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Acta Physica Polonica A

Realistic Measurement of Phase

Henryk Paul, Ulf Leonhardt

Published Jul 1, 1994• 3 citations

Abstract

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.

DOI: 10.12693/aphyspola.86.213

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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.

An overview is given on experimental schemes for measuring phase properties (in particular, phase distributions) discussed and partly even realized thus far. It is pointed out that all of them amount to measuring the Q function for the field. Moreover, experimental procedures for determining the Wigner function for the field more or less directly are described which thus allow to assign experimental significance to the concept of area of overlap in phase space when applied to the quantum mechanical description of phase.

The deteriorating effect of low-efficiency detectors in different schemes suitable for a direct measurement of the Q function and in optical homodyne tomography is studied in some detail. It turns out that this effect amounts to smoothing the respective quasiprobability distribution that would be measured with unit-efficiency detectors. Our main result is that those smoothed distributions can be identified with certain s-parametrized quasiprobability distributions. Thus the latter gain direct experimental significance as distributions measurable under realistic experimental conditions.

The deteriorating effect of low-efficiency detectors in different schemes suitable for a direct measurement of the Q function and in optical homodyne tomography is studied in some detail. It turns out that this effect amounts to smoothing the respective quasiprobability distribution that would be measured with unit-efficiency detectors. Our main result is that those smoothed distributions can be identified with certain s-parametrized quasiprobability distributions. Thus the latter gain direct experimental significance as distributions measurable under realistic experimental conditions.

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.

Generalizing a recent theoretical result obtained by Freyberger and Schleich [Phys. Rev. A 47, R30 (1993)], we show that the phase-measurement scheme proposed and realized by Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992)] amounts to measuring the Q function for the light under investigation, provided the reference beam (used for homodyne detection) is a very strong coherent field so that it can be described classically. The desired phase distribution follows from the Q function by averaging over the field amplitude. Since an analysis of an earlier proposal by Bandilla and Paul [Ann. Phys. (Leipzig) 23, 323 (1969)] to measure phase distributions via amplification led to just the same result, a perfect physical equivalence of those two approaches has thus been established.

We propose the reconstruction of the quantum state of light from the homodyne photocounting statistics of a single, realistic photodetector. Contrary to the development of homodyning over the last decade, our approach is based on unbalanced detection with a weak local oscillator. Representing the quantum state in terms of s-parametrized quasiprobability distributions, the method even allows one to determine the Wigner function provided the quantum efficiency of the detector is sufficiently large. We show that perturbing effects due to classical noise of the local oscillator are small. \textcopyright{} 1996 The American Physical Society.