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arXiv:2607.01796

Interferometric characterization of the relative phase between two X-ray free-electron laser pulses using long-lived Mössbauer resonances

Lukas Wolff, Jörg Evers

quant-phphysics.opticsSubmitted Jul 2, 2026• Announced Jul 3, 2026
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Abstract

Coherence-based spectroscopy methods are powerful tools to explore structure and dynamics of matter. However, towards higher photon energies, the generation of sequences of pulses with well-characterized relative delays and phases remains a challenge. Here, we introduce a method to measure the relative phase $\varphi$ between subsequent transform-limited pulses from high-repetition-rate x-ray free-electron lasers (XFELs). It is based on a Ramsey-type interference measurement, enabled by introducing long-lived Mössbauer resonances into the XFEL beam path up- or downstream a primary experiment, which allow one to bridge the temporal gap between the XFEL pulses. The measured phase can be used as additional input for the analysis of the primary experiment.

Comments
11 pages, 6 figures

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Coherence-based spectroscopy methods are powerful tools to explore structure and dynamics of matter. However, towards higher photon energies, the generation of sequences of pulses with well-characterized relative delays and phases remains a challenge. Here, we introduce a method to measure the relative phase φ\varphiφ between subsequent transform-limited pulses from high-repetition-rate x-ray free-electron lasers (XFELs). It is based on a Ramsey-type interference measurement, enabled by introducing long-lived Mössbauer resonances into the XFEL beam path up- or downstream a primary experiment, which allow one to bridge the temporal gap between the XFEL pulses. The measured phase can be used as additional input for the analysis of the primary experiment.
Up to now, experiments involving M\"ossbauer 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 M\"ossbauer 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.
The time-delayed scattering in 14.4 keV Mossbauer resonance at 57 Fe has been intensively studied using pulsed synchrotron radiation source and delayed coincidence technique. Here, results of x-ray interferometric experiments are shown, which were obtained at KEK with this nuclear resonance. There, the interference arises from the beams re-emitted by different nuclei with some time interval after the absorption of a single incident photon. High visibility interference oscillations verified the coherent superposition as well as the complete coincidence in the time domain of the beams after the nuclear resonance. The collective nuclear excitations from a single photon are discussed with regard to the no-reduction of the detection probability. In addition, the phase information transfer in successive absorption and re-emission with time-delay, and the dispersion effect at the nuclear resonance are shown.
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