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 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.
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.
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.
A parallel development of the semiclassical and quantum statistics of multispatiotemporal mode direct, homodyne, and heterodyne detection using an ideal (except for its subunity quantum efficiency) photon detector is presented. Particular emphasis is placed on the latter two coherent detection Configurations. The primary intent is to delineate the semiclassical theory's regime of validity and to show, within this regime of validity, how the quantum theory's signal quantum noise, local oscillator quantum noise, the quantum noise incurred because of subunity detector quantum efficiency, plus (for heterodyning only) image band quantum noise produce the quantitative equivalent of the semiclassical theory's local oscillator shot noise. The effects of classical fluctuations on the local oscillator, and the recently suggested dual-detector arrangement for suppressing these fluctuations, are treated. It is Shown that previous studies of this arrangement have neglected a potentially significant noise contribution.
A parallel development of the semiclassical and quantum statistics of multi-spatiotemporal mode direct, homodyne, and heterodyne detection using an ideal (except for its sub-unity quantum efficiency) photon detector is presented. Particular emphasis is placed on the latter two coherent detection configurations. The primary intent is to delineate the semiclassical theory's regime of validity and to show, within this regime of validity, how the quantum theory's signal quantum noise, local oscillator quantum noise, the quantum noise incurred because of sub-unity detector quantum efficiency, plus (for heterodyning only) image band quantum noise produce the quantitative equivalent of the semiclassical theory's local oscillator shot noise. The effects of classical fluctuations on the local oscillator, and the recently suggested dual- detector arrangement for suppressing these fluctuations, are treated. It is shown that previous studies of this arrangement have neglected a potentially significant noise contribution.
Squeezed states of light can be detected for precision measurements with a heterodyne detector by use of a bichromatic field as the local oscillator, due to the phase-sensitive nature of the device. However, divergence consists of the theoretical description of the quantum noise performance of this detector. Two existing theoretical models are briefly reviewed, with one model predicting a 3 dB quantum noise floor change in detection of two-mode squeezed light if the local oscillator is replaced by a monochromatic one, whereas the other model foretells the same noise floor no matter which local oscillator is used. An experiment on heterodyne detection of two-mode squeezed light is carried out to put the two models under test. No significant difference in the noise floor level is observed between the two detectors, showing the noiseless property of both detectors. This work should be of importance for the understanding of noise origin in optical detection and of great interest in practical application for squeezing-enhanced audio band gravitational-wave signal searching.
We present an experimental scheme that achieves ideal phase detection on a two-mode field. The two modes a and b are the signal and image band modes of a heterodyne detector, with the field approaching an eigenstate of the photocurrent Z^=a+${\mathit{b}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$. The field is obtained by means of a high-gain phase-insensitive amplifier followed by a high-transmissivity beam splitter with a strong local oscillator at the frequency of one of the two modes.