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Journal of Physics A Mathematical and General

Feasible phase detection with ideal sensitivity

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

Published Sep 7, 1996• 2 citations
Publisher

Abstract

We present a first feasible scheme for detecting the phase of a single mode of radiation with ideal RMS sensitivity versus the average number of photons . It involves pairs of alternate independent homodyne measurements of two conjugated quadratures on a weakly squeezed state at the input. Nonunit quantum efficiency of photodetectors degrades phase sensitivity leading to a power law , with increasing versus .

DOI
10.1088/0305-4470/29/17/030

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We present a first feasible scheme for detecting the phase of a single mode of radiation with ideal RMS sensitivity versus the average number of photons . It involves pairs of alternate independent homodyne measurements of two conjugated quadratures on a weakly squeezed state at the input. Nonunit quantum efficiency of photodetectors degrades phase sensitivity leading to a power law , with increasing versus .
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.
The problem of noise from external feedback control is addressed for homodyne detection of optical phase shifts. Two different schemes are analyzed: the conventional single-homodyne detection, and a two-quadrature scheme, which involves couples of independent measurements of two conjugated quadratures. The effectiveness of the feedback mechanism in both schemes is studied versus the feedback coupling parameter and high-sensitive detection regimes are discussed. The effect of nonunit quantum efficiency of photodetectors is also considered. It is shown that nonunit quantum efficiency affects phase sensitivity in the single-quadrature scheme to a larger extent than in the double-quadrature one. \textcopyright{} 1996 The American Physical Society.
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 performance of phase-sensing interferometers employing squeezed states and homodyne detection is analyzed and compared to the performance of systems employing direct detection. Standard differenced direct-detection Michelson and Mach-Zehnder interferometers are shown to be suboptimal in the sense that an observation/measurement-noise coupling occurs, which can degrade performance. Homodyne-detection interferometers in which the phase shift in one arm is the conjugate of that in the other arm do not suffer from the preceding drawback. Overall, however, the performance of differenced direct-detection and homodyne-detection interferometers is similar in single-frequency operation. In particular, both detection schemes reach the standard quantum limit on position-measurement sensitivity in single-frequency interferometric gravity-wave detectors at roughly the same average photon number. This limit arises from back action in the form of radiation pressure fluctuations entering through the energy-phase uncertainty principle. Multifrequency devices can circumvent this uncertainty principle, as illustrated by the conceptual design given for a two-frequency interferometer which can greatly surpass the standard quantum limit on position sensing. This configuration assumes that ideal photodetectors respond to photon flux rather than energy flux.
We propose a model for the measurement of an arbitrary multimode entangled state of the cavity field using two-photon correlated emission laser. We consider two cases: (a) The modes have different frequencies and are detected separately and (b) the modes consist of two orthogonal polarization states and are detected using a single balanced homodyne detector. The basic idea is to amplify the initial multimode state such that there is no-noise in the quadrature of interest and all the noise is fed into the conjugate quadrature component. The amplified noise-free quadrature is prepared in different phases and then corresponding quadrature distribution is measured. The Wigner function of the initial multimode entangled state is then reconstructed by using inverse Radon transformation. This scheme is insensitive to the noise associated with the nonunit efficiency of the detector in the homodyne detection measurement scheme.