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Bulletin of the American Physical Society

Tomographic reconstruction of the Wigner function of an itinerant microwave field

F. Mallet, Manuel A. Castellanos-Beltran, Hsiang‐Sheng Ku, K. D. Irwin, Leila R. Vale, Konrad W. Lehnert

Published Mar 19, 2010• 0 citations
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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 2010 Nature 464 1170), 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.
A wide range of experiments studying microwave photons localized in superconducting cavities have made important contributions to our understanding of the quantum properties of radiation. Propagating microwave photons, however, have so far been studied much less intensely. Here we present measurements in which we reconstruct the quantum state of itinerant single photon Fock states and their superposition with the vacuum by analyzing moments of the measured amplitude distribution up to fourth order. Using linear amplifiers and quadrature amplitude detectors, we have developed efficient methods to separate the detected single photon signal from the noise added by the amplifier. From our measurement data we have also reconstructed the corresponding Wigner function.