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Physical Review A

Comment on ‘‘Phase-difference operator’’

David Thomas Pegg, John A. Vaccaro

Published Jan 1, 1995• 14 citations
Publisher

Abstract

We examine the operator \ensuremath{\Phi}${\mathrm{^}}_{12}$ proposed by Luis and S\'anchez-Soto [Phys. Rev. A 48, 4702 (1993)] as a possible description of the phase difference between two electromagnetic fields. We show that there is a difficulty in directly representing the phase difference between arbitrary fields by this operator, but conclude that the incorporation of the operator into an alternative definition of phase difference constitutes a valuable alternative approach to the quantum description of phase.

DOI
10.1103/physreva.51.859

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We examine the operator \ensuremath{\Phi}{\mathrm{^}}_{12} proposed by Luis and S'anchez-Soto [Phys. Rev. A 48, 4702 (1993)] as a possible description of the phase difference between two electromagnetic fields. We show that there is a difficulty in directly representing the phase difference between arbitrary fields by this operator, but conclude that the incorporation of the operator into an alternative definition of phase difference constitutes a valuable alternative approach to the quantum description of phase.
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.
Interpretational difficulties of a quantum measurement [J. W. Noh, A. Foug\`eres, and L. Mandel, Phys. Rev. A 45, 424 (1992)] are pointed out. The nonunitarity of the respective exponential phase operator reflects an enhanced noise, which can be reduced by discarding ambiguous output data. Consequently, dispersion of such measurements does not represent a meaningful performance measure relevant to the input state. Moreover, the proposed model of quantum phase measurements provides a discrete spectrum of the quantum phase operator, whereas the phase-shift parameter is continuous.
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 $\stackrel{^}{\ensuremath{\varphi}}$.
We introduce a unitary operator representing the exponential of the phase difference between two modes of the electromagnetic field. The eigenvalue spectrum has a discrete character that is fully analyzed. We relate this operator with a suitable polar decomposition of the Stokes parameters of the field, obtaining a natural classical limit. The cases of weakly and highly excited states are considered, discussing to what extent it is possible to talk about the phase for a single-mode field. This operator is applied to some interesting two-mode fields.
The difficulties in formulating a natural and simple operator description of the phase of a quantum oscillator or single-mode electromagnetic field have been known for some time. We present a unitary phase operator whose eigenstates are well-defined phase states and whose properties coincide with those normally associated with a phase. The corresponding phase eigenvalues form only a dense subset of the real numbers. A natural extension to the definition of a time-measurement operator yields a corresponding countable infinity of eigenvalues.