Van Ryn, Nicholas.

### Abstract:

The motivation for this Masters thesis is to develop numerical algorithms to study the dynamical evolution of non-Markovian open quantum systems. Such systems are of importance if one is interested in modeling solid state systems which are candidates
for the qubit - the quantum analog of the binary digit. Such an example may be a trapped spin onto which is encoded a chosen spin state. In reality, such a spin is never completely isolated from the environment, and so from a practical point of view it is of interest to study the dynamics of this interaction between some open system with an environment. The goal here is to create a computer program to simulate this behaviour of all density matrix elements for the open system numerically. Many interesting quantum systems, spin chains as an example, do not behave as a Markovian process, and it is sometimes difficult or perhaps indeed impossible to derive exact analytical solutions. As such, the techniques used in this thesis are aimed at describing non-Markovian processes in a way that approaches the exact solution. The study begins by introducing the reader to important concepts and results in the general study of both closed and open quantum systems. Differences in the treatment of the two types of systems are pointed out, and the necessary standard
equations used generally are presented. Additionally, two different techniques are explained for the study of open quantum systems, namely the density matrix approach and the stochastic wavefunction approach. Important results from these two methods are presented and the section ends by convincing the reader of their equivalence.
The second chapter begins with an example of an open quantum system which exhibits non-Markovian behaviour. The model of the spin star system is described and important results are given from references. This chapter introduces the reader to the model, conceptually explaining the system, and going on to show its exact
analytical behaviour. This basic model, with minor changes, will be used throughout this study and the physics, interactions and symmetries, does not really change. This study then shows how one can use the stochastic wavefunction method to solve the dynamics of the spin star model. This chapter follows with deriving stochastic
equations for the same system as the preceding chapter, and using these equations a numerical algorithm is developed, the results of which provide a good comparison between the exact analytical and exact numerical techniques. As a further example, a similar but slightly more complex system is studied in exactly the same manner, with the only important difference being that the open quantum system to be modeled is now a spin-one particle. Important differences in the results are pointed out and explained, and important similarities are highlighted. In presenting the results of this second simulation, shortcomings of the numerical technique and areas of applicability are discussed. In the final chapter the author considers using this numerical technique's ability to completely map the dynamics for a density matrix to investigate a measure of
quantumness for an open system. This research has been submitted for publication to a peer reviewed journal.