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16th Congress of the International Commission for Optics: Optics as a Key to High Technology

Phase in quantum optics

Stephen Mark Barnett

Published Jul 26, 1993• 1 citation
Publisher

Abstract

The Hermitian optical phase operator realises Dirac's conception of a phase, conjugate to the photon number for a mode of the electromagnetic field. However, much confusion has resulted from the failure to recognise that the single word "phase" is used to describe distinct concepts in the description of a quantised electromagnetic field mode. We compare and contrast these distinct phase concepts.

DOI
10.1117/12.2308446

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Physical Review A•Feb 1, 1989•873 citations
The Hermitian optical phase operator realises Dirac's conception of a phase, conjugate to the photon number for a mode of the electromagnetic field. However, much confusion has resulted from the failure to recognise that the single word "phase" is used to describe distinct concepts in the description of a quantised electromagnetic field mode. We compare and contrast these distinct phase concepts.
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.
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 usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.