Although the reconstruction of the Wigner function from generalized quasiprobability distributions that can be interpreted as smoothed Wigner functions poses no problem for theoretical calculations, there are, however, difficulties in the reconstruction when those distributions are gained from measurements.
We propose the reconstruction of a smoothed Wigner function from quadrature distributions instead of the Wigner function itself to avoid the problems with the inverse Radon transformation. Starting from the well known relation between s -parametrized quasiprobability distributions W ( x , p , s ) and quadrature component distributions w ( x , ), we show that the sampling function needed to determine a smoothed Wigner function W ( x , p ,-| s |) is basically a shifted and scaled version of the well known pattern function necessary to reconstruct the probability of finding no photon in the signal field. An analogous result has been recently found in the literature. Here we obtain this result in a very concise and alternative way.
In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.
In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
We have recently derived a phase space approximation for wave propagation, and we have shown that this approximation is accurate especially compared to the stationary phase approximation. The method gives an approximate Wigner distribution, which raises the question, is the approximate Wigner distribution representable? If it is, then we can invert it to obtain the corresponding wave function. If it is not, can we obtain an approximate wave function from it, and how good is the approximation? Although the answer to the first question is no, we show that from the Wigner approximation one can recover the exact spatial spectrum magnitude and derivative of the phase. We also consider other methods for obtaining approximate wave functions from the Wigner approximation.