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Physical Review A

Detection of the density matrix through optical homodyne tomography without filtered back projection

Giacomo Mauro D’Ariano, Chiara Macchiavello, Matteo G. A. Paris

Published Nov 1, 1994• 218 citations
Publisher

Abstract

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)].

DOI
10.1103/physreva.50.4298

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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 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.
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.
Quantum tomography is a procedure to determine the quantum state of a physical system, or equivalently, to estimate the expectation value of any operator. It consists in appropriately averaging the outcomes of the measurement results of different observables, obtained on identical copies of the same system. Alternatively, it consists in maximizing an appropriate likelihood function defined on the same data. The procedure can be also used to completely characterize an unknown apparatus. Here we focus on the electromagnetic field, where the tomographic observables are obtained from homodyne detection.
A simple and efficient method for characterization of multidimensional Gaussian states is suggested and experimentally demonstrated. Our scheme shows analogies with tomography of finite-dimensional quantum states, with the covariance matrix playing the role of the density matrix and homodyne detection providing Stern-Gerlach-like projections. The major difference stems from a different character of relevant noises: while the statistics of Stern-Gerlach-like measurements is governed by binomial statistics, the detection of quadrature variances corresponds to ${\ensuremath{\chi}}^{2}$ statistics. For Gaussian and near Gaussian states the suggested method provides, compared to standard tomography techniques, more stable and reliable reconstructions. In addition, by putting together reconstruction methods for Gaussian and arbitrary states, we obtain a tool to detect the non-Gaussian character of optical signals.
Ultimately the properties of all optical fields are contained in the density matrix that specifies their quantum mechanical state. It has only recently been demonstrated that it is possible to actually reconstruct this density matrix from measured data, and to do so with temporal resolution of a few hundred femtoseconds.[1] This technique, known as Optical Homodyne Tomography (OHT), is unfortunately restricted to the measurement of only single mode radiation. In fact it selects out from the test field only those components which overlap temporally with the local oscillator pulse mode.