We revisit the quantum phase operator
Φ introduced by Garrison and Wong. Denoting by
N the number operator, we provide a detailed proof of the Heisenberg commutation relation
ΦN−NΦ=iI on the natural maximal domain
D(ΦN)∩D(NΦ) as well as the failure of the Weyl commutation relations, and discuss some further interesting properties of this pair.
How does the classical notion of ``phase'' apply to a quantum harmonic oscillator H=1/2(q${\mathrm{^}}^{2}$+p${\mathrm{^}}^{2}$), [q^,p^]=i\ensuremath{\Elzxh}, which cannot have sharp position and momentum? A quantum state \ensuremath{\rho}^ 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 \ensuremath{\varphi}^ is that the phase distribution P(\ensuremath{\Phi})= Tr{\ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi})\ensuremath{\rho}^} attribute the correct sharp phase to any such ``classical phase'' state. This requires that the Weyl symbol [\ensuremath{\varphi}^
]w(q,p) of \ensuremath{\varphi}^ tend to \ensuremath{\theta} mod2\ensuremath{\pi} as r\ensuremath{\rightarrow}\ensuremath{\infty}, where \ensuremath{\theta}=${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$(p/q) and r=(
q2+
p2)1/2. There are infinitely many such phase operators. Each is expressible as \ensuremath{\varphi}^=[${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$(p^/q^)${]}_{\mathrm{\ensuremath{\Omega}}}$, where \ensuremath{\Omega} specifies an ordering rule for q^ and p^. The commutator -i[H^,\ensuremath{\varphi}^]=1-2\ensuremath{\pi}[\ensuremath{\delta}(${\mathrm{tan}}^{\mathrm{\ensuremath{-}}1}$p^/q^)${]}_{\mathrm{\ensuremath{\Omega}}}$ corresponds to the Poisson bracket {H,${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cl}}$${\mathrm{}}}_{\mathrm{}\mathrm{PB}}$=1-2\ensuremath{\pi}\ensuremath{\delta}(\ensuremath{\theta}) for the single-valued classical phase ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cl}}$=\ensuremath{\theta} mod2\ensuremath{\pi}. Phase states \ensuremath{\Gamma}^(\ensuremath{\Phi}) are defined by the condition that their Weyl symbols [\ensuremath{\Gamma}^(\ensuremath{\Phi})
]w(r,\ensuremath{\theta})\ensuremath{\rightarrow}\ensuremath{\delta}(\ensuremath{\theta}-\ensuremath{\Phi}) as r\ensuremath{\rightarrow}\ensuremath{\infty}. If moreover ${\mathrm{\ensuremath{\int}}}_{0}^{2\mathrm{\ensuremath{\pi}}}$d\ensuremath{\Phi}\ensuremath{\Gamma}^(\ensuremath{\Phi})=1^, then \ensuremath{\Gamma}^(\ensuremath{\Phi}) is a phase probability operator measure (POM). In particular, \ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi}) is a phase POM.Normalizable approximate phase states \ensuremath{\Gamma}${\mathrm{^}}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi}) are defined by [\ensuremath{\Gamma}${\mathrm{^}}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi})
]w(r,\ensuremath{\theta}) =2\ensuremath{\pi}${\mathrm{\ensuremath{\varepsilon}}}^{2}$${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\varepsilon}}\mathit{r}}$[\ensuremath{\Gamma}^(\ensuremath{\Phi})
]w(r,\ensuremath{\theta}), \ensuremath{\varepsilon}\ensuremath{\ll}1. Phase states are not, in general, ``pure orthogonal'' in the sense \ensuremath{\Gamma}^(\ensuremath{\Phi})\ensuremath{\Gamma}^(\ensuremath{\Phi}\ensuremath{'})=\ensuremath{\delta}(\ensuremath{\Phi}-\ensuremath{\Phi}\ensuremath{'})\ensuremath{\Gamma}^(\ensuremath{\Phi}), unless they are of the form \ensuremath{\delta}(\ensuremath{\varphi}^-\ensuremath{\Phi}). However, any phase state \ensuremath{\Gamma}${\mathrm{^}}_{1}$(\ensuremath{\Phi}) is trace orthogonal to any phase POM \ensuremath{\Gamma}${\mathrm{^}}_{2}$(\ensuremath{\Phi}), in the sense that Tr{\ensuremath{\Gamma}${\mathrm{^}}_{2}$(\ensuremath{\Phi}\ensuremath{'})\ensuremath{\Gamma}${\mathrm{^}}_{1}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi})}\ensuremath{\rightarrow}\ensuremath{\delta}(\ensuremath{\Phi}-\ensuremath{\Phi}\ensuremath{'}) as \ensuremath{\varepsilon}\ensuremath{\rightarrow}0. This implies that measurement of the POM \ensuremath{\Gamma}${\mathrm{^}}_{2}$(\ensuremath{\Phi}\ensuremath{'}) on the state \ensuremath{\Gamma}${\mathrm{^}}_{1}^{\mathrm{\ensuremath{\varepsilon}}}$(\ensuremath{\Phi}) yields the outcome \ensuremath{\Phi} with probability 1 as \ensuremath{\varepsilon}\ensuremath{\rightarrow}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 \ensuremath{\Delta}H\ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\ge}0, the lower limit \ensuremath{\Delta}H\ensuremath{\Delta}\ensuremath{\varphi}=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\ensuremath{\pi}), the uncertainty relation is effectively \ensuremath{\Delta}H\ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\ge}1/2. \textcopyright{} 1996 The American Physical Society.
In a series of papers, Barnett, Pegg (1986, 1988, 1989, 1992) and various co-authors have proposed a description of quantum phase by means of a collection of s-dimensional states and operators, s>or=1. We analyse the limiting procedure they employ for large s, which is known not to be compatible with quantum mechanics in the usual sense. Further, we supply a rigorous demonstration of the asymptotic limits of the `mean` and `variance` of their system of operators in coherent states. These values had previously been given but not justified mathematically. Our analysis, based on the asymptotic analysis of certain random variables, shows that the physical deductions that can be drawn from these limits are limited. We also prove that the (s+1)-dimensional `pure phase` LHW-states they consider form a sequence of approximate eigenvectors for the Weyl-quantized angle operator Delta ( phi ) and the Toeplitz phase operator X proposed by Garrison and Wong (1970), Popov and Yarunin (1992), and others. These states X s ( theta ), to our knowledge first introduced by Lerner, Huang and Waiters (1970), can be used to construct a system of measurement as in the usual quantum theory, sensitive to certain qualities of phase, but not all. Indeed, a feature of the Barnett-Pegg method, when it gives finite answers, is the construction of associated measurement systems for different observables. We give examples of sequences of (s+1)-dimensional devices which represent measurements significantly closer to ideal for X. This serves as a model for corresponding devices for Delta ( phi ), or indeed, any observable with a continuous spectrum, contingent on its spectral decomposition being obtained explicitly.
The well-known difficulties of defining a phase operator of an oscillator are considered from the point of view of the canonical transformation to action and phase-angle variables. This transformation turns out to be nonbijective, i.e., it is not a one-to-one onto mapping. In order to make possible the unitarity of its representations in quantum optics we should enlarge the Hilbert space of the problem. In this enlarged space we find a phase operator that, after projection, reproduces previous candidates to represent a well-behaved phase operator in the quantum domain.
The phase of a simple-harmonic vibration or wave is a useful and easily-understood concept in classical mechanics. It is the angle that determines the sinusoidal development of the motion, and it varies linearly with time, increasing by 360°(or 2π radians) over a complete cycle of oscillation. By contrast, the concept of phase in the quantum mechanics of harmonic motion has been fraught with difficulties, which have hampered its use. Recently, D T pegg and S M Barnett (Europhys. Lett. 6 483) have proposed a new definition of the quantum-mechanical phase operator, which they suggest will lead to important and far-reaching consequences.