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Quantum Optics Journal of the European Optical Society Part B

Presumable solutions of quantum phase problem and their flaws

A. Lukš, Vlasta Peřinová

Published Jun 1, 1994• 57 citations
Publisher

Abstract

A combination of less frequent criticism with positive investigations has resulted in the substitution of group theoretic considerations by a simpler quantum mechanical model, has taken into account homodyne and heterodyne detection schemes, and proceeded by an analysis of phase data processing. Limiting procedures in s-phase formalisms have been provided concentrating on the Wigner function for number and ideal phase. The Wigner function for number and realistic phase has been expressed by closed formulae along with the antinormal phase distributions.

DOI
10.1088/0954-8998/6/3/003

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A combination of less frequent criticism with positive investigations has resulted in the substitution of group theoretic considerations by a simpler quantum mechanical model, has taken into account homodyne and heterodyne detection schemes, and proceeded by an analysis of phase data processing. Limiting procedures in s-phase formalisms have been provided concentrating on the Wigner function for number and ideal phase. The Wigner function for number and realistic phase has been expressed by closed formulae along with the antinormal phase distributions.
An overview is given on experimental schemes for measuring phase properties (in particular, phase distributions) discussed and partly even realized thus far. It is pointed out that all of them amount to measuring the Q function for the field. Moreover, experimental procedures for determining the Wigner function for the field more or less directly are described which thus allow to assign experimental significance to the concept of area of overlap in phase space when applied to the quantum mechanical description of phase.
Various probability distributions are calculated for photocurrent fluctuations for photoelectric detection and are expressed in terms of quantum averages of field operators. The calculation is based on an extended theory of Kelley and Kleiner for photoelectric counting [Phys. Rev. 136, A316 (1963)], which allows us to obtain the counting distributions for multiple time intervals and multiple detectors. The finite duration of the detector's response function is also considered in our formalism, as is the possibility of fluctuations in the response function itself. This generalized theory is applied to homodyne and heterodyne detection processes. We find that with ideal conditions (perfectly balanced detectors with unit quantum efficiency), the characteristic function of the photocurrent fluctuations from homodyne detection is connected to the Fourier transformation of the Wigner function of a single-mode field, while that from heterodyne detection is linked to the antinormally ordered characteristic function of the field. More specifically, if the two orthogonal quadratures of the photocurrent are recorded in the heterodyne detection, the joint probability distribution for the two quadratures is simply the Q function of the field without the need for optical tomography. Our formalism is applied to the multimode case and allows one to draw important conclusions about the possibility of measurements of the complete quantum state of a multimode field. In particular, the complete characterization of fields with intermode correlations is a nontrivial undertaking, as we demonstrate with our general formalism as well as by specific examples.
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.
It is well known that two quasi-probability distributions for an optical mode, the Wigner (W) and Husimi (Q) functions, can be measured by homodyne and heterodyne detection, respectively. These harmonic oscillator distribution functions can be generalized to apply to other algebras, in particular, the su(2) algebra of a two-level atom. In this paper I investigate the relationship between these quasi-probability functions on the Bloch sphere, and the probability distributions for the integrated homodyne and heterodyne photocurrents from the fluorescence of the atom. While there is no particularly close relation between the su(2) Wigner function and homodyne detection, the su(2) Husimi function is simply equal to the heterodyne distribution function multiplied by a weighting factor. I also show that the two-level atom heterodyne distribution function can be expressed in a form identical to the corresponding optical mode distribution function. This correspondence defines the natural analogue of optical coherent states for a two-level atom. In deriving these results the theory of effects (positive-operator-valued measures) is essential.