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Physical Review A

Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase

Karl Vogel, H. Risken

Published Sep 1, 1989• 1,311 citations
Publisher

Abstract

It is shown that the probability distribution for the rotated quadrature phase [${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$exp(i\ensuremath{\theta})+a exp(-i\ensuremath{\theta})]/2 can be expressed in terms of quasiprobability distributions such as P, Q, and Wigner functions and that also the reverse is true, i.e., if the probability distribution for the rotated quadrature phase is known for every \ensuremath{\theta} in the interval 0\ensuremath{\le}\ensuremath{\theta}<\ensuremath{\pi}, then the quasiprobability distributions can be obtained.

DOI
10.1103/physreva.40.2847

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It is shown that the probability distribution for the rotated quadrature phase [a\ifmmode∘\else\textdegree\fi{a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}a\ifmmode∘\else\textdegree\fiexp(i\ensuremath{\theta})+a exp(-i\ensuremath{\theta})]/2 can be expressed in terms of quasiprobability distributions such as P, Q, and Wigner functions and that also the reverse is true, i.e., if the probability distribution for the rotated quadrature phase is known for every \ensuremath{\theta} in the interval 0\ensuremath{\le}\ensuremath{\theta}<\ensuremath{\pi}, then the quasiprobability distributions can be obtained.
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.
We introduce the Wigner operator [Formula: see text] for the rotated quadrature phases and use the technique of integration within an ordered product of operators to derive its explicitly simpler form. Based on this, the mutual relations between [Formula: see text] and the corresponding marginal probability distribution operator can be easily revealed. The Wigner function theory is thus be recasted into a more elegant and concise formalism. The squeezing in rotated quadrature phase is discussed with the same method.
Phase properties of squeezed number states and squeezed thermal states are studied. Exact analytical formulae for phase distributions based on different phase approaches are derived and illustrated graphically. It is shown that the phase quasiprobability distribution P (W) ( theta ) associated with the Wigner function does not depend on the photon number of the initial number state and has the same form for both kinds of squeezed states under consideration while the Pegg-Barnett phase distribution and the phase quasiprobability distributions associated with the Q function and the Glauber-Sudarshan P function, approach P (W) ( theta ) in the limit of highly excited states.
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.