Complete experimental characterization of the quantum state of a light mode via the Wigner function and the density matrix: application to quantum phase distributions of vacuum and squeezed-vacuum states

TLDR

The density matrix provides a complete quantum mechanical characterization of a state. The authors use balanced homodyne detection to record quadrature distributions, then apply optical homodyne tomography to reconstruct the Wigner function and, from it, the density matrix with its complex phase. Using OHT, the authors experimentally reconstructed the Wigner function and density matrix of vacuum and squeezed‑vacuum states, obtaining the complex wavefunction for the vacuum and both Pegg–Barnett and Wigner optical phase distributions.

Abstract

We have used the recently demonstrated method of optical homodyne tomography (OHT) to measure the Wigner quasiprobability distribution (Wigner function) and the density matrix for both a squeezed-vacuum and a vacuum state of a single spatial-temporal mode of the electromagnetic field. This method consists of measuring a set of probability distributions for many different Hilbert-space representations of the field-quadrature amplitude, using balanced homodyne detection, and then using tomography to obtain the Wigner function. Once the Wigner function is obtained, one can acquire the density matrix, including its complex phase. In the case of a pure state, this technique yields an experimentally determined complex wavefunction, as demonstrated here for the vacuum. The density matrix represents a complete quantum mechanical characterization of the state. From the measured density matrix we have obtained the Pegg–Barnett optical phase distribution, and from the Wigner function, the Wigner optical phase distribution.