## 3 Application to the Quantum Two-Element Antenna Array
### Quantum Noise and Commutation Relations  
The incident field (incoming modes) and emitted field (outgoing modes) may be described by the 4D columns of creation-annihilation operators \(\{\hat{x}_{1},\hat{x}_{2},\hat{y}_{1},\hat{y}_{2}\}\) and \(\{\hat{x}_{1i},\hat{x}_{2i},\hat{y}_{1i},\hat{y}_{2i}\}\), which are coupled via the resolvent matrix (\(\overline{\hat{S}}\)-matrix) as  
\[ \begin{pmatrix}\hat{x}_{1}\\ \hat{x}_{2}\\ \hat{y}_{1}\\ \hat{y}_{2}\end{pmatrix} =\underbrace{ \begin{pmatrix}r&t_{1}&t_{2}&t_{3}\\ t_{1}&r&t_{3}&t_{2}\\ t_{2}&t_{3}&r&t_{1}\\ t_{3}&t_{2}&t_{1}&r\end{pmatrix} }_{\text{Resolvent matrix}}\cdot \begin{pmatrix} \hat{x}_{1i}\\ \hat{x}_{2i}\\ \hat{y}_{1i}\\ \hat{y}_{2i}\end{pmatrix} _{\text{Resolvent matrix}}. \tag{16}\]  
Here, \(\hat{x}_{1,2}\) is the pair of outgoing mode creation operators at the first and second feeding transmission lines, \(\hat{y}_{1,2}\) is the pair of outgoing creation operators at the first and  
Figure 2: The scheme of a two-element quantum antenna. (**a**) Configuration of antenna: Every emitting element of antenna is an electrical dipole. The dipoles are identical and identically oriented. (**b**) Equivalent electrical circuit of the antenna: The dipoles are fed by the incident field via the corresponding transmission line (operators \(\hat{x}_{1,2}\)). The feeding is symmetrical (tap is placed at the central point of the dipole). Antenna elements are coupled via reactive impedance \(X_{Coup}\) (no thermal losses at the antenna). A part of the incident power is reflected back; another part is distributed between the dipoles and emitted (operators \(\hat{y}_{1,2}\)). The feeding transmission lines are considered as an external environment. The element coupling produces the special type of noise (operators \(\hat{f}_{1,2}\)). second dipole emitters, and the additional subscript \(i\) indicates the incoming mode at the corresponding channel. We assume that all channels are identical; therefore, the reflection coefficient \(r\) is the same in all channels. A similar relation may be written for the annihilation operators via the Hermitian conjugation of Equation (16). The emitting properties of the antenna are characterized by the array factor [18]. For the quantum case, it becomes the photonic operator. Its positive frequency component in the \(yz\) plane is:  
\[\mathrm{AF}^{+}(\theta)=\hat{y}_{1}e^{i\phi}+\hat{y}_{2}e^{-i\phi}, \tag{17}\]  
where \(\phi=kd\cos\theta+\beta\), with \(\theta\) as the polar angle in the spherical system with origin placed at the antenna geometrical center, and \(\beta\) as the phase shift. We assume that the system is lossless and reciprocal; thus, matrix \(\overline{S}\) is unitary [46; 47]. It leads to the coupling relations between its elements:  
\[|r|^{2}+|t_{1}|^{2}+|t_{2}|^{2}+|t_{3}|^{2}=1, \tag{18}\]  
\[\mathrm{Re}(r_{1}^{*}+t_{2}t_{3}^{*})=0, \tag{19}\]  
\[\mathrm{Re}(r_{2}^{*}+t_{1}t_{3}^{*})=0, \tag{20}\]  
\[\mathrm{Re}(r_{3}^{*}+t_{1}t_{2}^{*})=0. \tag{21}\]  
The operators satisfy the ordinary bosonic commutation relations \(\left[\hat{x}_{1},\hat{x}_{1}^{+}\right]=\left[\hat{x}_{2},\hat{x}_{2}^{+} \right]=1\), \(\left[\hat{x}_{1i},\hat{x}_{1}^{+}\right]=\left[\hat{x}_{2i},\hat{x}_{2i}^{+} \right]=1\), \(\left[\hat{y}_{1},\hat{y}_{1}^{+}\right]=\left[\hat{y}_{2},\hat{y}_{2}^{+} \right]=1\), \(\left[\hat{y}_{1i},\hat{y}_{1i}^{+}\right]=\left[\hat{y}_{2i},\hat{y}_{2i}^{+} \right]=1\), and all other pairs of operators commute. It is convenient for antenna applications to transform the equivalent four-port network into a pair of coupled two-port networks. This entails the decomposition of the whole system into a pair of subsystems, each of which is described on a different basis (actually, the whole antenna decomposes into a pair of dipole emitters and feeding transmission lines).