Phase states and phase operators for the quantum harmonic oscillator

How does the classical notion of ‘‘phase’’ apply to a quantum harmonic oscillator H=1/2(q^2+p^2), [q^,p^]=iħ, which cannot have sharp position and momentum? A quantum state ρ^ can be assigned a definite classical phase only if it is a large-amplitude localized state. Our only demand, therefore, on a (Hermitian) phase operator φ^ is that the phase distribution P(Φ)= Trδ(φ^-Φ)ρ^ attribute the correct sharp phase to any such ‘‘classical phase’’ state. This requires that the Weyl symbol [φ^]w(q,p) of φ^ tend to θ mod2π as r→∞, where θ=tan-1(p/q) and r=(q2+p2)1/2. There are infinitely many such phase operators. Each is expressible as φ^=[tan-1(p^/q^)]Ω, where Ω specifies an ordering rule for q^ and p^. The commutator -i[H^,φ^]=1-2π[δ(tan-1p^/q^)]Ω corresponds to the Poisson bracket H,φcl PB=1-2πδ(θ) for the single-valued classical phase φcl=θ mod2π. Phase states Γ^(Φ) are defined by the condition that their Weyl symbols [Γ^(Φ)]w(r,θ)→δ(θ-Φ) as r→∞. If moreover ∫02πdΦΓ^(Φ)=1^, then Γ^(Φ) is a phase probability operator measure (POM). In particular, δ(φ^-Φ) is a phase POM.Normalizable approximate phase states Γ^ɛ(Φ) are defined by [Γ^ɛ(Φ)]w(r,θ) =2πɛ2e-ɛr[Γ^(Φ)]w(r,θ), ɛ≪1. Phase states are not, in general, ‘‘pure orthogonal’’ in the sense Γ^(Φ)Γ^(Φ′)=δ(Φ-Φ′)Γ^(Φ), unless they are of the form δ(φ^-Φ). However, any phase state Γ^1(Φ) is trace orthogonal to any phase POM Γ^2(Φ), in the sense that TrΓ^2(Φ′)Γ^1ɛ(Φ)→δ(Φ-Φ′) as ɛ→0. This implies that measurement of the POM Γ^2(Φ′) on the state Γ^1ɛ(Φ) yields the outcome Φ with probability 1 as ɛ→0. Phase measurements of the first kind are possible in principle; they would allow one to prepare (approximate) phase states and monitor their phase evolution in a quantum (phase) nondemolishing manner. Cases of special interest are the Susskind-Glogower and the Cahill-Glauber ordered phase states and operators. The energy-phase (or number-phase) uncertainty relation is ΔHΔφ≥0, the lower limit ΔHΔφ=0 being realized by pure number states; however, for states whose Wigner functions are localized away from the origin and from the extremities of the (single-valued) phase window (0,2π), the uncertainty relation is effectively ΔHΔφ≥1/2. © 1996 The American Physical Society.