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Physical Review A

Phase measurement andQfunction

Ulf Leonhardt, Henryk Paul

Published Apr 1, 1993• 131 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.47.r2460

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arXiv:quant-ph/9712032•Dec 16, 1997•1 citation
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.
Recently Noh, Fourgeres and Mandel (NFM) have improved the operational approach to the quantum phase problem substantially and measured the phase dispersion of coherent light down to very small means photon numbers of the order of 10(exp -2). This has prompted many other investigations and clarified some important questions in relation to what is actually measured. Although their treatment is rather general, we confine ourselves here to the case of a strong local oscillator (LO) and reproduce their measurement scheme. Surprisingly enough, this simultaneous measurement of the sine and the cosine of the phase difference is completely equivalent to an old proposal to measure the phase after strong linear amplification realized experimentally by the Welling group. The reason for this rests on the fact, that in both cases the results are determined by the Q function of the signal. This was shown for amplification in and for the measurement after beam splitting by Lai and Haus.
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.
We address the problem of identifying an operational prescription for quantum phase measurements. As is known, different experiments can lead to different measured phase operators. However, we show that ambiguities of interpretation can arise even if a single experiment, such as that of Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991)], is chosen as defining a phase measurement. We show by reference to a simple but fundamental example that it is not possible to deduce a unique phase difference from the measurements.
A beam splitter is assumed to separate an incident beam of light with indeterminate phase into a reflected and a transmitted beam. In analogy with classical physics, a homodyne approach similar to that suggested by Noh, Fougeres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991)] is used to measure the phase of each beam at two distant locations. The nonlocal collapse of the wave function due to such a phase measurement made on one of the two beams is shown to reduce the state of the remaining beam to a coherent state (to a good approximation) with the corresponding phase. However, these homodyne phase measurements also exhibit several nonclassical features, including an oscillatory fine structure, which complicate their interpretation as the phase of the field.
Recently, Torgerson and Mandel [Phys. Rev. Lett. 76, 3939 (1996)] have reported a disagreement between two schemes for measuring the phase difference of a pair of optical fields. We analyze these schemes and derive their associated phase-difference probability distributions, including both their strong and weak field limits. Our calculation confirms the main point of Torgerson and Mandel of the non-uniqueness of an operational definition of the phase distribution. We further discuss the role of postselection of data and argue that it cannot meaningfully improve the sensitivity.