Higher-order properties and Bell-inequality violation for the three-mode enhanced squeezed state

By extending the usual two-mode squeezing operator $S_{2}=\exp [ i\lambda (Q_{1}P_{2}+Q_{2}P_{1}) ]$ to the three-mode squeezing operator $S_{3}=\exp {i\lambda [ Q_{1}(P_{2}+P_{3}) +Q_{2}(P_{1}+P_{3}) +Q_{3}(P_{1}+P_{2}) ]}$, we obtain the corresponding three-mode squeezed coherent state. The state's higher-order properties, such as higher-order squeezing and higher-order sub-Possonian photon statistics, are investigated. It is found that the new squeezed state not only can be squeezed to all even orders but also exhibits squeezing enhancement comparing with the usual cases. In addition, we examine the violation of Bell-inequality for the three-mode squeezed states by using the formalism of Wigner representation

Orientation and Motion of Water Molecules at Air/Water Interface

Analysis of SFG vibrational spectra of OH stretching bands in four experimental configurations shows that orientational motion of water molecule at air/water interface is libratory within a limited angular range. This picture is significantly different from the previous conclusion that the interfacial water molecule orientation varies over a broad range within the vibrational relaxation time, the only direct experimental evidence for ultrafast and broad orientational motion of a liquid interface by Wei et al. [Phys. Rev. Lett. 86, 4799, (2001)] using single SFG experimental configuration

Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees

AbstractLet G be a simple graph, its Laplacian matrix is the difference of the diagonal matrix of its degrees and its adjacency matrix. Denote its eigenvalues by μ(G)=μ1(G)⩾μ2(G)⩾⋯⩾μn(G)=0. A vertex of degree one is called a pendant vertex. Let Tn,k be a tree with n vertices, which is obtained by adding paths P1,P2,…,Pk of almost equal the number of its vertices to the pendant vertices of the star K1,k. In this paper, the following results are given:(1) Let T be a tree with n vertices and k pendant vertices. Thenμ(T)⩽μ(Tn,k),where equality holds if and only if T is isomorphic to Tn,k.(2) Let G be a simple connected bipartite graph with degrees d1,d2,…,dn. Thenμ(G)⩾21n∑i=1ndi2,where equality holds if and only if G is a regular connected bipartite graph.(3) Let G be a simple connected bipartite graph with vertices v1,v2,…,vn and their degrees d1,d2,…,dn. Thenμ(G)⩾2+1m∑vi∼vj,i<j(di+dj-2)2,where m is the edge number of G and equality holds if and only if G is either a regular connected bipartite graph or a semiregular connected bipartite graph or the path with four vertices