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CRC Press eBooks

Phase Dynamics and Uncertainties

Stephen M. Barnett, John A. Vaccaro

Published Apr 27, 2007• 1 citation
Publisher

Abstract

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.

DOI
10.1201/b16006-7

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Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)] have proposed a formalism for a Hermitian optical phase operator, constructed from the so-called phase states. In this paper we give an alternative derivation. We begin by defining a unitary phase operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ from the relationship $\stackrel{^}{a}={e}^{i\stackrel{^}{\ensuremath{\varphi}}}R(\stackrel{^}{N})$ ($\stackrel{^}{a}$ is an annihilation operator for a single-mode boson field), but borrow the idea of a finite- (but arbitrarily large) dimensional space from Pegg and Barnett. Finally, we comment on an experiment to determine quantum phase uncertainty. Our analysis uses the operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$, which we feel is a more natural choice than $\stackrel{^}{\ensuremath{\varphi}}$. The use of ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ also avoids the multivalued problems associated with $\stackrel{^}{\ensuremath{\varphi}}$.

Physical Review A•Jul 1, 1992•11 citations
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.
We review some quantum-phase descriptions of optical fields. We focus on real fields that can be generated in practice in various nonlinear optical processes. Thus, we rather avoid discussions of phase formalisms as such and try to exploit their practical applicability in the description of optical fields. In the description of the phase properties we use two different, though related formalisms: the Pegg-Barnett Hermitian phase formalism and the formalism based on sss-parametrized phase distributions.
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.
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.
Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)] have proposed a formalism for a Hermitian optical phase operator, constructed from the so-called phase states. In this paper we give an alternative derivation. We begin by defining a unitary phase operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ from the relationship $\stackrel{^}{a}={e}^{i\stackrel{^}{\ensuremath{\varphi}}}R(\stackrel{^}{N})$ ($\stackrel{^}{a}$ is an annihilation operator for a single-mode boson field), but borrow the idea of a finite- (but arbitrarily large) dimensional space from Pegg and Barnett. Finally, we comment on an experiment to determine quantum phase uncertainty. Our analysis uses the operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$, which we feel is a more natural choice than $\stackrel{^}{\ensuremath{\varphi}}$. The use of ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ also avoids the multivalued problems associated with $\stackrel{^}{\ensuremath{\varphi}}$.