Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Modern Optics

Limiting Procedures for the Optical Phase Operator

Stephen Mark Barnett, David Thomas Pegg

Published Oct 1, 1992• 39 citations
Publisher

Abstract

We examine some of the attempts to describe the phase of a single field mode by a quantum operator acting in the conventional infinite Hilbert space. These operators lead to bizarre properties such as non-random phases for the number states and experience consistency difficulties when used to obtain a phase probability density. Moreover, in these approaches operator functions of phase are not simply functions of a phase operator. We show that these peculiarities do not arise when the Hermitian optical phase operator is employed. In our opinion, the problems associated with the descriptions of phase in conventional infinite Hilbert space arise from the nature of the limiting process.

DOI
10.1080/09500349214552141

Related papers

Quantum optical phase

David Thomas Pegg, Stephen Mark Barnett

The phase associated with a single mode of the electromagnetic field is complementary to the photon number. This simple idea leads us to construct a probability density for the phase. The phase operator cannot be represented exactly in the usual-infinite Hilbert space but can be constructed in a subspace of it and provides a valid representation of phase if used together with a suitable limiting procedure. We introduce the Hermitian optical phase operator and describe some of its most important properties. We also provide brief discussions of some alternative approaches to the phase problem and to the question of phase measurements. The paper concludes with an extensive bibliography.

Journal of Modern Optics•Feb 1, 1997•63 citations

Phase operators on Hilbert space

John A. Vaccaro

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 ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$. 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 ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ 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 ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ and so the correponding limit points are ordinary operators on ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$. 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.

Physical Review A•Apr 1, 1995•25 citations

Consistency of quantum descriptions of phase

Joan A. Vaccaro, David Thomas Pegg

In the Ψ-space limiting procedure limits of expectation values, rather than operator or states, are found as the state space dimensionality tends to infinity. This approach has been applied successfully to the calculation of the phase properties of various states of light, but its status as a valid quantum mechanical thory equivalent to the usual infinite Hilbert space (H-space) approach has not yet been fully accepted. Here we address this issue by investigating the formal relationship between the two approaches. We establish the consistency between the Ψ-space and H-space approaches for observables which are amenable to an H-space treatment. Such observables are represented in H by operators which are strong limits of Ψ-space operators and which obey the same algebra as the corresponding Ψ-space operators. The phase operator, however, exists in H only as a weak limit of a Ψ-space operator. For such limits the Ψ-space operator algebra is not preserved, which is the fundamental reason for the difficulties in constructing a consistent quantum description of phase in H. We show that for the phase observable the Ψ-space approach is consistent with the probability-operator measure (POM) method with the important distinction that, whereas the relation between non-orthogonal POMs and probability has to be accepted in the latter method as a postulate, the corresponding relation is derived in the Ψ-space approach. We conclude that the Ψ-space approach is not only equivalent to, but is also more fundamental than both the H-space and POM approaches.

Physica Scripta•Jan 1, 1993•38 citations

Comment on ‘‘Phase properties of the quantized single-mode electromagnetic field’’

Xin Ma, William Rhodes

We show that the approach by Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)] to construct a phase operator through a limiting procedure does not give a satisfactory description of the phase properties of the quantized electromagnetic field. We refute the claim that their approach predicts the same results of physical experiments on phase properties as the conventional Hilbert space approach. Their procedure could only be used as a definition of the phase, but it is not a good definition because the ``phase'' states they used to construct their ``phase'' operator are not of precisely defined phase contrary to their assertion.

Physical Review A•Mar 1, 1991•11 citations

Phase operator problem and infinitesimal analysis

Masanao Ozawa

To find an operator representation of the phase variable of a single-mode electromagnetic field, the Schrodinger representation is extended to a larger Hilbert space including states with infinite excitation by nonstandard analy- sis, so that the self-adjoint phase operator is acting on the extended Hilbert space.It is shown that this phase operator is a Naimark extension of the optimal probability operator-valued measure for the phase parameter already known in the quantum estimation theory, and is naturally considered as the controversial limit of the phase operators on finite dimensional spaces recently proposed by Pegg and Barnett.Eventually, two of recent promising attempts to obtain the correct statistics of the phase variable in quantum mechanics is synthesized in the present framework based on new mathematical ideas concerning the number system including infinite and infinitesimal numbers.

数理解析研究所講究録•Mar 1, 1994•0 citations

Quantum-optical Phase and Canonical Conjugation

David Thomas Pegg, Joan A. Vaccaro, Stephen Mark Barnett

We use a new limiting procedure, developed to study quantum-optical phase, to examine canonically conjugate operators in general. We find that Dirac's assumption that photon number and phase should be canonically conjugate variables, similar to momentum and position, is essentially correct. The difficulties with Dirac's approach are shown to arise through use of a form of the canonical commutator which, although the only possible form in the usual infinite Hilbert space approach, is not sufficiently general to be used as a model for a number-phase commutator. The approach in this paper unifies the theory of conjugate operators, which include photon number and phase, angular momentum and angle, and momentum and position as particular cases. The usual position-momentum commutator is regained from a more generally applicable expression by means of a domain restriction which cannot be used for the phase-number commutator.

Journal of Modern Optics•Nov 1, 1990•71 citations
We examine some of the attempts to describe the phase of a single field mode by a quantum operator acting in the conventional infinite Hilbert space. These operators lead to bizarre properties such as non-random phases for the number states and experience consistency difficulties when used to obtain a phase probability density. Moreover, in these approaches operator functions of phase are not simply functions of a phase operator. We show that these peculiarities do not arise when the Hermitian optical phase operator is employed. In our opinion, the problems associated with the descriptions of phase in conventional infinite Hilbert space arise from the nature of the limiting process.
The phase associated with a single mode of the electromagnetic field is complementary to the photon number. This simple idea leads us to construct a probability density for the phase. The phase operator cannot be represented exactly in the usual-infinite Hilbert space but can be constructed in a subspace of it and provides a valid representation of phase if used together with a suitable limiting procedure. We introduce the Hermitian optical phase operator and describe some of its most important properties. We also provide brief discussions of some alternative approaches to the phase problem and to the question of phase measurements. The paper concludes with an extensive bibliography.
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.
In the Ψ-space limiting procedure limits of expectation values, rather than operator or states, are found as the state space dimensionality tends to infinity. This approach has been applied successfully to the calculation of the phase properties of various states of light, but its status as a valid quantum mechanical thory equivalent to the usual infinite Hilbert space (H-space) approach has not yet been fully accepted. Here we address this issue by investigating the formal relationship between the two approaches. We establish the consistency between the Ψ-space and H-space approaches for observables which are amenable to an H-space treatment. Such observables are represented in H by operators which are strong limits of Ψ-space operators and which obey the same algebra as the corresponding Ψ-space operators. The phase operator, however, exists in H only as a weak limit of a Ψ-space operator. For such limits the Ψ-space operator algebra is not preserved, which is the fundamental reason for the difficulties in constructing a consistent quantum description of phase in H. We show that for the phase observable the Ψ-space approach is consistent with the probability-operator measure (POM) method with the important distinction that, whereas the relation between non-orthogonal POMs and probability has to be accepted in the latter method as a postulate, the corresponding relation is derived in the Ψ-space approach. We conclude that the Ψ-space approach is not only equivalent to, but is also more fundamental than both the H-space and POM approaches.
We show that the approach by Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)] to construct a phase operator through a limiting procedure does not give a satisfactory description of the phase properties of the quantized electromagnetic field. We refute the claim that their approach predicts the same results of physical experiments on phase properties as the conventional Hilbert space approach. Their procedure could only be used as a definition of the phase, but it is not a good definition because the ``phase'' states they used to construct their ``phase'' operator are not of precisely defined phase contrary to their assertion.
To find an operator representation of the phase variable of a single-mode electromagnetic field, the Schrodinger representation is extended to a larger Hilbert space including states with infinite excitation by nonstandard analy- sis, so that the self-adjoint phase operator is acting on the extended Hilbert space.It is shown that this phase operator is a Naimark extension of the optimal probability operator-valued measure for the phase parameter already known in the quantum estimation theory, and is naturally considered as the controversial limit of the phase operators on finite dimensional spaces recently proposed by Pegg and Barnett.Eventually, two of recent promising attempts to obtain the correct statistics of the phase variable in quantum mechanics is synthesized in the present framework based on new mathematical ideas concerning the number system including infinite and infinitesimal numbers.
We use a new limiting procedure, developed to study quantum-optical phase, to examine canonically conjugate operators in general. We find that Dirac's assumption that photon number and phase should be canonically conjugate variables, similar to momentum and position, is essentially correct. The difficulties with Dirac's approach are shown to arise through use of a form of the canonical commutator which, although the only possible form in the usual infinite Hilbert space approach, is not sufficiently general to be used as a model for a number-phase commutator. The approach in this paper unifies the theory of conjugate operators, which include photon number and phase, angular momentum and angle, and momentum and position as particular cases. The usual position-momentum commutator is regained from a more generally applicable expression by means of a domain restriction which cannot be used for the phase-number commutator.