# An illustration of the chosen coordinates

**Figure 1.** An illustration of the chosen coordinates. Position vectors within the electron density distribution of the sample are denoted by \boldsymbol{x^{\prime }}, and their projection onto the exit plane at x^{\prime }_z=0 directly behind the object is denoted by \boldsymbol{x}^{\prime }_\perp. The detector is placed at the position \boldsymbol{r}, at a distance *z* in the direction of propagation and at \boldsymbol{r}_\perp perpendicular to it. \boldsymbol{R}^{\prime } is the direction from the point of scattering to the detector.

**Abstract**

Time-resolved phase-contrast imaging using ultrafast x-ray sources is an emerging method to investigate ultrafast dynamical processes in matter. Schemes to generate attosecond x-ray pulses have been proposed, bringing electronic timescales into reach and emphasizing the demand for a quantum description. In this paper, we present a method to describe propagation-based x-ray phase-contrast imaging in nonrelativistic quantum electrodynamics. We explain why the standard scattering treatment via Fermi's golden rule cannot be applied. Instead, the quantum electrodynamical treatment of phase-contrast imaging must be based on a different approach. It turns out that it is essential to select a suitable observable. Here, we choose the quantum-mechanical Poynting operator. We determine the expectation value of our observable and demonstrate that the leading order term describes phase-contrast imaging. It recovers the classical expression of phase-contrast imaging. Thus, it makes the instantaneous electron density of non-stationary electronic states accessible to time-resolved imaging. Interestingly, inelastic (Compton) scattering does automatically not contribute in leading order, explaining the success of the semiclassical description.