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Journal of Optics B Quantum and Semiclassical Optics

Direct sampling of a smoothed Wigner function from quadrature distributions

Th. Richter

Published Oct 29, 1999• 11 citations
Publisher

Abstract

We propose the reconstruction of a smoothed Wigner function from quadrature distributions instead of the Wigner function itself to avoid the problems with the inverse Radon transformation. Starting from the well known relation between s -parametrized quasiprobability distributions W ( x , p , s ) and quadrature component distributions w ( x , ), we show that the sampling function needed to determine a smoothed Wigner function W ( x , p ,-| s |) is basically a shifted and scaled version of the well known pattern function necessary to reconstruct the probability of finding no photon in the signal field. An analogous result has been recently found in the literature. Here we obtain this result in a very concise and alternative way.

DOI
10.1088/1464-4266/1/6/306

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We propose the reconstruction of a smoothed Wigner function from quadrature distributions instead of the Wigner function itself to avoid the problems with the inverse Radon transformation. Starting from the well known relation between s -parametrized quasiprobability distributions W ( x , p , s ) and quadrature component distributions w ( x , ), we show that the sampling function needed to determine a smoothed Wigner function W ( x , p ,-| s |) is basically a shifted and scaled version of the well known pattern function necessary to reconstruct the probability of finding no photon in the signal field. An analogous result has been recently found in the literature. Here we obtain this result in a very concise and alternative way.
The sampling functions needed to reconstruct from quadrature distributions the density matrix elements in the displaced Fock-state basis are determined as scaled and shifted pattern functions fmn used to reconstruct the density matrix elements Q mn in the Fock-state basis. Having at hand the diagonal density matrix elements one can reconstruct any s-parametrized quasiprobability distribution via a simple weighted sum over these quantities. A smoothed Wigner function can be directly sampled from the measured quadrature distribution of the signal field. The corresponding sampling function is just a shifted and scaled version of f 00.
The sampling functions needed to reconstruct from quadrature distributions the density matrix elements in the displaced Fock-state basis are determined as scaled and shifted pattern functions fmn used to reconstruct the density matrix elements Q mn in the Fock-state basis. Having at hand the diagonal density matrix elements one can reconstruct any s-parametrized quasiprobability distribution via a simple weighted sum over these quantities. A smoothed Wigner function can be directly sampled from the measured quadrature distribution of the signal field. The corresponding sampling function is just a shifted and scaled version of f 00.
We present a new and quite simple representation of the realistic pattern functions needed to reconstruct the density matrix in the Fock basis from quadrature distributions measured with non-ideal detectors. We show that they can be expressed as a finite weighted sum over the ideal pattern functions of scaled argument. Moreover we propose an algorithm which takes the detection loss-affected density matrix elements and produces the Wigner function (s-parametrized quasiprobability) of the signal field.
We consider the Wigner quasiprobability distribution function of a single mode of an electromagnetic or matter-wave field to address the question of whether a direct stochastic sampling and binning of the absolute square of the complex field amplitude can yield a distribution function ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}$ that closely approximates the true particle number probability distribution Pn{P}_{n}Pn​ of the underlying quantum state. By providing an operational definition of the binned distribution ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}$ in terms of the Wigner function, we explicitly calculate the overlap between ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}$ and Pn{P}_{n}Pn​ and hence quantify the statistical distance between the two distributions. We find that there is indeed a close quantitative correspondence between ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}$ and Pn{P}_{n}Pn​ for a wide range of quantum states that have a smooth and broad Wigner function relative to the scale of oscillations of the Wigner function for the relevant Fock state. However, we also find counterexamples, including states with high mode occupation, for which ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{n}$ does not closely approximate Pn{P}_{n}Pn​.