10,875 research outputs found

    Quantum state engineering by a coherent superposition of photon subtraction and addition

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    We study a coherent superposition of field annihilation and creation operator acting on continuous variable systems and propose its application for quantum state engineering. Specifically, it is investigated how the superposed operation transforms a classical state to a nonclassical one, together with emerging nonclassical effects. We also propose an experimental scheme to implement this elementary coherent operation and discuss its usefulness to produce an arbitrary superposition of number states involving up to two photons.Comment: published version, 7 pages, 8 figure

    Generic Initial Ideals And Graded Artinian Level Algebras Not Having The Weak-Lefschetz Property

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    We find a sufficient condition that \H is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function =˝(h0,h1,...,hdβˆ’1>hd=hd+1)\H=(h_0,h_1,..., h_{d-1}>h_d=h_{d+1}) cannot be level if hd≀2d+3h_d\le 2d+3, and that there exists a level O-sequence of codimension 3 of type \H for hdβ‰₯2d+kh_d \ge 2d+k for kβ‰₯4k\ge 4. Furthermore, we show that \H is not level if Ξ²1,d+2(Ilex)=Ξ²2,d+2(Ilex)\beta_{1,d+2}(I^{\rm lex})=\beta_{2,d+2}(I^{\rm lex}), and also prove that any codimension 3 Artinian graded algebra A=R/IA=R/I cannot be level if \beta_{1,d+2}(\Gin(I))=\beta_{2,d+2}(\Gin(I)). In this case, the Hilbert function of AA does not have to satisfy the condition hdβˆ’1>hd=hd+1h_{d-1}>h_d=h_{d+1}. Moreover, we show that every codimension nn graded Artinian level algebra having the Weak-Lefschetz Property has the strictly unimodal Hilbert function having a growth condition on (hdβˆ’1βˆ’hd)≀(nβˆ’1)(hdβˆ’hd+1)(h_{d-1}-h_{d}) \le (n-1)(h_d-h_{d+1}) for every d>ΞΈd > \theta where h0...>hsβˆ’1>hs. h_0...>h_{s-1}>h_s. In particular, we find that if AA is of codimension 3, then (hdβˆ’1βˆ’hd)<2(hdβˆ’hd+1)(h_{d-1}-h_{d}) < 2(h_d-h_{d+1}) for every ΞΈ<d<s\theta< d <s and hsβˆ’1≀3hsh_{s-1}\le 3 h_s, and prove that if AA is a codimension 3 Artinian algebra with an hh-vector (1,3,h2,...,hs)(1,3,h_2,...,h_s) such that h_{d-1}-h_d=2(h_d-h_{d+1})>0 \quad \text{and} \quad \soc(A)_{d-1}=0 for some r1(A)<d<sr_1(A)<d<s, then (I≀d+1)(I_{\le d+1}) is (d+1)(d+1)-regular and \dim_k\soc(A)_d=h_d-h_{d+1}.Comment: 25 page
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