## 3 Application to the Quantum Two-Element Antenna Array
### Transformation of the Statistical Properties of Quantum Light via the Quantum Antenna  
Let us consider the calculation of the observable characteristics of quantum light. The antenna emission state is characterized by the wavefunction, which reads in the tensor form as  
(26)  
where denotes a state with \(n\) photons in mode number \(\alpha=1,2\), and \(C_{n_{1},n_{2}}\) are arbitrary coefficients which satisfy the normalization condition. The directional dependence of the field properties is defined by the array factor given by Relation (17). For our future purposes, it is convenient to move the directional dependence to the wavefunction and use the spatially independent array factor related to the antenna feed point. The situation is similar to the relation between the Schrodinger and Heisenberg pictures of quantum mechanics (with angle instead of time). Let us introduce the unitary operator  
(27)  
where \(\phi=(kd\cos\theta+\beta)/2\). The wavefunction of the field in the far zone may be introduced as  
(28)  
An arbitrary operator in such picture is transformed by the relation \(\hat{\mathbb{P}}\rightarrow\hat{\mathbb{Z}}^{+}\hat{\mathbb{P}}\hat{\mathbb{ Z}}\). In particular, the array factor will be transformed by the following relation  
(29)  
Let us turn now to the analysis of the antenna coupled with the feeding channels, as shown in Figure 2b. Here, we introduce the Hilbert space of a composite system defined as a tensor product \(\mathbb{S}=\mathbb{S}^{(x)}\otimes\mathbb{S}^{(y)}\) of Hilbert spaces of subsystems \(\mathbb{S}^{(x)}\) and \(\mathbb{S}^{(y)}\)(superscripts \((x)\) and \((y)\) are related to the feeding lines and emitting dipoles, respectively). We will introduce the pure quantum state of light in the form of the tensor product \(\ket{\Psi}=\ket{\Psi_{x}}\otimes\ket{\Psi_{y}}\). The states \(\ket{\Psi_{x}}\), \(\ket{\Psi_{y}}\) are given by  
\[ \begin{array}{c}\ket{\Psi_{x}}=\sum\limits_{n_{1}\,n_{2}}C_{n_{1},n_{2}}|n_{1}: \psi_{1}\rangle\otimes\ket{n_{2}:\psi_{2}},\\ \ket{\Psi_{y}}=\ket{0:\psi_{3}}\otimes\ket{0:\psi_{4}},\end{array}  \tag{30}\]  
where \(|n_{x}:\psi_{\alpha}\rangle\) denotes the state with \(n\) photons in the mode number \(\alpha=1,2,3,4\), similar to Equation (26), and \(C_{n_{1},n_{2}}\) are arbitrary coefficients which satisfy the normalization condition. The operator \(\hat{\mathbf{y}}\) defined by Relation (24) acts in the space \(\mathbb{S}^{(y)}\). It may be represented as an operator in \(\mathbb{S}\) using a form of the tensor product. We have  
\[\hat{\mathbf{y}}=\hat{I}_{x}\otimes\overline{B}\hat{\mathbf{x}}_{i}+\hat{ \mathbf{t}}\otimes\hat{I}_{y}, \tag{31}\]  
where \(\hat{I}_{x,y}\) are identity operators in space \(\mathbb{S}^{(x,y)}\). As an example, let us calculate the observable \(\left\langle\hat{\mathbf{y}}^{+}\hat{\mathbf{y}}\right\rangle_{\mathbb{S}}\), which may be associated with an analog of intensity for the scattering by a dielectric body considered in Section 3 (the symbol \(\langle\ldots\rangle_{\mathbb{S}}\) means averaging over the state \(\ket{\Psi}=\ket{\Psi_{x}}\otimes\ket{\Psi_{y}}\)). Taking into account that for the noise in the vacuum state \(\left\langle\hat{\mathbf{t}}\right\rangle=0\) and using Relations (30) and (31), we obtain  
\[\left\langle\hat{\mathbf{y}}^{+}\hat{\mathbf{y}}\right\rangle_{\mathbb{S}}= \left\langle\Psi_{y}\right|\overline{B}^{*}\hat{\mathbf{x}}_{i}^{+}\overline{B }\hat{\mathbf{x}}_{i}\ket{\Psi_{y}}+\langle 0|\hat{\mathbf{f}}^{+}\hat{\mathbf{f}}|0\rangle. \tag{32}\]  
The second term in Relation (32) corresponds to the support of fluctuations. It may be transformed to \(\langle 0|\hat{\mathbf{f}}^{*}\hat{\mathbf{f}}|0\rangle=|t_{2}|^{2}+|t_{3}|^{2} =1-|r|^{2}-|t_{1}|^{2}\). Let us consider an example of the antenna excitation by a single photon in each feeding line.