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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Tomographic measurement of the density matrix of the radiation field

Giacomo Mauro D’Ariano

Published Aug 1, 1995• 28 citations
Publisher

Abstract

Recently a novel homodine tomographic technique has been proposed which allows us to detect the density matrix experimentally in terms of averages on data. The method has been further extended to a general matrix element ( psi mod Q mod phi ) of the density operator Q, along with the possibility of using low-efficiency detectors. In this paper this technique is reconsidered as a new genuine quantum measurement. Numerical results for measured probabilities are given, and the mechanism underlying the generation of statistical errors in the measured matrix is illustrated. It is shown that the data processing of the novel technique can also be used as a new imaging algorithm for conventional tomography.

DOI
10.1088/1355-5111/7/4/022

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Recently a novel homodine tomographic technique has been proposed which allows us to detect the density matrix experimentally in terms of averages on data. The method has been further extended to a general matrix element ( psi mod Q mod phi ) of the density operator Q, along with the possibility of using low-efficiency detectors. In this paper this technique is reconsidered as a new genuine quantum measurement. Numerical results for measured probabilities are given, and the mechanism underlying the generation of statistical errors in the measured matrix is illustrated. It is shown that the data processing of the novel technique can also be used as a new imaging algorithm for conventional tomography.
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)].
We provide a simple analytic relation that connects the density operator \ensuremath{\rho}^ of the electromagnetic field with the tomographic homodyne probabilities for generic quantum efficiency \ensuremath{\eta} of detectors. The problem of experimentally ``sampling'' a general matrix element 〈\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}\ensuremath{\Phi}〉 is addressed in the statistically rigorous sense of the central-limit theorem. We show that experimental sampling is possible also for nonunit efficiency, provided that \ensuremath{\eta} satisfies a lower bound related to the ``resolutions'' of vectors \ensuremath{\Vert}\ensuremath{\psi}〉 and \ensuremath{\Vert}\ensuremath{\Phi}〉 in the quadrature representations. For coherent and number states the bound is \ensuremath{\eta}\ensuremath{\gtrsim}1/2. On the basis of computer-simulated experiments we show the feasibility of detecting delicate quantum probability oscillations, which otherwise would be smeared out by inefficient detection.
We propose a general method for reconstructing directly the density matrix of a single light mode in optical homodyne tomography. In our scheme the density matrix 〈a\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}${\mathit{a}}^{\ensuremath{'}}$〉 is obtained by averaging a set of pattern functions ${\mathit{F}}_{\mathit{aa}}^{\ensuremath{'}}$(${\mathit{x}}_{\mathrm{\ensuremath{\theta}}}$,\ensuremath{\theta}) with respect to the homodyne data ${\mathit{x}}_{\mathrm{\ensuremath{\theta}}}$. The functions show the typical features of the quadrature distributions for the corresponding density-matrix elements. It is also possible to compensate the effect of detection losses which requires, however, extra effort in both experimental and numerical precision. We calculate the pattern functions for the coherent-state and Fock representations and study their properties. We believe that our method is the most efficient way for reconstructing the density matrix from homodyne measurements.
We present a numerical algorithm for the maximum-likelihood estimation of the density matrix, and apply it to the homodyne tomography of a single-mode radiation field. The algorithm is based on a specific form of the Gauss decomposition for positive definite Hermitian matrices. Results from Monte Carlo simulated experiments are presented. PACS: 42.50.Dv, 03.65.Bz Over past several years a great deal of interest has been paid to the reconstruction of the complete quantum state from experimental data [1]. Theoretical research in this field, apart from designing new measurement schemes, explores also the problem of extracting the optimal amount of information from data available in a realistic experimental setup. In this communication, we present a numerical algorithm for reconstructing the density matrix using the maximum-likelihood method [2, 3]. We discuss this algorithm in the context of quantum homodyne tomography [4, 5], which is actually the most successful method for quantum state reconstruction, and, in fact, the unique technique that has been so far experimentally implemented to measure nonclassical states of light [4, 6]. The realistic homodyne measurement of the quadrature is described by the positive operator-valued measure