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EPJ Web of Conferences

A many-particle quantum-kinetic formalism for describing properties of light emitters in frozen dielectrics

M. G. Gladush

Published Dec 13, 2016• 4 citations
Publisher

Abstract

A many particle quantum-kinetic formalism is suggested to derive the Maxwell-Bloch-type equations which describe the interaction of quantum emitters with light in a frozen dielectric. It is shown that the quantum-kinetic formalism can meet the concept of local variations of dielectric properties and their influence on the emitter. The definitions of the local response and the effective refractive index in macroscopically homogeneous media are discussed.

DOI
10.1051/epjconf/201713203007

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A many particle quantum-kinetic formalism is suggested to derive the Maxwell-Bloch-type equations which describe the interaction of quantum emitters with light in a frozen dielectric. It is shown that the quantum-kinetic formalism can meet the concept of local variations of dielectric properties and their influence on the emitter. The definitions of the local response and the effective refractive index in macroscopically homogeneous media are discussed.
We obtained the system of Maxwell-Bloch equations (MB) that describe the interaction of cw laser with optically active impurity centers (particles) embedded in a dielectric material. The dielectric material is considered as a continuous medium with sufficient laser detuning from its absorption lines. The model takes into account the effects associated with both the real and the imaginary part of the dielectric constant of the material. MB equations were derived within a many-particle quantum-kinetic formalism, which is based on Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for reduced density matrices and correlation operators of material particles and the quantized radiation field modes. It is shown that this method is beneficial to describe the effects of individual and collective behavior of the light emitters and requires no phenomenological procedures. It automatically takes into account the characteristics associated with the presence of non-resonant and resonant particles filling the space between the optical centers.
Kinetics of a quantum system immersed in an equilibrium medium and completely described by its statistical operator is investigated. The case of a separable interaction is considered in details. A concept of the effective Hamilton operator of the system is discussed. The kinetic equation for a point particle in a spatially uniform external field placed in a medium of point particles is obtained in terms of the Wigner distribution function of a particle. The kinetic equation is applicable for strong spatially inhomogeneous (localized) states of the particle and strong external field. The elaborated theory is used also for a system of two-level emitters and photons interacting with an equilibrium phonon medium. The kinetic equation for the statistical operator of the system of the emitters and photons that describes the effect of emitter motion on the super radiance phenomenon is obtained.
The effect of macroscopic parameters of a substance on the optical characteristics of impurity particles is investigated. A generalized control equation is derived for two-level emitters forming an ensemble of optical centers in a transparent dielectric medium. In this equation, the effective values of the acting pump field and the radiative relaxation rate of an optical center are taken into account. The formalism developed here is a completely microscopic approach based on the chain of the Bogoliubov-Born-Green-Kirkwood-Yvon equations for reduced density matrices and correlation operators for material particles and modes of a quantized radiation field. The method used here makes it possible to take into account the effects of individual and collective behavior of emitters, which are associated with the presence of an intermediate medium, consistently without using phenomenological procedures. It is shown that the resultant analytic expression for the effective lifetime of the excited state of an optical center conforms with experimental data.
The macroscopic quantum theory of the electromagnetic field in a dielectric medium interacting with a dense collection of embedded two-level atoms fails to reproduce a result that is obtained from an application of the classical Lorentz local-field condition. Specifically, macroscopic quantum electrodynamics predicts that the Lorentz redshift of the resonance frequency of the atoms will be enhanced by a factor of the refractive index nnn of the host medium. However, an enhancement factor of (n2+2)∕3({n}^{2}+2)∕3(n2+2)∕3 is derived using the Bloembergen procedure in which the classical Lorentz local-field condition is applied to the optical Bloch equations. Both derivations are short and uncomplicated and are based on well-established physical theories, yet lead to contradictory results. Microscopic quantum electrodynamics confirms the classical local-field-based results. Then the application of macroscopic quantum electrodynamic theory to embedded atoms is proved false by a specific example in which both the correspondence principle and microscopic theory of quantum electrodynamics are violated.
The control of interactions among quantum emitters through nanophotonic structures offers significant opportunities for quantum technologies. However, a rigorous theoretical description of the interaction of multiple quantum emitters with complex, dispersive dielectric objects remains challenging. Here, we introduce an approach based on the modified Langevin noise formalism that unveils the roles of both the noise polarization currents of the dielectrics and the vacuum fluctuations of the electromagnetic field scattered by the dielectrics. This work extends Refs. \cite{miano_quantum_2025} and \cite{miano_spectral_2025} to the general case of an arbitrary number of emitters. The proposed approach allows us to describe the dynamics of the quantum emitters for arbitrary initial quantum states of the electromagnetic environment, consisting of two independent bosonic reservoirs, a medium-assisted reservoir and a scattering-assisted reservoir, each characterized by its own spectral density matrix. Specifically, we examine situations where both reservoirs are initially in thermal quantum states but have different temperatures. Understanding how these reservoirs shape the dynamics of the emitters is crucial for understanding light-matter interactions in complex electromagnetic environments and for improving intrinsic emitter properties within structured environments.
These notes, intended to be self contained and tutorial, present a direct, macroscopic approach to quantizing light inside a linear-response dielectric material when both spectral dispersion and spatial nonuniformity are present, but the spectral region of interest is optically transparent so that explicit treatment of the underlying physics of the medium is not needed. The approach taken is based on the macroscopic Maxwell equations and a corresponding Hamiltonian, without the use of Lagrangians or any dynamical model for the medium, and uses a standard mode-based quantization method. The treatment covers: energy density and flux in a dispersive dielectric; a summary of the inverse permittivity formalism; a new derivation of the mode normalization condition; a direct proof of the nonorthogonality of the modes; examples of quantized field expressions for the general case and various special cases; the relationship between group velocity and energy flux; the band approximation and the continuum limit; and quantum optical treatment of waveguide modes.