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arXiv:quant-ph/0502027 · International Journal of Modern Physics B

UNIFIED TREATMENT OF HETERODYNE DETECTION: THE SHAPIRO–WAGNER AND CAVES FRAMEWORKS

GIULIO LANDOLFI, GIOVANNA RUGGERI, GIULIO SOLIANI

Published Jun 10, 2005• 1 citation
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Abstract

A comparative study is performed on two heterodyne systems of photon detectors expressed in terms of a signal annihilation operator and an image band creation operator called Shapiro–Wagner and Caves' frame, respectively. This approach is based on the introduction of a convenient operator [Formula: see text] which allows a unified formulation of both cases. For the Shapiro–Wagner scheme, where [Formula: see text], quantum phase and amplitude are exactly defined in the context of relative number state (RNS) representation, while a procedure is devised to handle suitably and in a consistent way Caves' framework, characterized by [Formula: see text], within the approximate simultaneous measurements of noncommuting variables. In such a case RNS phase and amplitude make sense only approximately.

DOI
10.1142/s0217979205029833

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Modern Physics Letters B•Jun 20, 1999•4 citations
A comparative study is performed on two heterodyne systems of photon detectors expressed in terms of a signal annihilation operator and an image band creation operator called Shapiro–Wagner and Caves' frame, respectively. This approach is based on the introduction of a convenient operator [Formula: see text] which allows a unified formulation of both cases. For the Shapiro–Wagner scheme, where [Formula: see text], quantum phase and amplitude are exactly defined in the context of relative number state (RNS) representation, while a procedure is devised to handle suitably and in a consistent way Caves' framework, characterized by [Formula: see text], within the approximate simultaneous measurements of noncommuting variables. In such a case RNS phase and amplitude make sense only approximately.
A comparative study is performed of two heterodyne systems of photon detectors expressed in terms of a signal annihilation operator and an image band creation operator, called Shapiro-Wagner and Caves' frame, respectively. This approach is based on the introduction of a convenient operator hatpsi\\hat \\psihatpsi which allows a unified formulation of both cases. For the Shapiro-Wagner scheme, where [hatpsi,hatpsidag]=0[\\hat \\psi, \\hat \\psi^{\\dag}] =0[hatpsi,hatpsidag]=0, quantum phase and amplitude are exactly defined in the context of relative number state (RNS) representation, while a procedure is devised to handle suitably and in a consistent way Caves' framework, characterized by [hatpsi,hatpsidag]neq0[\\hat \\psi, \\hat \\psi^{\\dag}] \\neq 0[hatpsi,hatpsidag]neq0, within the approximate simultaneous measurements of noncommuting variables. In such a case RNS phase and amplitude make sense only approximately.
The quantum limits on simultaneous phase and squared-amplitude measurements made via optical heterodyne detection on a single-mode radiation field are established. The analysis proceeds from a fully quantum mechanical treatment of heterodyning with ideal photon detectors. A high mean field uncertainty principle is proven for simultaneous phase and squared-amplitude observations under the condition that the signal and image band states are independent, and the image band has zero mean. Operator representations are developed which show that no such principle applies when arbitrary signal/image band dependence is permitted, although the mean observations are no longer functions of the signal field alone. A multimode two-photon coherent state illustrating this behavior at finite energy is exhibited. Potential applications for the resulting improved accuracy measurements are briefly described.
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 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 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{\mathit{N}}^{2}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.
We study the minimum uncertainty relation obeyed by the phase operator [Formula: see text] in two-mode coherent state. The operator is suitable for Shapiro–Wagner heterodyne phase measurement scheme. It is due to the |ξ> representation (see Eq. (4)) that the difficulty brought by nonlinear square root operation in [Formula: see text] can be avoided in calculating miscellaneous expectation values. Just as the single-mode coherent state | z1> makes uncertainty relation, satisfied by S–G phase operator, minimum for large |z1|2, we show that |z1,z2> makes uncertainty relation obeyed by [Formula: see text] minimum when |z1|=|z2| is large enough. Some figures are plotted to support our conclusion.