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Journal of Physics A Mathematical and General

Phase in quantum optics

Stephen Mark Barnett, David Thomas Pegg

Published Dec 21, 1986• 206 citations
Publisher

Abstract

Dirac's prescription for quantisation does not lead to a unique phase operator for the electromagnetic field. The authors consider the commonly employed phase operators due to Susskind and Glogower (1964) and their extension to unitary exponential phase operators. However, they find that phase measuring experiments respond to a different operator. They discuss the form of the measured phase operator and its properties.

DOI
10.1088/0305-4470/19/18/030

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Dirac's prescription for quantisation does not lead to a unique phase operator for the electromagnetic field. The authors consider the commonly employed phase operators due to Susskind and Glogower (1964) and their extension to unitary exponential phase operators. However, they find that phase measuring experiments respond to a different operator. They discuss the form of the measured phase operator and its properties.
A new set of phase operators for the electromagnetic field has been defined by an extension of squeezed-state theory. In several cases these operators have similar or the same properties as the previously defined (Susskind–Glogower) operators. However, the new operators also have unique properties of their own. The expectation values of the phase operators have been found for a squeezed state and been shown to reduce to the expected classical values in the limit of small amounts of squeezing and large field excitation.
We address the problem of identifying an operational prescription for quantum phase measurements. As is known, different experiments can lead to different measured phase operators. However, we show that ambiguities of interpretation can arise even if a single experiment, such as that of Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991)], is chosen as defining a phase measurement. We show by reference to a simple but fundamental example that it is not possible to deduce a unique phase difference from the measurements.
We define a unitary phase operator for photons in a single momentum mode. The operator acts on a Hilbert space with basis consisting of all number states in both polarizations. The Susskind Glogower operator, E^=eiϕ^{\hat{E}} = e^{i \hatϕ}E^=eiϕ^​, thus defined is unitary and acts as a true ladder operator over all the states. It satisfies the commutation relation [E^,n^]=E^[{\hat{E}}, {\hat{n}}] = {\hat{E}}[E^,n^]=E^; consequently, the phase ϕ^\hatϕϕ^​ satisfies the canonical commutation relation [n^,ϕ^]=i[{\hat{n}}, {\hatϕ}] = i[n^,ϕ^​]=i. The eigenstates of the Susskind Glogower operator are found. Our model is able to directly account for phase measurements when polarization changes cannot be neglected and at the same time gives a well defined phase operator.
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.