Electromagnetic field quantization and quantum optical input-output relation for grating

A quantization scheme is developed for the radiation and higher order electromagnetic fields in one dimensional periodic, dispersive and absorbing dielectric medium. For this structure, the Green function is solved based on the plane wave expansion method, thus the photon operators, commutation relations and quantum Langevin equations are given and studied based on the Green function approach, moreover, the input-output relations are also derived. It is proved that this quantum theory can be reduced back to that of the predecessors’ study on the homogenous dielectric. Based on this method, we find that the transformation of the photon state through the lossy grating is non-unitary and that the notable non-unitary transformation can be obtained by tuning the imaginary part of the permittivity, we also discussed the excellent quantum optical properties for the grating which are similar to the classical optical phenomena. We believe our work is very beneficial for the control and regulation of the quantum light based on gratings.


Theory and Method
Solution of quantum Maxwell equations for 1D grating. We consider 1D periodic structure (1D photonic crystal) as shown in Fig. 1. In order to quantize the eletromagnetic field in this structure we will resolve the quantum Maxwell equations [22][23][24][25] using PWE method. Here the relative permittivity is periodic along the x direction and uniform along the y and z direction, we consider the transverse electric (TE) modes in which the electric field is polarized in the uniform y direction, all possible nonzero EM fields are denoted by ˆˆĤ E H ( , , ) jy represent expansion coefficients of the relative permittivity, noise current density and corresponding bosonic vector field. It is implicit in equations (1) that the Bloch theorem is satisfied for the one dimensional periodic medium.
The classical EM fields in the one dimensional periodic structures have already been studied in many previous works [29][30][31] . We borrow the ideas from these works to solve the quantum Maxwell equations, substitute the operator EM fields in quantum Maxwell equations for the plane waves expansion, thereby we find that the electric field ω E z ( , ) my obeys the partial differential equation In order to solve the electric field in this equation, we rewrite it as the following matrix form , , The corresponding Green function G(κ, κ′, ω) for this equation satisfies 2 here the Green function G(κ, κ′, ω) is also a M × M matrix which is the Fourier transform of G(z, z′, ω), I is the identity matrix, κ ω E ( , ) y and κ ω J ( , ) y are one column matrixes. The eigenvalues of the matrix P(ω) and the M × M matrix S(ω), whose σ th column (S 0σ (ω), S −1σ (ω), S 1σ (ω), …, S −Nσ (ω), S Nσ (ω)) T is the eigenvector corresponding to the eigenvalue −κ σ 2 (ω), can be obtained simultaneously, κ σ (ω) is the wavevector along z direction. The matrix where δ is a positive infinitesmal, the dependence on ω is implicit for κ σ (ω) and S(ω). The Green function G(z, z', ω) can be calculated by integrating G(κ, κ′, ω) over κ and κ′    where we assume the wave vector κ σ = β σ + iγ σ , β σ and γ σ are the real and imaginary parts of κ σ .
Annihilation and creation operators. Based on the explicit expressions of amplitude operators, we have also studied the spatial evolution of the amplitude operators which is governed by quantum Langevin equations (see Supplementary Material), where the quantum noise associated with the damping is taken into account by operator Langevin noise sources. After consideration of the commutation relations of the operator noise current densities, we can get the commutation relations of the amplitude operators (see Supplementary Material), from the results we can see that the commutation relations of the amplitude operators with different orders may not be zero. A special case is considered z = z' and then we define a matrix U mn (ω) in this case †^ The commutation relations of the photon annihilation operators with different orders should be equal to zero, so we introduce the photon annihilation operators  The matrixes of superposition coefficients X + (ω) and X − (ω) are determined by the commutation relations of the bosonic photon annihilation and creation operators www.nature.com/scientificreports www.nature.com/scientificreports/ It is clearly seen from Eq.
So far, The EM field quantization in one-dimensional photonic crystal is fully completed, the final form of the electric field can be written as in matrix Here ω a z ( , ) y 0 is the radiation order and the others are high orders. The matrix S connects the amplitude operators in different orders with the electric field operators in different orders, it is not unity matrix in the grating, which reveal that there is interaction between different orders in this case. When the model degenerates to the homogeneous case, the matrix S is unity and X is diagonal, then our theory can degenerate successfully to the the corresponding results of the previous work [36][37][38] of other authors who considered the EM quantization in the radiation order in the normal propagation case (see Supplementary Material).
Quantum optical input-output relation for grating. Now we turn to the problem of propagation of the quantized field 39 through 1D periodic dielectric slab-1D grating-embedded in two semi-infinite homogeneous dielectrics, which is shown in Fig. 2. The dielectric function is expressed as  to ensure that they commute with each other  and ε ω ( ) 3 . The semi-infinite orange area up the grating is indicated by 1 and the semi-infinite green area down the grating is indicated by 3, the grating is located between these two semi-infinite areas.    The new operators should fulfill the bosonic commutation relations Similarly, the coefficients of Y + (ω) and Y − (ω) can be determined from the above equations by substituting the Eqs. (20) into (21) ω ω ω = .
Finally, the quantum optical input-output relation expressed in the transfer matrix form can be transformed to the scattering matrix Q mn (mn = 11, 12, 21, 22)  The new operators ω + g ( ) and ω − g ( ) play the role of the noise sorces associated with the damping in the input-output relation. When we consider the special case of homogeneous dielectric, the input-output relation and related commutation relations can be also derived back to the previous study 37,38 .
Then we can derive the commutation relations between the output photon operators based on the input-output relation together with the known commutation relations between the input photon operators. After deliberate and straightforward calculation the results can be written in matrix form www.nature.com/scientificreports www.nature.com/scientificreports/ In the following we give the results about the values of commutation relations for the uniform slab and grating immersed in air. The grating refers to alternating dielectric bar and air, the period and bar width of the grating are denoted by Λ and w, the relative permittivity of the dielectric bar is marked by ε, the thicknesses of the uniform slab and grating are both denoted by l, the wavelength and incident angle of the EM field are represented by λ and θ. Because of the symmetry of this model, c 11,mn = c 22,mn , c 12,mn = c 21,mn .
The diagonal elements of matrixes c 11 , c 12 , c 21 and c 22 are real number which can be seen from Eq. (24). For the uniform dielectric the matrix X is diagonal, if the uniform dielectric is lossless, the coefficients X mn approach zero for higher orders (m ≠ 0), so only the radiated (m = 0) annihilation and creation operators are physically significant. Hence, we are interested in c 11,00 , c 12,00 , c 21,00 and c 22,00 in radiation order for our model.
In Fig. 3 we consider the transmission and values of commutation relations, c 11,00 and c 22,00 , for the lossless uniform slab and lossless grating as a function of the reduced wavelength λ/Λ. The corresponding case of lossy layer is shown in Fig. 4. For uniform slab, no matter it is lossless or lossy, c 11,00 = 1 and c 12,00 = 0 hold, which can be www.nature.com/scientificreports www.nature.com/scientificreports/ seen from Figs. 3(b), 4(b,c), that means the output photons satisfy bosonic commutation relation and the annihilation operators for different channels commute with each other, these results coincide with the former work. For grating, only when it is lossless, c 11,00 = 1 and c 12,00 = 0, which can be seen from Fig. 3(d), when it is lossy, these equations are not true in this case, that is to say, c 11,00 ≠ 1 and c 12,00 ≠ 0, which can be seen from 4(e,f). After comparing the four different models, we find that the physical origin of this inequality is that the excitations, ω , in different orders interact with each other. Not only that, from Fig. 4(d-f) we also find that near the guided resonance, which is the Fano resonance in our optical model, obvious resonance and deviation of c 11,00 and c 12,00 , the deviation means that the departure of c 11,00 value from 1 and c 12,00 value from 0. It can be also clearly seen that at the reduced wavelength λ/a > 1.5, there is no guided resonance, while there is also no resonance for c 11,00 and c 12,00 and the deviation decreases.
The Heisenberg picture is implied in the quantization theory, when the theory is converted to Schördinger picture, we can understand the phenomenon further, which is the deviation of the bosonic commutation relations for the output photon operators in the lossy grating. In the Schördinger picture, the evolution operator is no longer unitary with respect to the radiated order which can be derived from the input-output relation 40,41 , so the transformation of the quantum states is non-unitary. From Fig. 4(e,f) we can see that the phenomenon of deviation is very small (~10 −3 ). Now we tune c 11,00 and c 12,00 , which describe the transformation of the photon states, by change the parameters in our model. Compared to the classical optics the Fano resonance can appear in the grating for quantum light which can be seen from Fig. 5(a,d), near the resonant absorption is notable. The peak and valley of c 11,00 and c 12,00 are coincident with the resonant absorption, the sign of c 12,00 is opposite to that near the adjacent resonant curve. We can see that the deviations of c 11,00 from 1 and c 12,00 from 0 are closely related to the Fano resonance in the grating. It is clearly seen in Fig. 5 that these deviations are very weak (~10 −3 ) due to the weak absorption (the imaginary part of the relative permittivity is only ε = . 0 1 I ). In Fig. 6 it is implied that the deviation of c 11,00 from 1 rises by increasing the imaginary part of the relative permittivity which is closely related to the effect of absorption, this devation can reach about 0.4. When ε I is located at (0.3, 1.5) the deviation of c 12,00 from 0 can be increased about 1 order of magnitude (~10 −2 ). ε I variation leads to notable effect of ω in higher orders on the output photon operators in radiation order, so we can enhance the effect of the nonunitary evolution obviously by changing the imaginary part of the relative permittivity of the grating.
We have also calculated the transmission, reflection and absorption for photon number density, which are equal to that in classical optics. The sharp resonance, guided resonance, which appears in the grating for classical light can also emerge for quantum light. At the resonance, 100% relative numbers of the outgoing photons in output channels are exhibited and near 100% absorption is realized for the lossy grating, the Q factor is high and the lifetime is long. From the asymptotic behavior of the resonance, some embedded resonances with zero linewidth can be found, these embedded resonances possess infinite high Q factor and infinite long lifetime, and are called light bound states in the continuum (BICs) which have attracted much attention in recent years in classical optics [13][14][15] . In our work, we also find the light BICs in quantum optics in theory. In classical optics, many applications of the grating are developed because of their excellent optical properties 42,43 , we believe that the grating can be applied in various areas of quantum optics, such as propagation of non-classical light, quantum state transformation, spontaneous emission of a nearby scatter and so on, these will be our next tasks.

Conclusion
We give the Green function and the EM field quantization for 1D periodic, dispersive and absorbing dielectric bulk medium firstly. The EM field are expanded in plane waves and are inserted to the quantum Maxwell equations, the Green function is solved, furthermore the electric field is quantized and the amplitude annihilators are established. The commutation relations of these amplitude operators in our periodic bulk system are calculated out based on the previous known commutation relations of the operator noise current density, we find that the amplitude operator don't commute with the its Hermite operator with different order, which is quite different Figure 6. The absorption (a), c 11,00 (b) and c 12,00 (c) for the grating as functions of reduced wavelength λ/Λ and imaginary part of the relative permittivity ε I . The real part of the relative permittivity is fixed at 4.0, the thickness and width of the grating are chosen as l = 1.75Λ, and w = 0.6Λ, respectively, the incident photons propagate normally θ = 0°. www.nature.com/scientificreports www.nature.com/scientificreports/ from the homogeneous dielectric case. Then we construct the photon annihilation operators by linear superposition of the amplitude operators. The quantum Langevin equations which determine the spatial evolution of the amplitude operators in our bulk system are provided and studied.
The quantum input-output relation for the grating is also derived, the output field operators can be described in terms of input field operators and noise sources associated with the loss in the gratings. We find that the conventional commutation relations are satisfied, for uniform slab or lossless grating, but for lossy grating, these relations do not hold, these phenomena originate from the interaction between the output photon in radiation order and the excitations in higher orders. The excellent quantum optical properties of the grating are also found and discussed. We believe our work is very beneficial for the control and regulation of the quantum light based on gratings.