Phase Operator for the Photon Field and an Index Theorem

TL;DR

This paper explores the mathematical properties of phase operators in quantum optics, revealing an index-related anomaly that impacts their consistent definition and uncertainty characteristics, and draws parallels to gauge theory anomalies.

Contribution

It introduces an index theorem perspective to the phase operator problem, linking it to quantum anomalies and analyzing the limitations of hermitian phase operators in large-dimensional truncated theories.

Findings

Hermitian phase operator of Pegg and Barnett has zero index and deviates from minimum uncertainty at large s.

Susskind-Glogower phase operator has a unit index, leading to anomalous identities.

The phase operator problem is analogous to chiral anomaly, suggesting a new class of quantum anomaly.

Abstract

An index relation $dim\ ker\ a^{\dagger}a - dim\ ker\ aa^{\dagger} = 1$ is satisfied by the creation and annihilation operators $a^{\dagger}$ and $a$ of a harmonic oscillator. A hermitian phase operator, which inevitably leads to $dim\ ker\ a^{\dagger}a - dim\ ker\ aa^{\dagger} = 0$, cannot be consistently defined. If one considers an $s+1$ dimensional truncated theory, a hermitian phase operator of Pegg and Barnett 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 a new class of quantum anomaly. From an anomaly view point ,it is not surprising that the phase operator of Susskind and Glogower, which carries a unit index, leads to an anomalous identity and an anomalous commutator.