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arXiv:cond-mat/9907080 · Physical Review Letters

Radiative Transfer Theory for Vacuum Fluctuations

Eugene G. Mishchenko, C. W. J. Beenakker

Published Dec 27, 1999• 12 citations
arXivHTMLPDF

Abstract

A semiclassical kinetic theory is presented for the fluctuating photon flux emitted by a disordered medium in thermal equilibrium. The kinetic equation is the optical analog of the Boltzmann-Langevin equation for electrons. Vacuum fluctuations of the electromagnetic field provide a new source of fluctuations in the photon flux, over and above the fluctuations due to scattering. The kinetic theory in the diffusion approximation is applied to the super-Poissonian noise due to photon bunching and to the excess noise due to beating of incident radiation with the vacuum fluctuations.

DOI
10.1103/physrevlett.83.5475

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A semiclassical kinetic theory is presented for the fluctuating photon flux emitted by a disordered medium in thermal equilibrium. The kinetic equation is the optical analog of the Boltzmann-Langevin equation for electrons. Vacuum fluctuations of the electromagnetic field provide a new source of fluctuations in the photon flux, over and above the fluctuations due to scattering. The kinetic theory in the diffusion approximation is applied to the super-Poissonian noise due to photon bunching and to the excess noise due to beating of incident radiation with the vacuum fluctuations.
Kinetic equations have played a significant role in optics since the introduction of the radiative transport equation (RTE) by Schuster in the early 20th century. Now they are applied in many scientific fields such as atmospheric physics and biomedical imaging to understand how light interacts with a complex medium. In this thesis we investigate how these equations arise in: 1. models of classical optics in an inhomogeneous disordered scattering medium 2. in fully quantum real space models with a random distribution of atoms In certain spatial and temporal regimes it has been empirically observed that light interacting with a highly disordered scattering medium may be described by a radiative transport equation with a constant scattering coefficient. One way to derive such equations from microscopic principles is to model the medium as a random field which is statistically homogeneous. We show that in a quasi-homogeneous media, where statistical correlations vary rapidly on small length scales and slowly on large length scales, light obeys a radiative transport equation with variable scattering coefficient. In the second part of this thesis we introduce a new model in quantum optics which allows us to treat a quantized electromagnetic field coupled to a medium of two level atoms, both in real space. The rest of the chapter focuses on the interaction of atoms with a field consisting of at most one photon. First we recover a classical result due to Wigner and Weisskopf on the rate of single atom spontaneous emission. Next we show that in a statistically homogeneous distribution of two level atoms the field and atomic amplitude satisfy linear kinetic equations. Finally we show that at sufficiently long times and large distances these amplitudes are asymptotically governed by diffusion equations which may be solved analytically. In the final chapter we use the previous model to analyze a system in which there can be at most two photons. This allows us to study the time evolution of states that initially contain two entangled photons, a phenomenon which has no classical analogue. We show that the various probability amplitudes satisfy linear two particle kinetic equations which again may be asymptotically evaluated at large distances and long times.
A theory is developed, valid within the accuracy of the first-order Born approximation, of the scattering of electromagnetic fields that applies under much more general conditions than do the current theories. The incident field is assumed to be statistically homogeneous and stationary, of arbitrary state of coherence, of arbitrary state of polarization, and to have arbitrary spectrum. The medium is assumed to be linear and spatially and temporally random, with the randomness being characterized by an ensemble that is statistically homogeneous, isotropic, and stationary. The analysis is not restricted to scattering from a medium in thermal equilibrium and applies in both off-resonance and near-resonance regions of the medium.
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Abstract The atom-field interaction is treated in semiclassical radiation theory, starting from the transformation from the minimal coupling Hamiltonian to the electric dipole form used extensively in quantum optics. The Heisenberg and density matrix approaches are developed and applied to two-state atoms, Bloch equations, Rabi oscillations, Maxwell-Bloch equations, and transition rates for absorption and emission. Einstein’s theory of blackbody radiation based on momentum fluctuations and dissipation is reviewed. The Einstein fluctuation formula is derived and used to introduce wave-particle duality, Hanbury Brown-Twiss correlations, and photon bunching.
We consider a quantum Langevin kinetic equation for a system of fermions. We first derive the Langevin force noise correlation functions in Landau's Fermi-liquid kinetic theory from general considerations. We then use the resulting equation to calculate the equilibrium dynamic structure factor in the collisionless regime at low temperatures. The result is in agreement with the conventional many-body result. We then use the theory to derive both the fluctuating Navier-Stokes equations for a quantum fluid and the fluctuating hydrodynamic equations for fermions in the presence of quenched disorder. We also discuss the modifications needed for the fluctuating hydrodynamic equations to describe an electron fluid with long-ranged interactions, and we prove an HHH theorem for the nonlinear Landau kinetic equation.
We discuss the quantum fluctuations of the fields associated with a broad class of optical scattering and transmission problems by developing the quantum electrodynamics of an idealized linear, but nonuniform, dielectric medium. We present and compare two quantization schemes for this purpose. The first is based on the expansion of the field in terms of a set of single-frequency solutions of the Maxwell equations. The second involves expanding the field in the set of plane-wave solutions of the Maxwell equations in the vacuum. The relation between the two quantization schemes is discussed in the framework of the scattering theory that connects them. The methods presented are used to show that various field components within a dielectric medium may be either superfluctuant or subfluctuant relative to their fundamental uncertainties in the vacuum. These alterations of the fluctuation properties of the fields are shown to lead to changes in the spontaneous emission rates for both electric and magnetic dipole transitions of excited atoms within or near dielectric media. We also analyze the quantum properties of the transition radiation emitted by a fast charged particle in passing from one dielectric medium to another.