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arXiv:atom-ph/9608002 · Physical Review A

Quantized fields in a nonlinear dielectric medium: A microscopic approach

Mark S. Hillery, Leonard D. Mlodinow

Published Jan 1, 1997• 24 citations
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Abstract

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.

DOI
10.1103/physreva.55.678

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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.
The microscopic operators have been investigated for radiation and polarization fields within a discrete molecular medium. The medium comprises atoms (or molecules) each containing an arbitrary number of energy levels. Explicit mode expansions have been derived for the quantized microfields in terms of normal Bose operators for polariton creation and annihilation. These microscopic operators have been demonstrated to yield the correct macroscopic and local field operators presented in part I [Phys. Rev. A 53, 3543 (1996)]. On the other hand, the commutation relations between the expanded microfields differ from the exact commutation relationships. This happens because the microfields have been described in terms of a continuous refractive index. Hence the expanded field operators do not extend to modes with extremely high frequencies ranging over the photon umklapp frequencies cG. In spite of that, the proper commutation relations hold between the resultant macroscopic (averaged) fields. This justifies separate quantization of the slowly modulated (macroscopic) part of the radiation field in linear dielectrics: such an approach is utilized in phenomenological quantization schemes. Finally, although the microscopic operators presented are not complete, the mode expansions adequately represent the quantized microfields associated with the optical modes in condensed molecular media.
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 derive an effective Hamiltonian for the nonlinear process of parametric downconversion in the presence of absorption. Based upon the Green-function method for quantizing the electromagnetic field, we first set up Heisenberg's equations of motion for a single atom driven by an external electric field and in the presence of an absorbing dielectric material. The equations of motion are then solved to second order in perturbation theory which, in rotating-wave approximation, yields the standard effective interaction Hamiltonian known from free-space nonlinear optics. In a second step, we derive the local-field-corrected Hamiltonian for an atom embedded in a dielectric host medium, that is, a nonlinear crystal. Here we show that the resulting effective Hamiltonian is found to be trilinear in the electric and noise-polarization fields and is thus capable of describing nonlinear noise processes. Furthermore, it reduces to the phenomenological nonlinear Hamiltonian for the cases where absorption, and hence the noise-polarization field, vanishes.
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.
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.