Intensity correlations in the Wigner representation

Abstract

We derive a compact expression for the second-order correlation function [Formula: see text] of a quantum state in terms of its Wigner function, thereby establishing a direct link between [Formula: see text] and the state's shape in phase space. We conduct an experiment that simultaneously measures [Formula: see text] through direct photocounting and reconstructs the Wigner function via homodyne tomography. The results confirm our theoretical predictions.This article is part of the theme issue 'The quantum theory of light'.

Keywords: Wigner function; correlations; phase-space; squeezing.

Figure 1.

Second-order correlation function as a function of the mean photon number for the three typical states considered in this work. Note, that the red curve is for pure squeezed states and not for attenuated squeezed states.

Figure 2.

Experimental set-up (for more details, see Ref. [46]): a type-II OPO is continuously pumped with a 532 nm Nd:YAG laser. The output modes are mixed using a PBS and HWP. A two-photon detection is implemented on one output of the PBS via multiplexed superconducting nanowire single-photon detectors after frequency filtering. The other output is analysed via a homodyne detection for full quantum state tomography.

Figure 3.

Values of $g^{(2)}(0)$ as a function of the angle of the wave plate, for two different input powers indicated in the insets. In each figure, we plot the results obtained from both direct photon counting and via the Wigner function reconstructed from homodyne detection. The shadows indicate the errors associated with both methods.

Figure 4.

Contour plots of the reconstructed Wigner functions for several HWP angles showing the transition from a thermal state corresponding to the angle $0^{\circ }$ to a squeezed vacuum state corresponding to $22.5^{\circ }$. We also include the variances derived from the reconstruction. The top panel correspond to 2 mW and the bottom panel to 10 mW.