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IEEE Conference on Decision and Control (CDC)

Quantum control model for spatial propagation of electromagnetic fields in dielectrics

Wen-Bin Dong, Re-Bing Wu, Jing Zhang, Chunwen Li, Tzyh‐Jong Tarn

Published Dec 1, 2013• 0 citations
Publisher

Abstract

The design of many quantum optical devices is attributed to the determination of spatially distributed parameters (e.g., refraction index in photonic crystal), which calls for a theoretical description for the spatial propagation of quantum fields. This paper discusses the spatial evolution of quantized electromagnetic fields in waveguide with spatially variant dielectric parameters, which is in parallel with the time evolution in the standard input-output formalism [C. W. Gardiner and M. J. Collett, Phys.Rev.A 31, 3761 (1985)]. This formalism is non-trivial to quantum optics in nonlinear and dispersive medium, and provides a new perspective of modeling and analyzing the control of quantum fields by matter.

DOI
10.1109/cdc.2013.6760057

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The design of many quantum optical devices is attributed to the determination of spatially distributed parameters (e.g., refraction index in photonic crystal), which calls for a theoretical description for the spatial propagation of quantum fields. This paper discusses the spatial evolution of quantized electromagnetic fields in waveguide with spatially variant dielectric parameters, which is in parallel with the time evolution in the standard input-output formalism [C. W. Gardiner and M. J. Collett, Phys.Rev.A 31, 3761 (1985)]. This formalism is non-trivial to quantum optics in nonlinear and dispersive medium, and provides a new perspective of modeling and analyzing the control of quantum fields by matter.
In this paper, we explore quantum control systems tuned by spatially varying functions, which are different from the broadly studied quantum control systems with time-dependent controls. Such control problems exist in the design of quantum waveguides and materials (e.g., fibers, photonic crystals, and metamaterials) and have not been considered from a control theoretic point of view. The spatial quantum control model is built from quantized Maxwell equations for dielectric waveguides, where the spatially varying dielectric parameters are taken as controls. The model can be applied to both passive and active dielectrics or their mixtures such as metamaterials. We derive the iterative equation for slab structures, which is analogous to a discrete-time control system. For homogeneous waveguides, the model is parallel with the well-known Langevin equation in the time domain. However, for inhomogeneous and dispersive waveguides, the control model becomes bilinear even for linear dielectric materials and hence is more complicated. The resulting input-output relation is derived in terms of irrational transfer functions, and the statistical properties of quantum noises are analyzed. This work provides a new perspective on modeling and analyzing quantum control systems, on the basis of which further design and optimization with spatial parameters can be explored.
Using quantum theory, we study the propagation of an optical field in an inhomogeneous dielectric, and apply this scheme to traveling optical fields in a waveguide. We introduce a field-atom interaction Hamiltonian and derive the refractive index using quantum optics. We show that the transmission and reflection of optical fields at an interface between different materials can be described with normalized Fresnel coefficients and that this representation is related to the beam splitter operator. We then study the propagation properties of the optical fields for two types of slab waveguides: step-index and graded-index. The waveguides are divided into multiple layers to represent the spatial dependence of the optical field. We can evaluate the number of photons in an arbitrary volume in the waveguide using this procedure. Using the present method, the quantum properties of weak optical fields in a waveguide are revealed, while coherent states with higher amplitudes reduces to representation of classical waveguide optics.
An introduction to the quantum theory of nonlinear optics This article is provides an introduction to the quantum theory of optics in nonlinear dielectric media. We begin with a short summary of the classical theory of nonlinear optics, that is nonlinear optics done with classical fields. We then discuss the canonical formalism for fields and its quantization. This is applied to quantizing the electromagnetic field in free space. The definition of a nonclassical state of the electromagnetic field is presented, and several examples are examined. This is followed by a brief introduction to entanglement in the context of field modes. The next task is the quantization of the electromagnetic field in an inhomogeneous, linear dielectric medium. Before going on to field quantization in nonlinear media, we discuss a number of commonly employed phenomenological models for quantum nonlinear optical processes. We then quantize the field in both nondispersive and dispersive nonlinear media. Flaws in the most commonly used methods of accomplishing this task are pointed out and discussed. Once the quantization has been completed, it is used to study a multimode theory of parametric down conversion and the propagation of quantum solitons.
We have developed a quantum-mechanical formalism which permits the treatment of light propagation within the conceptual framework of quantum optics. The formalism rests on the calculation of the momentum operator for the radiation field, and yields directly a description for the spatial progression of the electromagnetic waves. In this paper we give a quantum-mechanical treatment for refraction and reflection by applying our formalism to propagation through a linear dielectric. The fidelity with which this formalism reproduces all results known from classical optics demonstrates its validity.
We present a fully canonical quantization scheme for the electromagnetic field in dispersive and lossy linear dielectrics. This scheme is based on a microscopic model, in which the medium is represented by a collection of interacting matter fields. We calculate the exact eigenoperators for the coupled system and express the electromagnetic field operators in terms of them. The dielectric constant of the medium is explicitly derived and is shown to satisfy the Kramers-Kronig relations. We apply these results to treat the propagation of light in dielectrics and obtain simple expressions for the electromagnetic field in the medium in terms of space-dependent creation and annihilation operators. These operators satisfy a set of equal-space commutation relations and obey spatial Langevin equations of evolution. This justifies the use of such operators in phenomenological models in quantum optics. We also obtain two interesting relationships between the group and the phase velocity in dielectrics.
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.