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New Interpretation of the Wigner Function

Jamil Daboul

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

I define a two-sided or forward-backward propagator for the pseudo-diffusion equation of the 'squeezed' Q function. This propagator leads to squeezing in one of the phase-space variables and anti-squeezing in the other. By noting that the Q function is related to the Wigner function by a special case of the above propagator, I am led to a new interpretation of the Wigner function.

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Journal of Physics A Mathematical and General•Aug 21, 1995•8 citations
I define a two-sided or forward-backward propagator for the pseudo-diffusion equation of the 'squeezed' Q function. This propagator leads to squeezing in one of the phase-space variables and anti-squeezing in the other. By noting that the Q function is related to the Wigner function by a special case of the above propagator, I am led to a new interpretation of the Wigner function.
I show that a usual propagator cannot be defined for the pseudo-diffusion equation of the Q-functions. Instead, a forward-backward propagator is defined, which motivated a generalization of Cahill-Glauber interpolating operator. Our generalized operator {\\bf Q}(p, q;\\sigma_p^{-1},\\sigma_q) depends on two squeezing parameters \\sigma_p and \\sigma_q, and is shown to obey a generalized pseudo-diffusion equation or a diffusion equation, depending on the curve (\\sigma_p(\\mu),\\sigma_q(\\mu)) along which one moves in the (\\sigma_p, \\sigma_q) plane. An algorithm is also given for squeezing Q functions directly, using one-dimensional diffusion propagators.
We show that the projection operator mod pq;yei phi )(yei phi ;pq mod , where mod pq;yei phi ) is a squeezed state, obeys a partial differential equation in which the squeeze parameter y plays the role of time. It follows that related functions, such as the probability distribution functions and the Wigner function are solutions of this equation. This equation will be called a pseudodiffusion equation, because it resembles a diffusion equation in Minkowski space. We give general solutions of the pseudo-diffusion equation, first by the method of separation of variables and then by the Fourier transform method, and discuss the limitations of the latter method. The Fourier method is used to introduce squeezing into the number states, the thermal light and the Wigner function.