## Two-level system coupled to a structured environment - a computational study.

##### Abstract

It is well known that analytical solutions to large complex problems are not tractable. Computational
or numerical simulations of these types of systems allow one to investigate regimes where analytical
methods fail. In this thesis the dynamics of an open quantum system which does not have an exact
analytical solution is calculated numerically. The model investigated is a two-level system coupled
to a non-trivial structured environment. There are two regimes of interaction between the system
and environment, which is of interest in this thesis, namely the weak and strong coupling regimes.
These different regimes are obtained by tuning the interaction strength parameter in the model that
also define whether the model is Markovian or non-Markovian. Other parameters of the model are
also varied in order to gauge thermalization and relaxation rates during interaction. Computational or
numerical simulations can become a long tedious process when one is running simulations on a single
core or even a dual core machine, since these simulations can run for days and in all cases require
multiple runs to obtain statistics. With the development of clusters which contain thousands of cores
and parallel computing software, running simulations is now very efficient and not a tedious process
as before since parts of the code can be run over multiple cores. In this thesis the code is written in
Python and uses the parallel computing software library called Message-Passing Interface (MPI).
In the first chapter two types of quantum systems, namely closed and open quantum systems, are
introduced. The standard equations generally used for these two types of systems are presented and
the differences in the treatment of these systems are pointed out. The Heisenberg and interaction
pictures are also described in this chapter. The model is then discussed in depth in chapter two where
the Hamiltonian of the total system is introduced and the interaction picture Hamiltonian is derived.
An overview of approximate analytical solutions to the model is discussed. Chapter three discusses
parallel coding specifically MPI and the code for the simulations are presented and explained. In
chapter four plots are done simulating the model and discussing the effects and outcome of changing
parameters which govern the interaction and thermalization of the system and the environment.