# Topology and Polarization of Optical Vortex Fields from Atomic Phased Arrays
## 4 Polarization Parameters of Atomic-Array Radiation  
The presence of a longitudinal electric field component in the vector vortices requires the definition of a \(3\times 3\) spin density matrix described by nine independent parameters(including the field's intensity). One possibility is to use expansion coefficients of nine Gell-Mann matrices [20,21]. It is also possible to describe the electromagnetic field in terms of state multipoles. Using this approach, it was shown in Ref. [22] that the polarization parameters of twisted photons are in one-to-one correspondence with the related parameters of the spin-density matrix of optically polarized atoms excited through a dominant electric dipole mechanism. We will briefly describe the polarization formalism here. An arbitrary (pure) state of a spin 1 particle \(|\chi_{1}\rangle\), or a three-dimensional vector field, can be expanded in terms of basis kets \(|\chi_{1i}\rangle\)\((i=x,y,z)\) in a Cartesian basis  
\[|\chi_{1}\rangle=\sum_{i}a_{i}|\chi_{1i}\rangle,\]  
or, equivalently, in a spherical basis  
\[|\chi_{1}\rangle=a^{+}|\chi_{1+}\rangle+a^{0}|\chi_{10}\rangle+a^{-}|\chi_{1-}\rangle,\]  
where \(a_{i}\) and \(a^{\pm}\) are complex amplitudes related by  
\[a^{\pm}=\frac{(\mp a_{x}+ia_{y})}{\sqrt{2}},\,a_{0}=a_{z},\,\sum_{i}|a_{i}|^{2 }=1.\]  
The spin density matrix is \(\rho_{ij}=a_{i}a_{j}^{*}\), and we will use the parameterization and normalization conventions from Refs. [22,23]:  
\[ \begin{array}{ll}\rho_{ij}=&\frac{1}{3}\Big{\{}I+\frac{3}{2}(p_{x}{\cal P}_ {x}+p_{y}{\cal P}_{y}+p_{z}{\cal P}_{z})+\\ &\frac{2}{3}(p_{xy}{\cal P}_{xy}+p_{yz}{\cal P}_{yz}+p_{xz}{\cal P}_{xz})+\\ &\frac{1}{6}(p_{xx}-p_{yy})({\cal P}_{xx}-{\cal P}_{yy})+\frac{1}{2}p_{zz}{ \cal P}_{zz}\Big{\}}_{ij},\end{array}  \tag{29}\]  
where \({\cal P}_{i}\), \({\cal P}_{ij}\) are the operators of the spin and quadrupole moments, and \(p_{i}\), \(p_{ij}\) are the corresponding vector and quadrupole polarizations, which can be expressed in terms of the above amplitudes \(a_{i}\) and \(a^{\pm,0}\),  
\[p_{i}=i\sum_{jk}\epsilon_{ijk}a_{j}a_{k}^{*};\,p_{ik}=-\frac{3}{2}(a_{i}a_{k}^ {*}+a_{k}a_{i}^{*}-\frac{2}{3}\delta_{ik}). \tag{30}\]  
In particular,  
\[ \begin{array}{ll}p_{xx}-p_{yy}=3(a^{+}a^{-*}+a^{-}a^{++});\\ p_{zz}=|a^{+}|^{2}+|a^{-}|^{2}-2|a^{0}|^{2};\\ p_{xy}=i\frac{3}{2}(a^{+}a^{-*}-a^{-}a^{++})\\ p_{xz}=\frac{3}{2\sqrt{2}}(a^{+}a^{0*}+a^{0}a^{+*}-a^{-}a^{0*}-a^{0}a^{-*})\\ p_{yz}=i\frac{3}{2\sqrt{2}}(a^{+}a^{0*}-a^{0}a^{++}+a^{-}a^{0*}-a^{0}a^{-*}). \end{array}  \tag{31}\]  
The components of polarization have the following bounds: \(-3\leq p_{xx}-p_{yy}\leq 3\), \(-2\leq p_{ii}\leq 1\), \(-\frac{3}{2}\leq p_{ij}\leq\frac{3}{2}\) (\(i\
eq j\)), and \(-1\leq p_{i}\leq 1\). These quantities are commonly referred to in atomic and nuclear physics as orientation \(p_{i}\) and alignment \(p_{ik}\). For a comprehensive formalism of spin 1 particle polarization, see also Ref. [24], noting that the definition of \(p_{nm}\) used here has an extra factor of 3. Another convention for the description of optical polarization in 3D fields uses an expansion in terms of Gell-Mann matrices [25] and is equivalent to the approach presented here. Let us derive expressions for the vector polarization \(p_{z}\) (or orientation) and quadrupole polarization \(p_{zz}\) (or alignment) of the vector vortex field generated by the phased ring array of atoms. The simplest expressions are obtained for \(N\rightarrow\infty\) with the electric field from the vector potential Equation (15):  
\[ \begin{split} E^{\mu}_{m_{z}}&=\frac{\partial}{ \partial t}A^{\mu}_{m_{z}}=-i\omega A^{\mu}_{m_{z}}=-i\omega Ae^{-i(\omega t-kz _{z})}\sqrt{\frac{\kappa}{2\pi}}\\ &(-\frac{i}{2}e^{i(l+m_{z}-1)\phi_{p}})_{l+m_{z}-1}(\kappa\rho)( 1+m_{z}\cos^{2}\theta_{k})\eta^{\mu}_{1}\\ &+\frac{i}{2}e^{i(l+m_{z}+1)\phi_{p}})_{l+m_{z}+1}(\kappa\rho)( 1-m_{z}\cos^{2}\theta_{k})\eta^{\mu}_{-1}\\ &\hskip 28.452756 pt+\frac{m_{z}}{2\sqrt{2}}e^{i(l+m_{z})\phi_{p}})_{ l+m_{z}}(\kappa\rho)\sin 2\theta_{k}\eta^{\mu}_{0}).\end{split}