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arXiv:1010.0584 · Journal of Physics A Mathematical and Theoretical

Wigner function evolution in a self-Kerr medium derived by entangled state representation

Liyun Hu, Zhenglu Duan, Xue-xiang Xu, Zisheng Wang

Published Apr 14, 2011• 8 citations
arXivHTMLPDF

Abstract

By introducing the thermo entangled state representation, we convert the calculation of Wigner function (WF) of density operator to an overlap between "two pure" states in a two-mode enlarged Fock space. Furthermore, we derive a new WF evolution formula of any initial state in self-Kerr Medium with photon loss and find that the photon number distribution for any initial state is independent of the coupling factor with Kerr Medium, where the number state is not affected by the Kerr nonlinearity and evolves into a density operator of binomial distribution.

DOI
10.1088/1751-8113/44/19/195304

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By introducing the thermo entangled state representation, we convert the calculation of Wigner function (WF) of density operator to an overlap between "two pure" states in a two-mode enlarged Fock space. Furthermore, we derive a new WF evolution formula of any initial state in self-Kerr Medium with photon loss and find that the photon number distribution for any initial state is independent of the coupling factor with Kerr Medium, where the number state is not affected by the Kerr nonlinearity and evolves into a density operator of binomial distribution.
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Using the thermal-entangled state representation and the operator-ordering method, we investigate Wigner function (WF) for the squeezed negative binomial state (SNBS) and the analytical evolution law of density operator in the amplitude decay channel. The results show that the analytical WF is related to the square of the module of single-variable Hermite polynomials, which leads to a new two-variable special function and its generating function, and the parameters s and γ play opposite roles in the WF distributions. Besides, after undergoing this channel, the initial pure SNBS evolves into a new mixed state related to two operator Hermite polynomials within normal ordering, and fully loses its nonclassicality and decays to vacuum at long decay time.
We adopt a new approach, thermo entangled representation, to study time evolution of density operator in thermal environment. We then investigate the analytical expressions of Wigner function (WF) evolution of arbitrary number excited coherent states (ECSs) and excited even (odd) coherent states (EECSs, EOCSs) in thermal environment, respectively. In addition, their nonclassicality is numerically discussed by exploring the negativity of WF with decay time in thermal channel, respectively. It is found that WF loses its non-Gaussian nature and becomes Gaussian after long times.
Master equations of density operators can be concisely solved by virtue of thermo entangled state representation and the technique of integration within an ordered product of operators.The solution of master equations of density operators is given.We get the Wigner function evolution formula in phase space and photon number distribution formula of the state,which is a superposition of vacuum state and single-photon state initially in laser process.By numerical calculation we plot its Wigner function and photon number distribution evolution curves,when cavity gain constant and cavity loss constant are both equal to certain values.The effect of decoherence on nonclassical property of the state is discussed by comparing negative area size and the depth of negativity of Wigner function.Numerical results show that the negativity area size and the depth of negativity of its Wigner function decrease and then vanish with dissipation.In addition,we calculate the variation of negative volume of Wigner function with dissipation time.We find that it decreases with dissipation time increasing.These results show that the nonclassical properties of the state are weakened and then disappear with dissipation time increasing.