spin angular momentum operator  (Genovis Inc)

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: