectroscopy of a RF-dressed trap are presented. The results are obtained in a dressed Ioffe-Pritchard trap in the resonant configuration.

Experimental procedure

The loading procedure into the adiabatic potential was described in detail in a previous paper [13]. Rubidium 87 atoms are initially confined in their internal state F = 2, m F = 2 in a static magnetic trap and evaporatively cooled to a temperature T 0 . The dressed trap is loaded into the dressed state m F = 2 with a 500 ms RF ramp from 800 kHz to the final value of ω 1 , ranging between 3 and 8 MHz. At the end of this loading stage, a second RF field with a fixed frequency ω 2 and a Rabi frequency Ω 2 is switched on for 2 seconds, to perform the spectroscopy of the adiabatic potential. The Rabi frequency Ω 1 of the dressing field is kept constant during the loading stage and the probing stage [20]. The RF fields are produced by two identical antennas (9 windings, 10 mm in diameter) placed face to face across the vacuum chamber, at a distance of 17 mm from the atomic cloud. At the input of each antenna, the RF amplitude lies between 160 and 640 mA pp for the dressing field, and between 0.8 and 5 mA pp for the probing field. Finally, the atom number and the optical density are recorded after a time of flight of 7 or 10 ms, and a spectrum is recorded as a function of the probing frequency ω 2 . This procedure is repeated for different values of Ω 1 , Ω 2 and ω 1 . As discussed in the previous section, resonances are expected at frequencies around Ω 1 , ω 1 -Ω 1 , ω 1 + Ω 1 , 2ω 1 -Ω 1 , etc. [15]. The number of atoms remaining after the application of the probe field is plotted (squares) as a function of the detuning ∆ 2 = ω 2 -ω 1 to the frequency ω 1 . The dressing amplitude is 400 mApp at the input of the RF antenna, corresponding to a Rabi frequency Ω 1 = 2π × 50 kHz as deduced from the spectrum. The probing amplitude is 1.6 mApp, smaller by a factor 250. The calculated number of remaining atoms for T = 11 µK is plotted as a full grey line, see text.

Experimental spectra

A typical spectrum recorded around ω 1 = 2π × 8 MHz is plotted in figure 4. For this data set, the dressing amplitude is set to 400 mA pp while the probing amplitude is 1.6 mA pp . The two resonances at ω 1 ± Ω 1 are visible, broadened by the energy distribution in the trap at temperature T . For probing frequencies between ω 1 -Ω 1 and ω 1 +Ω 1 , the atomic loss rate rapidly decreases to zero, and no transition is observed at a frequency ω 2 = ω 1 , as expected from the previous discussion. At ω 2 = ω 1 ±Ω 1 , the probing frequency is resonant with the bottom of the adiabatic potential and almost all the atoms are lost (80% for this particular choice of the probe coupling Ω 2 ).

Away from this value, the atoms satisfying Ω(r) > |ω 2 -ω 1 | are lost. As expected, the observed line has an asymmetric shape, with a dip at ω 2 = ω 1 ± Ω 1 and approximately exponential outer wings with a width related to the temperature. Indeed, in a harmonic trap, the lineshape should be exponential, with a 1/e-width k B T/(2 ) and a threshold at an energy corresponding to the trap bottom. Using an exponential trial function p(E) = (k B T ) -1 exp(-E/k B T ) for the energy distribution, the resulting expression for f rem derived from Eq.( 3), and the temperature of T = 11 µK deduced from an independent time-of-flight measurement, we obtain the predicted spectrum plotted with a grey line in figure 4. f rem is normalised to total the atom number, and the limited maximum contrast of 80% of atom loss for 2 s at resonance is also taken into account. The experimental spectrum is in good agreement with the predicted lineshape, indicating that the exponential model used to estimate the energy distribution is indeed sufficient.

The sharp edge of the spectrum allows a good determination of the dressing Rabi frequency Ω 1 at the bottom of the adiabatic potential, equal to 50 kHz in the spectrum of figure 4. The determination is more accurate when a smaller probe Rabi frequency is used. A larger amplitude Ω 2 leads to broadening of the resonance line. Experimentally, an amplitude of 1.6 to 3.2 mA pp was found to be a good compromise between a good probing efficiency (80% of atom loss after 2 s on resonance) and a negligible line broadening. The RF dressing amplitude was varied with a RF attenuator and the position of the peak atom loss was recorded to check the validity of the determination of Ω 1 by this technique. The result is plotted on figure 5. The detuning at the maximum atom loss is expected to be directly proportional to the RF amplitude. At a dressing frequency ω 1 = 2π × 8 MHz, the deduced Rabi frequency is Ω 1 = ηI 1 , with a scaling factor η = 2π × 135 kHz/A pp , I 1 being the RF amplitude in amperes peak-to-peak. This is in agreement with the value of 130 kHz/A pp calculated from the geometric characteristics of the antenna.

The determination of Ω 1 from the spectra around ω 2 = ω 1 ± Ω 1 is also confirmed by the data recorded at low probing frequency. Figure 6 presents a typical spectrum at frequency ω 2 Ω 1 . We recover a Rabi frequency of 75 kHz at 3 MHz of dressing frequency. Again, the anisotropic shape is due to energy distribution of the atoms confined in the dressed trap. The overall shape of the spectrum is in reasonable agreement with the calculated spectrum for a temperature of 4 µK.

Evaporative cooling

Garrido Alzar et al. [14] proposed to use this second RF in this trap to force evaporation and cool the atoms. Indeed, the energy sensitivity of the atom loss induced by the second RF can be used to selectively remove the most energetic atoms from the trap. We implemented this forced evaporative cooling inside the RF-dressed trap. For this purpose, the second RF was scanned from an initial frequency away from the resonance down to a frequency close to the resonance at the trap bottom. The experiments where performed with a dressing RF amplitude of Ω 1 = 2π × 50 kHz, at ω 1 = 2π × 8 MHz. To optimize the symmetry of the coupling through the two resonant surfaces S ± , the low frequency resonance was preferred. The frequency of the weak field ω 2 /2π was lowered from 600 kHz to a final value ν end . We observed a decrease of the cloud temperature and the atom number as ν end is lowered, see figure 7. The selective out-coupling of the atoms with a large vertical energy E z works as expected around the low frequency resonance, whereas almost no effect was obtained when scanning the frequency around ω 1 , where the out-coupling is asymmetric. In our dressed trap, the phase space density increases only slightly, due to a poor thermalisation efficiency with the low horizontal frequencies (5 Hz × 20 Hz). With larger horizontal frequencies in the dressed trap, as is the case in a dressed quadrupole trap for example [START_REF] Morizot | [END_REF], and thus with a higher collision rate, thermalisation in the trap would lead to an increase of the phase space density [21], as shown by molecular dynamics simulations. From the point of view of the out-coupling efficiency, our results show that evaporative cooling is feasible in a resonant dressed trap in the same conditions as in a conventional magnetic trap.

Conclusion and prospects

In this paper, we have demonstrated RF spectroscopy in a RF dressed trap operated in a regime where the trapped atoms are resonant with the dressing RF source. Thus we use two RF fields: a strong dressing RF source, and a second weak probing field which couples the atoms out of the dressed trap. Spectra are recorded either around the dressing RF frequency (a few MHz), or at low frequencies near the Rabi frequency (a few tens of kHz). The experimental data fit well with a simple model based on the minimum energy for a trapped atom to be outcoupled by the weak probe. The low frequency transition is used to selectively evaporate the hotter atoms from the dressed trap. A temperature decrease is demonstrated in the transverse direction of the dressed trap together with an atomic population decrease. The resulting phase space density increase is modest at the present stage of the experiment since the very low motional frequencies of this anisotropic trap imply a low collision rate and thus a poor thermalisation.

The results of this paper emphasise the interest in utilizing resonant RF fields to realise unusual atomic potentials and show that we can apply evaporative cooling inside these potentials. Unlike light based traps, a resonant RF-based trap is not affected by spontaneous emission. On the other hand it is sensitive to the noise of the RF source. However, heating, and the ensuing losses, can be kept at very low values provided there is a good control of the parameters of the source [13]. Finally, we conclude that with reasonable trap frequencies (in all directions) it should be possible to have an efficient RF evaporative cooling of an atomic population in a RF trap.

Figure 1 .

 1 Figure 1. Adiabatic potential energy (V m F 1

Figure 2 .

 2 Figure 2. Cut of the resonant surfaces S ± in the plane x = 0 in the three cases discussed in the paper. (a): |ω 2 -nω 1 | = 2Ω 1 > Ω 1 , (b): |ω 2 -nω 1 | = Ω 1 , (c): |ω 2 -nω 1 | = 0.7Ω 1 , i.e. Ω 1 ω 0 /ω 1 < |ω 2 -nω 1 | < Ω 1 . The isomagnetic surface S 0 to which the atoms are trapped is also represented. ω 1 is set to 2π × 3 MHz to show the potentials clearly and Ω 1 is set to 2π × 75 kHz. (d): Corresponding outer resonant surface S -in a 3D representation in the case (c).

Figure 3 .

 3 Figure 3. Minimum energy required for an atom to leave the dressed trap by reaching a resonant surface, as a function of the probe detuning |ω 2 -nω 1 |. The parameters are: ω 1 = 2π × 3 MHz and Ω 1 = 2π × 75 kHz.

2

 2

Figure 4 .

 4 Figure 4. Spectroscopy of the dressed trap, around the dressing frequency ω 1 = 2π × 8 MHz. The number of atoms remaining after the application of the probe field is plotted (squares) as a function of the detuning ∆ 2 = ω 2 -ω 1 to the frequency ω 1 . The dressing amplitude is 400 mApp at the input of the RF antenna, corresponding to a Rabi frequency Ω 1 = 2π × 50 kHz as deduced from the spectrum. The probing amplitude is 1.6 mApp, smaller by a factor 250. The calculated number of remaining atoms for T = 11 µK is plotted as a full grey line, see text.

Figure 5 .

 5 Figure 5. Rabi frequency Ω 1 , as deduced from the position of the edge of the line, plotted as a function of the dressing RF amplitude. RF dressing frequency: ω 1 = 2π × 8 MHz.

Figure 6 .

 6 Figure 6. Spectroscopy of the dressed trap, around the dressing amplitude Ω 1 = 2π × 75 kHz, for a dressing frequency ω 1 = 2π × 3 MHz. The horizontal axis indicates the probe frequency ω 2 . The dressing amplitude is 580 mApp at the input of the RF antenna, corresponding to a Rabi frequency Ω 1 = 2π×75 kHz (see text). The probing amplitude is 9 mApp at the input of the second antenna, but only the component parallel to Z couples to the atoms (see text). The calculated number of remaining atoms for T = 4 µK is plotted as a full grey line.

Figure 7 .

 7 Figure 7. Final atom number N and kinetic temperature Tz, deduced from the vertical size of the cloud after a time-of-flight expansion, after an evaporation ramp of the second RF field between 600 kHz and ν end .

Acknowledgments

We thank V. Dini for assistance in data processing. This work was supported by the Région Ile-de-France (contract number E1213 and IFRAF), by the PPF 'Manipulation d'atomes froids par des lasers de puissance', by the European Community through the Marie Curie Training Network 'Atom Chips' under contract number MRTN-CT-2003-505032, and by Leverhulme Trust. B. M. G. thanks CNRS for support. Laboratoire de physique des lasers is UMR 7538 of CNRS and Paris 13 University. LPL is member of the Institut Francilien de Recherche sur les Atomes Froids (IFRAF).