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arXiv:0708.4094 · Journal of Modern Optics

A note on the measurement of phase space observables with an eight-port homodyne detector

Jukka Kiukas, PEKKA J. LAHTI

Published Jul 1, 2008• 21 citations
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Abstract

It is well known that the Husimi Q-function of the signal field can actually be measured by the eight-port homodyne detection technique, provided that the reference beam (used for homodyne detection) is a very strong coherent field so that it can be treated classically. Using recent rigorous results on the quantum theory of homodyne detection observables, we show that any phase space observable, and not only the Q-function, can be obtained as a high amplitude limit of the signal observable actually measured by an eight-port homodyne detector. The proof of this fact does not involve any classicality assumption.

DOI
10.1080/09500340701864718

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It is well known that the Husimi Q-function of the signal field can actually be measured by the eight-port homodyne detection technique, provided that the reference beam (used for homodyne detection) is a very strong coherent field so that it can be treated classically. Using recent rigorous results on the quantum theory of homodyne detection observables, we show that any phase space observable, and not only the Q-function, can be obtained as a high amplitude limit of the signal observable actually measured by an eight-port homodyne detector. The proof of this fact does not involve any classicality assumption.
We consider the quantum optical eight-port homodyne detection scheme in the case that each of the associated photon detectors is assigned with a different quantum efficiency. We give a mathematically rigorous and strictly quantum mechanical proof of the fact that the measured observable (positive operator measure) in the high-amplitude limit is a smearing of the covariant phase space observable related to the ideal measurement, that is, the measurement performed with fully efficient detectors. The result is proved for an arbitary parameter field. Furthermore, we investigate some properties of the measured observable. In particular, we show that detector inefficiencies do not affect the observable's ability to distinguish between different states.
The eight-port homodyne detection apparatus is analysed in the framework of the operational theory of quantum measurement. For an arbitrary quantum noise leaking through the unused port of the beam splitter, the positive operator valued measure and the corresponding operational homodyne observables are derived. It is shown that such an eight-port homodyne device can be used to construct the operational quantum trigonometry of an optical field. The quantum trigonometry and the corresponding phase space Wigner functions are derived for a signal field probed by a classical local oscillator and a squeezed vacuum in the unused port.
The eight-port homodyne detection apparatus is analyzed in the framework of the operational theory of quantum measurement. For an arbitrary quantum noise leaking through the unused port of the beam splitter, the positive operator valued measure and the corresponding operational homodyne observables are derived. It is shown that such an eight-port homodyne device can be used to construct the operational quantum trigonometry of an optical field. The quantum trigonometry and the corresponding phase space Wigner functions are derived for a signal field probed by a classical local oscillator and a squeezed vacuum in the unused port.
We show that the phase shift of $\frac{\ensuremath{\pi}}{2}$ is crucial for the phase-space translation covariance of the measured high-amplitude limit observable in eight-port homodyne detection. However, for an arbitrary phase shift $\ensuremath{\theta}$ we construct explicitly a different nonequivalent projective representation of R2{\mathbb{R}}^{2}R2 such that the observable is covariant with respect to this representation. As a result we are able to determine the measured observable for an arbitrary parameter field and phase shift. Geometrically the change in the phase shift corresponds to the tilting of one axis in the phase space of the system.
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.