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arXiv:0903.3523 · Physical Review A

Effective nonlinear Hamiltonians in dielectric media

J. A. Crosse, Stefan Scheel

Published Mar 10, 2010• 16 citations
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Abstract

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.

DOI
10.1103/physreva.81.033815

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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.
An effective Hamiltonian for the second-order nonlinear process of parametric downconversion in the presence of absorption is derived. The electromagnetic field is quantized using the Green function method. Heisenberg's equations of motion for the interaction of a single atom with a driving electric field are then solved to second order in perturbation theory in the rotating-wave approximation. The real-cavity model is used to embed the interacting atom in a bulk medium and correct for local field effects of the material. The resulting Hamiltonian is found to be trilinear in the electric and noise polarization fields and reduces to the phenomenological nonlinear Hamiltonian for the cases where absorption vanishes.
A study of the effects of absorption on the nonlinear process of parametric down-conversion is presented. Absorption within the nonlinear medium is accounted for by employing the framework of macroscopic QED and the Green tensor quantization of the electromagnetic field. An effective interaction Hamiltonian, which describes the nonlinear interaction of the electric field and the linear noise polarization field, is used to derive the quantum state of the light leaving a nonlinear crystal. The signal and idler modes of this quantum state are found to be a superposition of the electric and noise polarization fields. Using this state, the expression for the coincident count rates for both type-I and type-II down-conversion are found. The nonlinear interaction with the noise polarization field was shown to cause an increase in the rate on the order of ${10}^{\ensuremath{-}12}$ for absorption of 10% per centimeter. This astonishingly small effect is found to be negligible compared to the decay caused by linear absorption of the propagating modes. From the expressions for the biphoton amplitude, it can be seen that the maximally entangled states can still be produced even in the presence of strong absorption.
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.
We present a quantization scheme for the electromagnetic field in locally responding nonlinear dielectric materials with dispersion and absorption. Based on the Green tensor expansion of the macroscopic electromagnetic field, we derive an expression for the nonlinear polarization that contains, besides the reactive part which is responsible for the nonlinear interaction, a noise contribution that has hitherto been disregarded. For the process of third-harmonic generation, we find that the magnitude of the nonlinear noise polarization is proportional to the intensity of the pump fields.
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.
The theory of quantum effects in nonlinear dielectric media is developed. The nonlinear dielectric media is influenced by an external pumping field. The diagonalization of the Hamiltonian of a quantized field is obtained by the canonical Bogoliubov transformations. The transformations allow us to obtain the general expressions for the number of created photons and for the degree of squeezing. In the case of a plane pumping wave, for example, the results are calculated by using the zero order of the secular perturbation theory, with small parameters characterizing the medium nonlinearity. The Heisenberg equations of motion are obtained for non-stationary case and a commonly used Hamiltonian is derived from the first principles of quantum electrodynamics.