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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Generalized measurements in eight-port homodyne detection

Alfredo Luis, Jan Peřina

Published Aug 1, 1996• 15 citations
Publisher

Abstract

We examine the eight-port homodyne detector when there is a vacuum at two of the input ports. It can then be described as a generalized measurement on the other two fields. We derive the class of two-field operators whose statistics can be deduced from the measurement for different choices of beamsplitters.
DOI
10.1088/1355-5111/8/4/009

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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.
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.
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.
Classically eight- and twelve-port homodyne detectors can be used to measure the Stokes parameters of a two-mode field simultaneously. We analyse these schemes in the quantum case as approximate or noisy joint measurements of the noncommuting Stokes operators. The statistics of the measurement are described in terms of these operators and noise. The twelve-port detector is also examined as allowing a noisy simultaneous measurement of the quadratures and intensity of a one-mode field. The vectors governing the statistics of the measurement are found. A simple procedure for obtaining the Wigner function emerges.