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John Wiley & Sons, Inc. eBooks

Quantum Multipole Radiation

A.S. Shumovsky

Published Apr 2, 2003• 5 citations
Publisher

Abstract

The author reviews investigations of the photon localization and quantum phase problems based on the use of the representation of multipole photons. Section II presents a general consideration of the field quantization, including a comparison of zero-point oscillations of the plane and multipole waves of photons. Section III discusses the atom-field interaction leading to the multipole radiation. The SU(2) quantum phase representation of atomic variables is considered, as is a connection between the SU(2) quantum phase states and entanglement phenomenon. In Section IV the author describes the quantum phase of multipole radiation caused by the angular momentum conservation in the process of radiation. This approach is compared with the Pegg–Barnett formalism and with Mandel's operational approach. Section V covers the quantum polarization properties of multipole radiation. Photon localization, quantum measurements, and causality are considered in Section VI.

DOI
10.1002/0471231479.ch8

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The author reviews investigations of the photon localization and quantum phase problems based on the use of the representation of multipole photons. Section II presents a general consideration of the field quantization, including a comparison of zero-point oscillations of the plane and multipole waves of photons. Section III discusses the atom-field interaction leading to the multipole radiation. The SU(2) quantum phase representation of atomic variables is considered, as is a connection between the SU(2) quantum phase states and entanglement phenomenon. In Section IV the author describes the quantum phase of multipole radiation caused by the angular momentum conservation in the process of radiation. This approach is compared with the Pegg–Barnett formalism and with Mandel's operational approach. Section V covers the quantum polarization properties of multipole radiation. Photon localization, quantum measurements, and causality are considered in Section VI.
We discuss the spatial properties of quantum radiation emitted by a multipole transition in a single atom. The qualitative difference between the representations of plane and spherical waves of photons is examined. In particular, the spatial inhomogeneity of the zero-point oscillations of multipole field is shown. We show that the vacuum noise of polarization is concentrated in a certain vicinity of atoms where it strongly exceeds the level predicted by the representation of the plane waves. A new general polarization matrix is proposed. It is shown that the polarization and its vacuum noise strongly depend on the distance from the source. 1 I.
The quantum polarization and phase properties of an electric dipole radiation are examined. It is shown that, unlike the classical picture, the quantum description of polarization needs nine independent Stokes operators, forming a representation of SU (3) sub-algebra in the Weyl-Heisenberg algebra of photons. A corresponding Cartan algebra determines the cosine and sine of the radiation phase operators. A new representation of dipole photons is proposed. The generators of the Cartan algebra are diagonal in this representation so that the corresponding number states describe the number of photons with given radiation phase. The eigenstates of the radiation phase are determined. They have a discrete spectrum and natural behaviour in the classical limit. The relation between the radiation phase, operational phases and the Pegg-Barnett approach is discussed.
multipole radiation, the local properties of the radiation field can be rigorously described in spite of the fact that photons are defined in whole space-time. In particular, the local intensity operators for the field components with different polarization can be introduced. The intensities associated with two different space-time points along the same direction can be measured together. A method of quantum measurement of correlation of polarizations at different distances based on the use of Aharonov-Bohm effect is proposed.
We will begin our journey through the background of quantum optics with an elementary, but quantitative, classical theory of radiative fields.Wewill first briefly outline the electromagnetic (EM) theory of radiation, and describe how the EM radiation may be understood as a wavewhich can be represented by a set of harmonic oscillators. We shall describe how the free or non-interacting EM field may be understood as a collection of harmonic oscillators which is quantized in the standard manner, and whose energy cannot be zero as a consequence of the basic non-commutability of the canonical field variables. This chapter discusses the properties of plane EM waves and normalization of the EM field in one dimension. This is followed by a description of the Hamiltonian and the amplitudes of the EM field in terms of the annihilation and creation operators. Based on this chapter, it is possible to considerably simplify the formulation of the physical basis for the mathematical description of the major problems of quantum optics.
For a long time, about 60 years, the phase as an observable of the quantized light fields has r emained a fundamental problem with considerable difficulties on quantum mechanics and also q uantum optics.Recently, however, the hermitian phase operator and its eigenstates in a finite H ilbert space are found by Pegg and Barnett.In this paper, we provide a historical review and t he background of the problem on the phase operator, explore the Pegg-Barnett formalism, and s how the recent developements of this fields and its application to various light fields.Further problems on this formalism and other approaches are also discussed.
A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ and so the correponding limit points are ordinary operators on Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.