The Preparation Of Multi-Mode Continuous-Variable Cluster States With Optical Cavity

The popularity and fast development of the internet and computer communication technology make people feel the rapid spread of information bring to our life. At the same time, people have been thinking and looking for more efficient and secure methods of communication, which has given rise to a new type of interdisciplinary-quantum information. Recently, continuous variable quantum information has attracted a lot of attention which has made researchers to explore it from theory to experiment and apply in practical areas such as continuous variable quantum teleportation, quantum entanglement transformation, quantum key distribution, quantum dense coding, quantum memory and quantum computation. The applications of quantum information need an entangled state as the resource and carrier of the information. It is well known that a continuous variable cluster state is a highly entangled multipartite pure state, which could be employed as the resource for one-way quantum computation. Several linear optical methods have been proposed for the preparation of continuous variable cluster states experimentally.In this paper, we discuss theoretical methods of how to prepare cluster states in a system composed of an array of coupled optical cavities. We propose two different theoretical methods for the preparation of continuous variable multimode cluster states. In the second chapter we illustrate the first method, which involves a scheme composed of an array of six single-mode optical cavities. Each cavity contains an ensemble of four-level atoms and short fibers are used to connect the cavities in order to realize the coupling between the modes. In the first step of the calculations, we transform the free Hamiltonian of the system to the interaction picture, then set a large detuning of frequencies between the cavity modes, the laser and the atomic transition, so we can perform adiabatic elimination of the excited states of the atoms. This allows us to get the simplified Hamiltonian, which involves transitions only between the atomic ground states and the cavity modes. Next, we introduce collective spin operators of the atoms and perform the Holstein-Primakoff transformation to get the bosonic representation of the collective operators. Then we perform a rotation transformation to get the final Hamiltonian describing the interaction between the atoms and the field. In the preparation of the cluster state, we need to find the unitary transformation matrix corresponding to the structure of the graph state, showing that all modes can be prepared in the squeezed state at the same time by a suitable choice of the Rabi frequency and phase of the driven laser field. We show that continuous variable six mode linear, hexagon and double-square cluster states can be realized because the system can evolve to a steady state with the cavity dissipation.Theoretically, the single step preparation method illustrated in the second chapter can be extended to the case of an arbitrary large number of modes and to arbitrary shape of the cluster state. In fact, we have also studied the feasibility of the single step method to prepare a continuous variable eight-mode cluster state. However, we have encountered several problems and could not solve it in a simple and transparent way. Instead, we put forward the multistep method of the preparation of continuous variable cluster states. The method is illustrated in the third chapter and involves a scheme composed of a single-mode circular cavity containing eight atomic ensembles. The ensembles are arranged according to the octagon and atoms are driven by two sets of lasers at the same time. In the calculations, we take a similar approach as in the first method. We transform the Hamiltonian of the system into the interaction picture as before, but because there is only one cavity, the method of transforming the free Hamiltonian is relatively simple. Then, we assume a large detuning between the light field and the atomic transition. Ignoring the spontaneous radiation of the atoms, we perform the adiabatic elimination of the upper atomic states and introduce collective spin operators. After performing the Holstein-Primakoff transformation and a series of simplified calculations, we get the effective Hamiltonian of the system. We then show how in eight steps we can make the cavity mode and the eight combination modes to achieve, with the help of the laser pulses, to the continuous variable octagon, linear and two-diamond cluster states. We prove that under the action of the cavity dissipation, all combination modes evolve into the squeezed vacuum state, which in turn is the continuous variables eight-mode cluster state. We check the condition that the target state satisfies the definition of the continuous-variable cluster state.In the fourth chapter we summarize our results and propose the possible direction for future work.