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Physical Review A

Lower bounds on phase sensitivity in ideal and feasible measurements

Giacomo Mauro D’Ariano, Matteo G. A. Paris

Published Apr 1, 1994• 58 citations
Publisher

Abstract

The detection of the phase shift between a single mode of the em field and a local reference oscillator is analyzed in the proper framework of quantum estimation theory. Such a fully quantum treatment clarifies the meaning of the ``operational approach'' suggested by Noh, Foug\'eres, and Mandel [Phys. Rev. A 45, 424 (1992)], in which different measurement schemes correspond to different phase operators. We show that the phase shift is actually measured in the form of the polar angle between two real output photocurrents, namely through a joint measurement of two conjugated quadratures of the field. This scheme is the only feasible one for detecting the quantum phase, and it is equivalently performed by either heterodyne detection or double-homodyne detection of the field. On the contrary, the customary homodyne detection (and, generally, any kind of measurement of a single phase-dependent variable) is more properly a zero-point phase measurement. As a definition of sensitivity we consider the output rms noise of the detection scheme, the only relevant one for actual experiments, in contrast with many other different notions currently adopted in the literature. We show that the rms phase sensitivity versus the average photon number n\ifmmode\bar\else\textasciimacron\fi{} is bounded by the ideal limit \ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\sim}n${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}1}$, whereas for the feasible schemes the bound is \ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\sim}n${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}2/3}$, in between the shot-noise level \ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\sim}n${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}1/2}$ and the ideal bound.The latter can acutally be achieved by single homodyne detection of the suitable squeezed states, but only in the neighborhood of a fixed zero-phase working point. The phase sensitivity bound \ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\sim}n${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}2/3}$ can be reached using coherent states with only 2% of squeezing photons, in contrast with the homodyne-detection bound \ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\sim}n${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}1}$ which is reached with 50%. The uncertainty product of two conjugated phase quadratures largely exceeds the Heisenberg limit, even for the ideal joint measurement. We also show that no sizeable improvement in sensitivity is found for detection schemes which involve wideband states, at least for the case of nonentangled states. At the end of the paper we give an extensive table of asymptotic sensitivities at large n\ifmmode\bar\else\textasciimacron\fi{} for both ideal and feasible schemes.

DOI
10.1103/physreva.49.3022

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Quantum phase measurement: A system-theory perspective

Jeffrey H. Shapiro, Scott Roger Shepard

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}$a^; its phase eigenkets generate the probability operator measure (POM) for maximum-likelihood phase estimation. When used with an optimum input state, the SG-POM produces a local phase accuracy \ensuremath{\delta}\ensuremath{\varphi}\ensuremath{\sim}1/${\mathit{N}}^{2}$, 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.

Physical Review A•Apr 1, 1991•253 citations
The detection of the phase shift between a single mode of the em field and a local reference oscillator is analyzed in the proper framework of quantum estimation theory. Such a fully quantum treatment clarifies the meaning of the ``operational approach'' suggested by Noh, Foug'eres, and Mandel [Phys. Rev. A 45, 424 (1992)], in which different measurement schemes correspond to different phase operators. We show that the phase shift is actually measured in the form of the polar angle between two real output photocurrents, namely through a joint measurement of two conjugated quadratures of the field. This scheme is the only feasible one for detecting the quantum phase, and it is equivalently performed by either heterodyne detection or double-homodyne detection of the field. On the contrary, the customary homodyne detection (and, generally, any kind of measurement of a single phase-dependent variable) is more properly a zero-point phase measurement. As a definition of sensitivity we consider the output rms noise of the detection scheme, the only relevant one for actual experiments, in contrast with many other different notions currently adopted in the literature. We show that the rms phase sensitivity versus the average photon number n\ifmmode\bar\else\textasciimacron\fi{} is bounded by the ideal limit \ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\sim}n\ifmmode\elseˉ\textasciimacron\fi\ensuremath−1{\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}1}\ifmmode\elseˉ​\textasciimacron\fi\ensuremath−1, whereas for the feasible schemes the bound is \ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\sim}n\ifmmode\elseˉ\textasciimacron\fi\ensuremath−2/3{\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}2/3}\ifmmode\elseˉ​\textasciimacron\fi\ensuremath−2/3, in between the shot-noise level \ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\sim}n\ifmmode\elseˉ\textasciimacron\fi\ensuremath−1/2{\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}1/2}\ifmmode\elseˉ​\textasciimacron\fi\ensuremath−1/2 and the ideal bound.The latter can acutally be achieved by single homodyne detection of the suitable squeezed states, but only in the neighborhood of a fixed zero-phase working point. The phase sensitivity bound \ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\sim}n\ifmmode\elseˉ\textasciimacron\fi\ensuremath−2/3{\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}2/3}\ifmmode\elseˉ​\textasciimacron\fi\ensuremath−2/3 can be reached using coherent states with only 2% of squeezing photons, in contrast with the homodyne-detection bound \ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\sim}n\ifmmode\elseˉ\textasciimacron\fi\ensuremath−1{\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}1}\ifmmode\elseˉ​\textasciimacron\fi\ensuremath−1 which is reached with 50%. The uncertainty product of two conjugated phase quadratures largely exceeds the Heisenberg limit, even for the ideal joint measurement. We also show that no sizeable improvement in sensitivity is found for detection schemes which involve wideband states, at least for the case of nonentangled states. At the end of the paper we give an extensive table of asymptotic sensitivities at large n\ifmmode\bar\else\textasciimacron\fi{} for both ideal and feasible schemes.
The double-homodyne and the heterodyne detection schemes for phase shifts between two synchronous modes of the electromagnetic field are analyzed in the framework of quantum estimation theory. The probability operator-valued measures (POM's) of the detectors are evaluated and compared with the ideal one in the limit of strong local reference oscillator. The present operational approach leads to a reasonable definition of phase measurement, whose sensitivity is actually related to the output r.m.s. noise of the photodetector. We emphasize that the simple-homodyne scheme does not correspond to a proper phase-shift measurements as it is just a zero-point detector. The sensitivity of all detection schemes are optimized at fixed energy with respect to the input state of radiation. It is shown that the optimal sensitivity can be actually achieved using suited squeezed states.
The standard single-shot estimate for the phase of a single-mode pulse of light is the argument of the complex amplitude of the field. This complex amplitude can be measured by heterodyne detection, in which the local oscillator is detuned from the system so that all quadratures are sampled equally. Because different quadratures do not commute, such a measurement introduces noise into the phase estimate, with a variance scaling as ${N}^{\ensuremath{-}1}$, where NNN is the maximum photon number. This represents the shot-noise limit or standard quantum limit (SQL). Recently, one of us [H.M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] proposed a way to improve upon this: a real-time feedback loop can control the local oscillator phase to be equal to the estimated system phase plus $\ensuremath{\pi}/2$, so that the phase quadrature of the system is measured preferentially. The phase estimate used in the feedback loop at time ttt is a functional of the photocurrent from time 000 up to time ttt in the single-shot measurement. In this paper we consider a very simple feedback scheme involving only linear electronic elements. Approaching the problem from semiclassical detection theory, we obtain analytical results for asymptotically large photon numbers. Specifically, we are able to show that the noise introduced by the measurement has a variance scaling as ${N}^{\ensuremath{-}3/2}$. This is much less than the SQL variance, but still much greater than the minimum intrinsic phase variance which scales as ${N}^{\ensuremath{-}2}$. We briefly discuss the effect of detector inefficiencies and delays in the feedback loop.
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 .
The phase of a single-mode field can be measured in a single-shot measurement by interfering the field with an effectively classical local oscillator of known phase. The standard technique is to have the local oscillator detuned from the system (heterodyne detection) so that it is sometimes in phase and sometimes in quadrature with the system over the course of the measurement. This enables both quadratures of the system to be measured, from which the phase can be estimated. One of us [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] has shown recently that it is possible to make a much better estimate of the phase by using an adaptive technique in which a resonant local oscillator has its phase adjusted by a feedback loop during the single-shot measurement. In a previous work [H. M. Wiseman and R. B. Killip, Phys. Rev. A 56, 944 (1997)] we presented a semiclassical analysis of a particular adaptive scheme, which yielded asymptotic results for the phase variance of strong fields. In this paper we present an exact quantum mechanical treatment. This is necessary for calculating the phase variance for fields with small photon numbers, and also for considering figures of merit other than the phase variance. Our results show that an adaptive scheme is always superior to heterodyne detection as far as the variance is concerned. However, the tails of the probability distribution are surprisingly high for this adaptive measurement, so that it does not always result in a smaller probability of error in phase-based optical communication.
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.