Probing arbitrary polarized photon pairs undergoing double Compton scatterings by a dedicated MC simulator validated with experimental data

TL;DR

This work addresses measuring polarization for high-energy (MeV) photons, where traditional polarizers are unavailable, by exploiting double Compton scattering and a $\Delta\hat{\Phi}$ observable. It develops a Kraus-operator based quantum-theory framework for multi-photon Compton processes and implements a Geant4-based Vienna-Warsaw (VW) Monte Carlo, capable of simulating arbitrary bipartite polarization states. The VW model is validated against a high-statistics J-PET dataset, yielding a consistent $\mathcal{V}^2$ around $0.27$–$0.28$ and an informative upper bound near $0.40$, demonstrating reliable polarization reconstruction in MeV photon pairs. The results support using polarization observables to enhance PET imaging and advance foundational studies of entanglement in high-energy photons, with implications for medical imaging and quantum information science.

Abstract

Quantum correlations in the polarization degrees of freedom of the two-photon system have been extensively studied and form our current understanding of the quantum nature of our world. Most of the studies are concentrated on the low-energy (optical) photon pairs, for which efficient polarization measurement devices exist. However, for high-energetic (MeV) pairs of photons, e.g. produced in the decay of positronium atoms, no polarizers are available. Partial information about the polarization degree of freedom can be extracted by exploiting the measurements of photon pairs that undergo double Compton scattering. We present a Geant4-based Monte Carlo Vienna-Warsaw model capable of simulating any initial polarization state of bipartite photons. This puts us in a position to derive the behavior of the experimental observable, the angular difference $Δ\hatΦ$ formed by the two scattering planes. We validate our Vienna-Warsaw simulator with the high-statistics experimental sample -- based on a total of $3 \times 10^5 $ event candidates -- of two-photon pairs measured with the J-PET Big Barrel detector. We deduce the value of the squared visibility (interference contrast) encoding the polarization in the angle difference of the two scattering planes, $Δ\hatΦ$. The simulated spectra are in good agreement with the experimental correlation spectra and behave as predicted by theory.

Figures (16)

• Figure 1: The scheme of the Compton scattering of a photon on a free electron. The $\hbox{\boldmath$\mathbf{k}$}$ and $\hbox{\boldmath$\mathbf{k'}$}$ are the wave vectors of the photon before and after scattering, respectively. The initial and final photon polarizations are marked as $\hbox{\boldmath$\mathbf{\epsilon}$}_{i}$ and $\hbox{\boldmath$\mathbf{\epsilon}$}_{f}$, respectively.
• Figure 2: (a) The envelope function $\mathcal{F}$ for the given incoming photon energy $k$ and the scattering angle $\tilde{\Theta}$ defines the maximum amplitude of the visible oscillations for polarized photons. The line colors represent different energies: black $k=0.1$, red $k=1 (511~\text{keV})$, green $k=2$ and blue $k=5$. (b) The visibility or interference contrast is the component that reduces the oscillation patterns depending on the incoming photon energy $k$ and the scattering angle $\tilde{\Theta}$. In particular, if the visibility is $0$, then no oscillation in $\Phi$ can be observed. This is the case for all energies if we consider small-angle scatterings or backwards scattering.
• Figure 3: Photon momentum direction determination. The cuboids represent four detector modules, and the red spheres correspond to points of interaction via Compton scattering. The momentum directions of the photon before first Compton scattering is calculated as $\hat{k}=\frac{\hbox{\boldmath$\mathbf{h}$}_1}{|\hbox{\boldmath$\mathbf{h}$}_1|}$ and before the second scattering as $\hat{k'}=\frac{\hbox{\boldmath$\mathbf{d}$}}{|\hbox{\boldmath$\mathbf{d}$}|}$ where $\hbox{\boldmath$\mathbf{d}$}=\hbox{\boldmath$\mathbf{h}$}_2-\hbox{\boldmath$\mathbf{h}$}_1$. The calculations assume that the photon source (yellow sphere) is at the center of the coordinate system.
• Figure 4: Estimation of the relative angle between the scattering planes of the photon pair. The vectors $\hbox{\boldmath$\mathbf{k}$}_{1/2}$ show the directions of the annihilation gamma's initial momentum. The vectors $\hbox{\boldmath$\mathbf{k'}$}_{1/2}$ are the momentum directions after the Compton interaction with the detector medium (scintillators). The yellow and blue planes represent the scattering plane of the first and second gamma, respectively. The relative angle $\Delta\hat{\Phi}$ is observable that relates to the initial joint polarisation state of the photon pair (see Eq. \ref{['eq:cross-section-probability-many-photons']}).
• Figure 5: Simulator scheme