X-ray phase measurements by time-energy correlated photon pairs

Abstract

The resolution of a measurement system is fundamentally constrained by the wavelength of the used wave packet and the numerical aperture of the optical system. Overcoming these limits requires advanced interferometric techniques exploiting quantum correlations. While quantum interferometry can surpass the Heisenberg limit, it has been confined to the optical domain. Extending it to x-rays enables sub-angstrom spatial and zeptosecond temporal resolution, unlocking atomic-scale processes inaccessible to existing methods. Here, we demonstrate x-ray quantum interferometry using 17.5-kilo-electron volt ( [Formula: see text] = 70 picometers) photon pairs. Our approach introduces a phase measurement technique with exceptional noise resilience, mitigating the impact of mechanical instabilities, vibrations, and photonic noise-key challenges in x-ray interferometry. By generating and using entangled x-ray photons, we lay the foundation for next-generation techniques with unprecedented phase precision. This breakthrough carries far-reaching consequences for fundamental physics, high-resolution imaging, and spectroscopy, bringing to light quantum optical effects never before accessed in the x-ray regime.

Fig. 1.. Comparison of Mach-Zehnder and SU(1,1) Interferometers.

(A) X-ray Mach-Zehnder interferometer: Laue diffraction in the leftmost lamella of a monolithic crystal interferometer generates two beams that are Laue reflected by the second lamella to interfere at the position of the third lamella, causing the resultant intensities to vary depending on the relative phase difference between the two waves. (B) SU(1,1) interferometer: Nonlinear crystals replace the beam splitters. These generate from the pump beam (purple) by parametric down-conversion two photons: idler (blue) and signal (red) detected by two detectors with coincidence discrimination. The intensity at the output (detectors) depends on the interference between the modes. A phase object placed between the two crystals introduces different phase shifts due to wavelength dispersion, yielding a corresponding change in the counted intensity. This allows for the detection of relative phase shifts by intensity measurements.

Fig. 2.. Experimental scheme of the x-ray SU(1,1) interferometer.

(A) A 35-keV beam enters the interferometer, constructed from a monolithic silicon crystal with two lamellae. The coincidence count rate at the detectors varies depending on the phase shift caused by a phase object inserted between the lamellae. (B) X-ray SPDC phase matching diagram where ${\overrightarrow{k}}_{\mathrm{p}}$ , ${\overrightarrow{k}}_{\mathrm{s}}$ , and ${\overrightarrow{k}}_{\mathrm{i}}$ are, respectively, the wave vectors of the pump, signal, and idler, and $\overrightarrow{G}$ is the lattice vector.

Fig. 3.. The process of noise filtering.

(A) Full spectrum as measured by a single detector. The blue shadow indicates the energy range selected for SNR optimization. (B to E) Time difference histograms: (B) with time coincidence but without energy conservation discrimination; (C) same as (B) but with energy conservation discrimination; (D) the data in (C) after the subtraction of the constant baseline spanned by the two outermost bins in (C). This baseline is generated by accidental coincidences, which are independent of the time difference. The red curve is a Gaussian fit yielding the actual temporal resolution of our system: 200-ns half width at half maximum. (E) Counts that do not conserve energy exhibit a nearly uniform distribution, confirming random arrival times of background photon pairs. a.u., arbitrary units.

Fig. 4.. Experimental results.

Normalized coincidence count rate at the output of the second lamella dependence on the thickness of the membranes. (A) Measured SPDC count rate after using temporal and energy filtering (green squares) and theoretical curve (red line). The measurement time for each point was $\sim$ 6 hours. Two separate measurements were performed on the 24-μm membrane at different times; both points are plotted but cannot be distinguished at this scale. (B) Count rate without applying energy or temporal filtering (black diamonds), under the same normalized theoretical model used in (A) (red line). In both figures, each dataset was normalized by its average count rate

Fig. 5.. Interferometer and membrane.

(A and B) The interferometer and (C) a typical membrane phase object