Correlated and X-Ray Quantum Dynamics

• D. Adigüzel (MSc student)
• Jörg Evers (group leader)
• Miriam Gerharz (PhD student)
• Robert Horn (PhD student)
• Ze-An Peng (PostDoc)
• Luis Yagüe Bosch (MSc student)

Mössbauer spectroscopy is used to study structure and dynamics across all the natural sciences, with high energy resolution enabled by the narrow nuclear resonances. However, these narrow resonances at the same time restrict the signal rate, such that experiments typically have to average over many x-ray excitations and extended measurement times. In principle, better x-ray sources could enable the measurement of full Mössbauer spectra using only a single x-ray excitation, and a proof-of-principle experiment has been reported. Here, we for the first time realize coherent nuclear forward scattering of self-seeded XFEL pulses, and observe up to 900 signal photons per excitation. We then introduce a new analysis approach, which allows one to identify different dynamics following an x-ray excitation in a given target without prior knowledge from the data. Subsequently, the data is sorted according to the identified dynamics classes. As a result, the dynamics of each class can be analyzed separately, thereby avoiding the detrimental averaging over different dynamics. Due to the sorting approach, also the vast majority of x-ray excitations which did not result in sufficient signal photons for an individual analysis can be gainfully used in the analysis. This approach could lead to novel applications for Mössbauer science, geared towards the study of out-of-equilibrium transient dynamics of the nuclei or their environment.

1. M. Gerharz, W. Hippler, B. Marx-Glowna, S. Sadashivaiah, K. S. Schulze, I. Uschmann, R. Lötzsch, K. Schlage, S. Velten, D. Lentrodt, L. Wolff, O. Leupold, I. Sergeev, H.-C. Wille, C. Strohm, M. Guetg, S. Liu, G. A. Geloni, U. Boesenberg, J. Hallmann, A. Zozulya, J.-E. Pudell, A. Rodriguez-Fernandez, Mohamed Youssef, A. Madsen,  L. Bocklage, G. G. Paulus, C. H. Keitel, T. Pfeifer, R. Röhlsberger, and Jörg Evers, submitted

Interference is a powerful tool for measuring and control. However, at x-ray energies, interferometry remains challenging due to the stringent stability requirements due to the low x-ray wavelength. Nevertheless, interference effects are essential to most applications in Mössbauer science, due to the coherent scattering nature. Here, we put forward and experimentally demonstrate a new interferometer design, which facilitates sensitive measurements as it features a vanishing signal intensity in its idle state ("dark fringe" setup). The two interfering pathways are not spatially separated, thereby improving the stability of the setup, but differ in polarization such that they can individually be addressed. The  relative phase between the two pathways can dynamically be tuned by displacing a Mössbauer target. We experimentally demonstrate the capabilities of this interferometer by controlling the transmitted x-ray intensity on  nanosecond time scales. We further demonstrate sensitive measurements by observing  the propagation of impulsively launched sound waves in the target over 10 microseconds.  The interferometer concept opens avenues towards polarization-sensitive phase measurements, the generation of coherent multi-pulse sequences for controlling nuclear dynamics, and the implementation of feedback loops to adaptively optimize the interferometer, thereby fueling the further development of nuclear quantum optics.

1. M. Gerharz, D. Lentrodt, L. Bocklage, K. Schulze, C. Ott, Rene Steinbrügge, O. Leupold, I. Sergeev, G. G. Paulus, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, submitted

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 is based on the scattering of spatially micro-structured 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. Interestingly, we find that the performance is optimized at qualitatively different operation conditions than considered so far. Our calculations predict more than 300 photons of 14.4 keV energy within a single nuclear linewidth of about 5 neV width per x-ray pulse, which renders the SPMR source competitive to state-of-the-art XFEL facilities.

1. Z.-A. Peng, C. H. Keitel, and J. Evers, submitted

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. As the basis of the analysis, we develop a comprehensive theory of nonlinear nuclear excitation in thin-film x-ray cavities by focused x-ray pulses. In particular, the effect of thin-film cavities on the pulse propagation is incorporated via a numerically efficient semianalytical algorithm. Using this new computational approach, we  identify cavity geometries with broad resonances that allow one to boost the nuclear excitation even at moderately tight focusing and study the effect of the cavity as function of XFEL parameters and focusing strength.

1. D. Lentrodt, C. H. Keitel, and J. Evers, Phys. Rev. Lett. 135, 033801 (2025)
2. D. Lentrodt, C. H. Keitel, and J. Evers, Phys. Rev. A 112, 013711 (2025)

Spectrally narrow resonances with high quality factors are the reference oscillators for the most precise operational clocks. Currently, the best clocks rely on optical transitions involving electronic resonances in atoms. Resonances in atomic nuclei are candidates for a further improvement of clocks, since they offer exceptionally high quality factors, operation with solid-state targets, and a higher resilience against external perturbations. Next to the ²²⁹Th isomer with transition energy in few-electronvolt range, ⁴⁵Sc is one of the most promising candidates. The transition energy to its isomeric state is 12.4 keV, and the natural lifetime of the resonance is 0.47s. However, so far, the resonance could not be directly excited. Here we report on the first direct x-ray excitation using the European x-ray free electron laser. As a result of the experiment, the resonance energy could be determined two orders of magnitude better than it was known before. In a second experiment, also the nuclear lifetime could be confirmed using measurements of the nuclear fluorescence in the time domain. 

1. Y. Shvyd'ko, R. Röhlsberger, O. Kocharovskaya, J. Evers et al., Nature 622, 471 (2023)
2. P. Liu, M. Gerharz et al., arXiv:2508.17538 [quant-ph]

Up to now, experiments involving Mössbauer nuclei have been restricted to the low-excitation regime. 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. However, in recent experiments at the European x-ray free electron laser, we already observed more than 900 resonant signal photons after a single x-ray excitation. X-ray free electron laser oscillators which have recently been demonstrated for the first time promise a further increase in source capabilities. This prompts the question how a potential excitation of Mössbauer nuclei beyond the linear regime could be experimentally detected. Here, we develop and explore a method to detect an excitation of nuclear ensembles beyond the low-excitation regime 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 linear regime, 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 linear regime.

1. L. Wolff and J. Evers, Phys. Rev. A 108, 043714 (2023)

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 quasimode theory, we develop a certification criterion for multi-mode effects in lossy resonators without the need for a fitting procedure or 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.

1. D. Lentrodt, O. Diekmann, C. H. Keitel, S. Rotter, J. Evers, Phys. Rev. Lett. 130, 263602 (2023)

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.

1. L. Woff and J. Evers, Phys. Rev. Research 5, 013071 (2023)

Numerous applications of Mössbauer spectroscopy are related to a unique resolution of absorption spectra of resonant radiation in crystals, when the nucleus absorbs a photon without a recoil. However, the narrow nuclear linewidth renders efficient driving of the nuclei challenging, restricting precision spectroscopy, nuclear inelastic scattering and nuclear quantum optics. Moreover, the need for dedicated x-ray optics restricts access to only few isotopes, impeding precision spectroscopy of a wider class of systems. Here, we put forward a novel Mössbauer source, which offers resonant photon flux for a large variety of Mössbauer isotopes with strongly suppressed electronic background. It is based on relativistic electrons moving through a crystal and emitting parametric Mössbauer radiation essentially unattenuated by electronic absorption. As a result, a collimated beam of resonant photons is formed, without the need for additional monochromatization. We envision the extension of high-precision Mössbauer spectroscopy to a wide range of isotopes at accelerator facilities, also using dumped electron beams.

1. O. D. Skoromnik, I. D. Feranchuk, J. Evers, and C. H. Keitel, Physical Review Accelerators and Beams 25, 040704 (2022)

Ensembles of Mössbauer nuclei embedded in thin-film cavities form a promising platform for x-ray quantum optics. A key feature is that the joint nuclei-cavity system can be considered as an artificial x-ray multilevel scheme in the low-excitation regime. Using the cavity environment, the structure and parameters of such level schemes can be tailored beyond those offered by the bare nuclei. However, so far, the direct determination of a cavity structure providing a desired quantum optical functionality has remained an open challenge. Here we address this challenge using an inverse design methodology. We show that the established fitting approach based on scattering observables in general is not unique, since the analysis may lead to different multilevel systems for the same cavity if based on observables in different scattering channels. Motivated by this, we distinguish between scattering signatures and the microscopic level scheme as separate design objectives, with the latter being uniquely determined by an ab initio approach. We find that both design objectives are of practical relevance and that they complement each other regarding potential applications. We demonstrate the inverse design for both objectives using example tasks, such as realizing electromagnetically induced transparency. Our results pave the way for future applications in nuclear quantum optics involving more complex x-ray cavity designs.

1. O. Diekmann, D. Lentrodt, and J. Evers, Physical Review A 105, 013715 (2022)
2. O. Diekmann, D. Lentrodt, and J. Evers, Physical Review A 106, 053701 (2022)

Coherent control of quantum dynamics is key to a multitude of fundamental studies and applications. In the visible or longer-wavelength domains, near-resonant light fields have become the primary tool with which to control electron dynamics. Recently, coherent control in the extreme-ultraviolet range was demonstrated, with a few-attosecond temporal resolution of the phase control. At hard-X-ray energies (above 5–10 kiloelectronvolts), Mössbauer nuclei feature narrow nuclear resonances due to their recoilless absorption and emission of light, and spectroscopy of these resonances is widely used to study the magnetic, structural and dynamical properties of matter. It has been shown that the power and scope of Mössbauer spectroscopy can be greatly improved using various control techniques. However, coherent control of atomic nuclei using suitably shaped near-resonant X-ray fields remains an open challenge. Here we demonstrate such control, and use the tunable phase between two X-ray pulses to switch the nuclear exciton dynamics between coherent enhanced excitation and coherent enhanced emission. We present a method of shaping single pulses delivered by state-of-the-art X-ray facilities into tunable double pulses, and demonstrate a temporal stability of the phase control on the few-zeptosecond timescale. Our results unlock coherent optical control for nuclei, and pave the way for nuclear Ramsey spectroscopy and spin-echo-like techniques, which should not only advance nuclear quantum optics, but also help to realize X-ray clocks and frequency standards. In the long term, we envision time-resolved studies of nuclear out-of-equilibrium dynamics, which is a long-standing challenge in Mössbauer science.

1. K. P. Heeg, A. Kaldun, C. Strohm, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers, Nature 590, 401-404 (2021)
2. MPIK press release: X-ray double flashes control atomic nuclei
3. MPG press release: Atomic nuclei in the quantum swing
4. Selection to the Physics World Top 10 Breakthroughs 2021

We develop two ab initio quantum approaches to thin-film x-ray cavity quantum electrodynamics with spectrally narrow x-ray resonances, such as those provided by Mössbauer nuclei. The first method is based on a few-mode description of the cavity, and promotes and extends existing phenomenological few-mode models to an ab initio theory. The second approach uses analytically known Green's functions to model the system. The two approaches not only enable one to ab initio derive the effective few-level scheme representing the cavity and the nuclei in the low-excitation regime, but also provide a direct avenue for studies at higher excitation, involving nonlinear or quantum phenomena. The ab initio character of our approaches further enables direct optimizations of the cavity structure and thus of the photonic environment of the nuclei, to tailor the effective quantum optical level scheme towards particular applications. To illustrate the power of the ab initio approaches, we extend the established quantum optical modeling to resonant cavity layers of arbitrary thickness, which is essential to achieve quantitative agreement for cavities used in recent experiments. Further, we consider multilayer cavities featuring electromagnetically induced transparency, derive their quantum optical few-level systems ab initio, and identify the origin of discrepancies in the modeling found previously using phenomenological approaches as arising from cavity field gradients across the resonant layers.

1. D. Lentrodt, K. P. Heeg, C. H. Keitel, and J. Evers, Physical Review Research 2, 023396 (2020)

Few-mode models are a cornerstone of the theoretical work in quantum optics, with the famous single-mode Jaynes-Cummings model being only the most prominent example. In this work, we develop an ab initio few-mode theory, a framework connecting few-mode system-bath models to ab initio methods. We first present a method to derive exact few-mode Hamiltonians for noninteracting quantum potential scattering problems and demonstrate how to rigorously reconstruct the scattering matrix from such few-mode Hamiltonians. We show that, upon inclusion of a background scattering contribution, an ab initio version of the well-known input-output formalism is equivalent to standard scattering theory. On the basis of these exact results for noninteracting systems, we construct an effective few-mode expansion scheme for interacting theories, which allows us to extract the relevant degrees of freedom from a continuum in an open quantum system. As a whole, our results demonstrate that few-mode as well as input-output models can be extended to a general class of problems and open up the associated toolbox to be applied to various platforms and extreme regimes. We outline differences of the ab initio results to standard model assumptions, which may lead to qualitatively different effects in certain regimes. The formalism is exemplified in various simple physical scenarios. In the process, we provide a proof of concept of the method, demonstrate important properties of the expansion scheme, and exemplify new features in extreme regimes.

1. D. Lentrodt and J. Evers, Physical Review X 10, 011008 (2020)

We introduce an analytical phase-reconstruction principle that retrieves atomic scale motion via time-domain interferometry. The approach is based on a resonant interaction with high-frequency light and does not require temporal resolution on the time scale of the resonance period. It is thus applicable to hard x rays and γ rays for measurements of extremely small spatial displacements or relative-frequency changes. Here, it is applied to retrieve the temporal phase of a 14.4 keV emission line of an 57Fe sample, which corresponds to a spatial translation of this sample. The small wavelength of this transition (λ=0.86  Å) allows for determining the motion of the emitter on sub-Ångström length and nanosecond timescales.

1. S. Goerttler, K. P. Heeg, A. Kaldun, P. Reiser, C. Strohm, J. Haber, C. Ott, R. Subramanian, R. Röhlsberger, J. Evers, and T. Pfeifer Physical Review Letters 123, 153902 (2019)

Time-domain interferometry (TDI) is a promising method to characterize spatial and temporal correlations at x-ray energies, via the so-called intermediate scattering function and the related dynamical couple correlations. However, so far, it has only been analyzed for classical target systems. Here, we provide a quantum analysis, and suggest a scheme that allows us to access quantum dynamical correlations. We further show how TDI can be used to exclude classical models for the target dynamics, and illustrate our results using a single particle in a double well potential. Interestingly, QTDI can access the full quantum-mechanical two-time correlations without backaction.

1. S. Castrignano and J. Evers, Physical Review Letters 122, 025301 (2019)
2. S. Castrignano and J. Evers, Physical Review A 101, 063842 (2020) (Editors' suggestion)

Spectroscopy of nuclear resonances offers a wide range of applications due to the remarkable energy resolution afforded by their narrow linewidths. However, progress toward higher resolution is inhibited at modern x-ray sources because they deliver only a tiny fraction of the photons on resonance, with the remainder contributing to an off-resonant background. We devised an experimental setup that uses the fast mechanical motion of a resonant target to manipulate the spectrum of a given x-ray pulse and to redistribute off-resonant spectral intensity onto the resonance. As a consequence, the resonant pulse brilliance is increased while the off-resonant background is reduced. Because our method is compatible with existing and upcoming pulsed x-ray sources, we anticipate that this approach will find applications that require ultranarrow x-ray resonances.

1. K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. R. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, C. H. Keitel, R. Röhlsberger, T. Pfeifer, and J. Evers Science 357, 375-378 (2017)

Modern x-ray light sources promise access to structure and dynamics of matter in largely unexplored spectral regions. However, the desired information is encoded in the light intensity and phase, whereas detectors register only the intensity. This phase problem is ubiquitous in crystallography and imaging and impedes the exploration of quantum effects at x-ray energies. In this experiment, we demonstrated phase-sensitive measurements characterizing the quantum state of a nuclear two-level system at hard x-ray energies. The nuclei are initially prepared in a superposition state. Subsequently, the relative phase of this superposition is interferometrically reconstructed from the emitted x rays. Our results form a first step towards x-ray quantum state tomography and provide new avenues for structure determination and precision metrology via x-ray Fano interference.

1. K. P. Heeg, C. Ott, D. Schumacher, H.-C. Wille, R. Röhlsberger, T. Pfeifer, and J. Evers, Phys. Rev. Lett. 114, 207401 (2015)
2. C. Ott, A. Kaldun, P. Raith, K. Meyer, M. Laux, J. Evers, C. H. Keitel, C. H. Greene, and T. Pfeifer, Science 340, 716 (2013)

We experimentally demonstrated group velocity control for x-ray photons of 14.4 keV energy via a direct measurement of the temporal delay imposed on spectrally narrow x-ray pulses. The subluminal light propagation was achieved by inducing a steep positive linear dispersion in the optical response of Fe Mössbauer nuclei embedded in a thin film planar x-ray cavity. To directly detect the temporal pulse delay, we developed a new method to generate frequency-tunable spectrally narrow x-ray pulses from broadband pulsed synchrotron radiation. Our theoretical model is in good agreement with the experimental data. This project is an important step towards the exploitation of non-linear phenomena with nuclei.

1. K. P. Heeg, J. Haber, D. Schumacher, L. Bocklage, H.-C. Wille, K. S. Schulze, R. Loetzsch, I. Uschmann, G. G. Paulus, R. Rüffer, R. Röhlsberger, and J. Evers, Phys. Rev. Lett. 114, 203601 (2015)

Quantum mechanical superpositions are important resources for future quantum technologies, but extremely fragile: already the interaction with the vacuum alone can destroy them. A promising solution in theory is already known for more than 40 years. At that time it was predicted that the interaction with the vacuum may be manipulated in such a way that it produces the desired superpositions instead of destroying them. Unfortunately, this is linked to rigorous conditions which so far hindered the experimental realization. We have circumvented these constraints, and demonstrated the spontaneous generation of coherences between the quantum states of a large ensemble of nuclei. Probing the system using nuclear resonance scattering, we observed clear signatures of the formed coherences, controlled by a weak external magnetic field. This opens up a variety of future perspectives for quantum optics with novel X-ray light sources.

1. K. P. Heeg, H.-C. Wille, K. Schlage, T. Guryeva, D. Schumacher, I. Uschmann, K. S. Schulze, B. Marx, T. Kämpfer, G. G. Paulus, R. Röhlsberger, and J. Evers, Phys. Rev. Lett. 111, 073601 (2013)

Quantum entanglement is one of the most intriguing properties of quantum mechanics, and is commonly visualized as a correlation of two or more quantum particles that form a single quantum object. But entanglement does not necessarily involve two quantum particles. Possibly even more fascinating, a single photon can entangle two modes of the vacuum. Here we show that this single-particle entanglement can also be generated by coherently controlling the quantum dynamics of nuclei. First, the nuclei are excited with a synchrotron radiation pulse. Then, by applying a sequence of weak magnetic fields to the nuclei, they are forced to emit a single photon in the x-ray regime such that it entangles field modes at energies about 10000 times higher than usual. This could pave the way for entanglement research in nuclear physics, based on experiments exploiting nuclear complexity with synchrotron radiation sources and free-electron lasers in an as-yet unexplored energy regime.

1. A. Pálffy, C. H. Keitel, and J. Evers, Phys. Rev. Lett. 103, 017401 (2009)
   (and July 2009 issue of Virtual Journal of Quantum Information)
   (and July 13, 2009 issue of Virtual Journal of Nanoscale Science & Technology)

The direct interaction of nuclei with super-intense laser fields is studied. We show that present and upcoming high-frequency laser facilities, especially together with a moderate acceleration of the target nuclei to match photon and transition frequency, do allow for resonant laser-nucleus interaction. These direct interactions may be utilized for the model-independent optical measurement of nuclear properties such as the transition frequency and the dipole moment, thus opening the field of nuclear quantum optics. As ultimate goal, one may hope that direct laser-nucleus interactions could become a versatile tool to enhance preparation, control and detection in nuclear physics.

1. T. J. Bürvenich, J. Evers and C. H. Keitel, Phys. Rev. Lett. 96, 142501 (2006)
   (and May 2006 issue of Virtual Journal of Ultrafast Science)
2. T. J. Bürvenich, J. Evers, and C. H. Keitel, Phys. Rev. C 74, 044601 (2006)
3. A. Pálffy, J. Evers, and C. H. Keitel, Phys. Rev. C 77, 044602 (2008)