Super-operator Linear Equations and their Applications to Quantum Antennas and Quantum Light Scattering

In this paper we developed the resolvent method for super-operator equations with their applications in quantum optics. Our approach is based on the novel concept of linear super-operator acting on the Hilbert subspace of vector or scalar linear operators satisfying physically reasonable commutation relations. The super-operator equations for the electromagnetic (EM) field operators are formulated for the problems of quantum antenna emission and quantum light scat-tering by a dielectric body. The general solution of super-operator equation is presented in terms of the classical resolvent. In contrast to the classical case, it includes the ancillary components associated with the quantum noise even in the absence of absorption. The reason for it lies in the energy exchange between different spatial regions with various bases for the field presentation (it looks like losses or gain from the point of view of the correspondent region). A number of examples (two-element dipole antenna, plane dielectric layer, and dielectric cylinder with circular cross section) which demonstrate the physical mechanism of the appearance of noise are considered. It is shown, that antenna emission or scattering transforms the coherent properties of quantum light. This opens a new way of controlling the coherence in a direction dependent manner, a feature that can be useful in various applications of quantum technologies, including, quantum radars and lidars, and quantum antennas.


Introduction
The scattering of quantum electromagnetic (EM)-field is related to the wide research area stimulated by the potential applications in quantum computing and quantum informatics [1][2][3][4], quantum communications [4,5], quantum circuits [6], quantum antennas [7][8][9][10][11][12][13], quantum radars and lidars [14][15][16], sensing and measurements [17].Currently this field is in the initial stages of its development.Probably, it will evolve along a path rather similar to classical electrodynamics and macroscopic antennas theory [18,19].On the other hand, from time to time it is expected to encounter fundamental problems, whose origin is completely quantum and who don't have any classical analog (as one of such examples, one may note the manifestation of photons entanglement [1][2][3][4][5]).One of the first stages in the development of the field consists of exact analytical solutions of some model problems.This phase has started; the quantization of some exact analytical solutions of Maxwell equations obtained via the separation of variables (spherically layered multimode systems [20] and parabolic mirrors [21]) have been given.
As a parallel way of development, one can mention the high-frequency physical diffraction theories (for example, quantum diffraction by apertures [22,23]) and their comparison with the physics of simple quantum-optical devices (beam-splitters, interferometers, etc. [24]).You can expect that this phase will stimulate the establishment of universal numerical techniques, based on the so-called "discretization" of Maxwell equations (such as the method of characteristic modes [25][26][27][28][29][30], integral equations [31], finite differences [32], finite elements [33], etc).In the process of their making, the methods of classical electrodynamics will be an important element of the framework.As promising step in this direction, one may note the quantization of characteristic modes [34,35], the preliminary version of integral equations [36], a numerical approach based on the synthesis of the time-domain finite element analysis with the canonical quantization technique [37].
Further progress in the design of quantum devices often necessitates accurate solution of the problems of quantum field scattering.From a naïve point of view, the canonical field quantization looks like a formal replacement of classical fields by the corresponding operators, which act in the Hilbert space of wavefunctions (quantum states) and satisfy appropriate commutation relations [38,39].However, the problem is not limited to the replacement of the classical values by the corresponding field operators.In classical electrodynamics, we note a widely used class of methods (including integral equations), in which the whole system is decomposed into a number of spatial regions.For example, for the scattering from a dielectric body, the integral equation is often formulated with respect to the field inside the body.The outside field is expressed through the inside one via an integral over the body volume.Another example is an antenna, defined as a device, which transforms the near field to the far field and vice versa.Such transformation is formally described by special integral relations based on the Green theorems [18].In such cases, the quantum field the situation changes dramatically.The quantum field alongside with the field operators (which satisfy Maxwell equations) is characterized by the quantum state (  -function).The introduction of partial regions requires the de- composition of the wavefunction of the whole space into those of the partial regions.The observable values are defined as expectation values of the corresponding operators with respect to the quantum state of the system [38,39].The field operators which are related to the given partial region act on the correspondent partial wavefunction.As a result, the scattering of quantum EM-field will be accompanied by the appearance of entanglement in the new basis.
The formal description in our paper will be given in terms of super-operators.The super-operator Α is defined as a mapping of one area of quantum operators onto another one (in contrast to an operator, which defines the correspondence between two areas of functions).The formation of the mathematical theory of superoperators [40] began simultaneously with the emergence of quantum mechanics (von Neumann equation and Lindblad master equation [41] are related to the typical class of superoperator equations for matrix of density).In this paper, we introduce another concept of linear super-operator, which allows applying the integral equations theory to the scattering of quantum light and quantum antennas.
It is important to make a note on the role of losses in our concept of super-operator.The losses in the conventional quantum language mean the exchange of energy between the considered system ("small" system) and large external system described as a bath [42+.The "small" system is considered from the point of view of an open system.If we speak about thermal losses, the energy exchange means the energy transformation to heat.In this case, the bath takes the form of thermal reservoir.Such exchange is accompanied by the production of noise, without which the correct commutation relations become unattainable.A similar situation takes place for an active medium: the gain is necessarily accompanied by the noise [43][44][45].
The modeling of quantum light scattering and quantum antenna emission in this general language leads to the description of every partial region as an open quantum system.It continues the concept formulated previously for the quantum antennas [7]: the emission of quantum antenna looks like a loss from its point of view (in spite of the absence of any absorption).Therefore, the external space (with the receiver and the environment is considered as a bath of non-thermal origin).It allows applying the general strongly developed theory of open systems [42] to quantum antennas.The subject of this paper is an application of the super-operator concept to the scattering of quantum light and quantum antennas.The paper is organized as follows: the general form of super-operator integral equation will be given in Section 2. The noise term will be considered and proved to have the form that guaranties the required form of commutation relations in general case.The application of the developed approach to the quantum antennas will be given in Section 3. The manifestations of the noise component in the transmission-reflection and scattering by the circular cylinder will be considered as examples in Sections 4 and 5, respectively.The conclusion and future perspectives are discussed in Section 6.

Preliminaries
The super-operator equation in our case may be written as where  is a spectral parameter (c-number), I is the unit super-operator, x and f are operators in the space of quantum states ( x is unknown, while f is given).
The formal solution of Equation ( 1) is , where is the resolvent, which in our case is a super-operator too.As mentioned above, the operators x and f must satisfy the required commutation relations.Therefore, we en- counter the main difference between the super-operator equations of the type (1) and their classical operator analogs.In contrast to the classical case, the resolvent    R acts not on the whole Hilbert space, but on the subspace limited by the commutation relations.As will be shown below, the action of the resolvent is able to remove the operator x out of the defined subspace.As a result, one may obtain a mathematically exact, but physically incorrect solution with inadequate commutation properties.In other words, equation ( 1) is related to ill-posed mathematical problems and should be regularized for its correct application.Its regularization will be given in this paper.It is based on the introduction of the ancillary components, which are associated with a special type of quantum noise.The configuration under consideration is shown in Figure 1.We decompose the operator of the longitudinal electric field into positive and negative frequency components [38,39] as and present it in the frequency domain as where k kc   , k is wavenumber, c is light velocity in free space.
where symbol  designates two-dimensional Laplacian and because just wavenumber will take a role of a variable of integration in different relations.
If the inside field is known, the outside field may be found via the integration over the body volume where S  X .For negative-frequency component, the equation may be found via Hermitian conjugation of Equation (6).Equations ( 5) and ( 6) look like the complete solution of the scattering problem: they satisfy Helmholtz equation over the whole space, boundary conditions and radiation conditions.
In contrast to the classical case, we need to add the commutation relation, which reflects the bosonic origin of EM-field.The commutation relation is local, thus it may be used in the ordinary form for homogeneous medium.As shown in Appendix B, it reads x x x (7) (In fact, it is obtained from relation for homogeneous dielectric space by replacing     x ).

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 don't satisfy the correct commutation rule (7).We will expand it next and make the suitable modifications, which lead to the physically correct solution.At first, let us explain the reason for such situation.To this end, we will calculate the commutator .
One can find the details of the calculations in Appendix B. As one can see, Equation (8) doesn't agree with the correct Equation (7), because of the appearance of the resolvent kernel, This is a manifestation of scattering, which means that the classical resolvent technique cannot be directly transferred to the quantum case.
We are going now to their special regularization, whose main idea is the addition of the ancillary noise components.As a result, Equations ( 5) and ( 6) are modified in the following way: where   ˆk F x is the noise-field.For the negative-frequency component inside the body the solution may be obtained via 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 noise field may be presented at the form of series over the eigen-states and similar for Hermitian conjugate operator.The pair ˆ, kn kn ff  is the crea- tion-annihilation operators for noise mode with index n at the given frequency kc   .
The commutation relations for them stem from Equation (10) and have the form

Observable values
We have considered the quantum light scattering from the point of view of open quantum systems (for example, applied to the modeling of lossy beam splitter in [44]).The concept of composite quantum systems is used in it as a main framework.The composite quantum system is considered as a set of two (or more) subsystems, whose statistical properties completely define the behavior of the total system.For example, the two subsystems may be two atoms in molecule, two modes in EM-field, and two elements in electrical circuit, etc.In our case, the fields in two geometrical regions (the area inside the dielectric body and the whole outside space, near-field and far-field zones of antennas, etc.) play the role of subsystems.As a starting point, the Hilbert space of a composite system defined as a tensor product


of Hilbert spaces of its subsystems (1) and (2) (superscripts 1 and 2 indicate the areas inside and outside the dielectric body, respectively) should be introduced.
The incident field is produced by the set of external sources distributed over the whole space.Such set may be considered as a composite quantum system and characterized by the density matrix, which is the operator in the state space .We will decompose the total source into two subsystems, in which the first one consists of sources outside the body and the second -of inside ones.We will assume, that the sources of different subsystems are uncorrelated.Therefore, the total density matrix takes the form of a tensor product of the density matrices of the partial regions: x for the field inside the body act on the total space .For calculation of observable values, the field operators are conveniently present in the form of the tensor products.Operator (9) will have a form

ˆˆˆˆˆˆ1
, ; , and similar for the negative-frequency field denote the identity superoperators in the subspaces (1,2) .
For an arbitrary operator Ô acting on the total space , the observable value is

 
ˆÔ Tr O  . Assuming that the fluctuations part is in vacuum state, we obtain for the observable field (with no support of fluctuations).For the intensity where x is the resolvent operator, and 0 ... denotes averaging over the vacuum state 00 .As one can see from Equation ( 15), the second term describes the vacuum fluctuations of the intensity of the field inside the body due to the exchang of its energy with the outside space.It shows, that the scattering of quantum light is necessarily accompanied by the fluctuations.It looks similar to the relation between fluctuations and dissipation ("fluctuation-dissipative theorem" *42]).Therefore, one can speak here about "fluctuation-scattering theorem".Its origin is completely quantum: the second term in Equation ( 15) vanishes in the classical limit.Another origin of the noise component is the thermal radiation of the body environment.In this case, we are speaking about thermal bath instead of photonic one.Note that the statement of the quantum light scattering problem without accounting for any type of reservoir becomes physically contradictive.

Application to the quantum two-element antenna array
The purpose of this section is to present an application of the super-operator concept to a quantum two-element array of dipole emitters shown at Figure 2.This type of antennas is widely used in classical radio engineering and its application in super-resolving quantum radars was proposed in [15].

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   x x y y , which are coupled via the resolvent matrix ( S -matrix) as Here x is the pair of outgoing mode creation operators at the first and second feeding transmission lines, ŷ is the pair of outgoing creation operators at the first and second dipole emitters, 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 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.It's positive-frequency component in the yz plane is: where cos kd     , with  being the polar angle in the spherical system with origin placed at the antenna geometrical center and  is the phase shift.We assume that the system is lossless and reciprocal; thus, matrix S is unitary [46,47].It leads to the coupling relations between its elements: The operators satisfy the ordinary bosonic commutation relations , and all other pairs of operators commute.
It is convenient for antenna applications to transform the equivalent 4-port network into a pair of coupled 2-port networks.This entails decomposition of the whole system into a pair of subsystems each of which is described in a different basis (actually, the decomposition of the whole antenna into a pair of dipole emitters and feeding transmission lines).To this end, Relation (16)  A describes the coupling between emitters with their corresponding feeding lines, as well as mutual transmission line coupling.Matrix B describes the coupling of every emitter with the free space, as well as mutual coupling of emitters due to their near fields.The vectors ˆˆˆ, , , ii x x y y are 2D columns of annihilation operators 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 ˆ0 i  y (it 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 number of photons (it means that ˆ0 i  y ).However, it is impossible to ignore them because they are responsible for thermal and vacuum EM-fluctuations, omitting of which contradicts the correct commutation relations.
To facilitate the application of the resolvent theory, we should transform the system ( 22), (23) in the following way.Relation (23) where î A  yf .The first term in the right-hand side describes the excitation of the di- pole emitters; while the second term corresponds to their coupling with the external space.From the 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 . The commutation relations for this operator stem from the commutation relations for ˆi y and using Equations ( 18), ( 19) may be written as The non-zero right-hand sides in Equations ( 25) manifest the non-unitarity of the matrix block B (in spite of unitarity of the total S -matrix in Relation ( 16)).The com- mutation Relations (25) completely coincide with the corresponding relations of lossy beam-splitter [44], where the right-hand sides are stipulated to produce the quantum noise due to dissipative processes.This is the reason to associate operator f with the quantum noise too, in spite of the absence of any dissipation.It is a result of the separate description of the 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.It leads to the modification of the Thevenin-like equivalent scheme [18]: in contrast to the classical case, this creates the pair of additional noise sources (see Figure 2b).

Transformation of the statistical properties of quantum light via the quantum antenna
Let us consider the calculation of observable characteristics of quantum light.The antenna emission state is characterized by the wavefunction, which reads in the tensor form as where : n  ψ denotes a state with n photons in mode number 1,2,   12 , nn C 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 An arbitrary operator in such picture is transformed by the relation ˆˆˆ  . In particular, the array factor will be transformed by the following relation Let us turn now to the analysis of antenna coupled with the feeding channels, as shown at Figure 2(b).Here, we introduce the Hilbert space of a composite system defined as a tensor product The operator ŷ defined by Relation (24) acts in the space () y .It may be represented as an operator in using the form of tensor product.We have where , ˆxy I are identity operators in spaces ( , ) xy .As an example, let us calculate the observable ˆ yy , which may be associated with an analog of intensity for the scat- tering by a dielectric body considered in Section 3 (the symbol ... means the aver- aging over the state xy      ).Taking into account that for the noise in the vacuum state ˆ0  f and using Relations ( 30) and ( 31), we obtain.ˆ0 0 .
The second term in Relation (32) 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 transformation of the wavefunction (35) Let us consider the special example where, we assume that antenna is perfectly matched to the feed lines ( 0 r  ) and the feed lines are uncoupled ( 1 0 t  ).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 for the wavefunction As one can see from Equation (37), in this special case two photons will be emitted with probability 0.5 from one of the emitters, while the probability to emit one photon from one of the emitters is equal to zero.It shows the fundamental property of quantum antennas: the process of antenna emission transforms the statistical properties of light (in particular, 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 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 quan- tum 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  .It opens the way of electrically angular scanning by means of quantum antennas.Such ability may be important for applications in quantum radars and lidars [15].

Planar dielectric layer: reflection-transmission
The problem geometry is shown at the Figure 3a.The system is excited by the external current   fx distributed inside the dielectric layer along the x-axis and vanish- ing outside the layer.Therefore, the field is dependent only on one variable, x.The equation for the total field becomes where the permittivity   x   inside the layer and   1 x   outside it.The boun- dary conditions for the unknown field stem from the boundary conditions for the tangential component of the electric field at the boundaries of the dielectric layer [19,46].The problem should be supplemented by the radiation condition at x < 0 and x>l.For the reflected-transmitted field inside the layer, we have equation
We can represent the unknown field as where A,B,C, and D are unknown constant coefficients.Using boundary conditions, we obtain the system of four linear equations where 43) is easily solvable.We need only two of unknowns, for which we have where is Fresnel coefficient for the normal incidence and   where the function may be associated with the resolvent kernel of integral operator and used for the analysis of commutation relation for the case of dielectric layer.In agreement with general Equation (A5), we have     , ; , ; . Therefore, the commutation relation may be written as ,  , ; .
The support of the first term in Equation ( 46) gives the correct commutation relation.For the analysis of all other terms let us consider the analytical continuation to the complex k-plane.All these terms are analytical functions with the exception of the poles defined by the equation 221 ik l re   .All these poles are located in the lower half-plane.
The functions are exponentially decreasing in the upper half-plane.Therefore, this component of the integral (47) may be closed in the upper half-plane gives zero contribution to the commutator.Thus, the noise components of the fields disappear in agreement with [44].

Scattering by the dielectric cylinder
The purpose of this subsection is to find the resolvent for the dielectric cylinder of circular cross section.To this end, we will solve the Helmholtz equation: 48) is associated with the external force arbitrarily distributed inside the cylinder.The unknown function satisfies the boundary conditions for the longitudinal component of electric field at the boundary of the cylinder [19,47] and radiation condition outside.
The scattered field may be represented similarly to Equation ( 42) by and where , nn AB are unknown coefficients, which should be founded from boundary con- ditions.We will omit here their calculations (see Appendix D).The final result for the field inside the cylinder is where Relation ( 51) expresses the scattered field through the incident field inside the cylinder in the canonical integral form.Therefore, Relation (52) may be associated with the resolvent kernel of integral operator and used for the analysis of commutation Relation (18) and the existence of noise.Let us transform Relation (18) for the dielectric cylinder to the form, similar Equation (47) for the 1D dielectric layer (integration over the whole axis instead of positive semiaxis with the subsequent integration in the complex plane).It is necessary to note, that here situation dramatically changes compared with the 1D case.The difficulty lies in the non-analyticity of the function   n Wk given by Relation (54) in the complex k -plane (this function is multi-valued with a branch point at 0 k  in contrast to the di- electric layer case).One can see from well-known identities


, that it is natural to place the cut in the complex k -plane along the negative semi-axis (jagged line in Figure 4).It easy to show, that k  in the upper half-plane.Thus, for the quantity (53), we obtain the relation between two branches of complex variable The integration in Relation (47) for the cylinder may be transformed into an integral over contour with its contour passing below branch-cut, as shown in Figure 4.
The contribution of the first term in Relation (52) to Relation (56) gives the conventional value of commutator without any noise components.However, in contrast to the 1D single-layer case (Section 4), the second term's contribution doesn't vanish.The reason being the impossibility to close the contour in the upper half-plane, where the function tends to zero (to do it one needs to cross the branch cut).This term is responsible for the noise current given by Equation (10).

Conclusion
In summary, we developed a novel method for the modeling of quantum light scattering and quantum antennas.It is formulated in terms of linear super-operator defined as an operator in the Hilbert space, which acts in the subspace of quantum state operators, satisfying the physically appropriate commutation relations.It is shown, that in the general case the naïve quantization via the formal exchange of the classical values by the corresponding operators leads to an ill-posed super-operator problem.The reason being, that the action of super-operator Α may throw the result outside of the required subspace, which violates the correct commutation relations.In order to satisfy them, one needs to do the regularization of super-operator equation.
The proposed regularization technique is based on the introduction of ancillary channels, which are associated with the novel type of quantum noise, whose origin is not related to any dissipation of energy.The regularized super-operator equation has the , where Q is a noise operator.Its form and commutation relations are determined in this paper.In general case, the super-operator equations may be solved numerically.Exact analytical solutions exist for special types of configurations.Sometime is promising the approximate analytical solution (for example, perturbation technique).The solution of the regularized super-operator equation may be expressed in terms of classical resolvent, which may be found by one of the methods of classical computational electromagnetics [48].
Simple applications of the theory are considered using several examples (two-element dipole quantum antenna, reflection-transmission in a 1D dielectric layer, and scattering by a dielectric cylinder).It is shown, that a special noise is produced via exchange of the energy between the body and the external space.From the "point of view" of the dielectric body, the process of multimode scattering looks like losses (or equivalent gain, if the energy enters from outside) in spite of the absence of any heating or real physical amplification.As a result, the exchange of energy is accompanied by the noise similar to that in systems with real physical dissipation.By the way, it becomes possible to speak about general "fluctuation-scattering theorem" similar to the well-known "fluctuation-dissipating theorem": the scattering of the quantum field is accompanied by fluctuations.In partial cases, a similar mechanism manifested itself in parallel-plane beam splitter [49] and time lenses [50].In [49], the initial photon state being a product of number states in both modes is considered.It was shown that the lossless due to beam-splitter coupling create a noise in the photon number of each mode in spite of the total photon number being invariant and noise-free.
The theory developed in this paper may be generalized for many other scattering problems (different types of quantum antennas, scattering from 3D dielectric bodies, or dielectric bodies with different types of environment, etc.).It opens the way to using the scattering analysis as a tool of controlling the statistical properties of quantum light.It may be applied to various problems posed by the quantum technologies, metamaterials [51], radars and lidars [14,16], quantum circuits, and nanoantennas [52][53][54][55].

Acknowledgments:
The authors acknowledge partial support by the H2020, project TERASSE 823878 and by the NATO Science for Peace and Security Programme under Grant id.G5860.

Appendix A. Resolvent theory for classical operators
The Fredholm integral equation of the second type in general form may be written as [56]  For the scattering problems the kernel of integral equation is the Green function of Helmholtz equation.For the two-dimensional (2D)-case   where Hx is the Hankel function of the zero order and the first type, As one can see, the integral operator kernel is symmetric: Regarding the validity of the resolvent approach for this class of integral equations, in principle, the eigen-functions for non-Hermitian operators may not generate the complete basis system, and may not exist at all.However, the operator   ( ) , ( ) x has some fundamental properties due to the spe- cial form of its kernel.First, the kernel has a weak singularity (integrable in the ordinary sense).Second, we can split the kernel   , g  xx into the real and imaginary parts and write , where subscripts R, I mean real and imaginary compo- nents, respectively.The singular component of operator ()  A is its real (Hermitian) part; while its imaginary part is regular.Thus, unbounded inverse   Combining these two facts means that operator ()  A is a weak perturbation of Hermi- tian operator () R  A [57].For weak perturbations of Hermitian operators, one can use the fundamental mathematical theory of pseudodifferential operators in Sobolev spaces [57] (in particular, the theorems on basis properties).For real k and     .
One more identity will be useful in our future consideration: .
It may be proven similarly to the ortogonality Relation (A3), however instead of Relation (A2) with index m its complex conjugated equation should be used.

Appendix B: Quantization of free electromagnetic field in the cylindrical basis
The purpose of this Appendix is the derivation of commutation Relation (7).We will consider the homogeneous space with permittivity  using the cylindrical basis.Let us present the positive-frequency component of the operator of vector potential in the form A similar relation may be written for the negative-frequency component of the electric field: x is the pair of Hermitially conjugate operators of spectral den- sity.The problem is to find the commutation relation for the operators of spectral density, which gives the canonical bosonic commutation relation for the EM-field.
Let us present the operator of spectral density as a superposition of cylindrical waves: Using Equation (A12), we transform it to Using the addition theorem for cylindrical functions [59]  x x , we obtain commuta- tion Relation (A16) in the final form One can see, that commutation Relation (A17) really gives the conventional commutation relation for the EM-field operators, which is  Appendix C: Derivation of Equation ( 8) Let us substitute the field representation (5) together with its Hermittian conjugate into the commutator of the operators.We have The second term in the right-hand-side of Equation (A21) may be transformed in the following way: ).For integration of the right hand sides in Equations (A29) and (A30), we use the addition theorem for cylindrical functions [60]                     Equations (A32), (A33) form a system of two linear algebraic equations with respect to the coefficients , nn AB.Finding  , nn AB, substituting them to Equations ( 49), (50)   using the Wronskian of cylindrical functions, we obtain Equations ( 52), (53).

Figure 1 .
Figure 1.Configuration of the problem; two-dimensional dielectric cylinder with arbitrary cross-section S is excited by the quantum light with spectral density and its Hermitian conjugate.The final result is

Figure 2 .
Figure 2. The scheme of 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 fed by the incident field via the corresponding transmission line (operators

.
. 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 The wavefunction of the field in the far zone may be in- of Hilbert spaces of its subsystems () x and() 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 xy      .The states , ψ denotes the state with n photons in the mode number 1

Figure 3 .U
Figure 3. Examples of the dielectric structures for resolvent analysis: (a) dielectric layer; (b) dielectric cylinder of circular cross section; the sources (colored grey) placed inside the dielectric body.The total field may be represented as a sum of incident and reflected-transmitted fields:

.
 outside.Our goal is to calculate the resolvent of the integral equation It is an analog of the general integral Equation (A1) for the 1D case (where   44) together with Relations (42), we obtain the solution inside the layer in the canonical form

Figure 4 .
Figure 4.The contour C (colored brown) is the integration contour in Equation (56).The cut is placed along the negative semi-axis and shown by the jagged line.

E
x is an unknown function,  is a given parameter, while ) are a priori given functions.The ei- gen-functions   n u x of integral Fredholm operator are defined by the homogeneous version of Equation (A1):

 x x x x x x (   2 S.
Lis a space of Lebesgue squared integrable functions).In such case the operator ()  A is a bounded operator acting Such operator is completely continuous(compact) of the eigenmodes (A3) and Hilbert-Schmidt Relation (A8), we can rewrite Equation (A22) as term in the right-hand-side of Equation (A21), we obtain by a similar argument term in Equation (A21), using Equations (A22)-(A24), we obtain the representation of the Green function in the eigenmode basis (Equation (A4)), as well as their orthogonality (Equation (A3)), we transform Equation (A26) to in Equation (A21) all obtained relations and using Hil- bert-Schmidt Relation (A8), we obtain Equation (8) after some elementary transformations.Appendix D: Derivation of Equations (52),(53).The boundary conditions for the scattering field on the cylinder yield may be rewritten as