| The advent of quantum computation and quantum information has led to the study of quantum computation based on molecular ro-vibrational states. The concept of quantum computation based on molecular ro-vibrations employs ro-vibrational ex-cited states to represent qubit, and the shaped femtosecond laser pulse in the IR regime could be adopted to implement quantum logic operation. An advantage of the molecule based quantum computation is that the ro-vibrational states are fair-ly stable and the number of available qubits is proportional to the ro-vibrational degrees of freedom for N-atomic molecule, which means the multi-qubit could be easily established. Several theoretical explorations for many molecules have shown that a quite high quantum gate fidelity can be realized. Thus, the vibrational quantum computation and the relevant questions, which include finding more suit-able molecular systems, designing quantum logic operation and the intramolecular entanglement, have attracted more and more attention.As one of most mystical features of the quantum formalism, the dynamical entanglement refers to the non-classical strong correlation exists in two or more quantum system. In the fields of quantum information and quantum computation, quantum entanglement is widely used as an important resource. By using quantum entanglement, one can accomplish some tasks which can not be realized by the classical computation, such as quantum algorithms which are based on the entan-glement, quantum teleportation, quantum dense coding and so on. Thus, the deep researches of entanglement have far-reaching influences on quantum information. In the recent studies, researchers have investigated the properties of quantum en-tanglement from different viewpoints, for example, the measurement, preparation, storage, transformation of entanglement and the evolution of entanglement under the decoherence process. Although dynamical entanglement itself is not of dynam-ical nature, entangled states are often generated and evaluated dynamically. The studies of dynamical properties of quantum entanglement in the real systems have became one subject of intense interest. The studies on the dynamical entanglement could help us to dynamically grasp the behavior of quantum entanglement in quan-tum information processing, and then lay the foundation for further control and use of entanglement.Since entanglement plays an important role in quantum information, dynam-ical entanglement in real molecular systems are becoming an important part of the molecular computation. The main topics of recent studies on the intramolec-ular entanglement are the dynamical properties of entanglement between different ro-vibrational modes and the entanglement between the electronic freedom and ro-vibrational freedom. Such studies could help one to deeply understand the molec-ular vibrations and its influence on the quantum information. Meanwhile, these studies also provide references for the appropriate use of quantum entanglement.The investigation of quantum chaos has shown that the quantum system’s clas-sical limit dynamics have great influences on generation and evolution of entangle-ment. The studies on many systems have shown that when the center of the wave packet is located on the chaotic trajectory, the generation rate of linear entropy is higher than the case of the center of the wave packet on the periodical orbit, and the generation rate is in a linear relationship with the Lyapunov expend. The quantum entanglement is considered to have the essence connection with quantum chaos. In other word, quantum entanglement could be a signature of quantum chaos. The vibration of polyatomic molecule, which contains many types of an-harmonic vibrations, is a complex system. In the field of highly excited states, due to the nonlinear coupling between different vibrations, the spectra is becoming extremely complex, and exhibits many unique features, including the local mod-e vibration and chaos. Since the traditional quantum method is extraordinarily complicated, the method and concept of classical dynamics can be introduced in describing the physical images of molecular vibration. It is shown that the semi-classical analysis of molecular vibrations could help one to identify the vibrational spectra, understand the intramolecular vibrational energy redistribution and so on. As a result, the analysis of the dynamical properties could be achieved through the classical-quantum correspondence.The decoherence of qubits is an important issue in restricting the advance of quantum computation. For a multipartite quantum system, decoherence leads to degradation of entanglement even sudden death. For the molecular vibrational qubits, the decoherence resources may come from the collision with other molecules and the intramolecular anharmonic resonances with the remaining modes. Regard-ing molecules in the gas phase, the number of collisions can be kept low. The studies on the intramolecular decoherence and robustness of entanglement against the remaining modes are thus important in selecting suitable molecules to apply quantum computation.The Lie algebraic model of molecules, which was introduced by Iachello and Levine et al. at1980s from the nuclear physics to molecular physics, has been proven to be an effective model in describing of vibrations in polyatomic molecules. The Lie algebraic model is based on the essence of dynamical symmetry. The dy-namical symmetry of a system means the invariant properties during evolution, and it shows that the Hamiltonian of the system can maintain unchanged under the time dependent operation. Therefore, the general process of constructing the algebraic Hamiltonian is as follows:one should find the dynamical symmetrical group firstly; then according to the dynamical properties the group chains are es-tablished; finally, the ro-vibrational Hamiltonian is expressed as the combination of Casimir operators of the subgroups in the group chains. On the corresponding basis, the matrix elements of the Hamiltonian are obtained, and the eigenvalues are the ro-vibrational energy levels which can be compared with the experimental results. Since the Lie algebraic is successfully applied, it’s superiority stands out day by day. For example, to obtain the perfect results, the Lie algebraic methods needs much few parameters in calculating the vibrational spectra. By applying this mode, the algebraic Hamiltonian of various molecules from small molecules to molecular chains could be established. Because of these advantages, the algebraic method has extensive applications. It can not only used to fit the vibrational spectra and determine the potential energy surface, but also to study the dynamical problems including:the molecular multiphoton process in IR laser field and its control, the molecular gas phase surface scattering and the intramolecular dynamical entangle-ment. In practice, the U(4) algebraic model is suitable to describe the vibrations of polyatomic molecules especially the small molecules, this is not only because the U(4) describes the3degrees of freedom, but also the Fermi interaction is intro-duced through the nondiagonal elements of Majorana without external algebraic. Moreover, the explicit classical Hamiltonian could be deduced by the U(3) alge-braic model. In the present work, the U(4) algebraic model is adopted to discuss the classical chaos and dynamical entanglement of triatomic molecule. Moreover, the explicit potential energy surface could be constructed, thus this model is also suitable to study the classical chaos of molecular vibrations.This thesis organized as followsIn chapter one, we give the background of our researches, and the brief summary of the advance of molecular vibrational quantum computation. We also review the development of the vibrational entanglement in the molecular systems.In chapter two, we mainly introduce the theoretical basis of this thesis. We firstly introduce the development of the Lie algebraic method in the molecular physics, and expound the general process of constructing the algebraic Hamiltonian and the relevant concepts. Then, by using U(4) algebraic model, we construct the algebraic Hamiltonian of the triatomic molecules, and the matrix elements are provided under the local mode basis. By employing the classical limit of Intensive Boson operator, we give the classical vibrational Hamiltonian in the intramolecular coordinates.In chapter three and four, by restricting the bending vibration at its ground state, we consider the classical dynamics and dynamical entanglement of stretching-stretching vibrations. In chapter three, we firstly discuss the phase structures of the coupling Morse oscillators, and explain the underlying classical dynamics of the tori on the Poincare sections. Then, we compare and discuss the phase structures and dynamical properties of two local mode molecules H20, H2S and three normal mode molecules O3,NO2and SO2. In chapter four, the dynamical entanglement be-tween two stretching vibrations of these five molecules are discussed, and in order to relate the dynamical properties of the system with dynamical entanglement, the relationship between energy transfer and dynamical entanglement are considered. The results show that there are close correlations between the dynamical entan-glement and the energy transfer between bonds when in the low energy levels. In the highly excited states, the evolution of dynamical entanglement of initial local mode character states in the local mode molecules shows the long period beats, and the dynamical behaviors are extremely different for the initial local and normal mode character states. But, the dynamical entanglement has the similar dynam-ical behavior of initial local and normal mode character states in normal mode molecules. Through the comparative study of the dynamical entanglement and classical dynamics, we find that the local mode vibration can decrease the degree of entanglement, but the classical chaos and nonlinear resonances can promote the generation of entanglement.In chapter five, We study the decoherence process caused by the bending vibra-tion when the bending vibration is released and the tripartite entanglement. By assuming the stretching-stretching vibrations as a bipartite qubit system, we firstly calculate the evolution of purity of the stretching-stretching qubit when bending vibration is initially prepared in different states. It is shown that bending vibra-tion can decrease the purity of the stretching-stretching system. For some special states the evolution of purity is in periodical, and the period is stable value which is not changed with different excitation in bending vibration. Then, we discuss the dynamical entanglement of the stretching-stretching vibrations under the affection of bending vibration. We have found that the bending vibration makes the period of energy transfer and dynamical entanglement even longer. For the local mode molecules, the correspondence between the energy transfer and entanglement is not changed, but the correspondence will be destroyed for the normal mode molecules. Finally, the dynamical behaviors of tripartite entanglement of the bending vibration and two stretching vibrations are discussed. The results show that the entanglement between the stretching-stretching vibrations are the main factor, and the tripartite entanglement of initial local bending states shows nice periodic.In the chapter six, we give a brief summary and an outlook. |
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