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Chinese Journal of Physics

Quantum Electrodynamics in a Linear Absorptive Dielectric Medium

Fardin Kheirandish, Majid Amooshahi

Published Dec 1, 2008• 0 citations
OpenAlex

Abstract

The electromagnetic field in a linear absorptive dielectric medium is quantized in the framework of the damped polarization model by a new minimal coupling scheme. A Hamiltonian containing a reservoir with continuous degrees of freedom is proposed. The reservoir minimally couples to the dielectric polarization and the electromagnetic field. The Langevin equation is obtained as the equation of motion of the polarization field. Finally, the equations of motion are solved for homogeneous dielectrics.

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The electromagnetic field in a linear absorptive dielectric medium is quantized in the framework of the damped polarization model by a new minimal coupling scheme. A Hamiltonian containing a reservoir with continuous degrees of freedom is proposed. The reservoir minimally couples to the dielectric polarization and the electromagnetic field. The Langevin equation is obtained as the equation of motion of the polarization field. Finally, the equations of motion are solved for homogeneous dielectrics.
The macroscopic electromagnetic field is quantized in a linear isotropic permeable-dielectric medium by associating a proper damped quantum-mechanical harmonic oscillator with each mode of the radiation field. We introduce a particular form of the Langevin equation with two memory functions and two random impulses to describe the motion of a damped oscillator. A fully canonical approach is used to quantize the damped oscillator, the conjugate momentum is defined and the quantum-mechanical Hamiltonian is introduced. This brings forth a direct method for the quantization of the radiation field and at the same time introduces an effective Hamiltonian operator. Special attention is paid to the vacuum field fluctuation in permeable dielectrics.
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.
We quantize the electromagnetic field in an absorbing resonant medium with translation symmetry broken along one direction. In this case the general polarization of the material system is driven by the field via a nonlocal complex susceptibility. We extend recently developed schemes for field quantization in absorbing dielectric media with local susceptibilities to dielectric systems described by a nonlocal susceptibility. Dissipation inside the medium is described by considering the coupling of polarization quanta of the system with the reservoir oscillators in the usual Langevin approach. The method is applied to the case of a semiconductor quantum well embedded in infinite barriers. Expressions for the electromagnetic-field operators are obtained in the different spatial regions and the spatial and frequency dependencies of the vacuum-field fluctuations are calculated.
We derive expressions for the quantum electromagnetic field in a dispersive and dissipative dielectric medium, treating the medium as a continuum. We compare the Langevin approach with the Fano diagonalization procedure for the coupled system composed of the electromagnetic field, the dielectric medium, and the reservoir. In particular, we show that the quantized electric and magnetic fields obtained in both methods have exactly the same form. More generally, in thermal equilibrium, correlation functions computed in both methods are identical.
After giving an outline of the quantization scheme based on the microscopic Hopfield model of a dielectric bulk material, we show how the classical phenomenological Maxwell equations of the electromagnetic field in the presence of dielectric matter of given space- and frequency-dependent complex permittivity can be transferred to quantum theory. Including in the theory the interaction of the medium-assisted field with atomic systems, we present both the minimal-coupling Hamiltonian and the multipolar-coupling Hamiltonian in the Coulomb gauge. To illustrate the concept, we discuss the input--output relations of radiation and the transformation of radiation-field quantum states at absorbing four-port devices, and the spontaneous decay of an excited atom near the surface of an absorbing body and in a spherical micro-cavity with intrinsic material losses. Finally, we give an extension of the quantization scheme to other media such as amplifying media, magnetic media, and nonlinear media.
Theories which have been used to describe the quantized electromagnetic field interacting with a nonlinear dielectric medium are either phenomenological or derived by quantizing the macroscopic Maxwell equations. Here we take a different approach and derive a Hamiltonian describing interacting fields from one which contains both field and matter degrees of freedom. The medium is modeled as a collection of two-level atoms, and these interact with the electromagnetic field. The atoms are grouped into effective spins and the the Holstein-Primakoff representation of the spin operators is used to expand them in inverse powers of the total spin. When the lowest order term of the interaction is combined with the free atomic and field Hamiltonians, a Hamiltonian describing a theory of noninteracting polaritons results. When higher order terms are expressed in terms of polariton operators standard nonlinear optical interactions emerge. These are then compared to the results of phenomenological and macroscopic theories. The theory is also used to derive an effective Hamiltonian describing counterpropagating modes in a nonlinear medium.