Determination of the Wigner Function from Photon Statistics *

We present an experimental realisation of the direct scheme for measuring the Wigner function of a single quantized light mode. In this method, the Wigner function is determined as the expectation value of the photon number parity operator for the phase space displaced quantum state. The Wigner function provides a complete representation of the quantum state in the form that is analogous to a classical phase space probability distribution [1]. An interesting and nontrivial problem is how to relate the Wigner function to directly measurable quantities so that it can be determined from data collected in a feasible experimental scheme. The first answer was given by Vogel and Risken [2] who noted that marginal distributions of the Wigner function can be measured with a balanced homodyne detector, and that the inverse transformation is possible by tomographic back-projection. This idea has been realized in a beautiful experiment by Smithey and co-workers [3], and it has quickly become a useful tool in studying quantum statistical properties of optical radiation [4]. In this contribution we briefly review our recent experimental realization [5] of the direct scheme for measuring the Wigner function of a light mode [6, 7]. This method, based on photon counting, provides the complete Wigner representation of the quantum state without using any numerical reconstruction algorithms. The basic idea underlying our experimental scheme is that the Wigner function W (α) at a given phase space point α is itself a well defined quantum observable. This * Contribution for the sixth central-european workshop on quantum optics, chateau Chudobin near Olomouc, 30 April-May 3, 1999.