Abstract
Collective behaviour is a characteristic feature in manybody systems, important for developments in fields such as magnetism, superconductivity, photonics and electronics. Recently, there has been increasing interest in the optically nonlinear response of collective excitations. Here we demonstrate how the nonlinear interaction of a manybody system with intense XUV radiation can be used as an effective probe for characterizing otherwise unresolved features of its collective response. Resonant photoionization of atomic xenon was chosen as a case study. The excellent agreement between experiment and theory strongly supports the prediction that two distinct poles underlie the giant dipole resonance. Our results pave the way towards a deeper understanding of collective behaviour in atoms, molecules and solidstate systems using nonlinear spectroscopic techniques enabled by modern shortwavelength light sources.
Introduction
Significant advancements in photonics^{1,2,3}, especially in electric field enhancement^{4,5} and harmonic generation^{6}, have been mostly triggered by the recent development in tailoring materials on the nanometre scale exploiting their resonant collective response to radiation^{7}. To optimize the coupling between the nanostructure and the electromagnetic field, a detailed understanding of the underlying resonant response is essential. To this end, atomic samples provide a valuable benchmark for understanding more complex systems, easing meaningful systematic investigations.
An illustrative example of a manybody system showing collective electronic behaviour is atomic xenon^{8}; its resonating character under extreme ultraviolet (XUV) radiation, known as the 4d giant dipole resonance, is interpreted as the collective response of many electrons to an external weakfield perturbation^{9,10}. The recent advent of highbrilliance light sources such as XUV and Xray freeelectron lasers (FELs) has opened a door to XUV and Xray studies beyond the linear regime. Exploiting this new highintensity technology renders it possible to investigate the collective response mechanisms of manybody systems through their nonlinear interaction with shortwavelength radiation. As shown here, this provides the possibility of unveiling substructures in the spectrum of collective excitations that cannot be resolved with linear spectroscopy.
The case of xenon ionization under the unprecedented conditions at FELs has been the subject of several investigations^{11,12,13,14,15,16,17}, which have stimulated speculations about the influence of collective effects on the process of multiple ionization^{12,14}; furthermore, a high harmonic generation experiment on xenon^{18} evidenced the impact of the 4d giant resonance on a nonlinear optical process^{19,20}. Yet, all these observations can be well understood, as far as collectiveness is concerned, in terms of the 1photon absorption crosssections of the various charge states of xenon^{13,17,21}, that is, in terms of the spectral characteristics of its linear response.
Employing nonlinear electron spectroscopy, namely through the study of xenon 2photon ionization, we demonstrate here that the nonlinear process unveils otherwise unresolved aspects of the collective behaviour of the system. Due to the photon energy selected, the 2photon process occurs through the giant resonance as an intermediate step (Fig. 1). We show, however, that a model assuming a single intermediate state cannot describe our results. Instead, the resonance feature in the predicted energy dependence of the 2photon process and its shape strongly suggest that more than one resonance state underlie the giant resonance^{22}; although these states are unresolved in the linear ionization of xenon, 2photon ionization turns out to be a sensitive process for their observation.
Results
Experiment and theory approach to nonlinear photoionization
Our findings are made possible by the combination of electron spectroscopy, which allows the disentanglement of photoemission processes from different orders of interaction, with firstprinciples calculations. We measured the relative yields of 1photon and 2photon ionization of the 4d shell of xenon (Fig. 1a,b) by electron spectroscopy and compare them with numerical solutions of the manyelectron Schrödinger equation for atomic xenon in the presence of an external XUV laser field. Our theoretical model captures manybody processes beyond linear response theory, allowing the selective inclusion of those electronic correlation effects that are responsible for collectiveness. For a system characterized by collective behaviour, the wavefunction is given by a coherent superposition of particle–hole states^{23}, due to the strong particle–hole interaction. We compare the experimental results with the full model, which describes the collective response of the system by accounting for the electron–hole interaction in all channels open to ionization (Fig. 1d), and a reduced model, which confines this interaction to the hole from which the electron was excited (Fig. 1c).
Electron spectroscopy of 1 and 2photon ionization
Our experiments were performed at the BL2 beamline of FLASH^{24}, the Freeelectron LASer in Hamburg, Germany. FEL pulses at photon energies of 105 and 140 eV were focused down to a few microns in front of the aperture of a magnetic bottle electron spectrometer. The spectrometer was used to measure the kinetic energy (KE) of the electrons produced by 1photon and 2photon absorption processes in an effusive jet of xenon atoms (see Methods).
Electron spectra (Fig. 2) were collected under different intensity conditions. The spectra include features caused by 1photon direct emission from the 5p, 5s and 4d shells as well as from NOO Auger decay^{25}. At higher kinetic energies, the 2photon ionization from the 4d shell is observed in a spectral feature that resembles in shape the 4d (1photon) emission lines and is separated from them by exactly the energy of one photon.
The relative yields from the 4d 1 and 2photon ionization processes are obtained by integrating the spectra over the corresponding kinetic energies regions (105 eV, 1photon: 33–39 eV; 140 eV, 1photon: 68–74 eV; 105 eV, 2photon: 136–146 eV; 140 eV, 2photon: 206–216 eV; see caption of Fig. 3) and are shown as a function of the FEL intensity in Fig. 3. At low intensities (I<10^{13} W cm^{−2}), the 1 and 2photon ionization yields show a linear and quadratic dependence, respectively. This confirms, on the basis of perturbation theory, the nature of the ionization processes. At higher intensities, the depletion of the neutral target induced by the enhanced 1photon ionization leads to a pronounced saturation effect.
Comparison between experimental and theoretical results
The experimental yields are compared with the results of calculations (Fig. 3) performed for the full and the reduced models, respectively. The theoretical yields are obtained from the numerical solutions of rate equations (see Methods). The comparison between experimental points and rate equation solutions employs a single normalization factor for all data sets (1photon and 2photon yields at 105 and 140 eV, respectively).
This comparison clearly shows that the full model reproduces the intensity dependence of the experimental yields, whereas the reduced model fails to do so. This means that the inclusion of Coulomb coupling between all possible electron–hole states, which is responsible for the collective electronic response of the system, is an essential ingredient for the correct description of the 2photon ionization process. The very good agreement is evident in the ratio between the 1photon and 2photon ionization yields at both photon energies over the whole intensity range as well as in the onset of the saturation due to neutral target depletion.
Discussion
Having validated our full model by the comparison with experimental yields at two photon energies, we investigate the influence of collective effects on the 1 and 2photon ionization crosssection over a wide photon energy range (Fig. 4). For the 1photon crosssection, the broadening is due to the wellknown broadening and blue shift of the giant resonance caused by the inclusion of coupling among different electron–hole states^{9}, which is reproduced by our calculations (Fig. 4).
As a mostly unexpected and counterintuitive result, the full model predicts a much broader 2photon crosssection curve than for the 1photon case. Considering the 2photon crosssection within perturbation theory, the 2photon crosssection curve produced by the reduced model can be qualitatively understood (Fig. 4a) in terms of a sequential process involving a single intermediate state, where the 2photon crosssection (reddotted curve) factorizes into two 1photon crosssections (one photon for exciting the giant resonance from the ground state (solid black curve), and the other photon for the transition from the resonance to the final state, which is modelled by a E^{−13/2} energy dependence (see Supplementary Discussion)). According to this twostep picture with a single intermediate state, one expects a narrower 2photon peak that is shifted to lower energy (dash–dotted blue curve), since the 1photon crosssection for exciting an electron from the intermediate state into the continuum decreases monotonically with increasing energy. This model captures qualitatively the behaviour of the 2photon crosssection in the reduced model case. In contrast, for the full model (Fig. 4b), the picture of a sequential process involving a single intermediate state does not hold: surprisingly, the 2photon crosssection curve is much broader than the 1photon crosssection curve and exhibits a kneetype structure. This substructure, which emerges in the nonlinear process, manifests the existence of more than one resonance state underlying the giant resonance^{22}. These states give rise to interference terms resulting in a broadening of the 2photon absorption crosssection curve (see Supplementary Discussion). Indeed, the experimental results cannot be explained, simultaneously at 105 and 140 eV, by the twostep picture with a single intermediate state (dash–dotted blue curve). In particular, at 140 eV the crosssection measured experimentally is ∼12 times larger than predicted by the single intermediate state model, while at 105 eV it is larger by a factor of 2.2. Further analysis within the timedependent configuration interaction singles scheme (TDCIS) reveals two underlying resonance states^{26}, which are indicated by arrows in the inset of Fig. 4b. The resonance positions are consistent with the substructure visible in the 2photon crosssection. Here for the first time, the agreement of a theoretical model with experimental results beyond the linear regime legitimizes the prediction of two resonances underlying the giant resonance^{22}.
Summarizing, we have shown that the nonlinear response of an electronic system to intense XUV radiation can be used to unveil information about the collective behaviour in manybody systems. The theoretical xenon 2photon crosssection exhibits a kneetype structure that is not visible in the 1photon crosssection. The present study demonstrates, employing xenon as a model system, how the nonlinear interaction regime can be utilized to investigate collective electronic behaviour. This stands only at the beginning of the way towards a deeper understanding of the collective response of manybody systems.
Methods
FEL beam transport and characteristics
The selfamplified spontaneous emission FEL pulses had a duration of about 80±20 fs and up to 40 μJ (at 105 eV) and 15 μJ (at 140 eV) energy per pulse. The bandwidth was about 1% at both photon energies. The FEL pulses were focused onto the sample by means of MoB_{4}C multilayer mirrors in a backreflecting geometry to produce a tight focusing of 5±1 μm full width at half maximum. The mirrors have a reflection bandwidth of ∼1 eV with peak reflectivity of ∼40% (at 105 eV) and ∼20% (at 140 eV) centred at the respective photon energy, thus enabling in addition a very effective filtering (>4 orders of magnitude) of any possible higher harmonic contamination (estimated <0.3%) that might be present in the FEL beam^{24}. FEL irradiance was tuned using a gas attenuator system and moving the focusing mirror along the beam direction in order to vary the beam crosssection within the interaction zone. The attenuator was used to control the energy per pulse delivered into the interaction region thereby providing a fine tuning of the intensity over a restricted range (∼1 order of magnitude). In addition, by varying the beam crosssection from the minimum value of 5 μm up to ∼190 μm, the intensity was altered over more than 4 orders of magnitude. The photon beam parameters were monitored online during the experiments. A calibrated gas monitor detector provided the energy of the FEL pulses on a singleshot basis^{24}. A chargecoupled device camera was used to record the singleshot FEL spectra from a variable line spacing grating spectrometer installed along the beam transport. The spectral information was used to normalize the beam intensity to the multilayer mirror reflection curve.
Electron spectroscopy for determining experimental yields
Electron spectra of 1photon and 2photon ionization of xenon were measured by means of a magnetic bottle electron spectrometer (MBES)^{27}. Technically, since the photon energies exceed the binding energy of the orbitals considered, the observed 2photon process is abovethreshold ionization.
The acceptance volume of the MBES, limited by the magnetic field lines of the system, had a size of ∼0.5 mm in the plane perpendicular to the spectrometer axis. The MBES enables 4π acceptance of the solid angle with an energy resolution for the detected electrons of 2%. By means of a retardation stage, it was possible to increase the resolution of the spectral features down to the FEL bandwidth limit.
The 1photon and 2photon signals were collected for different FEL intensities under different MBES settings as well as different conditions for the sample density. 2Photon electrons were collected under higher sample density conditions and applying a retarding field at the entrance of the MBES rejecting slow electrons, to avoid detector saturation induced by the 1photon signal. The intensityindependent normalization factors defining the relative yields (sample density, transmission of the analyser and detector gain) are calibrated by comparing the experimental and theoretical results obtained for the 1photon and 2photon ionization from the 3p orbital of argon, which is a much less complex system exhibiting negligible correlation effects, thereby providing a robust calibration reference.
The experimental intensity domains are not identical for the the 1photon and the 2photon yields, collected in subsequent measurements, because of the consistent varying of the selfamplified spontaneous emission FEL intensity during the shifts. For the 105eV case, where electron yields are more severely affected by saturation effects at high intensities, the experiment was performed under different focusing conditions to allow the investigation over a broader intensity range. The experimental yields are extracted by integrating the FEL intensityresolved electron spectra in the KE regions mentioned in the Results section, corresponding to the binding energy ranges from 66 to 72 eV and from 64 to 74 eV for the 1photon and the 2photon signals, respectively.
Data acquisition
Electron energy spectra were acquired by feeding the signal from the detector collection anode into a Lecroy WavePro 725ZiA digital oscilloscope (8 bit, 10 GSPS, 2.5 GHz bandwidth) triggered by a transistor–transistor logic signal synchronized with the FEL pulse. The DAQ server was controlled by a Labviewbased data acquisition (DAQ) client enabling the collection of singleshot spectra and their sorting according to the intensity and the spectral distribution of the FEL. Intensityresolved electron energy spectra can be extracted in two different ways depending on the energy region examined. LowKE spectra, produced by singlephoton processes, result from the analogue current signal collected by the detector anode. The 2photon direct ionization features, with yields that are some orders of magnitude lower than the 1photon features, result from the collection of only a few electrons per FEL shot by the detector. Their signal is time discriminated by software, and the histogram of their arrival time is taken in counting mode and suitably normalized, resulting in a virtually backgroundfree electron spectrum. This approach enables the extension of the dynamic range well beyond the limitation given by the digitizer.
Firstprinciples calculation of crosssections
Our model is based on the TDCIS^{28}. In this nonperturbative approach the full Nelectron Schrödinger equation is solved numerically
The wavefunction is expanded in the oneparticle–onehole basis:
where the index i denotes an initially occupied orbital, a stands for an unoccupied orbital and Φ_{0}〉 symbolizes the Hartree–Fock ground state. The crosssections for 1 and 2photon absorption are calculated via the population in the corresponding hole channels, which are distinguishable due to the different angular momenta of the ejected electron. The level of our calculations does not include any groundstate correlations. Within TDCIS it is possible to include and distinguish certain electronic correlation effects that are mediated by Coulomb interaction. In particular, for the description of a collective response, the system cannot be described by a single particle–hole state, but rather a superposition of particle–hole states is needed. The full model includes the coupling among the holes in the 4d, 5s and 5p orbitals and the electron. The corresponding Coulomb matrix elements are included for all different index pairs within the space of active orbitals. In this way, superpositions of particle–hole states, that is, collective states, may be described. In contrast, in the case of the reduced model the elements with i≠j are set to zero, which results in the description of coupling only with the very 4d orbital from which the electron was ionized.
Rate equations for theoretical yield calculation
The theoretical yields are obtained from the numerical solution of equations (3, 4, 5, 6, 7) valid for the electron yield from the neutral target (population N_{0}).
Rate equations are solved assuming a Gaussian pulse with 80 fs (full width at half maximum) duration. σ^{(1)} (1photon) and σ^{(2)} (2photon) ionization crosssections entering equations (4) and (5) are obtained for the full and the reduced model as described above. The rate equation solutions (Y_{1ph}, Y_{2ph}) are calculated over a very broad range (9 orders of magnitude) of laser intensities and the results are numerically integrated over the volume of acceptance of the electron analyser in order to account for the spatial distribution of the FEL fluence.
Additional information
How to cite this article: Mazza, T. et al. Sensitivity of nonlinear photoionization to resonance substructure in collective excitation. Nat. Commun. 6:6799 doi: 10.1038/ncomms7799 (2015).
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Acknowledgements
We thank the DESY staff for providing smooth machine conditions of FLASH during the beamtimes and Haiou Zhang and Enrico Ripani for their help in preparing and implementing the experiment. A.K., M.I., S.B., A.J.R., M.M. and R.S. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) under grant nos. SFB 925/A1, SFB 925/A3 and SFB 925/A5. A.K. and R.S. are grateful to YiJen Chen and Stefan Pabst for performing the calculations of the resonance energy positions. M.I. is furthermore grateful for funding by the Volkswagen foundation. The DCU group acknowledge support by SFI grant No. 12/IA/1742. The work was supported by the European COST Action CM1204 XLIC.
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The experiment was conceived by M.M., coordinated by T.M. and M.M. and performed by T.M., M.I., S.B., A.J.R., P.O.K., N.W., T.J.K., J.T.C. and M.M.; A.K. and R.S. performed the numerical calculations and elaborated the theoretical model; T.M. and A.K. analysed and combined the results from experiment and theory; the manuscript was prepared by T.M., A.K., M.M. and R.S. with contributions from all the authors.
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Supplementary Information
Supplementary Figure 1, Supplementary Discussion and Supplementary References (PDF 77 kb)
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Mazza, T., Karamatskou, A., Ilchen, M. et al. Sensitivity of nonlinear photoionization to resonance substructure in collective excitation. Nat Commun 6, 6799 (2015). https://doi.org/10.1038/ncomms7799
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