Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Quantum theory of light propagation: Linear medium

Izo I. Abram

Published Jun 1, 1987• 100 citations
Publisher

Abstract

We have developed a quantum-mechanical formalism which permits the treatment of light propagation within the conceptual framework of quantum optics. The formalism rests on the calculation of the momentum operator for the radiation field, and yields directly a description for the spatial progression of the electromagnetic waves. In this paper we give a quantum-mechanical treatment for refraction and reflection by applying our formalism to propagation through a linear dielectric. The fidelity with which this formalism reproduces all results known from classical optics demonstrates its validity.

DOI
10.1103/physreva.35.4661

Related papers

Propagation of Nonclassical Light

Izo I. Abram

Classical optics describes quite successfully the propagation of laser light, both in free-space and inside a transparent medium. On the other hand, neither classical optics nor the conventional formulation of quantum optics can address adequately the propagation of nonclassical light. Nonclassical states of light are usually described by second-quantization operators acting onto the vacuum state of the radiation field, and the corresponding expectation values of the electric and magnetic fields are often zero. Clearly, the description of propagation of nonclassical light requires a direct quantum mechanical treatment of the spatial progression of the field operators. This chapter outlines the main features of propagative quantum optics, and examines more particularly the treatment of propagation through a linear (refractive) medium, and across a refractive interface. In order to facilitate the discussion of the physical ideas, a plane-wave propagation geometry, which greatly simplifies the mathematical treatment, is used.

CRC Press eBooks•Aug 12, 2020•0 citations

Quantum theory for light propagation in a nonlinear effective medium

Izo I. Abram, E. Cohen

Starting with the canonical quantization procedure for the electromagnetic field inside an effective (linear or nonlinear) medium, we present a direct-space formulation of the theory of quantum optics. This approach does not use the conventional modal decomposition of the field, but relies on the electromagnetic momentum operator defined in terms of local electric- and magnetic-field operators. The momentum operator contains all the information on the spatial characteristics of the field, and can describe the translation of a short light pulse in a nonlinear medium, without a modal analysis of the pulse. Propagation is described through an operatorial wave equation that relates the temporal evolution of an electromagnetic pulse to its spatial progression. Through this equation, the direct-space approach to quantum optics can treat traveling-wave nonlinear-optical phenomena and, at the same time, account for their quantum statistics. The theory is applied to squeezed-light generation by the parametric down-conversion of a short laser pulse, as an illustration.

Physical Review A•Jul 1, 1991•52 citations

Quantum-mechanical description of a photon in refractive mediums and momentum operator

Toshio Inagaki

A quantum-mechanical description of a photon in refractive mediums is proposed. Here we assume that the medium is isotropic and linear, and the material constants (the dielectric constant and permeability) are both the functions of the position coordinates only and dispersionless. It is found that the momentum operator is generally ill defined as well as the position operator. The momentum operator, however, becomes approximately well defined when the extents of the appreciable changes of the material constants are very large compared to the photon wavelengths. Then, it is shown that the various expressions of the physical quantities in the classical electromagnetic theory, such as the energy and momentum, can be obtained from the expectation values of the corresponding operators. The Abraham-Minkowski controversy in the classical electromagnetic theory is glanced from the viewpoint of our quantum-mechanical approach.

Il Nuovo Cimento B•Apr 1, 2003•0 citations

Reflection and refraction of photons

Imants Bersons, Rita Veilande, Ojārs Balcers

Abstract Three linear equations are proposed as quantum-mechanical equations for the free propagating photons. The solution of these equations is the vector potential of the electromagnetic field and it is a product of the Gaussian functions for the transverse coordinates and the eigenfunction of the harmonic oscillator for the longitudinal coordinate. This solutions is used to describe the reflection and refraction of photons on the boundary between two dielectrics. The amplitude of the reflected field coincides with the Fresnel formulae for the plane waves, but the amplitude of the refracted field is different. These amplitudes are the probability’s amplitudes of reflection and refraction of photons—like in the quantum mechanics. The photons propagated from the first into the second medium increase (decrease) their transverse size in the plane of incidence, if the refraction index of the second medium is greater (smaller) than in the first medium. The increasing or decreasing grow when increasing the angle of incidence. The difference between the quantum mechanics of the particles with a mass and the photons, as well as the interpretation of the wave function of photons, is discussed.

Physica Scripta•Feb 1, 2022•5 citations

Quantum control model for spatial propagation of electromagnetic fields in dielectrics

Wen-Bin Dong, Re-Bing Wu, Jing Zhang +2 more

The design of many quantum optical devices is attributed to the determination of spatially distributed parameters (e.g., refraction index in photonic crystal), which calls for a theoretical description for the spatial propagation of quantum fields. This paper discusses the spatial evolution of quantized electromagnetic fields in waveguide with spatially variant dielectric parameters, which is in parallel with the time evolution in the standard input-output formalism [C. W. Gardiner and M. J. Collett, Phys.Rev.A 31, 3761 (1985)]. This formalism is non-trivial to quantum optics in nonlinear and dispersive medium, and provides a new perspective of modeling and analyzing the control of quantum fields by matter.

IEEE Conference on Decision and Control (CDC)•Dec 1, 2013•0 citations

Quantum theory of light in a dispersive structured linear dielectric: a macroscopic Hamiltonian tutorial treatment

Michael G. Raymer

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.

arXiv:1910.01746•Journal of Modern Optics•Jan 14, 2020•20 citations
We have developed a quantum-mechanical formalism which permits the treatment of light propagation within the conceptual framework of quantum optics. The formalism rests on the calculation of the momentum operator for the radiation field, and yields directly a description for the spatial progression of the electromagnetic waves. In this paper we give a quantum-mechanical treatment for refraction and reflection by applying our formalism to propagation through a linear dielectric. The fidelity with which this formalism reproduces all results known from classical optics demonstrates its validity.
Classical optics describes quite successfully the propagation of laser light, both in free-space and inside a transparent medium. On the other hand, neither classical optics nor the conventional formulation of quantum optics can address adequately the propagation of nonclassical light. Nonclassical states of light are usually described by second-quantization operators acting onto the vacuum state of the radiation field, and the corresponding expectation values of the electric and magnetic fields are often zero. Clearly, the description of propagation of nonclassical light requires a direct quantum mechanical treatment of the spatial progression of the field operators. This chapter outlines the main features of propagative quantum optics, and examines more particularly the treatment of propagation through a linear (refractive) medium, and across a refractive interface. In order to facilitate the discussion of the physical ideas, a plane-wave propagation geometry, which greatly simplifies the mathematical treatment, is used.
Starting with the canonical quantization procedure for the electromagnetic field inside an effective (linear or nonlinear) medium, we present a direct-space formulation of the theory of quantum optics. This approach does not use the conventional modal decomposition of the field, but relies on the electromagnetic momentum operator defined in terms of local electric- and magnetic-field operators. The momentum operator contains all the information on the spatial characteristics of the field, and can describe the translation of a short light pulse in a nonlinear medium, without a modal analysis of the pulse. Propagation is described through an operatorial wave equation that relates the temporal evolution of an electromagnetic pulse to its spatial progression. Through this equation, the direct-space approach to quantum optics can treat traveling-wave nonlinear-optical phenomena and, at the same time, account for their quantum statistics. The theory is applied to squeezed-light generation by the parametric down-conversion of a short laser pulse, as an illustration.
A quantum-mechanical description of a photon in refractive mediums is proposed. Here we assume that the medium is isotropic and linear, and the material constants (the dielectric constant and permeability) are both the functions of the position coordinates only and dispersionless. It is found that the momentum operator is generally ill defined as well as the position operator. The momentum operator, however, becomes approximately well defined when the extents of the appreciable changes of the material constants are very large compared to the photon wavelengths. Then, it is shown that the various expressions of the physical quantities in the classical electromagnetic theory, such as the energy and momentum, can be obtained from the expectation values of the corresponding operators. The Abraham-Minkowski controversy in the classical electromagnetic theory is glanced from the viewpoint of our quantum-mechanical approach.
Abstract Three linear equations are proposed as quantum-mechanical equations for the free propagating photons. The solution of these equations is the vector potential of the electromagnetic field and it is a product of the Gaussian functions for the transverse coordinates and the eigenfunction of the harmonic oscillator for the longitudinal coordinate. This solutions is used to describe the reflection and refraction of photons on the boundary between two dielectrics. The amplitude of the reflected field coincides with the Fresnel formulae for the plane waves, but the amplitude of the refracted field is different. These amplitudes are the probability’s amplitudes of reflection and refraction of photons—like in the quantum mechanics. The photons propagated from the first into the second medium increase (decrease) their transverse size in the plane of incidence, if the refraction index of the second medium is greater (smaller) than in the first medium. The increasing or decreasing grow when increasing the angle of incidence. The difference between the quantum mechanics of the particles with a mass and the photons, as well as the interpretation of the wave function of photons, is discussed.
The design of many quantum optical devices is attributed to the determination of spatially distributed parameters (e.g., refraction index in photonic crystal), which calls for a theoretical description for the spatial propagation of quantum fields. This paper discusses the spatial evolution of quantized electromagnetic fields in waveguide with spatially variant dielectric parameters, which is in parallel with the time evolution in the standard input-output formalism [C. W. Gardiner and M. J. Collett, Phys.Rev.A 31, 3761 (1985)]. This formalism is non-trivial to quantum optics in nonlinear and dispersive medium, and provides a new perspective of modeling and analyzing the control of quantum fields by matter.
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.