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The Wigner Function and its Marginal Distributions for Superposition of the Coherent State

Jiang Jun-qin

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The superposition state |αθ〉=C(|α〉+eiθ|-α〉) from the coherent state was constructed.The θ and α dependences of the Wigner function and its marginal distributions for the superposition state were investigated.It is shown that the Wigner function and its marginal distributions are evidently depended on θ and α.

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The quantum statistical properties of the superposition state, which involve two arbitrary coherent states |β〉 and |mβeiδ〉, are investigated from two aspects, such as the phase distribution and Winger function. The results show that the non-classical properties of the superposition state depend on β2, amplitude coefficient m, the phase difference between the coherent states δ, and the phase difference between the superposition coefficients . When selected suitable parameters the superposition state exhibits quantum effect. The phase distribution and Wigner function of an equally populated mixture of two coherent states are derived, too. By comparison the results show that the presence of the coherence terms makes the superposition state with full quantum-mechanical behavior.
For a tripartite entangled system, based on the newly constructed tripartite Einstein–Podolsky–Rosen entangled state, we find the entangled state representation of the Wigner operator. The corresponding Wigner function then leads us to obtain the physical marginal distributions. We emphasize that, for an entangled-particle system, the physical meaning of the Wigner distribution function should lie in that its marginal distributions give the probability of finding the particles in an entangled way.