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arXiv:quant-ph/9910092 · Journal of Optics B Quantum and Semiclassical Optics

Testing operational phase concepts in quantum optics

Jaroslav Řeháček, Z. Hradil, Miloslav Dušek, Ondřej Haderka, Martin Hendrych

Published May 11, 2000• 9 citations
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Abstract

An experimental comparison of several operational phase concepts is presented. In particular, it is shown that statistically motivated evaluation of experimental data may lead to a significant improvement in phase fitting upon the conventional procedure of Noh et al (1993 Phys. Rev. Lett. 71 2579). The analysis is extended to the asymptotic limit of large intensities, where a strong evidence in favour of multi-dimensional estimation procedures has been found.

DOI
10.1088/1464-4266/2/3/304

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Progress of Theoretical Physics•Oct 1, 2003•0 citations
An experimental comparison of several operational phase concepts is presented. In particular, it is shown that statistically motivated evaluation of experimental data may lead to a significant improvement in phase fitting upon the conventional procedure of Noh et al (1993 Phys. Rev. Lett. 71 2579). The analysis is extended to the asymptotic limit of large intensities, where a strong evidence in favour of multi-dimensional estimation procedures has been found.
Various phase concepts may be treated as special cases of the maximum likelihood estimation. For example, the discrete Fourier estimation that actually coincides with the operational phase of Noh, Foug\`eres, and Mandel is obtained for continuous Gaussian signals with phase modulated mean. Since signals in quantum theory are discrete, a prediction different from that given by the Gaussian hypothesis should be obtained as the best fit assuming a discrete Poissonian statistics of the signal. Although the Gaussian estimation gives a satisfactory approximation for fitting the phase distribution of almost any state, the optimal phase estimation offers in certain cases a measurably better performance. This has been demonstrated in a neutron-optical experiment.
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.
In papers by Lynch [ Phys. Rev. A 41 2841 (1990)] and Gerry and Urbanski [ Phys. Rev. A 42 662 (1990)] it has been argued that the phase-fluctuation laser experiments of Gerhardt, Büchler and Lifkin [ Phys. Lett. 49A 119 (1974)] are in good agreement with the variance of the Pegg–Barnett phase operator for a coherent state, even for a small number of photons. We argue that this is not conclusive. In fact, we show that the variance of the phase in fact depends on the relative phase between the phase of the coherent state and the off-set phase ϕ 0 of the Pegg–Barnett phase operator. This off-set phase is replaced with the phase of a reference beam in an actual experiment and we show that several choices of such a relative phase can be fitted to the experimental data. We also discuss the Noh, Fougères and Mandel [ Phys. Rev. A 46 2840 (1992)] relative phase experiment in terms of the Pegg–Barnett phase taking post-selection conditions into account.
It is shown that there exist two kinds of operational (or measured) phase operators, those of the Hradil type [Phys. Rev. A 47 (1993), 4532], defined on the entire Hilbert space, and those of the Noh-Fougères-Mandel type [Phys. Rev. A 45 (1992), 424], defined only on a subspace. As a consequence, strictly speaking, the Noh-Fougères-Mandel measured phase cannot be defined in the case of, for example, coherent states, squeezed states and the vacuum. The difference between these operators is made clear by using the concept of the Bargmann distance between a quantum state and its “reduced” state. Since Noh et al. did not use the correct variances of the Susskind-Glogower and Pegg-Barnett phase operators when they compared their data with the results predicted through use of these operators, the correct expressions in that case are presented. Comparing the measured operators with the Hermitian phase operator in the extended Susskind-Glogower formalism, we show that the quantum phase given by the Hermitian phase operator is more appropriate than the measured phases. The variance of the Hermitian operator in the vacuum is consistent with the complementarity of the photon number and phase, which implies that the vacuum has a uniform phase distribution. Hradil's measured operator does not have such a property. Although the Noh-Fougères-Mandel operator has such a property in a particular limit, this operator does not represent, in principle, a meaningful measure of the phase measurement in a typical quantum region, because the Bargmann distance becomes large in such a region.