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Optics Letters

Optical heterodyne imaging and Wigner phase space distributions

Adam P. Wax, John E. Thomas

Published Sep 15, 1996• 37 citations
Publisher

Abstract

We demonstrate that optical heterodyne imaging directly measures smoothed Wigner phase space distributions. This method may be broadly applicable to fundamental studies of light propagation and tomographic imaging. Basic physical properties of Wigner distributions are illustrated by experimental measurements.
DOI
10.1364/ol.21.001427

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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.
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.
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.