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Physics World

Quantum phase reinvigorated

R. Loudon

Published Jan 1, 1989• 2 citations
Publisher

Abstract

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.

DOI
10.1088/2058-7058/2/1/23

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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.
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.
A general approach to the description of an angle and phase is given on the basis of the kq\mathrm{kq}kq representation. It is shown that an angular coordinate in quantum mechanics has to be treated as a quasicoordinate in order to avoid inconsistencies. The kq\mathrm{kq}kq representation leads to a consistent definition of the angular-momentum-angle degree of freedom. Using the correspondence between classical and quantum mechanics for the phase of a harmonic oscillator, operators are defined that form a new quantum-mechanical representation. This representation clarifies the concept of the phase and sheds light on the general understanding of rotations in quantum mechanics.
One of the most commonly used quantum mechanical systems is that of the simple harmonic oscillator . The prototype system is the simple pendulum. Classically, the motion of the system exhibits harmonic, or sinusoidal oscillations. The sinusoidal motion of classical mechanics is the simplest system that produces oscillatory behavior, and therefore is found in almost an infinity of physical systems. Thus, the properties of the harmonic oscillator become important for their use in describing quite diverse physical systems. In a solid, the motion of the atoms is much like the motion of free particles, with the important exception that the atoms are forced on average to retain positions within the solid that specify a particular crystal structure. The motion remains that of a harmonic oscillator, but now the shift of the wave function and the energy levels are hybrids of the confinement harmonic oscillator and the magnetic harmonic oscillator.
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 physics, harmonic motion is among the most representative types of motion. A simple harmonic oscillator is often the source of any vibration with a restoring force proportional to Hooke’s law. Every minimum potential has a solution in the form of the harmonic oscillator potential. Little oscillations at the minimum are characteristic of almost all natural potentials and of many quanta mechanical systems. Harmonic motion is an essential building block for these more complex uses. The Schrödinger equation is a defining feature of the harmonic oscillator. Here, we demonstrate that the time-frequency plane is a useful tool for analyzing their dynamics. We numerically integrate several examples involving different input forces and demonstrate that the oscillations are clearly displayed and easily interpretable in the time-frequency plane.