## 3 Application to the Quantum Two-Element Antenna Array
### Transformation of the Statistical Properties of Quantum Light via the Quantum Antenna
The wave function of the incident field is \(\ket{\Psi}=\ket{\Psi_{x}}\otimes\ket{\Psi_{y}}=|11\rangle\otimes|00\rangle.\) Assuming \(t_{1}t_{2}+rt_{3}=0\) and \(t_{1}t_{3}+rt_{2}=0\), we obtain that the antenna interaction with the feed lines transforms the wavefunction to  
\[\ket{\Psi}=\ket{\Psi_{x}}\otimes\ket{00}+\ket{00}\otimes\ket{\Psi_{y}}, \tag{33}\]  
where  
\[\ket{\Psi_{x}}=\sqrt{2}(rt_{1})^{*}(\ket{20}+\ket{02})+\Big{(}r^{2}+t_{1}^{2} \Big{)}^{*}|11\rangle, \tag{34}\]  
\[\ket{\Psi_{y}}=\sqrt{2}(t_{2}t_{3})^{*}(\ket{20}+\ket{02})+\Big{(}t_{2}^{2}+t_{ 3}^{2}\Big{)}^{*}|11\rangle. \tag{35}\]  
As one can see from Equations (33)-(35), the feeding of the quantum antenna leads to an entanglement of quantum states of the external sources and the emitting dipoles. Following Equation (28) for the transformation of the wavefunction (35) to the radiative form, we obtain  
\[\ket{\Psi_{y}}_{rad}=\sqrt{2}(t_{2}t_{3})^{*}\Big{(}\ket{20}e^{2i\phi}+\ket{0 2}e^{-2i\phi}\Big{)}+\Big{(}t_{2}^{2}+t_{3}^{2}\Big{)}^{*}|11\rangle. \tag{36}\]  
Let us consider a special example where we assume that the antenna is perfectly matched to the feed lines (\(r=0\)) and the feed lines are uncoupled (\(t_{1}=0\)). In this case, the entanglement between the antenna emission and antenna feeding becomes suppressed, in spite of the entanglement between the two emitters. As a result, we have the wavefunction  
\[\ket{\Psi}_{rad}=\frac{i}{\sqrt{2}}\ket{00}\otimes\Big{(}\ket{20}e^{i\phi}+ \ket{02}e^{-i\phi}\Big{)}_{rad}. \tag{37}\]  
As one can see from Equation (37), in this special case, two photons will be emitted with a probability of 0.5 from one of the emitters, while the probability to emit one photon from one of the emitters is equal to zero. This shows the fundamental property of quantum antennas: the process of antenna emission transforms the statistical properties of light (in particular, it creates the inter-mode entanglement in the far field zone). The angular distribution of the emitted field is characterized by the first- and second-order correlation functions at the same point in the far zone. The corresponding expressions for the quantum state (37) read  
\[G^{(1)}(\theta)=\sin^{2}\theta(_{rad}\langle\Psi\left|\mathrm{AF}^{+}\cdot \mathrm{AF}\right|\Psi\rangle_{rad})=2\sin^{2}\theta, \tag{38}\]  
\[G^{(2)}(\theta)=\sin^{4}\theta(_{rad}\langle\Psi\left|\mathrm{AF}^{+}\cdot \mathrm{AF}^{+}\cdot\mathrm{AF}\cdot\mathrm{AF}\right|\Psi\rangle_{rad})=4\sin^ {4}\theta\cos^{2}\phi= \tag{39}\]  
The first-order coherence is equal to the intensity of the emitted field. Its directive properties are equal to those of a Hertzian dipole. This results from the summation of partial powers emitted by each emitter without any interference mechanism for the quantum state (37). In contrast, the second-order coherence exhibits the narrow lobe-like directive properties, as follows from Equation (39). The angular dependence of the second-order correlation exhibits a set of identical lobes separated by zeros. Such behavior was predicted for the first time for the \(N\) element antenna array in another quantum state in [10]. The number of lobes increases and their width decreases with increasing \(kd\). The direction of the lobes is controllable by varying the phase shift \(\beta\). It facilitates a method of electrically angular scanning by means of quantum antennas. Such ability may be important for applications in quantum radars and lidars [15].