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Electromagnetic field quantization in nonlinear and inhomogeneous dielectrics

L.-M. Duan, Guang‐Can Guo

Published Dec 1, 1996• 0 citations
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Abstract

The electromagnetic field is quantized in nonlinear media whose dielectric functions and nonlinear susceptibilities may vary with space. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg equations we prove that Maxwell's equations are satisfied for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure developed in Ref.[1] for linear inhomogeneous media. The general formalism is illustrated by applications to quantization of the second-order and third-order nonlinear quasi-steady-state processes.

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The electromagnetic field is quantized in nonlinear media whose dielectric functions and nonlinear susceptibilities may vary with space. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg equations we prove that Maxwell's equations are satisfied for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure developed in Ref.[1] for linear inhomogeneous media. The general formalism is illustrated by applications to quantization of the second-order and third-order nonlinear quasi-steady-state processes.
A simple approach is proposed for the quantization of the electromagnetic field in nonlinear and inhomogeneous media. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg's equations we derive Maxwell's equations for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure for linear inhomogeneous media [Phys. Rev. A 43, 467 (1991)]. The explicit Hamiltonians for the second-order and third-order nonlinear quasi-steady-state processes are obtained based on this quantization procedure.
The electromagnetic field is quantized in a material whose dielectric function varies with frequency and one spatial dimension. The dielectric function is assumed to have a known form and to be real over the range of frequencies important for a particular application, for example the propagation of an optical signal. General properties of the mode functions are derived and employed in the quantization procedure. Expressions are obtained for the energy and momentum density and current operators, and these are shown to satisfy the appropriate conservation and continuity relations. The general formalism is illustrated by application to the examples of a homogeneous dielectric and two different dielectrics with a sharp interface.
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.
A technique of canonical quantization in a general dispersive nonlinear dielectric medium is presented. The medium can be inhomogeneous and anisotropic. The fields are expanded in a slowly varying envelope approximation to allow quantization. An arbitrary number of envelopes is included, assuming lossless propagation in each relevant frequency band. The resulting Lagrangian and Hamiltonian agree with known propagation equations and expressions for the dispersive energy. The central result of the theory is an expansion of the quantum Hamiltonian in terms of annihilation and creation operators corresponding to group-velocity photon-polariton excitations in the dielectric.
A quantization scheme for the electromagnetic field in absorbing dielectrics developed previously is extended to cover more complicated arrangements of dielectric media and to investigate various limiting cases of the general formalism. The limiting cases include media that have vanishing imaginary parts in their dielectric functions, because either the refractive index or the extinction coefficient vanishes. The further limit of a unit real dielectric function establishes the connection of the formalism with the well-known quantized field expressions in free space. Detailed calculations are presented for the quantization in the system of two different absorbing dielectrics in contact at a plane interface and for the cavity formed in the free space between two separated absorbing dielectrics. The forms of the field operators are determined for both systems, the canonical commutation relations are verified, and the spectra of the vacuum field fluctuations are calculated and illustrated. The calculations are restricted throughout to fields that propagate perpendicular to the dielectric interfaces. \textcopyright{} 1996 The American Physical Society.
A quantization scheme for the phenomenological Maxwell theory of the full electromagnetic field in an inhomogeneous three-dimensional, dispersive, and absorbing dielectric medium is developed. The classical Maxwell equations with spatially varying and Kramers-Kronig consistent permittivity are regarded as operator-valued field equations, introducing additional current- and charge-density operator fields in order to take into account the noise associated with the dissipation in the medium. It is shown that the equal-time commutation relations between the fundamental electromagnetic fields $\mathbf{E}^$ and $\mathbf{B}^$ and the potentials $\mathbf{A}^$ and $\stackrel{^}{\ensuremath{\varphi}}$ in the Coulomb gauge can be expressed in terms of the Green tensor of the classical problem. From the Green tensors for bulk material and an inhomogeneous medium consisting of two bulk dielectrics with a common planar interface it is explicitly proven that the well-known equal-time commutation relations of QED are preserved.