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Journal of Optics B Quantum and Semiclassical Optics

Homodyning the exponential phase moments of smoothed Wigner functions

Thomas Richter, Matteo G. A. Paris

Published Feb 1, 2001• 1 citation
Publisher

Abstract

We show that the exponential phase moments associated with the s-ordered generalized Wigner function can be sampled from homodyne data for s 0. The tomographic sampling functions are evaluated for non-unit homodyne quantum efficiency, and the statistical reliability of the reconstruction is tested by Monte Carlo simulated experiments. The reconstruction of the s-ordered phase distribution from sampled moments is discussed, and the reduction of statistical fluctuations by adaptive techniques is also analysed.

DOI
10.1088/1464-4266/3/1/358

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We show that the exponential phase moments associated with the s-ordered generalized Wigner function can be sampled from homodyne data for s 0. The tomographic sampling functions are evaluated for non-unit homodyne quantum efficiency, and the statistical reliability of the reconstruction is tested by Monte Carlo simulated experiments. The reconstruction of the s-ordered phase distribution from sampled moments is discussed, and the reduction of statistical fluctuations by adaptive techniques is also analysed.
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States with explicit quantum character, such as squeezed vacuum and bright squeezed light, as well as coherent states and incoherent superpositions of coherent states were generated and analysed by tomographical methods. Wigner functions, photon-number distributions, density matrices and phase distributions were reconstructed with high accuracy. Features such as photon number oscillations, sub-Poissonian and super-Poissonian photon statistics, bifurcations of the phase distribution, and loss of coherence were observed, demonstrating the usefulness of quantum state reconstruction as an analysing tool in quantum optics experiments.
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In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on R2\mathbb R^2R2 which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter 1/2<η≤11/2 < η\leq 11/2<η≤1, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the L∞\mathbb L_\inftyL∞​-risk over a class of infinitely differentiable. We compute also the lower bound for the L2\mathbb L_2L2​-risk. We construct adaptive estimator, i.e. which does not depend on the smoothness parameters, and prove that it attains the minimax rates for the corresponding smoothness class functions. Finite sample behaviour of our adaptive procedure are explored through numerical experiments.