Quantum Interference in Two-Atom Resonant X-ray Scattering of an Intense Attosecond Pulse

TL;DR

The paper develops a QED-based, time-dependent framework to model resonant x-ray scattering from two non-interacting Ne$^{+}$ ions driven by intense attosecond pulses, incorporating Rabi dynamics, photoionization, and Auger decay. By solving coupled equations for bound and scattered-state amplitudes, it separates elastic scattering and resonance fluorescence channels and analyzes their interference across pulse areas, interatomic geometries, and initial states. It finds that the total resonant yield can exceed non-resonant yields and displays a structure-factor–like angular dependence, with fringe visibility highly sensitive to pulse area and state indistinguishability; in the linear regime, fringe contrasts are maximized and damage is minimized, suggesting near-damage-free, element-specific imaging potential with ultrafast resonant scattering. The results provide guidance on parameter regimes for ultrafast SPI-like imaging using resonant x-ray pulses, including the role of higher charge states and initial-state coherences in shaping observable interference patterns.

Abstract

We theoretically investigate resonant x-ray scattering from two non-interacting Ne+ ions driven by an intense attosecond pulse using a non-relativistic, QED-based time-dependent framework. Our model includes Rabi oscillations, photoionization, Auger decay, and quantum interference among elastic scattering and resonance fluorescence pathways. We analyze how the total scattering signal depends on pulse intensity, atomic configuration, and initial electronic state. We find that the total resonant scattering yield exceeds its non-resonant counterpart; the angular dependence of the signal qualitatively resembles a two-atom structure factor; and the visibility of interference fringes is sensitive to pulse area and the initial electronic state. Only a subset of final states reached via resonance fluorescence exhibits interference, determined by the indistinguishability of photon emission pathways. Fringe visibility is maximized in the linear scattering regime, where ionization is minimal and resonance fluorescence pathways can be largely indistinguishable. These results highlight optimal conditions for applying ultrafast resonant x-ray scattering to single-particle imaging.

Figures (12)

• Figure 1: Schematic diagram of resonant scattering from a collection of atoms including both resonance fluorescence and elastic scattering pathways for scattering. The incident x-ray is chosen to be linearly polarized along the z-axis and propagating along the x-axis. We note that only two atoms are included in our quantum-mechanical treatment, and the additional atoms are shown only as visual context representing a typical experimental environment.
• Figure 2: Electronic states and processes of a single Ne+ exposed to an intense x-ray pulse. The figure includes photoionization (purple arrow) and Auger decay (black arrows) pathways and electronic transitions associated with Rabi oscillations (blue lines) and resonance fluorescence (gray dashed lines). The elastic scattering processes (orange lines) do not induce electronic transition.
• Figure 3: Resonant scattering photon yield from two atoms for a 0.25 fs ($t_{wid}$), Q = 2$\pi$ pulse for the case of planar scattering ($\phi_s = 90 \degree$) when the initial state of each atom is $\ket{1}$. (a) Scattering angle dependence for the two scattering pathways of resonance fluorescence (RF) and elastic scattering (ES), their incoherent sum (ES + RF), coherent sum (Coh sum). The black and purple solid curves are fits [Eq. (\ref{['fit_function']})] which are proportional to the structure factor and $\boldsymbol{R} =10 \lambda_{in} \hat{z}$. (b) Angle dependence for resonance fluorescence contribution from different final scattered states for the same position of two atoms. (c), (d) Position dependence of resonance fluorescence yields from different final scattered states. (e) Position dependence of the coherent sum. R0, RX, RY and RZ corresponds to $\boldsymbol{R}$=0, $\boldsymbol{R} =10 \lambda_{in} \hat{x}$, $\boldsymbol{R} =10 \lambda_{in} \hat{y}$, and $\boldsymbol{R} =10 \lambda_{in} \hat{z}$, respectively.
• Figure 4: Interference pathways for resonance fluorescence from two atoms when the initial state of the system is $\ket{11}$. (a) The states on the left describe the system before emission (unscattered states, purple circles) and the states on the right describe the scattered electronic states (gold circles) of the system entangled with the emitted photon. The blue solid lines and the gray dashed lines depict Rabi oscillation and resonance fluorescence processes respectively. (b), (c) shows two example pathways to obtain a final scattered state $\ket{11}$ (marked red in (a)) for strong and weak incident intensities, respectively.
• Figure 5: Resonant scattering photon yield from two atoms using a 0.25 fs ($t_{wid}$) pulse with $Q = \pi$. (a) Scattering angle dependence from the different channels. Note that ES yield is multiplied by a factor of 25 to show its angular dependence. (b) Scattering angle dependence of resonance fluorescence for different final scattered states. (c) Position dependence of resonance fluorescence yields from different final scattered states for $\theta_s = 3\degree$. The other parameters are the same as Fig. \ref{['Fig_2Pi_signal_initialstate2pz']}.