The Shapiro-Wagner phase measurement is addressed here. It is clarified that this concept includes an infinite number of operator realizations of quantum phase measurements, which may be characterized using polar and spectral decomposition. For accurate phase measurement, the regimes of ideal and optimized phase resolution are specified. The former is equivalent to the Susskind-Glogower or Hermitian phase concepts and needs infinite energy on the auxiliary input port. The ultimate phase resolution cannot surpass the limit 1/n, n being the total average number of photons entering both the signal and image input ports.
The Shapiro-Wagner phase measurement is addressed here. It is clarified that this concept includes an infinite number of operator realizations of quantum phase measurements, which may be characterized using polar and spectral decomposition. For accurate phase measurement, the regimes of ideal and optimized phase resolution are specified. The former is equivalent to the Susskind-Glogower or Hermitian phase concepts and needs infinite energy on the auxiliary input port. The ultimate phase resolution cannot surpass the limit 1/n, n being the total average number of photons entering both the signal and image input ports.
The Susskind-Glogower (SG) phase operator is shown to be the maximum-likelihood (ML) quantum measurement of optical phase for all input states. The ML performance of the SG measurement is optimized, at average photon number N, by an input state with number-ket representation of the form ${\mathrm{\ensuremath{\psi}}}_{\mathrm{n}}$=A/(1+n), for 0\ensuremath{\le}n\ensuremath{\le}M<\ensuremath{\infty}. The optimum phase-error behavior, measured by the reciprocal peak-likelihood, is proportional to 1/N2, as opposed to the 1/ \ensuremath{\surd}N and 1/N dependences of coherent-state and optimum squeezed-state interferometers.
The isometry U${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$,U${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$U^\ensuremath{\ne}U^U${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=1^, which relates the ideal phase measurement to the feasible phase measurement based on heterodyne detection, is specified here. Consequently, the Shapiro-Wagner phase measurement [IEEE J. Quantum Electron. QE-20, 803 (1984)] incorporates the ideal phase concepts of Susskind-Glogower [Physics 1, 49 (1964)] or Pegg-Barnett [Phys. Rev. A 39, 1665 (1989)].
Phase measurements on a single-mode radiation field are examined from a system-theoretic viewpoint. Quantum estimation theory is used to establish the primacy of the Susskind-Glogower (SG) phase operator; its phase eigenkets generate the probability operator measure (POM) for maximum likelihood phase estimation. A commuting observables description for the SG-POM on a signal x apparatus state space is derived. It is analogous to the signal-band x image-band formulation for optical heterodyne detection. Because heterodyning realizes the annihilation operator POM, this analogy may help realize the SG-POM. The wave function representation associated with the SG POM is then used to prove the duality between the phase measurement and the number operator measurement, from which a number-phase uncertainty principle is obtained, via Fourier theory, without recourse to linearization. Fourier theory is also employed to establish the principle of number-ket causality, leading to a Paley-Wiener condition that must be satisfied by the phase-measurement probability density function (PDF) for a single-mode field in an arbitrary quantum state. Finally, a two-mode phase measurement is shown to afford phase-conjugate quantum communication at zero error probability with finite average photon number. Application of this construct to interferometric precision measurements is briefly discussed.
Phase measurements on a single-mode radiation field of annihilation operator a^ are examined from a system-theoretic viewpoint. Quantum estimation theory is used to establish the primacy of the Susskind-Glogower (SG) phase operator, exp^(i\ensuremath{\varphi})==(a^a${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${)}^{\mathrm{\ensuremath{-}}1/2}$a^; its phase eigenkets generate the probability operator measure (POM) for maximum-likelihood phase estimation. When used with an optimum input state, the SG-POM produces a local phase accuracy \ensuremath{\delta}\ensuremath{\varphi}\ensuremath{\sim}1/N2, where N is the average photon number of the state. This performance is far superior to the \ensuremath{\delta}\ensuremath{\varphi}\ensuremath{\sim}1/ \ensuremath{\surd}N and \ensuremath{\delta}\ensuremath{\varphi}\ensuremath{\sim}1/N behaviors of coherent-state and optimized squeezed-state interferometers, respectively. A commuting-observables description for the SG-POM on an extended, signal\ifmmode\times\else\texttimes\fi{}apparatus, state space is derived. It is analogous to the signal-band\ifmmode\times\else\texttimes\fi{}image-band formulation for optical heterodyne detection. Because heterodyne detection realizes the a^-POM, this analogy may help realize the SG-POM. Two new classes of nonclassical states are identified through study of the SG operator---the coherent phase states and the squeezed phase states. These are direct analogs of the familiar coherent and squeezed states associated with the annihilation operator. Linear system theory is used to elicit a broader class of states---the rational phase states---which includes the coherent and squeezed phase states as special cases, and affords a filter-theory derivation of the minimum average-energy quantum state with prescribed phase-measurement statistics. Fourier theory is also used to prove a number-phase uncertainty relation without recourse to the usual linearization procedures, and to show that the Pegg-Barnett phase operator has the same measurement statistics as the SG-POM for arbitrary quantum states.
In a shot-noise-limited (coherent state) phasesensing interferometer, the root-mean-square (rms) phase error is ~ 1 / N av 1 / 2 where Nav is the average number of detected photons. In an optimized squeezed-state interferometer, the rms phase error goes as 1/Nav. We have considered the phase measurement problem from the viewpoint of quantum estimation theory.1 In particular, for the problem of optimum phase-shift estimation from observation of a single mode of the radiation field under an average photon number constraint, we have found both the optimum quantum state and the optimum quantum measurement. The former turns out to have a truncated number-state expansion whose coefficients, out to the truncation point, fall off as 1/n. The latter turns out to be the Susskind-Glogower phase operator. More important, the resulting phase estimation performance improves as 1 / N av 2 .
It is shown that there exist two kinds of operational (or measured) phase operators, those of the Hradil type [Phys. Rev. A 47 (1993), 4532], defined on the entire Hilbert space, and those of the Noh-Fougères-Mandel type [Phys. Rev. A 45 (1992), 424], defined only on a subspace. As a consequence, strictly speaking, the Noh-Fougères-Mandel measured phase cannot be defined in the case of, for example, coherent states, squeezed states and the vacuum. The difference between these operators is made clear by using the concept of the Bargmann distance between a quantum state and its “reduced” state. Since Noh et al. did not use the correct variances of the Susskind-Glogower and Pegg-Barnett phase operators when they compared their data with the results predicted through use of these operators, the correct expressions in that case are presented. Comparing the measured operators with the Hermitian phase operator in the extended Susskind-Glogower formalism, we show that the quantum phase given by the Hermitian phase operator is more appropriate than the measured phases. The variance of the Hermitian operator in the vacuum is consistent with the complementarity of the photon number and phase, which implies that the vacuum has a uniform phase distribution. Hradil's measured operator does not have such a property. Although the Noh-Fougères-Mandel operator has such a property in a particular limit, this operator does not represent, in principle, a meaningful measure of the phase measurement in a typical quantum region, because the Bargmann distance becomes large in such a region.