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arXiv:hep-th/9411066 · Physical Review A

Phase operator for the photon field and an index theorem

Kazuo Fujikawa

Published Oct 1, 1995• 33 citations
arXivHTMLPDF

Abstract

An index relation dim ker ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a-dim ker ${\mathrm{aa}}^{\ifmmode^\circ\else\textdegree\fi{}}$=1 is satisfied by the creation and annihilation operators ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ and a of a harmonic oscillator, where dim ker ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a, for example, stands for the number of normalizable basis vectors ${\mathit{u}}_{\mathit{n}}$ that satisfy ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{au}}_{\mathit{n}}$=0. A Hermitian phase operator, which inevitably leads to dim ker ${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a-dim ker ${\mathrm{aa}}^{\ifmmode^\circ\else\textdegree\fi{}}$=0, cannot be consistently defined. If one considers an (s+1)-dimensional truncated theory, a Hermitian phase operator of Pegg and Barnett [Europhys. Lett. 6, 483 (1988)] which carries a vanishing index can be defined. However, for arbitrarily large s, we show that the vanishing index of the Hermitian phase operator of Pegg and Barnett causes a substantial deviation from minimum uncertainty in a characteristically quantum domain with small average photon numbers. We also mention an interesting analogy between the present problem and the chiral anomaly in gauge theory which is related to the Atiyah-Singer index theorem. It is suggested that the phase operator problem related to the above analytic index may be regarded as an alternative class of quantum anomaly. From an anomaly viewpoint, it is not surprising that the phase operator of Susskind and Glogower [Physics 1, 49 (1964)], which carries a unit index, leads to an anomalous identity and an anomalous commutator.

DOI
10.1103/physreva.52.3299

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An index relation dim ker a\ifmmode∘\else\textdegree\fi{\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}a\ifmmode∘\else\textdegree\fia-dim ker aa\ifmmode∘\else\textdegree\fi{\mathrm{aa}}^{\ifmmode^\circ\else\textdegree\fi{}}aa\ifmmode∘\else\textdegree\fi=1 is satisfied by the creation and annihilation operators a\ifmmode∘\else\textdegree\fi{\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}a\ifmmode∘\else\textdegree\fi and a of a harmonic oscillator, where dim ker a\ifmmode∘\else\textdegree\fi{\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}a\ifmmode∘\else\textdegree\fia, for example, stands for the number of normalizable basis vectors un{\mathit{u}}_{\mathit{n}}un​ that satisfy {\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{au}}_{\mathit{n}}=0. A Hermitian phase operator, which inevitably leads to dim ker a\ifmmode∘\else\textdegree\fi{\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}a\ifmmode∘\else\textdegree\fia-dim ker aa\ifmmode∘\else\textdegree\fi{\mathrm{aa}}^{\ifmmode^\circ\else\textdegree\fi{}}aa\ifmmode∘\else\textdegree\fi=0, cannot be consistently defined. If one considers an (s+1)-dimensional truncated theory, a Hermitian phase operator of Pegg and Barnett [Europhys. Lett. 6, 483 (1988)] which carries a vanishing index can be defined. However, for arbitrarily large s, we show that the vanishing index of the Hermitian phase operator of Pegg and Barnett causes a substantial deviation from minimum uncertainty in a characteristically quantum domain with small average photon numbers. We also mention an interesting analogy between the present problem and the chiral anomaly in gauge theory which is related to the Atiyah-Singer index theorem. It is suggested that the phase operator problem related to the above analytic index may be regarded as an alternative class of quantum anomaly. From an anomaly viewpoint, it is not surprising that the phase operator of Susskind and Glogower [Physics 1, 49 (1964)], which carries a unit index, leads to an anomalous identity and an anomalous commutator.
An index relation dim ker a−dim ker a†=1dim\ ker\ a - dim\ ker\ a^{\dagger} = 1dim ker a−dim ker a†=1 is satisfied by the creation and annihilation operators a†a^{\dagger}a† and aaa of a harmonic oscillator. Implications of this analytic index on the possible form of the phase operator are discussed. A close analogy between the present phase operator problem and chiral anomaly in gauge theory, which is associated with Atiyah-Singer index theorem, is emphasized.
The notion of index is applied to analyze the phase operator problem associated with the photon. We clarify the absence of the hermitian phase operator on the basis of an index consideration. We point out an interesting analogy between the phase operator problem and the chiral anomaly in gauge theory, and an appearance of a new class of quantum anomaly is noted. The notion of index, which is invariant under a wide class of continuous deformation, is also shown to be useful to characterize the representations of Q-deformed oscillator algebra.
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 $\stackrel{^}{\ensuremath{\varphi}}$.
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.
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.