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arXiv:quant-ph/9606034 · Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Reconstructing the density operator by using generalized field quadratures

G M D'Ariano, S Mancini, V I Man'ko, P Tombesi

Published Oct 1, 1996• 106 citations
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Abstract

The Wigner function for one- and two-mode quantum systems is explicitly expressed in terms of the marginal distribution for the generic linearly transformed quadratures. Then, the density operator of those systems is also written in terms of the marginal distribution of these quadratures. Some applications and a reduction to the usual optical homodyne tomography are considered.

DOI
10.1088/1355-5111/8/5/007

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The Wigner function for one- and two-mode quantum systems is explicitly expressed in terms of the marginal distribution for the generic linearly transformed quadratures. Then, the density operator of those systems is also written in terms of the marginal distribution of these quadratures. Some applications and a reduction to the usual optical homodyne tomography are considered.
We report measurements of the Wigner distribution and the density matrix of an electromagnetic field mode, for both vacuum and quadrature-squeezed states. As proposed by Vogel and Risken,1 we obtained the Wigner distribution by tomographic inversion of a set of probability distributions of field-quadrature amplitudes measured by using balanced homodyne detection. A Fourier transform of the Wiener function yields the density matrix, which, according to the standard interpretation of quantum mechanics, contains all knowable information about a given quantum system.
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 $s=\ensuremath{-}(1\ensuremath{-}\ensuremath{\eta})/\ensuremath{\eta}.$