The Application Of Laplace Transformation In The Open Quantum System Mechanics

The dynamic of multi-body open quantum system is a hot topic in quantum information science.The environment of an open quantum system is composed of infinite freedom degrees.In order to solve the dynamic of the system accurately,it is necessary to solve the dynamic equations with infinite variables.In principle,we can regard the quantum system and its environment as a closed system,and solve its dynamics by the Schr¨odinger equation.However,it is not accessible due to the fact that an environment composed of infinite degrees of freedom will quickly exhaust the computation source.In order to exactly solve the dynamics of the open quantum system,it is necessary to employ more advanced methods than the Schr¨odinger equation.The open quantum system inevitably interacts with the surrounding environment,which will lead to the leaking of its energy and quantum information out of the system into the environment.To describe this process,the traditional methods,such as quantum master equation,are based on the Born-Markovian approximation,which assumes that the system-environment interaction strength is very small.The dynamics of the open quantum system is then Markovian.When the interaction strength is comparable to the eigenenergy of the system,the information that leaks into the environment will has finite probability to come back to the system in a finite time,which means the dynamics of open quantum system becomes then non-Markovian.In this thesis,we will introduce both the quantum Markovian master equation and the non-Markovian quantum-state-diffusion equation.With respect to the quantum control field,which deals with the detrimental influence of the environment,we will also introduce the dynamical decoupling method that is used to suppress the decoherence effect.The open quantum system is proposed to explore both the dynamics and control over the reduced density matrix out of a composed system.Various analytical and numerical techniques have been devoted to this interesting field.In some situations,we hope to know the exact solution of the system dynamics,which is often presented by an integro-differential equation,with a rapid technique.In this thesis,we employ the Laplace transformation in the following open quantum system models,which is popular in literatures:(1)a three-level atomic system interacting with two environments,(2)a JC model in a dissipative environment,(3)a two-qubit system trapped in a dissipative cavity;(4)a multiple-qubit model in a zero-temperature reservoir.The exact solutions of these four physical models manifest the role of the Laplace transformation in the dynamics of open quantum systems.We can extend this method to deal with more complex quantum models in the future.