The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.
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.
The phase operators P ± of the electromagnetic field, whose classical analogues are exp (∓ iθ), θ being the classical phase, and the number operator N associated with a harmonic oscillator mode are used to construct a class of operators T m = N(P −)m and T + m = (P +)m N, m = 1, 2,…. It is shown that T m, T + m and N satisfy a simple algebra. T m and T + m are used to construct model Hamiltonians H m describing the coupling of a two-level atom to a field that leads to decay and exact revivals in the atomic dynamics. In the second part of the paper, a family of unitary operators U m is constructed in terms of T m and T + m. For m = 1 and m = 2 the states generated by U m from the vacuum are non-Gaussian squeezed states. The variance properties of the squeezed states so constructed are analysed and compared with those of the Gaussian squeezed states. It is shown that the non-Gaussian squeezed states constructed in this paper have small values for the variance product that rise only logarithmically with for large values of , the mean number of quanta.
It has been argued that the positive-frequency part of the quantized electromagnetic field is the "observable" that one would most naturally associate with field measurements using quantum photodetectors. However, since it is possible in principle to make field measurements via the process of stimulated emission, the question of the possible solutions of the characteristic-value equation for the creation operator $a\ifmmode\dagger\else\textdagger\fi{}$ is examined. Various proofs are given to demonstrate that the characteristic kets of $a\ifmmode\dagger\else\textdagger\fi{}$ are not physically admissible states of the radiation field. The possible existence of other useful basis states besides
∣n〉, $|\ensuremath{\alpha}〉$, and states generated from these by unitary transformations is then considered. It is shown that when certain restrictions are placed on the correspondence between Hermitian combinations of the arbitrary non-normal operators
b and $b\ifmmode\dagger\else\textdagger\fi{}$ and the harmonic-oscillator variables
x,
p, and
H, then the only possible basis states are the coherent states $|\ensuremath{\alpha}〉$ and the number states
∣n〉. A $\ensuremath{\lambda}$-dependent variation on the photon annihilation operator
a is also considered. Its characteristic states for $\ensuremath{-}1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1$ are derived, and shown to form a complete set.
The difficulties in formulating a natural and simple operator description of the phase of a quantum oscillator or single-mode electromagnetic field have been known for some time. We present a unitary phase operator whose eigenstates are well-defined phase states and whose properties coincide with those normally associated with a phase. The corresponding phase eigenvalues form only a dense subset of the real numbers. A natural extension to the definition of a time-measurement operator yields a corresponding countable infinity of eigenvalues.