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Scientific Reports

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

Tiecheng Wang

Published Dec 27, 2019• 2 citations
Publisher

Abstract

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.

DOI
10.1038/s41598-019-56197-1

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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.
We present a one-dimensional quantization scheme for the electromagnetic field in arbitrary planar absorbing and dispersive dielectrics, taking into account explicitly the finite extent of media. The complete form of the electric field operator includes a part that corresponds to the free fields incident from the vacuum towards the medium and a particular solution which can be expressed by using the classical Green-function integral representation of the electromagnetic field. By expressing the classical Green function in terms of the light modes of the dielectric structure, we obtain a natural generalization to absorbing media of the familiar method of mode expansion which strictly applies to non-absorbing media. As an application of our mode expansion, we obtain general quantum-optical input-output relations relating the output photon operators in vacuum to the input photon operators and to the reservoir noise operators. The input-output relations obtained are uniquely determined by the knowledge of the classical light modes of the planar system. We finally show that our quantization scheme for arbitrary planar dielectrics satisfies the prescribed equal-time commutation relations of QED.
We present a three-dimensional quantum theory of light scattering, absorption, and emission in arbitrary planar dielectric systems. The theory is based on a scheme for quantization of the electromagnetic field in dispersive and lossy arbitrary planar dielectrics that takes explicitly into account the finite extent of the medium. We present general quantum-optical input–output relations of the output TE and TM photon operators to the corresponding input photon operators and to the reservoir noise operators. These three-dimensional input–output relations are uniquely determined by knowledge of the classical light modes of the planar dielectric structure.
A quantization scheme for the radiation field in dispersive and absorptive linear dielectrics is developed, which applies to both bulk material and multilayer dielectric structures. Starting from the phenomenological Maxwell equations, where the properties of the dielectric are described by a permittivity consistent with the Kramers-Kronig relations, an expansion of the field operators is performed that is based on the Green function of the classical Maxwell equations and preserves the equal-time canonical field commutation relations. In particular, in frequency intervals with approximately vanishing absorption the concept of quantization through mode expansion for dispersive dielectrics is recognized. The theory further reveals that weak absorption gives rise to space-dependent mode operators that spatially evolve according to quantum Langevin equations in the space domain. To illustrate the applicability of the theory to inhomogeneous structures, the quantization of the radiation field in a dispersive and absorptive one-interface dielectric is performed. \textcopyright{} 1996 The American Physical Society.
Using the Green-function approach to the problem of quantization of the phenomenological Maxwell theory, the propagation of quantized radiation through dispersive and absorptive multilayer dielectric plates is studied. Input-output relations are derived, with special emphasis on the determination of the quantum noise generators associated with the absorption of radiation inside the dielectric matter. The input-output relations are used to express arbitrary correlation functions of the outgoing field in terms of correlation functions of the incoming field and those of the noise generators. To illustrate the theory, photons at dielectric tunneling barriers are considered. It is shown that inclusion in the calculations of losses in the photonic band gaps may substantially change the barrier traversal times. \textcopyright{} 1996 The American Physical Society.
Standard macroscopic QED is built on the second-order Green's function for the electric field and discards open-system boundary terms. Here we develop a first-order electromagnetic operator approach that retains both E\mathbf{E}E and H\mathbf{H}H and keeps those boundary terms, naturally leading to a quantum input-output formalism. We recast Maxwell's equations as an operator equation for the dual field E\mathit{E}E=[E,H]T[\mathbf{E},\mathbf{H}]^T[E,H]T, whose first-order Green operator ggg propagates the electromagnetic state between surfaces. Symmetries of the Maxwell operator under energy and reciprocal inner products yield the propagation formula, Lorentz reciprocity, and a generalized optical theorem, with minimal vector calculus. Quantizing via a Heisenberg-Langevin approach for absorptive, dispersive media yields two independent quantum noise sources: bulk Langevin operators from material absorption and input-output field operators at the boundary. Expressing the interior field in terms of these operators and the Green propagator yields an exact closed commutation relation [E,E†]∝Im g[{\mathit{E}},{\mathit{E}}^\dagger]\propto \mathrm{Im}\,g[E,E†]∝Img, consistent with the fluctuation-dissipation theorem. This identity holds even when dielectrics extend to the boundary, as in waveguide input-output problems, and enables quantum input-output descriptions of complex photonic structures where the Green's function is obtained numerically, extending the framework beyond cavities and waveguides.
Using the Green function approach to the problem of quantization of the phenomenological Maxwell theory, the propagation of quantized radiation through dispersive and absorptive multilayer dielectric plates is studied. Input--output relations are derived, with special emphasis on the determination of the quantum noise generators associated with the absorption of radiation inside the dielectric matter. The input--output relations are used to express arbitrary correlation functions of the outgoing field in terms of correlation functions of the incoming field and those of the noise generators. To illustrate the theory, the effect of a single-slab plate on the mean photon-number densities in the frequency domain is discussed in more detail.