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CRC Press eBooks

Mathematical Elaborations

Stephen M. Barnett, John A. Vaccaro

Published Apr 27, 2007• 0 citations
Publisher

Abstract

The development of the theory of quantum phase had been forestalled by the lack of a consistent and physically meaningful definition for the phase angle. However, once the Pegg-Barnett phase formalism was introduced this impasse was removed and the theoretical and experimental research into quantum phase began with vigour. From the outset there was a need to further develop the more mathematical aspects of the formalism. These included exploring a new Wigner representation of the state of a harmonic oscillator, the nature of the canonical conjugate relationship between Nˆ and φˆθ , the implications of the special limiting procedure s → ∞ and the consistency of the formalism with the traditional description of a harmonic oscillator. In this chapter we review the main results of the investigations into these issues.

DOI
10.1201/b16006-6

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The development of the theory of quantum phase had been forestalled by the lack of a consistent and physically meaningful definition for the phase angle. However, once the Pegg-Barnett phase formalism was introduced this impasse was removed and the theoretical and experimental research into quantum phase began with vigour. From the outset there was a need to further develop the more mathematical aspects of the formalism. These included exploring a new Wigner representation of the state of a harmonic oscillator, the nature of the canonical conjugate relationship between Nˆ and φˆθ , the implications of the special limiting procedure s → ∞ and the consistency of the formalism with the traditional description of a harmonic oscillator. In this chapter we review the main results of the investigations into these issues.
As soon as it was introduced, the Pegg-Barnett phase formalismwas applied to the analysis of quantum phase in a variety of situations. We review the seminal studies in this chapter. We beginwith the analyses of special classes of stateswhose phase properties are optimised in some way, for example, by having the minimum variance of the phase angle subject to various constraints. We then look at the connection between the phase observable and amplitude quadrature observables of an optical field. We end with the dynamics of the phase of a single-mode optical field under diffusive and non-diffusive evolution. These studies paved the way for the application of the new formalism to the theoretical analysis of quantum phase in general.
The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.
In a series of papers, Barnett, Pegg (1986, 1988, 1989, 1992) and various co-authors have proposed a description of quantum phase by means of a collection of s-dimensional states and operators, s>or=1. We analyse the limiting procedure they employ for large s, which is known not to be compatible with quantum mechanics in the usual sense. Further, we supply a rigorous demonstration of the asymptotic limits of the `mean` and `variance` of their system of operators in coherent states. These values had previously been given but not justified mathematically. Our analysis, based on the asymptotic analysis of certain random variables, shows that the physical deductions that can be drawn from these limits are limited. We also prove that the (s+1)-dimensional `pure phase` LHW-states they consider form a sequence of approximate eigenvectors for the Weyl-quantized angle operator Delta ( phi ) and the Toeplitz phase operator X proposed by Garrison and Wong (1970), Popov and Yarunin (1992), and others. These states X s ( theta ), to our knowledge first introduced by Lerner, Huang and Waiters (1970), can be used to construct a system of measurement as in the usual quantum theory, sensitive to certain qualities of phase, but not all. Indeed, a feature of the Barnett-Pegg method, when it gives finite answers, is the construction of associated measurement systems for different observables. We give examples of sequences of (s+1)-dimensional devices which represent measurements significantly closer to ideal for X. This serves as a model for corresponding devices for Delta ( phi ), or indeed, any observable with a continuous spectrum, contingent on its spectral decomposition being obtained explicitly.
The phase of a simple-harmonic vibration or wave is a useful and easily-understood concept in classical mechanics. It is the angle that determines the sinusoidal development of the motion, and it varies linearly with time, increasing by 360°(or 2π radians) over a complete cycle of oscillation. By contrast, the concept of phase in the quantum mechanics of harmonic motion has been fraught with difficulties, which have hampered its use. Recently, D T pegg and S M Barnett (Europhys. Lett. 6 483) have proposed a new definition of the quantum-mechanical phase operator, which they suggest will lead to important and far-reaching consequences.
Quantum mechanics is arguably one of the most successful scientific theories ever and its applications to chemistry, optics, and information theory are innumerable. This book provides the reader with a rigorous treatment of the main mathematical tools from harmonic analysis which play an essential role in the modern formulation of quantum mechanics. This allows us at the same time to suggest some new ideas and methods, with a special focus on topics such as the Wigner phase space formalism and its applications to the theory of the density operator and its entanglement properties. This book can be used with profit by advanced undergraduate students in mathematics and physics, as well as by confirmed researchers.