Method for Direct Measurement of the Wigner Function in Cavity QED and Ion Traps

Abstract

Phase space representations are very useful in quantum mechanics since they allow the calculation of correlation functions of operators as classical-like integrals, and are also helpful for the study of the transition to classical physics. The oldest of such representations is due to Wigner [1], who used it as a convenient tool to calculate quantum corrections to classical statistical mechanics. It was shown by Moyal [2] that the quantum average of a Weyl-ordered (symmetric-ordered) function of the momentum and position operators could be expressed as a classical-like average of the corresponding classical function (in which the operators are replaced by c numbers), with the Wigner distribution acting as a weight measure in phase space [3]. The uncertainty principle forbids, however, the interpretation of this function as a probability distribution, since it is not possible to determine simultaneously the momentum and the position of a particle. In fact, it is easy to find examples of states for which the Wigner distribution assumes negative values. This fact may lead to the idea that it does not correspond to any directly measured quantity. Up until now, this notion has been upheld by the fact that the different schemes proposed so far to determine the Wigner distribution of a quantum system rely either on tomographic reconstructions from data obtained in homodyne measurements [4,5] or on convolutions obtained by photon counting [6]. It is the purpose of this Letter to show that the Wigner function can be directly measured, through a very simple scheme, especially suitable to experiments in cavity quantum electrodynamics and in ion traps. This is especially important in view of recent experimental results concerning the production and detection of coherent superpositions of localized mesoscopic states [7,8]. In these experiments, the existence of coherence is inferred from partial information obtained about the system. A method yielding more complete knowledge onf the quantum states involved would be highly desirable. Quantum tomography schemes for determining the vibrational state of a trapped ion were proposed in [5]. In cavity QED, information on the field must be obtained from probe atoms, since the high Q of the cavity and the weak intensities involved do not allow the direct measurement of the field. A method for realizing the “quantum endoscopy” of a field in a cav-