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Interferometric measurement of nuclear resonant phase shift with a nanoscale Young double waveguide

This paper demonstrates a nanoscale Young double-waveguide interferometer that successfully measures the dispersive phase shift of 14.4 keV x-rays interacting with a $^{57}$ Fe Mössbauer resonance, utilizing Bayesian inference to extract microscopic coupling parameters inaccessible through intensity data alone.

Original authors: Leon M. Lohse, Ankita Negi, Markus Osterhoff, Paul Meyer, Sergey Yaroslavtsev, Aleksandr I. Chumakov, Lars Bocklage, Ralf Röhlsberger, Tim Salditt

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Leon M. Lohse, Ankita Negi, Markus Osterhoff, Paul Meyer, Sergey Yaroslavtsev, Aleksandr I. Chumakov, Lars Bocklage, Ralf Röhlsberger, Tim Salditt

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a very specific, tiny whisper in a noisy room. Usually, to hear a whisper, you need a lot of time and a very quiet environment. But what if you could hear that whisper with just a single word, and figure out exactly how the sound changed as it passed through a specific object?

That is essentially what this team of scientists achieved, but instead of sound, they used X-rays, and instead of a whisper, they measured a tiny shift in the phase (the timing or "rhythm") of the light wave.

Here is a breakdown of their experiment using simple analogies:

1. The Challenge: Measuring the Invisible "Rhythm"

Light travels in waves. When a wave hits an atom, it doesn't just bounce off; it gets slightly delayed, like a runner slowing down for a split second when stepping on a patch of mud. This delay is called a phase shift.

The Problem: In the world of visible light (like a laser pointer), measuring this delay is easy. But with X-rays, the waves are so tiny and fast that measuring this delay is incredibly hard. It's like trying to measure the exact moment a hummingbird flaps its wings using a stopwatch that only ticks once a year.
The Goal: They wanted to measure this delay caused by a specific type of iron atom (called Mössbauer iron) that acts like a perfect, ultra-precise "tuning fork" for X-rays.

2. The Solution: The "Double-Slit" for X-Rays

To solve this, the scientists built a microscopic version of a famous experiment called Young's Double-Slit, but instead of slits in a wall, they used two tiny tunnels (waveguides) stacked on top of each other.

The Setup: Imagine two parallel highways for X-rays.
- Highway A (The Signal): This tunnel contains a super-thin layer of the special iron atoms.
- Highway B (The Reference): This tunnel is empty of iron; it's just a clean path.
The Race: They shoot single X-ray photons into both tunnels at the same time.
- The photon in the Reference tunnel runs at full speed.
- The photon in the Signal tunnel hits the iron atoms. Even though the iron absorbs some light, the ones that do pass through get "delayed" (their phase shifts) because of the interaction with the iron.

3. The Magic Trick: Interference

When these two streams of X-rays exit the tunnels and meet at a detector 1 meter away, they crash into each other like ripples in a pond. This is called interference.

If the two waves are perfectly in step, they make a bright spot.
If one is delayed (shifted), the bright spot moves to the side.

Think of it like two people clapping. If they clap at the exact same time, the sound is loud. If one person is slightly late, the rhythm changes, and you can tell exactly how late they were by listening to the pattern of the sound.

By measuring exactly how much the pattern of bright and dark spots shifted, the scientists could calculate the exact delay (phase shift) caused by the iron atoms.

4. The Breakthrough: Seeing What Absorption Hides

Usually, scientists only look at how much light is absorbed (blocked) by the material. It's like looking at a foggy window and guessing how thick the glass is just by how much light gets through.

The Limitation: Sometimes, a thick piece of glass with a weak effect looks the same as a thin piece of glass with a strong effect. You can't tell the difference just by looking at the light that gets through.
The Discovery: This team found that by measuring the phase shift (the timing delay) along with the absorption, they could solve the puzzle.
- They found that the iron atoms were interacting with the X-rays much more strongly than they thought.
- The phase shift gave them a "secret code" that revealed the true strength of the connection between the light and the atoms, which the absorption data alone could not show.

5. Why It Matters (According to the Paper)

Efficiency: They managed to do this with very few photons (just a few hundred). It's like hearing a whisper with only a handful of words.
Precision: They used a special technique called "Bayesian inference" (a fancy way of using probability math) to extract the answer from the noisy data.
The Future: This proves that we can build tiny, chip-sized X-ray sensors that can measure the internal structure of materials with extreme precision. The paper suggests this could lead to "X-ray optics on a chip," combining waveguides and sensors into small devices.

In Summary:
The scientists built a microscopic racetrack for X-rays. By timing how much the "racers" were delayed by a tiny layer of iron, they could measure a property of the atoms that was previously invisible to standard X-ray cameras. They did this by turning a delay in time into a visible shift in a pattern of light, proving that even a few single photons can tell a very detailed story.

Technical Summary: Interferometric Measurement of Nuclear Resonant Phase Shift with a Nanoscale Young Double Waveguide

Problem Statement
The phase shift of an electromagnetic wave, imprinted by its interaction with atomic scatterers, is a fundamental quantity encoding information about optical resonances and photon-matter interactions. While interferometric phase measurements are routine in the optical regime, they remain notoriously challenging in the hard x-ray frequency regime due to the extreme stability and translation precision (picometer scale) required by short wavelengths. Existing hard x-ray interferometers, typically based on bulk crystal reflections (e.g., Bonse-Hart interferometers), are unsuitable for nanoscopic systems. Furthermore, indirect methods relying on Kramers-Kronig relations or time-domain reconstructions from macroscopic samples require large photon counts and cannot resolve phase shifts in nanostructures where waveguide quantum electrodynamics (QED) phenomena occur. Specifically, the direct measurement of the dispersive phase shift induced by the 14.4 keV nuclear resonance of the Mössbauer isotope $^{57}$ Fe coupled to an x-ray waveguide has been elusive.

Methodology
The authors developed a nanoscale double-waveguide interferometer inspired by Young's double-slit experiment to measure this phase shift directly.

Device Fabrication: A pair of planar x-ray waveguides (0.5 mm long) was fabricated via magnetron sputtering. Each waveguide consists of a 20 nm boron carbide ( $B_4C$ ) core surrounded by 30 nm molybdenum cladding, creating a 50 nm baseline separation. One waveguide (Signal Waveguide, SWG) contains a 0.6 nm-thin film of isotope-enriched $^{57}$ Fe (approx. two atomic layers), while the other (Reference Waveguide, RWG) is free of resonant nuclei. The structure is enclosed in germanium wafers to block the tails of the illuminating focus.
Experimental Setup: Experiments were conducted at the ID18 beamline of the European Synchrotron Radiation Facility (ESRF) using the "Synchrotron Mössbauer Source" (SMS). The SMS provides a monochromatic beam with a bandwidth of a few neV, tuned across the $^{57}$ Fe resonance ( $\hbar\omega_0 = 14.41$ keV) via a Doppler velocity transducer.
Detection and Analysis: Photons transmitted through the waveguides were detected by a Timepix3 time-resolving pixel array detector located 1 m downstream. The detector's synchronization with the Doppler transducer allowed the photon time-of-arrival to encode the energy detuning ( $\Delta = \omega - \omega_0$ ).
Data Processing: Far-field interference patterns were recorded for three configurations: (1) both waveguides active (with resonant nuclei in SWG), (2) only the SWG active (RWG blocked), and (3) both waveguides active without resonant nuclei. The phase shift $\phi(\Delta)$ was extracted from the interference patterns using Bayesian inference (Markov Chain Monte Carlo sampling) to account for the Poissonian statistics of the sparse photon data (approx. 100 photons per detuning point).

Key Contributions

Nanoscale X-ray Interferometry: The work demonstrates the first direct interferometric measurement of a nuclear resonant phase shift in the hard x-ray regime using a nanoscale double-waveguide geometry.
Single-Photon Sensitivity: The method successfully extracts phase information from a very small number of detected photons (on the order of $10^2$ per energy point) by employing Bayesian inference, overcoming the limitations of traditional intensity-based measurements.
Disambiguation of Microscopic Parameters: The study shows that combining phase shift data with absorbance spectra allows for the unique extraction of microscopic coupling parameters that are otherwise ambiguous when using absorption data alone.

Results

Phase Shift Extraction: The measured phase shifts exhibit the expected resonance behavior near the $^{57}$ Fe transition. The phase is precisely localized (standard deviation of 0.13 rad) outside the immediate resonance region ( $\pm 6\Gamma$ ), where absorption is nearly total and phase information is lost.
Model Validation: The experimental data (both absorption and phase) were compared against a theoretical model describing the propagation of the guided mode interacting with the nuclear layer.
- An absorption-only fit yielded a coupling strength $|\zeta_m| = 4.5 \times 10^{-3}$ and a line broadening of $11\Gamma$ .
- A combined fit (absorption + phase) yielded a coupling strength $|\zeta_m| = 7.1 \times 10^{-3}$ and a line broadening of $9\Gamma$ .
Significance of Phase Data: The combined model's coupling strength ( $7.1 \times 10^{-3}$ ) closely matches theoretical design calculations, whereas the absorption-only model significantly underestimates it. This demonstrates that the phase shift provides critical, independent information that resolves the anticorrelation between line broadening and coupling strength inherent in absorption spectra. The phase shift decays asymptotically as $\Delta^{-1}$ , offering quantitative information over a wider frequency range than the absorbance, which decays as $\Delta^{-2}$ .

Significance and Claims
The paper claims that this work lays the basis for integrated x-ray interferometric sensors. By directly measuring the phase shift, the authors demonstrate a method to quantify the coupling strength between waveguide modes and nuclear ensembles, a parameter inaccessible via intensity data alone. The authors note that this technique is not restricted to nuclear resonances and can be analogously applied to electronic resonances. They suggest this paves the way for "x-ray photonics on a chip," integrating components such as curved waveguides, beam splitters, and interferometric sensors. The authors also propose that the demonstrated control over phase shifts could enable active control of narrowband pulses in hard x-ray nanophotonic devices, potentially through the manipulation of magnetic hyperfine spectra in thicker resonant layers.