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Springer proceedings in physics

Wigner Distributions for Quantum States with Minimum Phase Uncertainty

B. Daeubler, H. Risken, L. Schoendorff

Published Jan 1, 1994• 0 citations
Publisher
DOI
10.1007/978-3-642-79101-7_25

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Experiments have been performed to determine the Wigner distribution and the density matrix (and for pure states the wave function) of a light mode, by using tomographic inversion of a set of measured probability distributions for quadrature amplitudes.From these measurements the quantum distributions of optical phase and photon number have been obtained.The measurements of quadrature-amplitude distributions for a temporal mode of the electromagnetic field are carried out using balanced homodyne detection.We refer to this new method as optical homodyne tomography.Given the measured density matrix, one can experimentally infer any of the various quantum distributions of optical phase, in particular the Pegg-Barnett (or, equivalently, Shapiro-Shepard) phase distribution, the marginal Wigner distribution, and the Vogel-Schleich operational phase distribntion.We have used this approach to make measurements of the number-plase uncertainty relation for coherent-state fields.The coherent states do not attain the minimum value for the number-phase uncertainty product, as set by the expectation value of the commutator of the number and phase operators; this is true theoretically and experimentally.
With the correspondence between operator ordering rules and probability distribution functions, moments can be calculated in many ways. We show here the independence of minimum uncertainty relations on the choice of Hermitian ordering rule. Generalized Wigner functions were expressed in terms of ordering real-valued function ϴ(x) for x ϵ R satisfying ϴ(0)=1. Uncertainties of position and momentum are obtained for a Gaussian state. Results confirm that the minimum uncertainty relation holds even when the Hermitian ordering rule is modified.