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Journal of Modern Optics

Determination ofs-ordered moments and moment generating function from quadrature distributions

T. Richter

Published Dec 1, 1999• 2 citations
Publisher

Abstract

The sampling functions needed to reconstruct from quadrature distributions both the s-ordered moments of the field and the generalized moment generating function are determined. The sampling function in the latter case turns out to be surprisingly simple. It is proportional to the well-known pattern function f 0k needed to reconstruct from homodyne data the density matrix element e 0k in the Fock basis. As a by-product a useful sum relation for the pattern functions f mn is found.

DOI
10.1080/09500349908231396

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Determination of s-ordered moments and moment generating function from quadrature distributions

Thomas Richter

Journal of Modern Optics•Dec 15, 1999•1 citation
The sampling functions needed to reconstruct from quadrature distributions both the s-ordered moments of the field and the generalized moment generating function are determined. The sampling function in the latter case turns out to be surprisingly simple. It is proportional to the well-known pattern function f 0k needed to reconstruct from homodyne data the density matrix element e 0k in the Fock basis. As a by-product a useful sum relation for the pattern functions f mn is found.
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.
Within the framework of a unified approach, we derive several identical representations of the realistic (and ideal) pattern functions fk,l{f}_{k,l}fk,l​ necessary to reconstruct the density-matrix element ${\ensuremath{\varrho}}_{\mathrm{kl}}$ in the Fock basis from realistic quadrature distributions measured with a nonideal balanced homodyne detector of overall efficiency $\ensuremath{\eta}.$ To this end we first establish a biorthogonality relation for the pattern functions from which the form of the Fourier-transformed pattern functions can be easily read off. Then a different handling of the inverse Fourier transform yields several identical representations for the desired pattern functions provided that $\ensuremath{\eta}>1/2.$ In particular, we derive a representation of the realistic pattern functions in terms of a finite weighted sum over scaled ideal ones. Compared to a previously given form, this expression has a nice analytical structure, reveals some general features of the realistic pattern functions, and is particularly well suited for their numerical computation. From this representation sum relations for the pattern functions follow. With their help we show that both a smoothed Wigner function and a generalized moment generating function can be directly sampled from realistic quadrature distributions. The corresponding sampling functions are basically shifted and/or scaled versions of the ideal pattern functions f0,0{f}_{0,0}f0,0​ and f0,k,{f}_{0,k},f0,k​, respectively. As a by-product we derive the sampling functions needed to reconstruct the density matrix from the Fourier-transformed realistic quadrature distribution, i.e., from the s-ordered characteristic function with $s=\ensuremath{-}(1\ensuremath{-}\ensuremath{\eta})/\ensuremath{\eta}.$
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 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.