A reconciliation of the Pryce-Ward and Klein-Nishina statistics for semi-classical simulations of annihilation photons correlations

Abstract

Two photons from the ground state para-positronium annihilation are emitted in a maximally entangled singlet state of orthogonal polarizations. In case of the Compton scattering of both photons the phenomenon of quantum entanglement leads to a measurable increase in the azimuthal correlations of scattered photons, as opposed to a classical description treating the two scattering events as independent. The probability of the scattering of the system of the entangled photons is described by the Pryce-Ward cross section dependent on a difference of the azimuthal scattering angles in the fixed coordinate frame, while the independent scattering of single photons is described by the Klein-Nishina cross section dependent on the azimuthal angle relative to each photon's initial polarization. Since the singlet state of orthogonal polarizations is rotationally invariant, it does not carry any physical information on the initial polarizations of the single annihilation photons. In such bipartite state the angular origin for the Klein-Nishina cross section is undefined, making the Pryce-Ward and Klein-Nishina descriptions mutually exclusive. However, semi-classical simulations of the joint Compton scattering of entangled photons - implementing the Pryce-Ward cross section, but still treating the two photons as separate entities - can reconcile the Pryce-Ward correlations with the Klein-Nishina statistics for single photons by implementing a modified version of a scattering cross section presented in this work.

Figures (3)

• Figure 1: The Compton scattering geometry. Two annihilation photons propagate in the opposite directions along the $z$-axis. Their mutually orthogonal initial polarizations -- indicated by the blue planes -- are here assumed to be well defined. This assumption is the core of a semi-classical treatment. Violet arrows show the scattered photons' momenta, without any indication of their post-scattering polarizations. The green scattering planes provide a spatial indication of the azimuthal scattering angles $\varphi_1,\varphi_2$ relative to the fixed coordinate axis, and of $\phi_1,\phi_2$ relative to the initial photons' polarizations. They are related via one photon's initial polarization angle ${\color{black}\Phi}$ relative to the fixed axis. For visual purposes the azimuthal angles are shown in a left-handed convention (measured from the $x$-axis towards the negative direction of the $y$-axis).
• Figure 2: Reduced distributions of the azimuthal scattering angle $\boldsymbol{\phi_2}$ for the second annihilation photon. Distributions are marginalized over the scattering angle $\theta_2\in[0,\pi]$ relative to the photon's initial direction. Azimuthal $\phi_2$ is defined relative to the photon's initial polarization. Each distribution is obtained by the Monte-Carlo integration, i.e. a random sampling of the Klein-Nishina (KN) and/or Pryce-Ward (PW) cross section, using $4\cdot10^7$ samples. Labels indicate whether the first $\gamma_1$ and the second $\gamma_2$ photon were generated according to the KN or the PW distribution. The analytic form of these distributions is given by Eqs. (\ref{['f_kn_kn']}), (\ref{['f_kn_pw']}), (\ref{['f_pw_pw']}). The probability distributions $f(\phi_2)$ are scaled by a factor $2\pi$.
• Figure B1: Reduced (marginalized over $\chi_1$ and $\chi_2$) distributions from Eqs. (\ref{['reduced1']}) and (\ref{['reduced2']}). See the main text for details.