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Physical Review A

Analytical expression for the angular correlation function of two Lyman- α photons in the photodissociation of hydrogen molecules

Kouichi Hosaka, Yutaro Torizuka, Philipp Schmidt, Arno Ehresmann, Takeshi Odagiri, M. Kitajima, Noriyuki Kouchi

Published Jun 23, 2021• 2 citations
Publisher

Abstract

An analytical expression of the angular correlation function of a pair of Lyman-$\ensuremath{\alpha}$ photons in the photodissociation of a hydrogen molecule is derived theoretically in a manner based on both atomic and molecular physics and quantum optics. The angular correlation function turns out to be expressed in terms of cosine (sine) functions of four angular variables of detectors with five coefficients. The angular correlation function is expanded in terms of the spherical harmonics for investigating which terms are involved, and we discuss the reason why they are involved. Interesting features are revealed in the expansion. We then search for a special detector arrangement where the angular-dependent terms vanish in the angular correlation function expressed in terms of the spherical harmonics. It turns out that no such detector arrangement in fact exists, but there is a special pair of detector arrangements where the angular-dependent terms vanish in the summation of two values of the angular correlation function expressed in terms of the spherical harmonics. We refer to such a pair of detector arrangements as a magic pair.

DOI
10.1103/physreva.103.062809

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宮城晴英, Haruhide Miyagi, 淳 市村 +3 more

•Jul 1, 2007•0 citations
An analytical expression of the angular correlation function of a pair of Lyman-\ensuremathα\ensuremath{\alpha}\ensuremathα photons in the photodissociation of a hydrogen molecule is derived theoretically in a manner based on both atomic and molecular physics and quantum optics. The angular correlation function turns out to be expressed in terms of cosine (sine) functions of four angular variables of detectors with five coefficients. The angular correlation function is expanded in terms of the spherical harmonics for investigating which terms are involved, and we discuss the reason why they are involved. Interesting features are revealed in the expansion. We then search for a special detector arrangement where the angular-dependent terms vanish in the angular correlation function expressed in terms of the spherical harmonics. It turns out that no such detector arrangement in fact exists, but there is a special pair of detector arrangements where the angular-dependent terms vanish in the summation of two values of the angular correlation function expressed in terms of the spherical harmonics. We refer to such a pair of detector arrangements as a magic pair.
The angular correlation functions (ACFs) of a pair of Lyman-\ensuremath{\alpha} photons emitted from H fragments in the photodissociation of a hydrogen molecule are measured at a 33.66-eV incident photon energy and at hydrogen gas pressures of approximately 0.1 and 1 Pa. The ACFs are measured for both opposite and nonopposite arrangements of the two photon detectors. It turns out that the experimental ACFs involve neither the contribution of the reactions H(n=2)+H2\mathrm{H}(n\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}2)\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}{\mathrm{H}}_{2}H(n=2)+H2​ nor the contribution of the cascade from H(n\mathrm{H}(nH(n \ensuremath{\geqslant} 3) to H(2p)2p)2p) fragments. Thus the experimental ACFs are those for primary H(2p)2p)2p) pairs following the photodissociation of H2{\mathrm{H}}_{2}H2​. The experimental ACFs are compared with (i) the theoretical ACF for entangled pairs of H(2p)2p)2p) atoms, where the magnetic quantum number of each hydrogen atom is indefinite, and (ii) the theoretical ACF for H(2p)2p)2p) pairs with definite magnetic quantum number of each hydrogen atom relative to the internuclear axis [the former entangled state of H(2p)2p)2p) pairs is a sum of the latter pair states with definite magnetic quantum number]. In the theoretical ACF in (ii), the disentanglement in H(2p)2p)2p) pairs during the dissociation is considered. The experimental ACFs show a similar tendency in angular dependence to the theoretical ACF for entangled H(2p)2p)2p) pairs. However, there still remains a considerable difference in the variation magnitude between those experimental and theoretical ACFs. The experimental ACFs show the reverse tendency in angular dependence to the theoretical ACF for H(2p)2p)2p) pairs with definite magnetic quantum number of each hydrogen atom relative to the internuclear axis. We thus conclude that the pair of H(2p)2p)2p) atoms in the photodissociation of H2{\mathrm{H}}_{2}H2​ is unlikely to be in the definite states of magnetic quantum number of each hydrogen atom relative to the internuclear axis, i.e., unlikely to be in the components of the entangled state of H(2p)2p)2p) pairs.
The angular correlation functions (ACFs) of a pair of Lyman-$\ensuremath{\alpha}$ photons in photodissociation of H2{\mathrm{H}}_{2}H2​ and D2{\mathrm{D}}_{2}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 2p2p2p 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 2p2p2p atom pairs turn out to be in the 1u{1}_{u}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{1}_{u}1u​ symmetry properties, which are invariant during the dissociation from the Franck-Condon region towards infinite internuclear distance.
The angular correlation of two Lyman-α photons generated by the linearly polarized single-photon absorption of H 2 (X 1 Σ g + ) followed by the neutral dissociation of the doubly excited Q 2 1 Π u (1) state has been theoretically investigated with the second-order correlation function in quantum optics. A strong angular correlation has been obtained even for the molecules randomly oriented in space, due to the complicated entanglement of two H(2p) atoms in the intermediate state. It has been revealed how the symmetry properties in the electronic state are transferred to the two-photon state and how they manifest themselves in the angular correlation function.
The angular distribution of two Lyman-alpha photons, i.e., the probability density that two Lyman-alpha photons are emitted in given directions, in the photodissociation of a hydrogen molecule have been measured at the hydrogen gas pressures of 0.40 and 0.13 Pa. We have found that the experimental angular distributions seem to approach the theoretical one by our group [J. Phys. B 40, 617 (2007)] with decreasing pressure, which indicates the generation of the entangled pair of H(2p) atoms shown in the theory and the role of the reaction of the entangled pair of H(2p) atoms with an H2 molecule that efficiently changes the entanglement.
An angular correlation function (ACF) of a pair of Lyman-$\ensuremath{\alpha}$ photons produced in photodissociation of H2{\mathrm{H}}_{2}H2​ with linearly polarized incident light is measured on a whole sphere at a 33.66-eV photon energy so that we may distinctly substantiate that the 2p2p2p atom pairs are in the state previously identified by Torizuka et al. as the 2p2p2p atom-pair state [Phys. Rev. A 99, 063426 (2019)], who measured the ACF only on a circle in the plane perpendicular to the incident light beam. The 2p2p2p atom-pair state determined with much less uncertainty than before is investigated in terms of whether or not the state is entangled. We discover that the pair of fragment H(2p)\mathrm{H}(2p)H(2p) atoms is certainly entangled in terms of the electron-electron entanglement, and obtain the entanglement entropy of the 2p2p2p atom-pair state. The similar discussion is made in terms of the electronic spatial-spin entanglement too, where the spatial motion and spin motion of electrons are considered nonidentical partial systems at variance with the former electron-electron entanglement. It turns out that the pair of fragment H(2p2p2p) atoms is entangled in terms of the electronic spatial-spin entanglement as well as the electron-electron entanglement.