The Garrison–Wong phase operator and the rotational covariance of phase states

Every mode of a quantized boson (photon) field is described as a quantum harmonic oscillator. In this description one meets the quantum phase problem, which has been reviewed in the paper [1]. Using the photon annihilation and creation operators â and â†, respectively, which obey the boson commutation relation [â, â†] = 1̂, with 1̂ an identity operator, the number operator is expressed as n̂ = â†â. Its eigenstates are the number states |n⟩, n̂|n⟩ = n|n⟩. These states are rotationally invariant, exp(i∆φn̂)|n⟩ = exp(i∆φn)|n⟩, with ∆φ a phase shift. The usual phase states |φ⟩, |φ⟩ = 1 √ 2π ∑∞ n=0 e |n⟩, are not orthogonal. They are “rotationally covariant”, exp(i∆φ)|φ⟩ = |φ + ∆φ⟩. As they yield an identity resolution, 1̂ = ∫ θ0+2π θ0 |φ⟩⟨φ|dφ, with θ0 a reference phase, they enable one to “quantize” any function of the phase M(φ), i.e., to introduce an operator M̂ = ∫ θ0+2π θ0 M(φ)|φ⟩⟨φ|dφ, where, e.g., M(φ) = cosφ, sinφ,φ. The Susskind-Glogower cosine and sine operators are mentioned in the paper [1] and the Garrison-Wong (GW) phase operator φ̂θ0 (notation by [2]) has been introduced in the paper [3] for θ0 = −π, in general it is studied in the paper [4]. For the determination of eigenstates of these operators corresponding to a real M(φ), the general result of the mathematical paper [5] can be utilized. For the connection with the literature, we introduce the eigenstates of the GW phase operator, |θ⟩GW, φ̂−π|θ⟩GW = θ|θ⟩GW with the property GW⟨θ|θ⟩GW = δ(θ − θ′), θ, θ′ ∈ (−π, π), where