Jörg Evers

Parametric x-ray radiation (PXR) is a tunable and versatile x-ray production mechanism that relies on relativistic electrons traversing through a crystal. When the crystal contains Mössbauer nuclei, they may also contribute to the x-ray generation, giving rise to spectrally narrow parametric Mössbauer radiation (PMR) as a promising alternative to undulator-based x-ray sources. It is well known that the x-ray generation strongly depends on the diffraction geometry. Motivated by this, here, we develop a framework for parametric x-ray generation in general diffraction geometries, encompassing the entire x-ray emission range from the conventional grazing extremely asymmetric diffraction (EAD) geometry with a large emission direction to the crystal surface up to the grazing specular diffraction (SD) geometry with emission near parallel to the surface. The framework further incorporates a range of physical effects arising from more realistic modeling of both the Mössbauer crystal and the electron beam. We show that the PMR generation in the grazing SD geometry qualitatively differs from the previously considered grazing EAD case, and explain the physical origins of these differences. We further apply our framework to analyze the geometry dependence of superradiant parametric Mössbauer radiation (SPMR) generated from microbunched electron beams provided by x-ray free-electron lasers. Our results set the stage for the exploration of PXR, PMR, superradiant amplification of parametric x-ray radiation, and SPMR beyond the established diffraction geometries.

Future applications of the spectrally narrow x-ray resonances in Mössbauer nuclei require x-ray sources with exceptionally high peak and average spectral flux. Here, we introduce a superradiant parametric Mössbauer radiation (SPMR) source, which utilizes the scattering of spatially microstructured electron bunches produced in x-ray free-electron laser (XFEL) accelerators on crystals. The spatial structuring leads to a coherent add-up of the radiation from different electrons, thereby enhancing the Mössbauer radiation by many orders of magnitude. We find that the performance is optimized at qualitatively different operation conditions than considered so far. For a focused electron beam with parameters based on the European XFEL facility, our approach predicts the generation of more than 900 SPMR photons per pulse within one linewidth of the Fe 26 57 Mössbauer resonance. This opens up new opportunities for precision metrology, and for hard x-ray pump-probe spectroscopy techniques via the parallel use of the bunched XFEL electron and photon beams.

A method to compute the excitation of narrow transitions at hard-x-ray energies by short focused x-ray pulses is developed. In particular, the effect of thin-film cavities on the pulse propagation is incorporated via a semianalytical algorithm requiring the numerical evaluation of only one two-dimensional Fourier transform. We investigate various limiting cases to confirm the reliability of the algorithm. As an application, we show how a focused x-ray pulse propagates in cavity structures utilized in previous theoretical studies and experiments with collimated beams.

Strong excitation of nuclear resonances, particularly of Mössbauer nuclei, has been a longstanding goal and the advance of novel x-ray sources is promising new options in this regard. Here, we map out the necessary experimental conditions for the more general goal of entering the nonlinear optics regime with nuclei and compare with available technology. In particular, we present a comprehensive theory of nonlinear nuclear excitation in thin-film x-ray cavities by focused x-ray pulses. We thereby identify cavity geometries with broad resonances that allow one to boost the nuclear excitation even at moderately tight focusing and offer the possibility to mitigate radiation damage.

Up to now, experiments involving Mössbauer nuclei have been restricted to the low-excitation regime (LER). The reason for this is the narrow spectral linewidth of the nuclei. This defining feature enables Mössbauer spectroscopy with remarkable resolution and convenient control and measurements in the time domain, but at the same time implies that only a tiny part of the photons delivered by accelerator-based x-ray sources with orders-of-magnitude larger pulse bandwidth are resonant with the nuclei. X-ray free-electron lasers promise a substantial enhancement of the number of nuclear-resonant photons per pulse, such that excitations beyond the LER come within reach. This raises the question of how the onset of nonlinear excitations could be experimentally verified. Here, we develop and explore a method to detect an excitation of nuclear ensembles beyond the LER for ensembles of nuclei embedded in x-ray waveguides. It relies on the comparison of the x rays coherently and incoherently scattered off of the nuclei. As a key result, we show that the ratio of the two observables is constant within the LER, essentially independent of the details of the nuclear system and the characteristics of the exciting x rays. Conversely, deviations from this equivalence serve as a direct indication of excitations beyond the LER. Building upon this observation, we develop a variety of experimental signatures both, for near-instantaneous impulsive and for temporally extended nonimpulsive x-ray excitation. Correlating coherently and incoherently scattered intensities further allows one to compare theoretical models of nonlinear excitations more rigorously to corresponding experiments.

Quantum models based on few-mode master equations have been a central tool in the study of resonator quantum electrodynamics, extending the seminal single-mode Jaynes-Cummings model to include loss and multiple modes. Despite their broad application range, previous approaches within this framework have either relied on a Markov approximation or a fitting procedure. By combining ideas from pseudomode and quasinormal mode theory, we develop a certification criterion for multi-mode effects in lossy resonators. It is based on a witness observable, and neither requires a fitting procedure nor a Markov approximation. Using the resulting criterion, we demonstrate that such multi-mode effects are important for understanding previous experiments in x-ray cavity QED with Mössbauer nuclei and that they allow one to tune the nuclear ensemble properties.

Quantum models based on few-mode Master equations have been a central tool in the study of resonator quantum electrodynamics, extending the seminal single-mode Jaynes-Cummings model to include loss and multiple modes. Despite their broad application range, previous approaches within this framework have either relied on a Markov approximation or a fitting procedure. By combining ideas from pseudomode and quasinormal mode theory, we develop a certification criterion for multi-mode effects in lossy resonators. It is based on a witness observable, and neither requires a fitting procedure nor a Markov approximation. Using the resulting criterion, we demonstrate that such multi-mode effects are important for understanding previous experiments in X-ray cavity QED with M\"ossbauer nuclei and that they allow one to tune the nuclear ensemble properties.

Owing to their extremely narrow line-widths and exceptional coherence properties, Mössbauer nuclei form a promising platform for quantum optics, spectroscopy, and dynamics at energies of hard x-rays. A key requirement for further progress is the development of more powerful measurement and data analysis techniques. As one approach, recent experiments have employed time- and frequency-resolved measurements, as compared to the established approaches of measuring time-resolved or frequency-resolved spectra separately. In these experiments, the frequency-dependence is implemented using a tunable single-line nuclear reference absorber. Here, we develop spectroscopy and analysis techniques for such time- and frequency-resolved nuclear resonant scattering spectra in the frequency-frequency domain. Our approach is based on a Fourier-transform of the experimentally accessible intensities along the time axis, which results in complex-valued frequency-frequency correlation (FFC) spectra. We show that these FFC spectra not only exhibit a particularly simple structure, disentangling the different scattering contributions, but also allow one to directly access nuclear target properties and the complex-valued nuclear resonant part of the target response. In a second part, we explore the potential of an additional phase control of the x-rays resonantly scattered off of the reference absorber for our scheme. Such control provides selective access to specific scattering pathways, allowing for their separate analysis without the need to constrain the parameter space to certain frequency or time limits. All results are illustrated with pertinent examples in nuclear forward scattering and in reflection off of thin-film x-ray cavities containing thin layers of Mössbauer nuclei.