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Chinese Journal of Quantum Electronics

Wigner function of superposition of the excited coherent states

Jiang Jun-qin

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The superposition state C(a~(+m)|α+e~(iθ)α~(+m)|-α) from the excited coherent state was constructed. Dependences of Wigner function onθand m and its marginal distributions for the superposition state were numerically investigated.It is shown that the superposition state is always able to exhibit the non-classical properties for anyθ,Wigner function and its marginal distributions evidently depend onθand m.

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The superposition state C(a~(+m)|α+e~(iθ)α~(+m)|-α) from the excited coherent state was constructed. Dependences of Wigner function onθand m and its marginal distributions for the superposition state were numerically investigated.It is shown that the superposition state is always able to exhibit the non-classical properties for anyθ,Wigner function and its marginal distributions evidently depend onθand m.
The superposition of excited coherent states was constructed by superpositioning two excited coherent states of the same intensity and mutual phase conjugation,and their Wigner functions,second-order coherence degree and Mandel Q factors were calculated,upon which their non-classical properties were analyzed.The numerical results show that,the Wigner functions of the superposition of excited coherent states may become negative,and their second-order coherence degree and Mandel Q factors are both less than 1.Hence,the research shows that the quasi-probability distribution of the superposition of excited coherent states may become negative,and their light field has anti-bunching effects,with the distribution of photon number being of sub-Poisson distribution.Therefore,these superposition states are quantum states with non-classical properties.
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.
The Wigner distribution function for the time-dependent quadratic Hamiltonian system in the coherent Schrodinger cat state is investigated. The type of state we consider is a superposition of two coherent states, which are by an angle of π out of phase with each other. The exact Wigner distribution function of the system is evaluated under a particular choice of phase, δc, q. Our development is employed for two special cases, namely, the Caldirola–Kanai oscillator and the frequency stable damped harmonic oscillator. On the basis of the diverse values of the Wigner distribution function that were plotted, we analyze the nonclassical behavior of the systems.
We consider the quantum superposition of two coherent states shifted in phase by 3π/2, |α⟩ and |− i α⟩, as well as special states that are characterized solely with respect to amplitude α. According to analysis of nonclassical properties such as oscillatory and sub-Poissonian distribution of photon numbers, photon antibunching, quadrature squeezing, and negativity of Wigner function, it is concluded that the considered superposition state is a nonclassical state. Also, the special state is found to exhibit even stronger nonclassical features when relative phase ϕ of the superposition is equal to the average photon number |α| 2 .