| The continuous-variable （CV） cluster entangled state is the vital resource for construction of quantum information network and operation of one-way quantum computation. It can be used in the fields of quantum information communication, quantum computation and quantum measurement widely,and then continuing to promote the development of quantum information science and technology. Therefore, how to obtain scheme of generating the CV cluster states is one of the focuses for present researches. Nowadays,there are some theoretical and experimental studies on the generation of multipartite entanglement based on frequency domain and time domain,which can applied in wavelength division multiplexing and time division multiplexing.In this thesis, a large-Fresnel-number degenerate optical parametric oscillation （DOPO） operating below threshold and self-imaging optical parametric oscillator （OPO） are used as the basic model of research, which is pumped by two Laguerre-Gaussian modes. Then, the CV cluster entangled states based on spatial mode comb can be generated through process of parametric down-conversion. The starting point of the calculation is based on the Hamilton and Langevin equations, we have calculated the steady-state solution and the quadrature fluctuations of the output fields, in the experimental feasible parameters regime, obtained the entanglement degree versus different system parameters. Thus, we have gotten the optimal parameters of the system and proved that the CV cluster entangled states is entangled with each other. This is a theoretical scheme from frequency domain to spatial domain, which has strong operability. The CV cluster entangled states generated not only can be used in spatial division multiplexing, but also can establish a solid foundation for future practical quantum computation and quantum secret communication of large capacity optical fiber.The main works are as follows:（1） The large-Fresnel-number DOPO is pumped by two Laguerre-Gaussian modes (lg1p,1g-1p) with the same frequency and polarization,which can generate eleven-partite CV cluster entangled states of Laguerre-Gaussian modes based on optical spatial mode comb. In the scheme,the large-Fresnel-number DOPO can allow simultaneous and sustainable nonlinear interaction and resonance of all down-converted modes. The nonlinear interaction must satisfy energy, phase-match and orbit angular momentum conservation. The CV cluster entangled states based on optical spatial mode comb have important applications in measurement-based quantum computation, quantum information processing and quantum image.（2） The self-imaging OPO is used as the theory model and two nonlinear crystals PPKTP with different phase-matching zzz and yyy parametric down conversion are placed in the cavity, respectively. This scheme can double the number of down-converted modes by considering the different polarization.The OPO pumped by two Laguerre-Gaussian modes (lg1pz,1g-1py) can generate 20 Laguerre-Gaussian modes with the same frequency and different polarization （z, y） through the process of PDC. Then, the 20 down-converted modes turn into CV dual-rail cluster entangled state after passing through a 50/50 beam splitter. This scheme This scheme has strong operability and controllability in experiment, and provides the theoretical basis and experimental guidance for the large-scale spatial CV cluster entangled states in the future. |
