Nonlinear coherence effects in transient-absorption ion spectroscopy with stochastic extreme-ultraviolet free-electron laser pulses

TL;DR

This study demonstrates time-resolved nonlinear EUV absorption spectroscopy on multiply charged neon ions using phase-locked free-electron laser pulses, revealing transient spectral features and AC Stark shifts explained by a quantum model.

Contribution

It introduces a novel approach to observe nonlinear coherence effects in transient ions with phase-locked FEL pulses and models the enhanced ionic coupling during pulse overlap.

Findings

Observation of time-dependent spectral changes in doubly charged neon ions.

Detection of 10-meV-scale spectral shifts due to the AC Stark effect.

Explanation of the phenomena using a time-dependent quantum model.

Abstract

We demonstrate time-resolved nonlinear extreme-ultraviolet absorption spectroscopy on multiply charged ions, here applied to the doubly charged neon ion, driven by a phase-locked sequence of two intense free-electron laser pulses. Absorption signatures of resonance lines due to 2$p$--3$d$ bound--bound transitions between the spin-orbit multiplets $^3$P$_{0,1,2}$ and $^3$D$_{1,2,3}$ of the transiently produced doubly charged Ne$^{2+}$ ion are revealed, with time-dependent spectral changes over a time-delay range of $(2.4\pm0.3)\,\text{fs}$. Furthermore, we observe 10-meV-scale spectral shifts of these resonances owing to the AC Stark effect. We use a time-dependent quantum model to explain the observations by an enhanced coupling of the ionic quantum states with the partially coherent free-electron-laser radiation when the phase-locked pump and probe pulses precisely overlap in time.

Equations

$$A(\tau,\omega)=-\text{log}_{10}\bigg{[}\frac{I(\tau,\omega)}{I_{0}(\omega)}\bigg{]},$$

Full text

In interaction with matter the oscillating electric field of a laser not only induces transitions between bound electronic states but also affects the states and transitions themselves. It splits Autler and Townes (1955), shifts Fedorov (1998); Delone and Krainov (1999) and modifies the width Citron et al. (1977); Allen and Eberly (2012) and the shape Ott et al. (2013); Kaldun et al. (2014); Ott et al. (2014) of spectral transition lines depending on the amount of detuning out of resonance with the laser frequency and the field strength. Only for sufficiently high field strengths, at which more than one photon can interact with the quantum system on its intrinsic time and energy scale, these phenomena are accessible. Modern ultrafast lasers are effective driver and control tools for nonlinear effects at visible frequencies and have become the “working horses” for nonlinear coherent spectroscopies Mukamel (1995), in time domain and frequency domain, including the quantum control of bound–bound electronic transitions (see, e.g., Brif et al. (2010) and references therein).

    Since the advent of short-wavelength free-electron lasers (FELs) Ackermann et al. (2007); McNeil and Thompson (2010) the field of nonlinear spectroscopy is being extended into the extreme-ultraviolet (XUV) and x-ray spectral ranges Chapman et al. (2007); Sorokin et al. (2007); Young et al. (2010); Kanter et al. (2011); Doumy et al. (2011); Rudek et al. (2012); Rohringer et al. (2012); Weninger et al. (2013); Fuchs et al. (2015); Rudek et al. (2018); Young et al. (2018); Foglia et al. (2018). One advantage of employing x-rays is the ability to access bound–bound electronic transitions associated with the spatially localized inner-electronic shell and the potential to probe site-specific spectroscopic information of a sample. Since experimental studies on the impact of XUV/x-ray nonlinear effects on inner-shell-excited resonances are often hampered by the extremely short Auger decay times, yet, such research is rare. Nonetheless, first x-ray nonlinear line-shape modifications of inner-shell transitions have been studied experimentally Kanter et al. (2011) by employing Auger-electron spectroscopy. By contrast, we here address the valence electrons of the doubly charged neon ion, Ne2+, and manipulate—in the absence of any competing ultrafast decay channel—the ground state to excited state transitions between spin-orbit multiplets with intense XUV-FEL radiation. Being sensitive to the atomic/ionic dipole response and associated spectral line-shape modifications, our work demonstrates a direct view on XUV nonlinear effects occurring in the strongly driven system involving bound–bound transitions.

    In this Letter, we observe XUV-nonlinear physics near resonances in atomic ions. This is achieved by employing a time-resolved all-XUV nonlinear absorption spectroscopy method, based on the intense stochastic FEL pulses provided by the SASE (self-amplified spontaneous emission) Free-Electron Laser in Hamburg (FLASH). As a main result, we access the excited-state coherent response of the Ne2+ ions with spin-orbit state-specific spectroscopic resolution and identify the 10-meV-scale resolved discrete ionic multiplet transitions. Taking advantage of the partial temporal coherence of the stochastic light fields, this approach allows one to capture transient effects of nonlinear coupling in the XUV spectra on a few-femtosecond timescale, well below the average FEL pulse duration, but within the eV-scale reciprocal (Fourier-inverse) average spectral bandwidth. Furthermore, we observe and quantify XUV-intensity-dependent spectral shifts of ionic resonances. The presented all-XUV-optical transient absorption experiment with combined state-resolved spectroscopic and time-resolved dynamic access to the nonlinear absorption response (at least third order) can be regarded as a precursor study to nonlinear multidimensional spectroscopy in the XUV/x-ray spectral domain Schweigert and Mukamel (2007); Bennett et al. (2016) with currently available SASE FEL technology.

    In the experiment, we geometrically split the FEL beam into two approximately equal parts with average intensities in the mid $10^{13}\,\text{W/cm}^{2}$ region using the split-and-delay unit at beamline BL2 Wöstmann et al. (2013). Due to the beam’s spatial coherence properties, a phase-locked linearly polarized pulse pair is produced on a shot-to-shot basis, which has been demonstrated and employed in several previous experimental campaigns Jiang et al. (2010a, b); Moshammer et al. (2011); Roling et al. (2011); Meyer et al. (2012); Usenko et al. (2017). Both pulses, denoted by pump and probe, respectively, are focused with a spot size of ($20\pm 5$)$\,\upmu\text{m}$ Tiedtke et al. (2009) into a neon-filled gas cell ($\sim 50\,\text{mbar}$ backing pressure) with $2\,\text{mm}$ interaction length. Further downstream, in the optical far field, the transmitted pump and probe pulses are simultaneously detected. They are separated via an offset in space, and spectrally resolved via a flat-field variable-line-spacing (VLS) grating in combination with a CCD camera, obtaining a resolving power $E/\Delta E\approx 1500$. Figure 1(a) provides a schematic illustration of the experimental setup. The FEL was operated in single-bunch mode at a $10\,\text{Hz}$ repetition rate and centered at 50.6 eV photon energy with $\sim 0.8\,\text{eV}$ full width at half maximum (FWHM) spectral bandwidth of the average spectrum. The FEL pulse energy was measured shot by shot by the parasitic gas monitor detector (GMD) Tiedtke et al. (2008) upstream of the experiment. This FEL operation mode allows for the (post-)analysis and sorting of the individually taken photon spectra with respect to the pulse energies. Averaging over all data points, the mean pulse energy was $47.5\,\upmu\text{J}$ with 28% standard deviation due to the statistical shot-to-shot fluctuations. The average FEL temporal pulse duration was estimated to $\sim 100\,\text{fs}$ based on the measurement of the electron bunch duration Röhrs et al. (2009); Düsterer et al. (2014).

The physical mechanism of our experiment is illustrated in Fig. 1(b). The given FEL photon energy is sufficient to overcome the ionization thresholds Kramida et al. (2015) of the Ne atom at $21.6\,\text{eV}$, and its singly charged ion Ne*+* at $41.0\,\text{eV}$, respectively, via the sequential absorption of two XUV photons. The transiently created doubly charged Ne2+ ions represent the actual target of this experiment, which are (near-)resonantly excited and identified via the 2$p$–3$d$ electronic bound–bound transitions between the spin-orbit multiplets 3P0,1,2 and 3D1,2,3. With the relatively large cross section for the sequential single-photon ionization processes ($\text{Ne}\to\text{Ne}^{+}$ with $\sigma_{01}=9.32$ Mb Cav and $\text{Ne}^{+}\to\text{Ne}^{2+}$ with $\sigma_{12}=9.36$ Mb Cav at 50 eV) we estimate a fractional Ne2+ ion abundance of $\sim$70%, based on coupled linear rate equations Lambropoulos and Tang (1987). For this estimate, we assumed Gaussian pulses with 100 fs FWHM temporal duration and a peak intensity of $2\times 10^{13}\,\text{W/cm}^{2}$ to model the average FEL pulse shape, relating to an incoming fluence of approximately $2\,\text{J/cm}^{2}$. One goal of this work is to investigate the coherent nonlinear few-femtosecond dynamics around the identified resonant transitions in Ne2+ when both pump and probe pulses interfere and precisely overlap in time. Here it should be noted that the relevant ionic transitions are relatively long lived (on the timescale of the FEL pulse duration), hence their intrinsic decay dynamics are stationary during the 20-fs observation window. This enables an undisturbed access to the coherence effects of overlapping pump and probe pulses in the spectral vicinity of the resonances. We also identify enhanced plasma diffraction of XUV-optical pump photons into the direction of the detected probe beam, which is due to a nonlinearly enhanced Ne2+ abundance in temporal overlap of pump and probe pulses, as another contribution to affect the measured probe absorption spectrum.

    The measured absorbance $A$ is given in terms of the optical density (OD) following Beer–Lambert’s law,

[TABLE]

where $I(\tau,\omega)$ is the transmitted photon flux through the gas target, depending on both time delay and photon energy, and $I_{0}(\omega)$ is the incoming photon flux without the target, which was measured under the same experimental conditions. For weak-field optical transmission through the target, the quantity in Eq. 1 is directly proportional to the single-photon absorption cross section Gaarde et al. (2011); Santra et al. (2011). In order to minimize spectral irregularities due to the intrinsic FEL shot-by-shot fluctuations, the signal $I(\tau,\omega)$ was averaged over 200 consecutive single spectra at each time-delay-setting $\tau$, while the delay-independent incoming signal $I_{0}(\omega)$ was determined from the average over 3,300 FEL pulses. Note that spectral interferences according to the time delay of the two pulses are visible in the single-shot spectra but washed out in the averaged representation of $I(\tau,\omega)$ due to the 0.28 fs Wöstmann et al. (2013) shot-to-shot jitter of the interferometer. For details, see the Supplemental Material (section IV).

    Figure 2(a) shows the time-delay-resolved absorbance of the probe pulse as it was scanned over the temporal pulse overlap (i.e., the delay setting $\tau=0\,\text{fs}$) with an incremental step size of $\Delta\tau=0.4\,\text{fs}$. Embedded in the continuum absorption background, dominant resonant signatures are observed at around 49.25 eV, 49.29 eV and 49.37 eV photon energy which can be identified as 2$p$–3$d$ bound–bound transitions between the spin-orbit multiplets 3P0,1,2 and 3D1,2,3 of Ne2+, as schematically indicated by the dashed arrows in Fig. 1(b). Depending on $\tau$, the probe spectrum exhibits spectral changes including a decrease in the absorbance which is spectro-temporally localized near $49.5\,\text{eV}$ and when pump and probe pulses overlap perfectly in time around $\tau=0\,\text{fs}$. The measured temporal width of this transient bleach is $(2.4\pm 0.3)\,\text{fs}$ (FWHM), as obtained from a Gaussian fit along the time-delay trace, cf. Fig. 2(b). Time-delay-dependent changes of the continuum absorption background are not observed within the 20-fs observation window of the data shown in Fig. 2. Such behavior can be observed over a longer time-delay range ($>50~{}\textrm{fs}$), as a step-like bleaching of the optical density, indicative of a reduction of the neutral species when the pump pulse fully precedes the probe pulse. See Supplemental Material (section V) for details. It should be noted that the observed spectral lines occur slightly out of resonance with the blue-detuned $(50.6\pm 0.4)$-eV FEL photons of the probe beam and are thus measured only within the spectral wings.

In the following, we aim to develop a first understanding of the transient changes in the vicinity of the 2$p$–3$d$ resonances. For this purpose, we consider a few-level quantum model based on the approach previously described in the Supplemental Material of Ref. Ott et al. (2014), which we now employ with stochastic XUV fields to nonlinearly drive the coupled ${}^{3}\text{P}$–${}^{3}\text{D}$ $2p$–$3d$ multiplet transitions of Ne2+. Pump and probe electric fields are treated in the framework of the partial-coherence method Pfeifer et al. (2010) as two identical copies (perfect phase lock) with identical random-phase characteristics and delayed by $\tau$ with respect to each other. The starting point of this few-level model simulation is the 3P0,1,2 ionic ground state which we assume to be initially incoherently populated with equal probability across the spin-orbit substates owing to the statistical nature of the FEL pulses. This assumption also suppresses the formation of any spin-orbit wave-packet dynamics of the ionic ensemble in agreement with the experimental observation. Although we neglect any effect of the slow, 100-fs-timescale $\tau$-dependent buildup of the Ne2+-ion itself, we combine the few-level model with rate equations Meyer et al. (2012) to account for the few-fs-timescale nonlinear coherence enhancement effect of the peaking Ne2+-population yield at around $\tau=0\,\text{fs}$. This leads to an increased diffraction of pump photons in the direction of the probe beam and effectively reduces the absorbance of the probe. The effect is most pronounced in the 49.5 eV energy region where a shift to slightly lower photon energy of the pump is most significant (see Fig. 1c). A detailed description of the numerical approach can be found in the Supplemental Material (Sections I and II).

    The computational results are shown in Fig. 3. As in the experiment (cf. Fig. 2), the simulated pump–probe absorbance trace exhibits a spectral bleach in the vicinity of the resonances near 49.5 eV photon energy at around $\tau=0\,\text{fs}$, when pump and probe pulses precisely overlap in time and interact coherently. This is due to (i) additional (enhanced) nonlinear coupling between the near-resonantly driven transitions and (ii) enhanced plasma diffraction of the pump pulse as a consequence of the constructive interference of the “spiky” temporal sub-structure of the replica pulses (cf. Fig. S1 of the Supplemental Material). The qualitative agreement between experiment and simulation proves direct access to nonlinear coherence effects around these ionic transitions with this all-XUV-optical experimental method.

The observed trend of an enhanced coherent coupling in the pump–probe absorbance trace at zero time delay is similar to those discussed in early time-dependent transient-grating experiments with optical lasers Vardeny and Tauc (1981); Eichler et al. (1984), where the transient peaking in the measurement signal is explained by the nonlinear (wave-mixing) nature of the light–matter interaction. When combined with temporally incoherent fields, the accessible time resolution is essentially determined by the fields’ coherence time Morita and Yajima (1984); Tomita and Matsuoka (1986). The coherence time is quantified by the temporal width of the signal enhancement, provided that the replica pulses share identical spectro-temporal properties. The here observed $(2.4\pm 0.3)\,\text{fs}$ timescale correlates with the measured $\sim 0.8$ eV FWHM spectral bandwidth of the average spectrum according to the time–bandwidth product of Fourier-limited Gaussian pulses. It thus serves as an order-of-magnitude estimate for the coherence time.

    Finally, to systematically study the role of the nonlinear mechanisms that contribute to the changes of the electronic structure of the Ne2+ ion, we now perform static XUV absorption spectroscopy and vary the FEL-pulse intensity. In this case, the fields are temporally separated in the target ($\tau>100$ fs, probe first) and the transmitted spectra of the succeeding (pump) pulse are analyzed. While the preceding probe pulse already produces a high abundance of the Ne2+ ions, the succeeding pump pulse acts as a moderately strong dressing field of the ionic states and its optical absorption response is directly measured. The role of pump and probe are thus interchanged in this static setting, where the slightly weaker “probe” pulse is still strong enough to create a substantial amount of Ne2+ ions, while the temporally separated and stronger “pump” pulse is used to induce nonlinear effects. In Fig. 4 we show the measured XUV absorption spectra of the previously discussed ionic resonance features near 49.3 and 49.4 eV photon energy, now resolved for different FEL pulse energy regimes as selected by the GMD shot-by-shot measurement. Compared to the “natural” neon ionic spectrum, as obtained from high-power discharge-source measurements Livingston et al. (1997), the results presented here reveal considerable light-induced spectral modifications including clear intensity-dependent spectral shifts with a maximum relative resonance shift of about 50 meV observed for the ${}^{3}\text{P}_{0}$–${}^{3}\text{D}_{1}$ transition when subjected to an average FEL pulse energy of $>55\upmu\text{J}$ (marked by the red vertical lines in Fig. 4). This observation suggests nonlinear coupling dynamics of ionic resonances with strong XUV electric fields, pertaining to the dynamic Stark shift. It is interesting to see that the three resonances of the spin-orbit multiplet actually shift in different directions, effectively increasing their energy splitting, while the FEL photon energy centered at $\sim 50.6$ eV is blue-detuned to all resonances. Besides the repulsion of all resonances, we observe a shift in the mean position of the multiplet structure toward a lower transition energy, which is evidenced by the stronger redshift of the lowest resonance (red) with a comparatively weaker blueshift of the highest resonance (blue), while the central line (green) remains unchanged. Furthermore, we observe the lowest transition (red line) to nonlinearly increase the strength of its shift. We expect the nearby ${}^{3}\textrm{S}_{1}$ state at 52 eV, for which the FEL photon energy appears red-detuned, to play a major role in this seemingly complicated AC-stark-shifted multiplet structure. A numerical analysis of the observed shifts, including the coupling to the ${}^{3}\textrm{S}_{1}$ state, is presented in the Supplemental Material (Section III). This first experimental observation of such non-trivial light-induced level shifts clearly highlights the prospect of the here presented method of strong-field XUV absorption spectroscopy even in the static case, with direct access to distinct relativistic spin-orbit transitions driven by strong electric fields.

In conclusion, we have presented first experimental measurements of XUV-induced nonlinear coherence effects near excited-state ionic resonances with overlapping FEL pump and probe pulses in transient absorption geometry. The measured time scale of this transient effect is $(2.4\pm 0.3)\,\text{fs}$. The sensitivity to transient changes combines high spectral (limited by the grating spectrometer) with high temporal (limited by the FEL coherence bandwidth) resolution. The observed level shifts in strong XUV electric fields demonstrate the broad applicability of this nonlinear spectroscopy method with no fundamental limitation on spectral resolution, despite the much larger FEL bandwidth. The experiment thus represents a significant step towards the implementation of coherent multi-dimensional spectroscopy of XUV-excitation and decay dynamics in atoms and molecules with broadband intense SASE FELs.