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Physical Review A

Phase operators on Hilbert space

John A. Vaccaro

Published Apr 1, 1995• 25 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.51.3309

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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 $\stackrel{^}{\ensuremath{\varphi}}$.
The Pegg-Barnett phase-operator formalism utilizes a finite basis set to represent operators of the harmonic oscillator; this enables the phase to be represented by a Hermitian operator, but rests on taking the dimensionality of the basis set to infinity for observable quantities. Simultaneously, in their approach Pegg and Barnett consider quantum states of a harmonic oscillator which are normalized in the Fock space, i.e., the dimensionality of the basis set in which the states of the harmonic oscillator are defined is supposed to be infinite, while the phase operator is defined in the finite-dimensional basis set. In this paper we address the problem of a consistent definition of a coherent state within a finite state basis. We employ displacement operators to define such coherent states and numerically evaluate observables as a function of the size of the basis set. We investigate phase properties of these coherent states. We find that if the dimensionality of the state space is much larger than the mean occupation number of the coherent states, then the results obtained in the finite-dimensional basis are applicable in the case of a ordinary quantum-mechanical harmonic oscillator. These coherent states are minimum uncertainty states with respect to quadrature operators (i.e., the position and momentum operators) and do not exhibit quadrature squeezing. A weakly excited (compared with the dimensionality of the state space) coherent state in finite-dimensional basis is not strictly speaking a minimum uncertainty state with respect to the number and phase operators. We give definitions of amplitude and phase squeezing and show that weakly excited coherent states can be amplitude squeezed. In the high-intensity limit (again compared with the dimensionality of the state space) these states exhibit phase squeezing.
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.
Using the Pegg-Barnett phase operator formalism we have introduced the coherent phase state and the displaced phase state in a finite dimension. Using the definition of phase creation and annihilation operators we have constructed a displacement operator which yields coherent phase states and displaced phase states within a finite-dimensional basis and numerically evaluated observables as a function of the size of the basis set. We have shown that, when the dimension of the state space is much larger than the average excitation of phase quanta, then these states are the coherent phase state and displaced phase state of a harmonic oscillator which has an infinite number of eigenstates and exhibits phase squeezing up to 100%. The coherent phase state is a minimum-uncertainty state with respect to the phase quadrature operators but not the minimum-uncertainty state with respect to number and phase operators. The properties of coherent phase states for a two-state system are also investigated. It is interesting to realize that these coherent phase states and displaced phase states are exactly the same as the phase vacuum state and different phase states respectively for the single-mode light field case.
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.