The Non-classical Properties Of The Nonlinear Circular State And The Quantum Dynamics Of The Nonlinear Trapped Ion

Non-classical properties of quantum states and their preparation has been much attention. By building a new quantum states, people not only analyze their non-classical properties, also produce the quantum states by using interaction of atom or trapped ion with light fields. In this paper, on the basis of previous research, we mainly investigate influence of nonlinear effects on quantum states by using method of theoretical analysis and numerical calculation. The main contents are as follows:The first chapter introduces some non-classical properties of quantum states and quantum state engineering.The second chapter introduces the basic theories and typical models such as Wigner function, criterion of non-classical effects, etc.In the third chapter,we study a new quantum state,namely nonlinear circular state. The nonclassical properties of the state is analyzed such as time evolution of average photon number, sub-Possion distribution, and squeezing effect, and its Wigner functions is calculated. Through numerical analysis, the results show that with increasing of the Lamb-Dicke parameter value and superposition number of quantum states, the average photon number of nonlinear circular state is increased and the sub-Poisson distribution and squeezing effect of nonlinear circular state are both weakened.In the fourth chapter, we study the influence of nonlinear parameters on the cross-correlation function of two vibrational modes of the trapped ion along mutually orthogonal directions.The exact solution of the nonlinear double photons Jaynes-Cummings model which describes the interaction of a single two-dimension three-level trapped ion interacted with two classical standing wave lasers is obtained, and the cross-correlation function of two vibrational modes of the trapped ion is numerically analyzed. The results show that for small initial average quantum number of vibrational motion of the trapped ion, the anticorrelated effect is first srenghtened then weakened with increasing of the value of the Lamb-Dicke parameters, but the correlated effect is always srenghtened. The anticorrelated effect will be weakened then disappear increasing of initial average quantum number.The fifth chapter contains summary and outlook of the research work. In the future, I intend to study quantum properties of nonlinear added-photon circular state.