## 3 Application to the Quantum Two-Element Antenna Array
### Quantum Noise and Commutation Relations
To this end, Relation (16) may be rewritten as  
\[\hat{\mathbf{x}}=\overline{A}\hat{\mathbf{x}}_{i}+\overline{B}\hat{\mathbf{y} }_{i}, \tag{22}\]  
\[\hat{\mathbf{y}}=\overline{B}\hat{\mathbf{x}}_{i}+\overline{A}\hat{\mathbf{y} }_{i}, \tag{23}\]  
where \(\overline{A}= \begin{pmatrix}r&t_{1}\\ t_{1}&r\end{pmatrix} \) and \(\overline{B}= \begin{pmatrix}t_{2}&t_{3}\\ t_{3}&t_{2}\end{pmatrix} \) are 2D blocks of the \(\overline{S}\)-matrix in Relation (16). Matrix \(\overline{A}\) describes the coupling between emitters with their corresponding feeding lines, as well as mutual transmission line coupling. Matrix \(\overline{B}\) describes the coupling of every emitter with the free space, as well as the mutual coupling of emitters due to their near fields. The vectors \(\hat{\mathbf{x}},\hat{\mathbf{x}}_{i},\hat{\mathbf{y}},\hat{\mathbf{y}}_{i}\) are 2D columns of annihilation operators \(\hat{\mathbf{x}}= \begin{pmatrix}\hat{x}_{1}\\ \hat{x}_{2}\end{pmatrix} \), \(\hat{\mathbf{x}}_{i}= \begin{pmatrix}\hat{x}_{1i}\\ \hat{x}_{2i}\end{pmatrix} \), and similar for \(\hat{\mathbf{y}},\hat{\mathbf{y}}_{i}\). For the classical field, we obtain Equations (22) and (23) for the corresponding classical values instead of operators. In our case, the incoming field appears only at the first and second channels. In this case, for classical field we should take \(\hat{\mathbf{y}}_{i}=0\) (this means that the second subsystem is excited only via the coupling with the first one). However, such assumption becomes unphysical in the quantum case. The reason being the fundamental absence of empty channels in the quantum field. The non-excited channels in the quantum case should be considered as channels with zero photons (which means that \(\langle\hat{\mathbf{y}}_{i}\rangle=0\)). However, it is impossible to ignore them because they are responsible for thermal and vacuum EM fluctuations, the omission of which contradicts the correct commutation relations. To facilitate the application of the resolvent theory, we should transform the systems (22) and (23) in the following way. Relation (23) is equivalent to  
\[\hat{\mathbf{y}}=\overline{B}\hat{\mathbf{x}}_{i}+\hat{\mathbf{f}} \tag{24}\]  
where \(\overline{A}\hat{\mathbf{y}}_{i}=\hat{\mathbf{f}}\). The first term on the right-hand side describes the excitation of the dipole emitters, while the second term corresponds to their coupling with the external space. Fromthe point of view of the emitters it looks like energy is being lost, while it is not related to dissipation. Such \"effective losses\" may be positive and negative (the \"effective gain\" for receiving antenna). The operator commutes with \(\hat{\mathbf{x}},\hat{\mathbf{x}}_{i}\), and. The commutation relations for this operator stem from the commutation relations for \(\hat{\mathbf{y}}_{i}\), and using Equations (18) and (19), may be written as  
(25)  
The non-zero right-hand side of Equation (25) manifests the non-unitarity of the matrix block \(\overline{B}\) (in spite of the unitarity of the total \(\overline{S}\)-matrix in Relation (16)). The commutation Relation (25) completely coincides with the corresponding relations of the lossy beam-splitter [44], where the right-hand sides are stipulated to produce quantum noise due to the dissipative processes. This is the reason to associate operator \(\hat{\mathbf{f}}\) with quantum noise too, in spite of the absence of any dissipation. This is a result of the separate description of antenna emission and antenna feeding taking into account their interaction only at the final step. It is clear that the noise operator acts on the wavefunction of the second subsystem. This leads to the modification of the Thevenin-like equivalent scheme [18]: in contrast to the classical case, this creates a pair of additional noise sources (see Figure 2b).