LAUSR

The angular correlation functions (ACFs) of a pair of Lyman-$\ensuremath{\alpha}$ photons in photodissociation of ${\mathrm{H}}_{2}$ and ${\mathrm{D}}_{2}$ 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 ${1}_{u}$ 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 ${1}_{u}$ symmetry properties, which are invariant during the dissociation from the Franck-Condon region towards infinite internuclear distance.