Research Lindblad And Redfield Equation Spin Chain Balance In Nature

The study of physical phenomena of the open system is always a hot topic, in quantum theory, master equation is a common method used to study open systems. Master equation is an evolution equation of the reduced density operator in an environment or a bath in quantum system. In this paper, we mainly introduce the famous Lindblad Equation and Redfield Equation, both of which are under different approximations of Quantum Master Equation. In addition, we use both of Equations to study the properties of equilibrium of a one-dimensional quantum spin chain, and compare the results with the result of analytical method.Firstly, we introduce the model of one-dimension quantum spin chain, and the method proposed by Lieb et al which is an analytic solution of equilibrium problem. Also, we introduce the equilibrium property of spin chain, and use this method to be the basis comparison of Lindblad equation and Redfield equation.Secondly, we introduce the "Third Quantization" which is raised by Prosen. We use this method to derive the concrete form of Lindblad equation. We talk about two cases:The first case is only both ends of the spin chain are connected with two baths that have the same temperatures.The second case is every lattice point of the chain is connected with bath that has the same temperature. By means of calculating average magnetic moment and nearest neighbor correlation functions, we find that when the external field h is far greater than the spin-spin exchange interaction Jm, the results of the two physical quantities calculated in two cases accord well with the analytical solution. But when h is close to Jm, both of the two cases is not good enough when compared with the analytical result.Lastly, we use the third quantization to do the further research about Redfield equation.Firstly, we derive the concrete form of Redfield equation under one-dimension quantum XY spin chain, and study the equilibrium property of spin chain. Because of the Redfield equation is the promotion form of Lindblad equation, we prove that in the given parameters, Redfield equation can return to Lindblad equation. Secondly, through the numerical calculate, we also find that when h is far greater than Jm,the average magnetic moment and nearest neighbor correlation functions counted by Redfield equation and Lindblad equation is well accorded with the analytical result. But when h is close to Jm, both of the two equations has a bit of errors, the mainly reason is both of the two equations are obtained by the approximate of kinetic equation.Moreover, when in tight coupling, by comparing Redfield equation equation. When in tight coupling, by comparing Redfield equation with Lindblad equation, we find that the higher the temperature is, Redfield equation is more accord with the analytical result.