In this paper, the entanglement properties of a system of two atoms inside an optical cavity in a stochastic interaction with field are studied by the Jaynes-Cummings Model. The phase telegraph noise is considered as a noise term and an exact solution to the model is obtained. The solution reveals the resulting decoherence effects of the noise on the entanglement properties of the system. It shows that under the noise the individual atoms do not entangle with the cavity field. However, a strong atom-atom entanglement is observed in a stationary state. It is seen that a relatively strong noise is cooperative in the construction of the steady state atom-atom entanglement.
Using Buchberger-Shirshov Algorithm and Composition-Diamond lemma we obtain the reduced Gröbner-Shirshov bases of An and classify all reduced words of the affine Weyl group An.
We discuss the calculation of integral cohomology ring ofLG/TandΩG. First we describe the root system and Weyl group ofLG, then we give some homotopy equivalences on the loop groups and homogeneous spaces, and calculate the cohomology ring structures ofLG/TandΩGfor affine groupA^2. We introduce combinatorial integers(m,nj)which play a crucial role in our calculations and give some interesting identities among these integers. Last we calculate generators for ideals and rank of each module of graded integral cohomology algebra in the local coefficient ringℤ[1/2].
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