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Lecture notes in physics

A Two-Level Atom: Heterodyne and Homodyne Spectra

Alberto Barchielli, Matteo Gregoratti

Published Jan 1, 2009• 0 citations
Publisher
DOI
10.1007/978-3-642-01298-3_9

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Two-level atoms

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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.
We present results of a study of four-wave mixing in Rb vapour with highly nonlinear susceptibility, using both homodyne and heterodyne detection. We demonstrate that the spectra have different appearances for media possessing electromagnetically induced transparency and electromagnetically induced absorption, and for different relative polarizations of the drive and probe fields. We show that these differences allow the contributions of different processes responsible for the enhanced Kerr nonlinearity of the media to be distinguished.
Quantum-mechanical calculations of the mean-square fluctuation spectra in optical homodyning and heterodyning are made for arbitrary input and local-oscillator quantum states. In addition to the unavoidable quantum fluctuations, it is shown that excess noise from the local oscillator always affects homodyning and, when it is broadband, also heterodyning. Both the quantum and the excess noise of the local oscillator can be eliminated by coherent subtraction of the two outputs of a 50-50 beam splitter. This result also demonstrates the fact that the basic quantum noise in homodyning and heterodyning is signal quantum fluctuation, not local-oscillator shot noise.