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Physical Review A

Realistic optical homodyne measurements and quasiprobability distributions

Ulf Leonhardt, Henryk Paul

Published Dec 1, 1993• 200 citations
Publisher

Abstract

The deteriorating effect of low-efficiency detectors in different schemes suitable for a direct measurement of the Q function and in optical homodyne tomography is studied in some detail. It turns out that this effect amounts to smoothing the respective quasiprobability distribution that would be measured with unit-efficiency detectors. Our main result is that those smoothed distributions can be identified with certain s-parametrized quasiprobability distributions. Thus the latter gain direct experimental significance as distributions measurable under realistic experimental conditions.

DOI
10.1103/physreva.48.4598

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The deteriorating effect of low-efficiency detectors in different schemes suitable for a direct measurement of the Q function and in optical homodyne tomography is studied in some detail. It turns out that this effect amounts to smoothing the respective quasiprobability distribution that would be measured with unit-efficiency detectors. Our main result is that those smoothed distributions can be identified with certain s-parametrized quasiprobability distributions. Thus the latter gain direct experimental significance as distributions measurable under realistic experimental conditions.
The deteriorating effect of low-efficiency detectors in different schemes suitable for a direct measurement of the Q function and in optical homodyne tomography is studied in some detail. It turns out that this effect amounts to smoothing the respective quasiprobability distribution that would be measured with unit-efficiency detectors. Our main result is that those smoothed distributions can be identified with certain s-parametrized quasiprobability distributions. Thus the latter gain direct experimental significance as distributions measurable under realistic experimental conditions.
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.
From the experimental measurement of probability distributions of quadrature-field amplitudes, followed by numerical inversion (optical homodyne tomography), we have determined distributions and/or moments of the optical phase of small-photon-number fields for several definitions of the phase variable, including those based on Hermitian operators and on quasiprobability distributions. These measurements were performed on a vacuum field, and a weakly squeezed field. It is found that each definition of phase yields different distributions and/or moments for the experimental data. In addition, all of the experimentally determined quantities agree with the corresponding theoretical predictions.
We propose the reconstruction of the quantum state of light from the homodyne photocounting statistics of a single, realistic photodetector. Contrary to the development of homodyning over the last decade, our approach is based on unbalanced detection with a weak local oscillator. Representing the quantum state in terms of s-parametrized quasiprobability distributions, the method even allows one to determine the Wigner function provided the quantum efficiency of the detector is sufficiently large. We show that perturbing effects due to classical noise of the local oscillator are small. \textcopyright{} 1996 The American Physical Society.
A novel experimental scheme is proposed that allows us to avoid the deterioration of homodyne detection measurements due to nonideal detectors. The basic idea is to "preamplify" the signal by means of antisqueezing. Experimentally, we would employ a squeezer, e.g., a degenerate optical parametric amplifier, that squeezes just the nonobserved quadrature component of the electric field while antisqueezing the conjugate component which is measured. It is shown that for sufficiently strong antisqueezing one achieves the same measurement accuracy as with perfectly efficient detectors. In particular, in this way the actual Wigner function can be reconstructed in cptical homodyne tomography.