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arXiv:1107.0526 · Physical Review A

Optical homodyne tomography with polynomial series expansion

Hugo Benichi, Akira Furusawa

Published Sep 6, 2011• 9 citations
arXivHTMLPDF

Abstract

We present and demonstrate a method for optical homodyne tomography based on the inverse Radon transform. Different from the usual filtered back-projection algorithm, this method uses an appropriate polynomial series to expand the Wigner function and the marginal distribution, and discretize Fourier space. We show that this technique solves most technical difficulties encountered with kernel deconvolution-based methods and reconstructs overall better and smoother Wigner functions. We also give estimators of the reconstruction errors for both methods and show improvement in noise handling properties and resilience to statistical errors.

DOI
10.1103/physreva.84.032104

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We present and demonstrate a method for optical homodyne tomography based on the inverse Radon transform. Different from the usual filtered back-projection algorithm, this method uses an appropriate polynomial series to expand the Wigner function and the marginal distribution, and discretize Fourier space. We show that this technique solves most technical difficulties encountered with kernel deconvolution-based methods and reconstructs overall better and smoother Wigner functions. We also give estimators of the reconstruction errors for both methods and show improvement in noise handling properties and resilience to statistical errors.
This chapter begins with a review of classical tomography, which reconstructs hidden structures inside an object by successive transmissions of an X-ray beam at different angles and lateral displacements. By interpreting the Wigner distribution as an analogue of the density distribution in a physical object, the mathematical methods of classical tomography — the Radon transform, the Fourier slice theorem, and the inverse Radon transform — are adapted to perform a high resolution determination of a quantum state of light. In optical homodyne tomography, the classical transmission angle is replaced by the adjustable phase of a local oscillator, and the successive lateral displacements are replaced by a series of measurements of the quadrature operator defined by the local oscillator phase. Experimental data is presented showing various properties of the Wigner distribution for a coherent state.
The problem of optical homodyne tomography is considered in the context of a Bayesian model-fitting or inverse problem approach. An algorithm is formulated, based on matrix computation rather than the numerical approximation of an analytic inverse transform. This automatically takes into account the effects of noise, detector inefficiencies and incomplete sampling of the data. The relationships with conventional reconstruction schemes, methods for including various forms of prior information and for calculating error estimates are discussed. The process of reconstructing the photon number distribution and the density matrix are illustrated using both simulated and experimental data.
In this work, a continuous variable quantum tomography is presented. This is done using two approaches, one using simulated data and another one using data from measurements performed in the laboratory. The Radon inverse transform, under the approximation of filtered back projection, alongside the details on the implementa- tion of balanced homodyne detection needed to perform the tomography are clearly explained. Density matrices in the quadrature and number representation are also obtained from the Wigner function. Additionally, complete documentation of the computational implementation and a user-friendly interface for the tomography is reported
We propose an alternative method for detecting the density matrix of a radiation field via optical homodyne tomography. The method needs no assumption on the state, and the density matrix is recovered directly from averages on data. The tomographic reconstruction is very fast and provides very reliable statistics on line with detection. As a test of the method we present numericla results for a set of computer-simulated experiments, corresponding to different quantum states of the radiation source. Then a tomographic reconstruction is given for experimental data provided by Smithey etal. [Phys. Rev. Lett. 70, 1244 (1993)].
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 $s=\ensuremath{-}(1\ensuremath{-}\ensuremath{\eta})/\ensuremath{\eta}.$