We extend previous work on macroscopic canonical quantization leading to a multipolar Hamiltonian appropriate for application to quantum optics and cavity QED situations involving classical optical devices. In particular, we show that the electric displacement is the negative of the conjugate momentum field and that the Coulomb and polarization energies are equal to the sum of intra-atomic Coulomb and polarization energies and interatomic contact energies. The quantum Hamiltonian is now in a form in which the theory is manifestly gauge invariant.
We extend previous work on macroscopic canonical quantization leading to a multipolar Hamiltonian appropriate for application to quantum optics and cavity QED situations involving classical optical devices. In particular, we show that the electric displacement is the negative of the conjugate momentum field and that the Coulomb and polarization energies are equal to the sum of intra-atomic Coulomb and polarization energies and interatomic contact energies. The quantum Hamiltonian is now in a form in which the theory is manifestly gauge invariant.
A macroscopic, canonical quantization of the EM field and radiating atom system in quantum optics and cavity QED involving classical, linear optical devices, based on expanding the vector potential in terms of quasi mode functions is presented. The quasi mode functions approximate the true mode functions for the device, and are obtained by solving the Helmholtz equation for an idealized spatially dependent electric permittivity function describing the device. The Hamiltonian for the EM field and radiating atom system is obtained in multipolar form and the quantum EM field is found to be equivalent to a set of quantum harmonic oscillators, one oscillator per quasi mode. However, unlike true mode theory where the quantum harmonic oscillators are uncoupled, in the quasi mode theory they are coupled and photon exchange processes can occur. Explicit expressions for the coupling constants are obtained. The interaction energy between the radiative atoms and the quantum EM field depends on the amplitudes of the quasi mode functions at the positions of the radiating atoms, similar to that for the true mode approach. The simpler forms for the quasi mode functions enable the atom-field interaction energy to be written in a form in which the atoms are only coupled to certain types of modes—for example cavity quasi modes, which are large inside the optical cavity. In such cases the escape of energy from excited atoms in the cavity can be pictured in quasi mode theory as a two step process—the atom de-excites and creates a photon in a cavity quasi mode, the photon in the cavity quasi mode is then lost and appears as a photon in an external quasi mode. In this process the first step occurs via the atom-cavity quasi mode interaction, the second through coupling between cavity and external quasi modes. This may be contrasted with the true mode approach, where the excited atom loses its energy and the photon is created in one of the true modes. As all true modes have non-zero amplitudes outside as well as inside the cavity, the escape of energy from excited atoms in the cavity is seen as a one step process. An application of the quasi mode theory to the quantum theory of the beam splitter is outlined. The unitary operator used to describe this device is a scattering operator, relating initial and long time values of annihilation, creation operators for pairs of incident and reflected modes, interpreted here as quasi modes.
The quasi mode theory of macroscopic quantization in quantum optics and cavity QED developed by Dalton, Barnett and Knight is generalized. This generalization allows for cases in which two or more quasi permittivities, along with their associated mode functions, are needed to describe the classical optics device. It brings problems such as reflection and refraction at a dielectric boundary, the linear coupler, and the coupling of two optical cavities within the scope of the theory. For the most part, the results that are obtained here are simple generalizations of those obtained in previous work. However the coupling constants, which are of great importance in applications of the theory, are shown to contain significant additional terms which cannot be ‘guessed’ from the simpler forms. The expressions for the coupling constants suggest that the critical factor in determining the strength of coupling between a pair of quasi modes is their degree of spatial overlap. In an accompanying paper a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary is given as an illustration of the generalized theory. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes.
Canonical quantization of the electromagnetic (EM) field is carried out for the situation where the total charge and current densities are the sum of contributions from neutral dielectric atoms whose effect is to be described purely classically in terms of spatially dependent electric permittivity and magnetic permeability functions, and neutral, stationary radiative atoms whose interaction with the EM field is to be treated quantum mechanically. The coefficients for the expansion of the vector potential in terms of mode functions determined from a generalized Helmholtz equation are chosen as independent generalized coordinates for the EM field. The spatially dependent electric permittivity and magnetic permeability appear in a generalized Helmholtz's equation and the former also occurs in the mode function orthogonality and normalization conditions. The quantum Hamiltonian is derived in a generalized multipolar form rather than the minimal coupling form obtained in other work. The radiative energy is the sum of quantum harmonic oscillator terms, one for each mode. The modes are independent in the present case of exact mode functions associated with the spatially dependent electric permittivity and magnetic permeability, there being no direct mode-mode coupling terms. In the electric dipole approximation the electric interaction energy contribution for each mode and radiative atom is proportional to the scalar product of the dipole operator with the mode function evaluated at the atom, times the annihilation operator, plus the Hermitian adjoint. This form has been widely used in studies of radiative processes for atomic systems in dielectric media, and it is justified here via the canonical quantization procedure. The results apply to the theoretical treatment of numerous quantum optical experiments involving such interactions in the presence of passive, lossless, dispersionless, linear classical optics devices such as resonator cavities, lenses, beam splitters, and so on. An illustrative application of the theory for atomic decay in a one-dimensional Fabry-Perot cavity is given. \textcopyright{} 1996 The American Physical Society.
Canonical quantization of the electromagnetic (EM) field is carried out for the situation where the total charge and current densities are the sum of contributions from neutral dielectric atoms whose effect is to be described purely classically in terms of spatially dependent electric permittivity and magnetic permeability functions, and neutral, stationary radiative atoms whose interaction with the EM field is to be treated quantum mechanically. The coefficients for the expansion of the vector potential in terms of mode functions determined from a generalized Helmholtz equation are chosen as independent generalized coordinates for the EM field. The spatially dependent electric permittivity and magnetic permeability appear in a generalized Helmholtz's equation and the former also occurs in the mode function orthogonality and normalization conditions. The quantum Hamiltonian is derived in a generalized multipolar form rather than the minimal coupling form obtained in other work. The radiative energy is the sum of quantum harmonic oscillator terms, one for each mode. The modes are independent in the present case of exact mode functions associated with the spatially dependent electric permittivity and magnetic permeability, there being no direct mode-mode coupling terms. In the electric dipole approximation the electric interaction energy contribution for each mode and radiative atom is proportional to the scalar product of the dipole operator with the mode function evaluated at the atom, times the annihilation operator, plus the Hermitian adjoint. This form has been widely used in studies of radiative processes for atomic systems in dielectric media, and it is justified here via the canonical quantization procedure. The results apply to the theoretical treatment of numerous quantum optical experiments involving such interactions in the presence of passive, lossless, dispersionless, linear classical optics devices such as resonator cavities, lenses, beam splitters, and so on. An illustrative application of the theory for atomic decay in a one-dimensional Fabry-Perot cavity is given. \textcopyright{} 1996 The American Physical Society.
We present an overview of the framework of macroscopic quantum electrodynamics from a quantum nanophotonics perspective. Particularly, we focus our attention on three aspects of the theory that are crucial for the description of quantum optical phenomena in nanophotonic structures. First, we review the light-matter interaction Hamiltonian itself, with special emphasis on its gauge independence and the minimal and multipolar coupling schemes. Second, we discuss the treatment of the external pumping of quantum optical systems by classical electromagnetic fields. Third, we introduce an exact, complete, and minimal basis for the field quantization in multiemitter configurations, which is based on the socalled emitter-centered modes. Finally, we illustrate this quantization approach in a particular hybrid metallodielectric geometry: two quantum emitters placed in the vicinity of a dimer of Ag nanospheres embedded in a SiN microdisk.
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.