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Journal of Modern Optics

Determination of Density Matrices from Field Distributions and Quasiprobabilities

H. Kühn, D.‐G. Welsch, W. Vogel

Published Aug 1, 1994• 72 citations
Publisher

Abstract

It is shown that the density matrix elements of a radiation mode in a chosen field strength representation can be obtained from the field strength distributions (with the phase being varied in a π interval) by a twofold Fourier transformation, and from the quasiprobability distributions P(α; s) (s < 0) by a threefold one. The method also applies, in principle, to the smoothed distributions observed in balanced homodyne (four-port and eight-port) detections with detector efficiencies less than unity.

DOI
10.1080/09500349414552441

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It is shown that the density matrix elements of a radiation mode in a chosen field strength representation can be obtained from the field strength distributions (with the phase being varied in a π interval) by a twofold Fourier transformation, and from the quasiprobability distributions P(α; s) (s < 0) by a threefold one. The method also applies, in principle, to the smoothed distributions observed in balanced homodyne (four-port and eight-port) detections with detector efficiencies less than unity.
Using coherent phase states, parameterized phase state distributions for a single-mode radiation field are introduced and their integral relation to the phase-parameterized field-strength distributions is studied. The integral kernel is evaluated and the problem of direct sampling of the coherent phase state distributions using balanced homodyne detection is considered. Numerical simulations show that when the value of the smoothing parameter is not too small the coherent phase state distributions can be obtained with sufficiently well accuracy. With decreasing value of the smoothing parameter the determination of the coherent phase state distributions may be an effort, because both the numerical calculation of the sampling function and the measurement of the field-strength distributions are required to be performed with drastically increasing accuracy.
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.
We have used the recently demonstrated method of optical homodyne tomography (OHT) to measure the Wigner quasiprobability distribution (Wigner function) and the density matrix for both a squeezed-vacuum and a vacuum state of a single spatial-temporal mode of the electromagnetic field. This method consists of measuring a set of probability distributions for many different Hilbert-space representations of the field-quadrature amplitude, using balanced homodyne detection, and then using tomography to obtain the Wigner function. Once the Wigner function is obtained, one can acquire the density matrix, including its complex phase. In the case of a pure state, this technique yields an experimentally determined complex wavefunction, as demonstrated here for the vacuum. The density matrix represents a complete quantum mechanical characterization of the state. From the measured density matrix we have obtained the Pegg–Barnett optical phase distribution, and from the Wigner function, the Wigner optical phase distribution.
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.