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 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.
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.
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, 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.
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.
Quantum noise in optical heterodyning and homodyning, usually analyzed under the assumption that the photodetector output is conditionally Poisson,1,2 is frequently supposed to arise from local oscillator (LO) shot noise. However, a fully quantum mechanical analysis from Yuen and Shapiro3 has shown that it actually arises from the signal quantum fluctuation; in particular homodyning and heterodyning are realizations of abstract quantum measurements of the field quadratures. We analyze the effect of LO noise, both quantum and excess noise, on homodyne and heterodyne detection. For conventional input light, i.e., fields in coherent states or their random superpositions, we demonstrate how a conditionally Poisson calculation would yield the correct answer but incorrect interpretation for the detection quantum noise.
The experimental schemes for measuring quantum-mechanical phase properties of light suggested and partly also realized thus far, namely (i) amplification, with the help of a quantum amplifier, of the microscopic field before phase measurement, (ii) heterodyning the field with a strong local oscillator, and (iii) performing two separate homodyne measurements on the field after beam splitting, are compared from a theoretical point of view.They share the common feature that undesired noise enters the experimental setup, which makes the measurement fuzzy.It will be pointed out that all three schemes amount to measuring the Q function of the original field, and hence are fully equivalent.Since the Q function can be interpreted as a smoothed Wigner function, one may associate with the introduced noise a smoothing process in which intrinsically quantum-mechanical features displayed by finer details of the Wigner function -especially by the occurrence of negative values -are lost.As a consequence, the measured phase distribution will be broader than the "true" one based on the concept of a quantum-mechanical phase operator.In realistic experiments, the nonunit detection efficiency further deteriorates the measuring results.It will be shown that also this echect can be properly described by an (additional) smoothing process leading to a certain s-parametrized quasiprobability distribution, with a parameter s that is connected with the detection efficiency in a simple way, as the distribution that is actually measurable.