In this paper, as a continuation of our recent article demonstrating that the 2p atom pair is entangled in the H2 photodissociation [Phys. Rev. A 111, 023116 (2025)], we have discovered the origins of the entanglement for the 2p atom-pair state and obtained the entanglement entropies by origins. The same investigation has been carried out for the precursor molecular state of the 2p atom pair. The reason for the increase of the entanglement entropy during the dissociation from the precursor molecule into the H(2p) + H(2p) atom pair has been made clear.
The angular correlation functions (ACFs) of a pair of Lyman-$\ensuremath{\alpha}$ photons in photodissociation of
H2 and
D2 are measured with linearly polarized incident light at a 33.66-eV incident photon energy in a wider angular range at a narrower angular step with smaller distortion than before [Y. Nakanishi et al., Phys. Rev. A 90, 043405 (2014)] so that we identify the atom-pair state emitting the pair of Lyman-$\ensuremath{\alpha}$ photons and find out whether the atom pair is entangled or not. Searching for reasonable
2p atom-pair states that reproduce the experimental ACFs to solve the issue, we show that hydrogen molecules are photoexcited to the ${Q}_{2}{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}(1)$ state in the Franck-Condon region and then the ${Q}_{2}{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}(1)$ state comes to superpose with the ${Q}_{2}{\phantom{\rule{0.16em}{0ex}}}^{3}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}(2)$ state as the internuclear distance increases to infinity. The superposition is brought about by the spin-orbit coupling, which is effective around infinite internuclear distance because the potential-energy curves of those states are close to each other, but is negligibly small around the Franck-Condon region because they are apart from each other. The
2p atom pairs turn out to be in the
1u superposition state, which is entangled. We therefore conclude that an entangled pair of hydrogen atoms is spontaneously produced through the photodissociation of a hydrogen molecule, and the entanglement originates from the
1u symmetry properties, which are invariant during the dissociation from the Franck-Condon region towards infinite internuclear distance.