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arXiv:quant-ph/0508166 · Physical Review A

Binomial states and the phase distribution measurement of weak optical fields

K. L. Pregnell, David Thomas Pegg

Published Jun 24, 2003• 18 citations
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Abstract

We show that the eight-port interferometer used by Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 71, 2579 (1993)] to measure their operational phase distribution of light can also be used to measure the canonical phase distribution of weak optical fields, where canonical phase is defined as the complement of photon number. A binomial reference state is required for this purpose, which we show can be obtained to an excellent degree of approximation from a suitably squeezed state. The proposed method requires only photodetectors that can distinguish among zero photons, one photon, and more than one photon and is not particularly sensitive to photodetector imperfections.

DOI
10.1103/physreva.67.063814

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We show that the eight-port interferometer used by Noh, Foug`eres, and Mandel [Phys. Rev. Lett. 71, 2579 (1993)] to measure their operational phase distribution of light can also be used to measure the canonical phase distribution of weak optical fields, where canonical phase is defined as the complement of photon number. A binomial reference state is required for this purpose, which we show can be obtained to an excellent degree of approximation from a suitably squeezed state. The proposed method requires only photodetectors that can distinguish among zero photons, one photon, and more than one photon and is not particularly sensitive to photodetector imperfections.
We present a quantum analysis of the measurement of a relative optical phase using a many-port method that can be implemented using arrays of photodetectors in the limit of a large number of detector elements. Its application to coherent states and states not describable by the semiclassical theory shows that the many-port method has in principle several advantages compared to an eight-port method previously studied. In some cases, especially for very weak fields, the many-port method provides a more faithful characterization (i.e., a smoother distribution) of the distribution of the relative phase than does the eight-port method. The definition used here for the relative phase between two fields is an operational one, not corresponding to any presently known Hermitian phase operator. We discuss the conditions under which our phase definition corresponds to the best estimate in the maximum-likelihood sense. We find that if the many-port data set is analyzed in a way that retains the contributions from zero photon counts, the many-port phase distribution takes on a character that resembles other phase distributions, such as the Pegg-Barnett or Wigner phase distributions, that show very small modulation depth in the case of weak fields.
The operational approach to the measurement of phase studied by Noh, Fouger\`es, and Mandel [Phys. Rev. A 45, 424 (1992); 46, 2840 (1992); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 47, 4535 (1993)] is applied to the measurement of the state of polarization of fully polarized light. Operational counterparts of the quantum Stokes parameters are introduced and their fluctuations are examined. It is shown that if the polarized field is weak, the measured fluctuations are influenced not only by the quantum properties of the source field but also that of the measurement. This character is reflected on the measured probability distributions of the parameters of polarization, which are also investigated independently for the fully polarized coherent states and the Fock states as the initial field strength is varied. Finally, connection between the operational approach to the measurement of polarization and the su(2) interferometry is examined.
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 reconsider the determination of the operational defined phase difference between two weak optical fields based on photon counting when the small number of possible experimental outcomes makes it harder to derive a continuous phase distribution. We show that the experimental procedure introduced to overcome this difficulty, namely, an external phase shifting of one input state, is not necessary. Indeed, all measuring data conditioned on the phase shift can be obtained from the data recorded without the external phase shift. Moreover, we investigate the effect of retaining ambiguous data or discarding even more data than only the ambiguous ones, respectively, and discuss the influence of a nonunit efficiency of the photodetectors.
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.