Quantum optics with X-rays has long been a somewhat exotic activity, but it is now rapidly becoming relevant as precision x-ray optics and novel X-ray light sources, and high-intensity lasers are becoming available. This article gives an overview of the current state of the field and an outlook to future prospects

Line intensities and oscillator strengths for the controversial 3C and 3D astrophysically relevant lines in neonlike Fe 16+ ions are calculated. We show that, for strong x-ray sources, the modeling of the spectral lines by a peak with an area proportional to the oscillator strength is not sufficient and non-linear dynamical effects have to be taken into account. Furthermore, a large-scale configurationinteraction calculation of oscillator strengths is performed with the inclusion of higher-order electroncorrelation effects. The dynamical effects give a possible resolution of discrepancies of theory and experiment found by recent measurements, which motivates the use of light-matter interaction models also valid for strong light fields in the analysis and interpretation of astrophysical and laboratory spectra.PACS numbers: 31.15.am,32.30.Rj,42.50.Ct Astrophysical spectra recorded by space observatories provide the only means to determine the element composition, temperature, density, and velocity of distant celestial objects such as stars, x-ray binaries, black hole accretion discs, or active galactic nuclei [1][2][3][4][5][6][7][8][9]. Such x-ray (or optical) spectra are often composed of a series of peaks associated with a range of elements, ionic charge states, and transitions. Therefore, a large amount of reliable atomic data is needed to disentangle the physical properties of the emitting objects. Such data-transition energies and probabilities, oscillator strengths, collisional and recombination cross sections, etc.-may be obtained from laboratory astrophysics experiments (see e.g. [3, 10-16]) or, more economically, from theoretical calculations (see e.g. [17][18][19][20]).The x-ray emission lines of highly charged Fe ions are among the brightest in astrophysical spectra. Within the last decade, several observations were performed with the space laboratories Chandra and XMM-Newton (see e.g. [5,6,21,22]). The line-strength ratio of two 2p → 3d lines in Fe 16+ , customarily denoted as 3C [2p 6 (J = 0) → (2p 5 ) 1/2 3d 3/2 (J = 1), transition energy of 826 eV] and 3D [2p 6 (J = 0) → (2p 5 ) 3/2 3d 5/2 (J = 1), at 812 eV], was observed, but the results disagreed with theoretical predictions [17][18][19][20]. Initially this disagreement was considered to originate from the co-existence of different charge states of Fe, and later, after laboratory measurements, as an effect of electron-impact excitation of the ion. Furthermore, since several theoretical calculations of transition probabilities in highly charged ions agreed well with the experiments (see, e.g. [23][24][25]), there was no reason to assume that essential contributions had not been included in the predictions. The question was out of focus until the first laser spectroscopic experiment in the x-ray regime [3], enabled by the advent of x-ray freeelectron-laser (XFEL) facilities [26]. This experiment at the Linac Coherent Light Source (LCLS, Ref.[27]) gave hints for an incorrect atomic structure theory: a disagreement between all state-of-the-art ...

Optical frequency combs have had a remarkable impact on precision spectroscopy [1][2][3] . Enabling this technology in the x-ray domain is expected to result in wide-ranging applications, such as stringent tests of astrophysical models and quantum electrodynamics 4 , a more sensitive search for the variability of fundamental constants 5 , and precision studies of nuclear structure 6 . Ultraprecise xray atomic clocks may also be envisaged 7 . In this work, an x-ray pulse-shaping method is put forward to generate a comb in the absorption spectrum of an ultrashort high-frequency pulse. The method employs an opticalfrequency-comb laser, manipulating the system's dipole response to imprint a comb on an excited transition with a high photon energy. The described scheme provides higher comb frequencies and requires lower optical-comb peak intensities than currently explored methods [8][9][10] , preserves the overall width of the optical comb, and may be implemented by presently available x-ray technology 11 .The spectrum of an optical frequency comb consists of equally spaced, precisely known peaks, centred at an optical frequency . X-ray frequency combs would enable the aforementioned applications in the x-ray range. Stringent tests of fundamental physics may be pursued, e.g., accurate measurements of transition energies in highly charged ions, which are predicted to be more sensitive to the variability of fundamental constants 5 than currently investigated species. Presently, XUV combs (∼ 30 eV) are generated 10 via intracavity high-order harmonic generation (HHG) 8,9 . A femtosecond enhancement cavity is utilized to reach the required peak intensities [8][9][10] , up to ∼ 10 14 W/cm 2 . However, relativistic effects limit the efficiency of HHG at high harmonic orders 20. The investigation of schemes to further increase the carrier frequency of the comb at accessible driving intensities is therefore required.Short-wavelength light sources with improved brilliance and bandwidth 11 enable studies of x-ray quantum optics, e.g., in highly charged ions or nuclei 4,6,21 . Recently, an amplitude-shaping scheme was put forward to imprint a comb onto narrowband x rays 22. Comb generation was also suggested 23 via quantum phase modulation 24. However, these schemes are conditioned either by demanding requirements on the x-ray source Figure 1. Three-level scheme used to describe the interaction between the model system and the driving fields. (a) A low-density ensemble of ions, modelled as a three-level system, is driven by an ultrashort, broadband x-ray pulse (X1, solid, blue), exciting the fast decaying level 2, followed by an optical pulse (L1, dashed, red) coupling this excited state to the metastable state 3. Thereby the system is prepared in an initial state which is a superposition of states 1 and 3. (b) An optical frequency comb (L2, solid, red) is subsequently used to periodically drive the optical transition 2 ↔ 3. The emitted x rays (Xout, dashed, wavy, blue) lead to either gain or attenuation of the incident pulse X...

The spectrum of resonance fluorescence is calculated for a two-level system excited by an intense, ultrashort x-ray pulse made available for instance by free-electron lasers such as the Linac Coherent Light Source. We allow for inner-shell hole decay widths and destruction of the system by further photoionization. This two-level description is employed to model neon cations strongly driven by x rays tuned to the 1s 2p-1 --> 1s-1 2p transition at 848 eV; the x rays induce Rabi oscillations which are so fast that they compete with Ne 1s-hole decay. We predict resonance fluorescence spectra for two different scenarios: first, chaotic pulses based on the self-amplified spontaneous emission principle, like those presently generated at x-ray free-electron-laser facilities and, second, Gaussian pulses which will become available in the foreseeable future with self-seeding techniques. As an example of the exciting opportunities derived from the use of seeding methods, we predict, in spite of above obstacles, the possibility to distinguish at x-ray frequencies a clear signature of Rabi flopping in the spectrum of resonance fluorescence

The evolution of a V-type three-level system is studied, whose two resonances are coherently excited and coupled by two ultrashort laser pump and probe pulses, separated by a varying time delay. We relate the quantum dynamics of the excited multilevel system to the absorption spectrum of the transmitted probe pulse. In particular, by analyzing the quantum evolution of the system, we interpret how atomic phases are differently encoded in the time-delay-dependent spectral absorption profiles when the pump pulse either precedes or follows the probe pulse. This scheme is experimentally applied to atomic Rb, whose fine-structure-split 5s  (2)S{1/2}→5p(2)P{1/2} and 5s(2)S_{1/2}→5p(2)P{3/2} transitions are driven by the combined action of a pump pulse of variable intensity and a delayed probe pulse. The provided understanding of the relationship between quantum phases and absorption spectra represents an important step towards full time-dependent phase reconstruction (quantum holography) of bound-state wave packets in strong-field light-matter interactions with atoms, molecules, and solids.

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The influence of nonlinear dynamical effects is analyzed on the observed spectra of controversial 3C and 3D astrophysically relevant x-ray lines in neonlike Fe 16+ and the A, B, C lines in natriumlike Fe 15+ ions. First, a large-scale configuration-interaction calculation of oscillator strengths is performed with the inclusion of higher-order electron-correlation effects. Also, quantum-electrodynamic corrections to the transition energies are calculated. Further considered dynamical effects provide a possible resolution of the discrepancy between theory and experiment found by recent x-ray freeelectron-laser measurements of these controversial lines. We find that, for strong x-ray sources, the modeling of the spectral lines by a peak with an area proportional to the oscillator strength is not sufficient and nonlinear dynamical effects have to be taken into account. Thus, we advocate the use of light-matter-interaction models also valid for strong light fields in the analysis and interpretation of the associated astrophysical and laboratory spectra. We investigate line-strength ratios distinguishing between the coherent and incoherent parts of the emission spectrum. In addition, the spectrum of Fe 15+ , an autoionizing ion which was also present in the recent laboratory experiment, is also analized.

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 2p-3d boundbound transitions between the spin-orbit multiplets 3 P0,1,2 and 3 D1,2,3 of the transiently produced doubly charged Ne 2+ ion are revealed, with time-dependent spectral changes over a time-delay range of (2.4 ± 0.3) 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. PACS numbers: .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 [1], shifts [2, 3] and modifies the width [4, 5] and the shape [6-8] 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 [9], in time domain and frequency domain, including the quantum control of boundbound electronic transitions (see, e.g., [10] and references therein).Since the advent of short-wavelength free-electron lasers (FELs) [11,12] the field of nonlinear spectroscopy is being extended into the extreme-ultraviolet (XUV) and x-ray spectral ranges [13][14][15][16][17][18][19][20][21][22][23][24]. One advantage of employing x-rays is the ability to access bound-bound electronic transitions associated with the spatially localized innerelectronic 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 ham-pered 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 [16] by employing Augerelectron spectroscopy. By contrast, we here address the valence electrons of the doubly charged neon ion, Ne 2+ , 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 i...