We quantize the electromagnetic field in the presence of a nonmoving dielectric sphere in vacuum. The sphere is assumed to be lossless, dispersionless, isotropic, and homogeneous. The quantization is performed using normalized eigenmodes as well as plane-wave modes. We specify two useful alternative bases of normalized eigenmodes: spherical eigenmodes and scattering eigenmodes. A canonical transformation between plane-wave modes and normalized eigenmodes is derived. This formalism is employed to study the scattering of a single photon, coherent squeezed light, and two-photon states off a dielectric sphere. In the latter case, we calculate the second-order correlation function of the scattered field, thereby unveiling the angular distribution of the Hong–Ou–Mandel interference for a dielectric sphere acting as a three-dimensional beam splitter. Our results are analytically derived for a dielectric sphere of arbitrary refractive index and size with a particular emphasis on the small-particle limit. As shown in Phys. Rev. A 108 , 033714 ( 2023 ) PLRAAN 1050-2947 10.1103/PhysRevA.108.033714 , this work sets the theoretical foundation for describing the quantum interaction between light and the motional, rotational, and vibrational degrees of freedom of a dielectric sphere.
We quantize the electromagnetic field in the presence of a nonmoving dielectric sphere in vacuum. The sphere is assumed to be lossless, dispersionless, isotropic, and homogeneous. The quantization is performed using normalized eigenmodes as well as plane-wave modes. We specify two useful alternative bases of normalized eigenmodes: spherical eigenmodes and scattering eigenmodes. A canonical transformation between plane-wave modes and normalized eigenmodes is derived. This formalism is employed to study the scattering of a single photon, coherent squeezed light, and two-photon states off a dielectric sphere. In the latter case, we calculate the second-order correlation function of the scattered field, thereby unveiling the angular distribution of the Hong–Ou–Mandel interference for a dielectric sphere acting as a three-dimensional beam splitter. Our results are analytically derived for a dielectric sphere of arbitrary refractive index and size with a particular emphasis on the small-particle limit. As shown in Phys. Rev. A 108 , 033714 ( 2023 ) PLRAAN 1050-2947 10.1103/PhysRevA.108.033714 , this work sets the theoretical foundation for describing the quantum interaction between light and the motional, rotational, and vibrational degrees of freedom of a dielectric sphere.
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.
We investigate the quantum optical scattering of two-photon wave packets by a macroscopic lossy sphere by means of macroscopic quantum electrodynamics in the form of modified Langevin noise formalism. Building on our recently developed general scattering framework, we show that under the far-field approximation, the probability amplitudes for photon detection exhibit a direct correspondence to the classical electric field. This allows us to effectively map the quantum interference calculation for the spherical geometry onto a solvable classical electromagnetic problem, specifically by expressing the quantum-scattering amplitudes in terms of the classical Mie coefficients. The two ingoing photons with an arbitrary frequency-polarization spectrum impinge onto the sphere along two different directions and, as a consequence of matter losses, their scattering involves the three independent processes where two, one, and zero outgoing photons survive. Noncollinearity of ingoing photons causes the existence of two different quantum paths they can follow upon scattering, producing interference effects in the detection of the above three processes, which is governed by the wave-packet spectral symmetry. Furthermore, we demonstrate that the sphere's rotational and reflectional symmetries can be exploited to identify specific scattering geometries that enforce perfect constructive or destructive (Hong-Ou-Mandel) interference, both for symmetric and antisymmetric wave packets. To assess the impact of matter dispersion or losses on quantum interference effects accompanying photon detection, we analyze the scattering of narrow-band two-photon wave packets by high-index dielectric lossy spheres. We show that classical Mie resonance peaks, due to their Fano-like traits, yield very strong constructive and destructive interference effects, occurring when the wave packet carrier frequency matches the resonance frequency and side Fano dip frequency, respectively. In addition, we consider the overall scattering probabilities of two, one, and zero photons, and we prove that, at the Mie resonance frequencies, they exhibit quantum interference effects that are extremely sensitive to the spectral symmetry of the input wave packet, thus suggesting an efficient spectral technique assisted by matter losses to identify entanglement.
We propose a numerical technique for modeling quantum multimodal light scattering by a perfectly conducting body. Using the special quantization technique, we develop the quantum adaptation of the characteristic mode approach widely used in classical electrodynamics. The method is universal with respect to the body's configuration, as well as its dimensions relative to the wavelength. Using this method and calculating the first- and second-order field correlation functions, we demonstrate how scattering affects quantum-statistical features of the field. As an example, we consider the scattering of two single-photon incident Gaussian beams on a cylinder with a circular cross section. We show that the scattering is accompanied by two-photon interference and demonstrates the Hong-Ou-Mandel effect. It is shown that the scattered two-photon field and its correlations manifest directive propagation, which is controllable by various means (angles of incidence, configuration of the body, relations between its dimensions and the wavelength). We expect that this method will be useful for designing quantum-optical devices.
The electromagnetic field is quantized in dielectric media that show both loss and dispersion. The complex dielectric function of the medium is assumed to be a known function and the loss is modeled by Langevin forces in the forms of noise current operators. The noise current correlation function is related to the assumed dielectric function by the fluctuation-dissipation theorem. Field quantization is carried out for the infinite homogeneous dielectric, the semi-infinite dielectric, and the dielectric slab, where the fields in the second and third cases are restricted to propagation perpendicular to the dielectric surfaces. The forms of the vector potential operator are obtained in the different spatial regions for all three geometries, and in each case the required canonical commutation relation for the vector potential and its conjugate generalized momentum operator is verified. The spatial dependence of the vacuum field fluctuations is calculated for the two dielectric geometries that have surfaces.
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.