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Physica Scripta

Realistic measurement of phase

Ulf Leonhardt, Henryk Paul

Published Jan 1, 1993• 17 citations
Publisher

Abstract

An overview is given on experimental schemes for measuring phase properties (in particular, phase distributions) discussed and partly even realized thus far. It is pointed out that all of them amount to measuring the Q function for the field. Moreover, experimental procedures for determining the Wigner function for the field more or less directly are described which thus allow to assign experimental significance to the concept of area of overlap in phase space when applied to the quantum mechanical description of phase.

DOI
10.1088/0031-8949/1993/t48/006

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An overview is given on experimental schemes for measuring phase properties (in particular, phase distributions) discussed and partly even realized thus far. It is pointed out that all of them amount to measuring the Q function for the field. Moreover, experimental procedures for determining the Wigner function for the field more or less directly are described which thus allow to assign experimental significance to the concept of area of overlap in phase space when applied to the quantum mechanical description of phase.
The experimental schemes for measuring quantum-mechanical phase properties of light suggested and partly also realized thus far, namely (i) amplification, with the help of a quantum amplifier, of the microscopic field before phase measurement, (ii) heterodyning the field with a strong local oscillator, and (iii) performing two separate homodyne measurements on the field after beam splitting, are compared from a theoretical point of view.They share the common feature that undesired noise enters the experimental setup, which makes the measurement fuzzy.It will be pointed out that all three schemes amount to measuring the Q function of the original field, and hence are fully equivalent.Since the Q function can be interpreted as a smoothed Wigner function, one may associate with the introduced noise a smoothing process in which intrinsically quantum-mechanical features displayed by finer details of the Wigner function -especially by the occurrence of negative values -are lost.As a consequence, the measured phase distribution will be broader than the "true" one based on the concept of a quantum-mechanical phase operator.In realistic experiments, the nonunit detection efficiency further deteriorates the measuring results.It will be shown that also this echect can be properly described by an (additional) smoothing process leading to a certain s-parametrized quasiprobability distribution, with a parameter s that is connected with the detection efficiency in a simple way, as the distribution that is actually measurable.
One of the most delicate problems in quantum optics is the proper description of the phase of a single-mode radiation field. One has first to devise an experimental scheme for phase measurement, thus giving an operational definition of phase, and afterwards search for the proper quantum-mechanical description of the experimental procedure. Several such schemes which differ distinctly in their experimental setup have already been proposed and even partly realized. In this paper, I will describe them in some detail and present the outlines of their theoretical analysis.>
A combination of less frequent criticism with positive investigations has resulted in the substitution of group theoretic considerations by a simpler quantum mechanical model, has taken into account homodyne and heterodyne detection schemes, and proceeded by an analysis of phase data processing. Limiting procedures in s-phase formalisms have been provided concentrating on the Wigner function for number and ideal phase. The Wigner function for number and realistic phase has been expressed by closed formulae along with the antinormal phase distributions.
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.