Tomographic reconstruction of the Wigner function on the Bloch sphere

TL;DR

This paper introduces a filtered backprojection algorithm for reconstructing the Wigner function of large angular momentum systems from Stern-Gerlach measurements, improving noise handling and eliminating the need for data binning.

Contribution

The method allows for robust Wigner function reconstruction from experimental data without full density matrix determination, accounting for measurement uncertainties and noise.

Findings

Successfully reconstructed the Wigner function of a spin-squeezed BEC state

Demonstrated that measurements in a single plane suffice for tomography

Method is insensitive to fluctuations in angular momentum j

Abstract

We present a filtered backprojection algorithm for reconstructing the Wigner function of a system of large angular momentum j from Stern-Gerlach-type measurements. Our method is advantageous over the full determination of the density matrix in that it is insensitive to experimental fluctuations in j, and allows for a natural elimination of high-frequency noise in the Wigner function by taking into account the experimental uncertainties in the determination of j, its projection m, and the quantization axis orientation. No data binning and no arbitrary smoothing parameters are necessary in this reconstruction. Using recently published data [Riedel et al., Nature 464:1170 (2010)] we reconstruct the Wigner function of a spin-squeezed state of a Bose-Einstein condensate of about 1250 atoms, demonstrating that measurements along quantization axes lying in a single plane are sufficient for performing this tomographic reconstruction. Our method does not guarantee positivity of the reconstructed density matrix in the presence of experimental noise, which is a general limitation of backprojection algorithms.