Analytical Solutions Of The Coupled Oscillators And The Ground-State Entanglement

In this thesis, analytical solutions of the n-coupled oscillators are studied by using a Vector Coherent State (VCS) type Bethe ansatz. Analytical expressions of excitation ener-gies and the corresponding eigenstates of the system with n=2and n=3are presented. As special examples, unitary transformation conditions for n=2and n=3cases, with which the coupled systems can be transformed into uncoupled systems, are derived. Under the unitary transformation conditions for n=2and n=3cases, the ground state energy of the systems as functions of coupling parameters λx, λp are analyzed. For two coupled oscillators, the reduced von Neumann entropy is used as the entanglement measure of the system, with which the entanglement of the system as a function of the coupling parameters λx, λp is determined. The results indicate that the entanglement as a function of the control parameters of the system clearly shows the quantum phase transitions in the system.