Journal: Scientific Reports

Article Title: Representation theory of coherent photons and application to CNOT operation for spin and orbital angular momentum

doi: 10.1038/s41598-025-23755-9

Figure Lengend Snippet: Single-qubit operation for SU(4) states of coherent photons for the spin angular momentum state of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\boldsymbol{\alpha }} \rangle _{\textrm{S}}$$\end{document} and the orbital angular momentum state of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\boldsymbol{\beta }} \rangle _{\textrm{O}}$$\end{document} . The identity operator of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{1}}$$\end{document} stands for the unit operation for SU(2) states.

Article Snippet: We have obtained the spin angular momentum operator in the many-body state, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\bf{S}}=\hbar \hat{\boldsymbol{\psi }}^{\dagger } {\boldsymbol{\sigma }} \hat{\boldsymbol{\psi }}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}^{\dagger }=({\hat{a}}_{\textrm{H}}^{\dagger } , {\hat{a}}_{\textrm{V}}^{\dagger } )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}=({\hat{a}}_{\textrm{H}}, {\hat{a}}_{\textrm{V}} )^{\textrm{t}}$$\end{document} are the spinor representation of creation and annihilation operators in HV bases to create and annihilate a photon, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\sigma }} = (\sigma _3, \sigma _1, \sigma _2)$$\end{document} are generators of spin angular momentum.

Techniques:

Journal: Scientific Reports

Article Title: Representation theory of coherent photons and application to CNOT operation for spin and orbital angular momentum

doi: 10.1038/s41598-025-23755-9

Figure Lengend Snippet: CNOT operation for coherent photons. ( a ) The weight diagram of SU(4) states of coherent photons with spin and orbital angular momentum. ( b ) NOT operator for orbital angular momentum. Two cylindrical lenses are separated with the twice of the focal length to achieve the phase-shift of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi$$\end{document} for converting the left vortex to the right vortex, and vice versa . ( c ) Poincaré sphere for orbital angular momentum. Far-field images are shown for left (L) and right (R) twisted states and their superposition states of horizontal (H), diagonal (D), vertical (V), and anti-diagonal (A) dipole states, respectively. The red circle shows how the half-wave phase-shift of (b) changes the twisted states. ( d ) Experimental set-up for CNOT operation to coherent photons with spin and orbital angular momentum. The system is made of three units, a classical entanglement generator, an operation unit, and a measurement unit. The generator is made of a Poincaré rotator, which allows the arbitrary rotation of polarisation states, and vortex lenses to allow spin-to-orbit converter. The generated entangled light is subject to the optional Bell projection, to allow changes in orbital angular momentum states by projection of spin state. The CNOT operation is achieved by splitting the spin state by a polarisation dependent beam splitter and apply the NOT operation to vertically polarised beam, while the horizontally polarised beam is preserved, and then recombined. Cyl cylindrical lens, HWPS half-wave phase-shifter, LD laser diode, CL collimator lens, PH pin hole, PL polariser, HWP half-wave plate, QWP quarter-wave plate, PBS polarisation beam splitter, NPBC non-polarisation beam combiner, NPBS non-polarisation beam splitter, M mirror, VL vortex lens, PM polarimeter, CMOS camera. PL2 and PL3 are optional and shown by the dotted lines.

Article Snippet: We have obtained the spin angular momentum operator in the many-body state, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\bf{S}}=\hbar \hat{\boldsymbol{\psi }}^{\dagger } {\boldsymbol{\sigma }} \hat{\boldsymbol{\psi }}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}^{\dagger }=({\hat{a}}_{\textrm{H}}^{\dagger } , {\hat{a}}_{\textrm{V}}^{\dagger } )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}=({\hat{a}}_{\textrm{H}}, {\hat{a}}_{\textrm{V}} )^{\textrm{t}}$$\end{document} are the spinor representation of creation and annihilation operators in HV bases to create and annihilate a photon, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\sigma }} = (\sigma _3, \sigma _1, \sigma _2)$$\end{document} are generators of spin angular momentum.

Techniques: Generated

Journal: Scientific Reports

Article Title: Representation theory of coherent photons and application to CNOT operation for spin and orbital angular momentum

doi: 10.1038/s41598-025-23755-9

Figure Lengend Snippet: CNOT operations for spin and orbital angular momentum. Polarisers (PL2 and PL3) were not inserted. ( a1 )–( s1 ) Inputs of macroscopically entangled states, which were superposition states between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\textrm{H} \rangle _{\textrm{S}}|\textrm{R} \rangle _{\textrm{O}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\textrm{V} \rangle _{\textrm{S}}|\textrm{L} \rangle _{\textrm{O}}$$\end{document} , and HWP2 was rotated from 0 to 90 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document} with a step of 5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document} . ( a1 ) is purely \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\textrm{H} \rangle _{\textrm{S}}|\textrm{R} \rangle _{\textrm{O}}$$\end{document} , while (j1) is purely \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\textrm{V} \rangle _{\textrm{S}}|\textrm{L} \rangle _{\textrm{O}}$$\end{document} . ( a2 )–( s2 ) Output far-field images after the CNOT operation. ( a2 ) should be preserved to being \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\textrm{V} \rangle _{\textrm{S}}|\textrm{L} \rangle _{\textrm{O}}$$\end{document} . ( j2 ) should be reverted for orbital angular momentum, such that the state must be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\textrm{V} \rangle _{\textrm{S}}|\textrm{R} \rangle _{\textrm{O}}$$\end{document} .

Article Snippet: We have obtained the spin angular momentum operator in the many-body state, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\bf{S}}=\hbar \hat{\boldsymbol{\psi }}^{\dagger } {\boldsymbol{\sigma }} \hat{\boldsymbol{\psi }}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}^{\dagger }=({\hat{a}}_{\textrm{H}}^{\dagger } , {\hat{a}}_{\textrm{V}}^{\dagger } )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}=({\hat{a}}_{\textrm{H}}, {\hat{a}}_{\textrm{V}} )^{\textrm{t}}$$\end{document} are the spinor representation of creation and annihilation operators in HV bases to create and annihilate a photon, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\sigma }} = (\sigma _3, \sigma _1, \sigma _2)$$\end{document} are generators of spin angular momentum.

Techniques:

Journal: Scientific Reports

Article Title: Representation theory of coherent photons and application to CNOT operation for spin and orbital angular momentum

doi: 10.1038/s41598-025-23755-9

Figure Lengend Snippet: CNOT operations for the singlet state with spin and orbital angular momentum. ( a1 )–( s1 ) Input images were taken after rotating PL2 from 0 to 180 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document} with a step of 10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document} . This changes the direction of the Bell projection for polarisation states. At the diagonal polarisation of ( e1 ), the dipole was aligned along the anti-diagonal direction, such that the phase was adjusted to be singlet. ( a2 )–( s2 ) Output far-field images for horizontal polarisation, projected by PL3, after the CNOT operation. ( e2 ) was controlled to keep the anti-diagonal dipole. ( a3 )–( s3 ) Output far-field images for vertical polarisation, projected by PL3, after the CNOT operation. ( e3 ) was rotated to be the diagonal dipole, which corresponds to the NOT operation. We can also confirm the expected CNOT operation for the input of ( n1 ) with the controlled preservation in ( n2 ), while the NOT operation worked properly for ( n3 ).

Article Snippet: We have obtained the spin angular momentum operator in the many-body state, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\bf{S}}=\hbar \hat{\boldsymbol{\psi }}^{\dagger } {\boldsymbol{\sigma }} \hat{\boldsymbol{\psi }}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}^{\dagger }=({\hat{a}}_{\textrm{H}}^{\dagger } , {\hat{a}}_{\textrm{V}}^{\dagger } )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}=({\hat{a}}_{\textrm{H}}, {\hat{a}}_{\textrm{V}} )^{\textrm{t}}$$\end{document} are the spinor representation of creation and annihilation operators in HV bases to create and annihilate a photon, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\sigma }} = (\sigma _3, \sigma _1, \sigma _2)$$\end{document} are generators of spin angular momentum.

Techniques: Preserving

Journal: Scientific Reports

Article Title: Representation theory of coherent photons and application to CNOT operation for spin and orbital angular momentum

doi: 10.1038/s41598-025-23755-9

Figure Lengend Snippet: CNOT operations for the triplet state with spin and orbital angular momentum. ( a1 )–( s1 ) Input images were taken after rotating PL2 from 0 to 180 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document} with a step of 10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document} . At the diagonal polarisation of ( e1 ), the dipole was aligned along the diagonal direction, such that the phase was adjusted to be triplet. ( a2 )–( s2 ) Output far-field images for horizontal polarisation, projected by PL3, after the CNOT operation. ( e2 ) was controlled to keep the diagonal dipole. ( a3 )–( s3 ) Output far-field images for vertical polarisation, projected by PL3, after the CNOT operation. ( e3 ) was rotated to be the anti-diagonal dipole, which corresponds to the NOT operation. We can also confirm the expected CNOT operation for the input of ( n1 ) with the controlled preservation in ( n2 ), while the NOT operation worked properly for ( n3 ).

Article Snippet: We have obtained the spin angular momentum operator in the many-body state, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\bf{S}}=\hbar \hat{\boldsymbol{\psi }}^{\dagger } {\boldsymbol{\sigma }} \hat{\boldsymbol{\psi }}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}^{\dagger }=({\hat{a}}_{\textrm{H}}^{\dagger } , {\hat{a}}_{\textrm{V}}^{\dagger } )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}=({\hat{a}}_{\textrm{H}}, {\hat{a}}_{\textrm{V}} )^{\textrm{t}}$$\end{document} are the spinor representation of creation and annihilation operators in HV bases to create and annihilate a photon, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\sigma }} = (\sigma _3, \sigma _1, \sigma _2)$$\end{document} are generators of spin angular momentum.

Techniques: Preserving

Journal: Scientific Reports

Article Title: Representation theory of coherent photons and application to CNOT operation for spin and orbital angular momentum

doi: 10.1038/s41598-025-23755-9

Figure Lengend Snippet: CNOT operations for spin and orbital angular momentum, obtained from the inputs, made of diagonally and anti-diagonally polarised states. PL2 was not employed in this measurement. ( a1 )–( s1 ) Inputs of macroscopically entangled states, which were superposition states between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\textrm{D} \rangle _{\textrm{S}}|\textrm{R} \rangle _{\textrm{O}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\textrm{A} \rangle _{\textrm{S}}|\textrm{L} \rangle _{\textrm{O}}$$\end{document} . Input images were taken after rotating HWP2 from 0 to 90 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document} with a step of 5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document} . ( e1 ) was made of the sum of these states. ( a2 )–( s2 ) Output far-field images for horizontal polarisation, projected by PL3, after the CNOT operation. ( e2 ) was controlled to keep the diagonal dipole. ( a3 )–( s3 ) Output far-field images for vertical polarisation, projected by PL3, after the CNOT operation. The input for ( e3 ) is anti-diagonal dipole, which was successfully reverted to be the diagonal dipole.

Article Snippet: We have obtained the spin angular momentum operator in the many-body state, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\bf{S}}=\hbar \hat{\boldsymbol{\psi }}^{\dagger } {\boldsymbol{\sigma }} \hat{\boldsymbol{\psi }}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}^{\dagger }=({\hat{a}}_{\textrm{H}}^{\dagger } , {\hat{a}}_{\textrm{V}}^{\dagger } )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\boldsymbol{\psi }}=({\hat{a}}_{\textrm{H}}, {\hat{a}}_{\textrm{V}} )^{\textrm{t}}$$\end{document} are the spinor representation of creation and annihilation operators in HV bases to create and annihilate a photon, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\sigma }} = (\sigma _3, \sigma _1, \sigma _2)$$\end{document} are generators of spin angular momentum.

Techniques:

Journal: Scientific Reports

Article Title: Anisotropic chemical bonding of lanthanide-OH molecules

doi: 10.1038/s41598-025-06281-6

Figure Lengend Snippet: Potential energies of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\text{f}^{n-1}6\text{s}^2+2\text{p}^6$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\text{f}^{n}6\text{s}+2\text{p}^6$$\end{document} electronic configurations of DyOH (panel ( a )) and ErOH (panel ( b )) near their equilibrium geometries as functions of the X -O separation with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X=\text{Dy}$$\end{document} or Er for a linear geometry and at a fixed O-H separation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.80a_0$$\end{document} . For each configuration the curves correspond to states with different \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} (or more precisely \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega |$$\end{document} ). Curves with the same color have the same \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega |$$\end{document} . The zero of energy is at the equilibrium geometry of energetically lowest potential. The potentials have been obtained with self-consistent-field calculations using basis sets that do not include excitations into 6p and 5d molecular orbitals. Panel ( c ) shows the splittings among the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} states of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\text{f}^{n-1}6\text{s}^2+2\text{p}^6$$\end{document} configuration (colored circles). The energies have been obtained with self-consistent-field calculations using basis sets that do include excitations into 6p and 5d molecular orbitals. The dashed curves through the markers are fits to these data and are described in the text.

Article Snippet: Moreover, tensor operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{kq}(\cdot ,\cdot )$$\end{document} of rank \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,3,\ldots$$\end{document} are constructed from rank-1 total electronic angular momentum operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{j}$$\end{document} and, finally, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{kq}(\hat{R})$$\end{document} are spherical harmonic functions, where unit vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R}=(\theta ,\varphi )$$\end{document} describes the orientation of the symmetry axis of the linear triatomic molecule in the laboratory-fixed coordinate system.

Techniques:

Journal: Scientific Reports

Article Title: Anisotropic chemical bonding of lanthanide-OH molecules

doi: 10.1038/s41598-025-06281-6

Figure Lengend Snippet: Isosurfaces of electronic Kohn-Sham molecular orbitals (MOs) for DyOH (panels ( a,b )) and ErOH (panels ( c,d )) at their equilibrium, linear geometries. In all panels the small, nearly hidden cyan, red, and gray balls correspond to the locations of the lanthanide, oxygen, and hydrogen atom, respectively. Panels ( a,c ) show some of the MOs for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\text{f}^{n-1}6\text{s}^2+2\text{p}^6$$\end{document} configuration where the 4f and 2p orbitals overlap. The chosen 4f orbitals for DyOH and ErOH resemble the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {f}_{xyz}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {f}_{yz^2}$$\end{document} cubic or tesseral harmonic, respectively, while the 2p orbital resembles the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {p}_z$$\end{document} cubic harmonic with the lobes aligned along the z or Ln-O axis. Similarly, panels ( c,d ) show some of the MOs for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\text{f}^{n-1}6\text{s}6\text{p}+2\text{p}^6$$\end{document} configuration, where now the largest feature corresponds to the 6p orbital resembling the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {p}_x$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_y$$\end{document} cubic harmonic.

Article Snippet: Moreover, tensor operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{kq}(\cdot ,\cdot )$$\end{document} of rank \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,3,\ldots$$\end{document} are constructed from rank-1 total electronic angular momentum operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{j}$$\end{document} and, finally, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{kq}(\hat{R})$$\end{document} are spherical harmonic functions, where unit vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R}=(\theta ,\varphi )$$\end{document} describes the orientation of the symmetry axis of the linear triatomic molecule in the laboratory-fixed coordinate system.

Techniques:

Journal: Scientific Reports

Article Title: Anisotropic chemical bonding of lanthanide-OH molecules

doi: 10.1038/s41598-025-06281-6

Figure Lengend Snippet: Electronic eigenenergies (colored circles) of DyOH at its equilibrium, linear geometry as a function of electron projection quantum number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} . The zero of energy is set at that of the lowest eigenstate. The excitation energies have been obtained with self-consistent-field calculations using basis sets that do include excitations into 6p and 5d molecular orbitals. For the lowest energies, lines connecting the colored circles correspond to states of the same configuration. For the higher energies, the lines although sorted by energy, are only guides for the eye. The blue arrow highlights the transition from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega =15/2$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega '=17/2$$\end{document} between the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm 6s^2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm 6s6p$$\end{document} states. It has an electric dipole moment of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.79 ea_0$$\end{document} .

Article Snippet: Moreover, tensor operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{kq}(\cdot ,\cdot )$$\end{document} of rank \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,3,\ldots$$\end{document} are constructed from rank-1 total electronic angular momentum operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{j}$$\end{document} and, finally, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{kq}(\hat{R})$$\end{document} are spherical harmonic functions, where unit vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R}=(\theta ,\varphi )$$\end{document} describes the orientation of the symmetry axis of the linear triatomic molecule in the laboratory-fixed coordinate system.

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Journal: Scientific Reports

Article Title: Anisotropic chemical bonding of lanthanide-OH molecules

doi: 10.1038/s41598-025-06281-6

Figure Lengend Snippet: Permanent dipole moment in atomic units \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ea_0$$\end{document} as function of projection quantum number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} of states of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\text{f}^{n-1}6\text{s}^2+2\text{p}^6$$\end{document} electronic ground-state configuration of DyOH (orange filled circles) and ErOH (grey filled circles) at their equilibrium linear geometry. Solid black curves are polynomial fits to the data as described in the text.

Article Snippet: Moreover, tensor operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{kq}(\cdot ,\cdot )$$\end{document} of rank \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,3,\ldots$$\end{document} are constructed from rank-1 total electronic angular momentum operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{j}$$\end{document} and, finally, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{kq}(\hat{R})$$\end{document} are spherical harmonic functions, where unit vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R}=(\theta ,\varphi )$$\end{document} describes the orientation of the symmetry axis of the linear triatomic molecule in the laboratory-fixed coordinate system.

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Journal: Scientific Reports

Article Title: Anisotropic chemical bonding of lanthanide-OH molecules

doi: 10.1038/s41598-025-06281-6

Figure Lengend Snippet: Molecular g factors, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\text{mol}$$\end{document} as defined in the text, ( a ) and transition magnetic moments ( b ) as functions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} for DyOH (orange filled circles) and ErOH (grey filled circles) at their linear equilibrium geometries along the body-fixed Ln-O axis. G -factors of the pseudo-spin are defined in the text. The DyOH and ErOH molecules are in their \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {4f}^9\hbox {6s}^2$$\end{document} + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {2p}^6$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {4f}^{11}\hbox {6s}^2$$\end{document} + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {2p}^6$$\end{document} ground-state configuration, respectively. In panel ( a ) the dashed orange and grey lines are the experimental g factors of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=15/2$$\end{document} level of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {4f}^9\hbox {6s}^2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {4f}^{11}\hbox {6s}^2$$\end{document} configurations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Dy}^+$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Er}^+$$\end{document} , respectively. In panel ( a ) the solid black curve is a polynomial in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^2$$\end{document} found from a fit to the data for DyOH, while the solid black curves in panel b) correspond to a fit to adjustable parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_1$$\end{document} times matrix elements of the angular momentum raising operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_{+1}/\hbar$$\end{document} .

Article Snippet: Moreover, tensor operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{kq}(\cdot ,\cdot )$$\end{document} of rank \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2,3,\ldots$$\end{document} are constructed from rank-1 total electronic angular momentum operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{j}$$\end{document} and, finally, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{kq}(\hat{R})$$\end{document} are spherical harmonic functions, where unit vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R}=(\theta ,\varphi )$$\end{document} describes the orientation of the symmetry axis of the linear triatomic molecule in the laboratory-fixed coordinate system.

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