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Quantum Optics Journal of the European Optical Society Part B

Characteristic functions and quantum measurements of optical observables

Yinchieh Lai, H. A. Haus

Published Dec 1, 1989• 84 citations
Publisher

Abstract

Three characteristic functions are reviewed along with the quantum theory of linear systems and used to derive the P-representation, Wigner distribution and quasi-probability density of coherent states and squeezed states. The quasiprobability density is interpreted as the outcome of measurements utilising a beamsplitter with homodyne detection. The quasiprobability density concept is generalised and its interpretation in terms of a measurement is presented. The measurement utilises a beam splitter with one of its inputs in a squeezed state followed by two homodyne detectors.

DOI
10.1088/0954-8998/1/2/003

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Three characteristic functions are reviewed along with the quantum theory of linear systems and used to derive the P-representation, Wigner distribution and quasi-probability density of coherent states and squeezed states. The quasiprobability density is interpreted as the outcome of measurements utilising a beamsplitter with homodyne detection. The quasiprobability density concept is generalised and its interpretation in terms of a measurement is presented. The measurement utilises a beam splitter with one of its inputs in a squeezed state followed by two homodyne detectors.
We have used the recently demonstrated method of optical homodyne tomography (OHT) to measure the Wigner quasiprobability distribution (Wigner function) and the density matrix for both a squeezed-vacuum and a vacuum state of a single spatial-temporal mode of the electromagnetic field. This method consists of measuring a set of probability distributions for many different Hilbert-space representations of the field-quadrature amplitude, using balanced homodyne detection, and then using tomography to obtain the Wigner function. Once the Wigner function is obtained, one can acquire the density matrix, including its complex phase. In the case of a pure state, this technique yields an experimentally determined complex wavefunction, as demonstrated here for the vacuum. The density matrix represents a complete quantum mechanical characterization of the state. From the measured density matrix we have obtained the Pegg–Barnett optical phase distribution, and from the Wigner function, the Wigner optical phase distribution.
Information carried by an optical beam of light can usually be conveyed in the form of a temporal modulation of the intensity, phase, frequency or the polarization of the constituent mode(s). In that regard, homodyne detection is one of the most popular and standard method to measure the quantum mechanical properties of light Yuen & Shapiro (1978); Yuen et al. (1979); Yuen & Shapiro (1980); Yurke (1985). The quantum theory of homodyne detection originated principally from the works of Yuen and Shapiro et al. Yuen & Shapiro (1978); Yuen et al. (1979). In this method, a weak signal field is combined with a strong local oscillator field at a beam splitter, and the resulting signal is measured as a photocurrent. With the homodyne detection method, the quantum state of the signal field, such as the quadrature amplitude of the squeezed state, is easily measured Slusher et al. (1985); Wu et al. (1986 ); Polzik et al. (1992), and the quasiprobability distribution function can be measured by using the so-called optical homodyne tomography Smithey et al. (1993); Banaszek & W’odkiewicz (1996); Wallentowitz & Vogel (1996); Youn et al. (2001). Furthermore, optical homodyne detection is used to eavesdrop on the quantum key in quantum cryptography Hirano et al. (2000).
We report measurements of the Wigner distribution and the density matrix of an electromagnetic field mode, for both vacuum and quadrature-squeezed states. As proposed by Vogel and Risken,1 we obtained the Wigner distribution by tomographic inversion of a set of probability distributions of field-quadrature amplitudes measured by using balanced homodyne detection. A Fourier transform of the Wiener function yields the density matrix, which, according to the standard interpretation of quantum mechanics, contains all knowable information about a given quantum system.
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.