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^]=i{}, 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})= Tr{{δ}({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} mod2{π} 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}^{{{-}}1}p^/q⁾{]}_{{{Ω}}}corresponds to the Poisson bracket H,{{{φ}}}clPB=1-2{π}{δ}({θ}) for the single-valued classical phase φcl={θ} mod2{π}. 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 ∫₀^2πd{Φ}{Γ}^({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}^^ε({Φ}) are defined by [{Γ}^^ε({Φ})]w(r,{θ}) =2{π}ε²{e}^{{{-}}{{ε}}r}[Γ⁽Φ){]}w$(r,{θ}), {ε}{}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 Tr{{Γ}^₂({Φ}{'}){Γ}^₁^ε({Φ})}{→}{δ}({Φ}-{Φ}{'}) as {ε}{→}0. This implies that measurement of the POM {Γ}^₂({Φ}{'}) on the state {Γ}^₁^ε({Φ}) yields the outcome {Φ} with probability 1 as {ε}{→}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 {Δ}H{Δ}{φ}{≥}0, the lower limit {Δ}H{Δ}{φ}=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 (0,2{π}), the uncertainty relation is effectively {Δ}H{Δ}{φ}{≥}1/2. {} 1996 The American Physical Society.