Direct measurement of the Wigner function by photon counting

Among many representations of the quantum state, the Wigner function offers an appealing possibility to describe quantum phenomena using the classical-like concept of phase space [1]. The Wigner function provides complete information on the state of a system, and it allows one to evaluate any quantum observable by phase space integration with an appropriate Wigner-Weyl ordered expression. Recently, the Wigner function has gained experimental significance due to the development of the optical homodyne tomography, a beautiful technique for measuring the quantum state of light pioneered by Smithey et al. [2] and further applied by Breitenbach et al. [3] In this method, rooted in the domain of image processing, the Wigner function is a natural representation of the quantum state reconstructed from experimental data. However, the route from raw experimental results to the Wigner function is not straightforward. First, a sample of homodyne events is collected and stored. Statistics of these events for a fixed local oscillator phase is described by a marginal projection of the Wigner function. In order to retrieve the complete Wigner function, a family of homodyne statistics measured for a sufficiently dense set of local oscillator phases has to be processed using the sophisticated filtered back-projection algorithm. In this Communication we report a direct measurement of the Wigner function of a light mode. This technique, based on photon counting, avoids the detour via complex numerical reconstruction algorithms. The principle of our measurement is entirely different from optical homodyne tomography. The Wigner function at a given phase space point is itself a well defined quantum observable [4]. Furthermore, the measurement of this observable can be implemented for optical fields using an arrangement employing an auxiliary coherent probe beam [5,6]. The amplitude and the phase of the probe field define the point in the phase space at which the Wigner function is measured. This allowed us to scan the phase space point-by-point, simply by changing the parameters of the probe field. A variation of this idea has been applied by Leibfried et al. [7] to determine the vibrational state of a trapped ion. Here we present an experiment, which to the best of our knowledge is the first direct measurement of the Wigner function for optical fields. Our experiment is based on the representation of the Wigner function at a complex phase space point denoted by α as the expectation value of the following operator: