## 2 Super-Operator Integral Equation for Scattering of Quantum Light by a Dielectric Body
### Commutation Relation and Noise Component  
As one can see, it is impossible to consider Equations (5) and (6) as a solution for the field operators because they do not satisfy the correct commutation rule (7). We will expand it next and make the suitable modifications, which will lead to the physically correct solution. First, let us explain the reason for such a situation. To this end, we will calculate  
Figure 1: Configuration of the problem: two-dimensional dielectric cylinder with arbitrary cross section \(S\) is excited by the quantum light with spectral density \(U_{k0}(\mathbf{X});\mathbf{x}\in S,\mathbf{X}\
ot\in S\).  
the commutator \(\left[\hat{U}_{k}(\mathbf{x}),\hat{U}_{k^{\prime}}^{+}(\mathbf{x}^{\prime})\right]\) for operators \(\hat{U}_{k}(\mathbf{x})\) and \(\hat{U}_{k^{\prime}}^{+}(\mathbf{x}^{\prime})\) defined by Relation (5) and its Hermitian conjugate. The final result is  
\[\left[\hat{U}_{k}(\mathbf{x}),\hat{U}_{k^{\prime}}^{+}(\mathbf{x}^{\prime}) \right]=\frac{\hbar k^{2}c}{\pi\epsilon}\text{Im}\left(\Gamma\left(\mathbf{x}, \mathbf{x}^{\prime};k\right)\right)\delta(k-k^{\prime}). \tag{8}\]  
One can find the details of the calculations in Appendix C. As one can see, Equation (8) does not agree with the correct Equation (7) because of the appearance of the resolvent kernel \(\Gamma\big{(}\mathbf{x},\mathbf{x}^{\prime};k\big{)}\) instead of the Green function \(g\big{(}\mathbf{x},\mathbf{x}^{\prime}\big{)}\). This is a manifestation of scattering, which means that the classical resolvent technique cannot be directly transferred to the quantum case.  
We will now consider their special regularization, the main idea of which is the addition of the ancillary noise components. As a result, Equations (5) and (6) are modified in the following way:  
\[\hat{U}_{k}(\mathbf{x})=\hat{U}_{k0}(\mathbf{x})-k^{2}(\epsilon-1)\int\limits _{\mathbb{S}}\Gamma\big{(}\mathbf{x},\mathbf{x}^{\prime};k\big{)}\hat{U}_{k0} \big{(}\hat{\mathbf{x}}^{\prime}\big{)}d\mathbf{x}^{\prime}+\hat{F}_{k}( \mathbf{x}), \tag{9}\]  
where \(\hat{F}_{k}(\mathbf{x})\) is the noise field. For the negative frequency component inside the body, the solution may be obtained via the Hermitian conjugation of Equation (9). The concept of noise looks similar to the lossy beam-splitter [44]; however, in our case, their physical origin is different (the thermal losses are absent). We have for the noise field (the symbol \(\hat{F}_{k}^{+}(\mathbf{x})\) here and in future denotes the Hermitian conjugation of the noise field). The noise field operators commute with field operators \(\hat{U}_{k0}(\mathbf{x})\), \(\hat{U}_{k0}^{+}(\mathbf{x})\) and satisfy the commutation relation  
\[\left[\hat{F}_{k}(\mathbf{x}),\hat{F}_{k^{\prime}}^{+}(\mathbf{x}^{\prime}) \right]=-\frac{\hbar ck^{2}}{\pi\epsilon}\delta\big{(}k-k^{\prime})\text{Im} \left(\Gamma\big{(}\mathbf{x},\mathbf{x}^{\prime};k\big{)}-g\big{(}\mathbf{x},\mathbf{x}^{\prime}\big{)}\right). \tag{10}\]  
The noise field may be presented in the form of a series over the eigenstates \(u_{n}(\mathbf{x})\) defined by Equation (A2):  
\[\hat{F}_{k}(\mathbf{x})=\sum_{n}\hat{f}_{kn}u_{n}(\mathbf{x}) \tag{11}\]  
and similar for the Hermitian conjugate operator. The pair \(\hat{f}_{kn}\), \(\hat{f}_{kn}^{+}\) is the creation-annihilation operator for noise mode with index \(n\) at the given frequency \(\omega=kc\). The commutation relations for them stem from Equation (10) and have the form  
\[\left[\hat{f}_{kn},\hat{f}_{k^{\prime}n^{\prime}}^{+}\right]=-\frac{\hbar ck^{ 2}\
u}{\pi\epsilon}\delta_{nn^{\prime}}\delta\big{(}k-k^{\prime}\big{)}\text{ Im}\left(\frac{1}{(\
u_{n}-\
u)\
u_{n}}\right). \tag{12}\]