The Quantum Phase Operator: A Review, 1st Edition (Hardback) book cover

The Quantum Phase Operator

A Review, 1st Edition

Edited by Stephen M. Barnett, John A. Vaccaro

CRC Press

504 pages | 71 B/W Illus.

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Hardback: 9781584887607
pub: 2007-04-27
eBook (VitalSource) : 9780429149160
pub: 2007-04-27
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Describing the phase of an electromagnetic field mode or harmonic oscillator has been an obstacle since the early days of modern quantum theory. The quantum phase operator was even more problematic with the invention of the maser and laser in the 1950s and 1960s. This problem was not solved until the Pegg-Barnett formalism was developed in the 1980s. Edited by one of the scientists who created this key solution, The Quantum Phase Operator: A Review charts the development of phase and angle operators from their first appearance to modern theory.

Bringing together vital works that have been published on the subject, the book presents the ideas that led to the current theory of the phase operator and provides a complete picture of the progress that has followed since then. With introductions by the editors to put the papers in context and unify the content of the book, each section focuses on a different aspect of phase operators. The editors also chronologically organize the papers within the sections to highlight how scientific thought has evolved, if at all, over time.

A collection of important relevant material that is scattered throughout the literature, this volume chronicles the history of the various facets of the quantum phase operator, promoting a solid foundation in quantum theory.


". . . useful and valuable collection of papers . . ."

— K.-E. Hellwig, in Zentralblatt Math, 2009

Table of Contents


The quantum theory of the emission and absorption of radiation

Amplitude and phase uncertainty relations

On the uncertainty relation for Lz and ø

On the commutator [Lz, ø]

Quantum mechanical phase and time operator

The quantum theory of light

Phase in quantum optics


Unitary phase operator in quantum mechanics

Hermitian phase operator ? in the quantum theory of light

On the Hermitian optical phase operator

Phase properties of the quantized single-mode electromagnetic field

Quantum theory of rotation angles

Quantum optical phase


Wigner function for number and phase

Quantum optical phase and canonical conjugation

Limiting procedures for the optical phase operator

Consistency of quantum descriptions of phase

Canonical and measured phase distributions

Number-phase Wigner function on Fock space

Antinormal ordering of phase operators and the algebra of weak limits

Pegg-Barnett operators of infinite rank

Phase operators on Hilbert space


Phase properties of squeezed states of light

A new approach to optical phase diffusion

Quantum theory of optical phase correlations

Physical number phase intelligent and minimum uncertainty states of light

Phase optimized quantum states of light

Phase fluctuations and squeezing

Phase properties of linear optical amplifiers


Phase measurements

On measuring extremely small phase fluctuations

Adaptive phase measurements of optical modes: Going beyond the marginal Q distribution

Phase measurements by projection synthesis

Quantum phase distribution by projection synthesis

Measuring the phase variance of light

Quantum phase distribution by operator synthesis

Single-shot measurement of quantum optical phase

Binomial states and the phase distribution measurement of weak optical fields


Experimental determination of number-phase uncertainty relations

Measurement of number-phase uncertainty relations for optical fields

Adaptive homodyne measurement of optical phase

Uncertainty principle for angular position and angular momentum

Minimum uncertainty states of angular momentum and angular position


Time in a quantum mechanical world

Complement of the Hamiltonian



About the Series

Series in Optics and Optoelectronics

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Subject Categories

BISAC Subject Codes/Headings:
SCIENCE / Optics
SCIENCE / Quantum Theory