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Unitary realizations of the ideal phase measurement

Buscemi, F, D'Ariano, G M, Sacchi, M F

Published Apr 10, 2003• 2 citations
OpenAlex

Abstract

We explicitly construct a large class of unitary transformations that allow to perform the ideal estimation of the phase-shift on a single-mode radiation field. The ideal phase distribution is obtained by heterodyne detection on two radiation modes after the interaction.

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One of the most delicate problems in quantum optics is the proper description of the phase of a single-mode radiation field. One has first to devise an experimental scheme for phase measurement, thus giving an operational definition of phase, and afterwards search for the proper quantum-mechanical description of the experimental procedure. Several such schemes which differ distinctly in their experimental setup have already been proposed and even partly realized. In this paper, I will describe them in some detail and present the outlines of their theoretical analysis.>

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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 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.
One of the most delicate problems in quantum optics is the proper description of the phase of a single-mode radiation field. One has first to devise an experimental scheme for phase measurement, thus giving an operational definition of phase, and afterwards search for the proper quantum-mechanical description of the experimental procedure. Several such schemes which differ distinctly in their experimental setup have already been proposed and even partly realized. In this paper, I will describe them in some detail and present the outlines of their theoretical analysis.>