Qualitative analysis of behaviors of quantum systems

Taking advantages of the Pechukas-Yukawa formalism, our research explores the dynamic behavior of small-scale quantum systems ranging from 2 to 5 qubits with the presence of decoherence, specifically investigating the conditions that favor the system's occupation of the ground state. Inspired by the fundamental insights of Landau-Zener transitions which show the relation between anti-crossings and transitions among adjacent energy levels, our study leverages the Pechukas-Yukawa formalism to explore the manipulation of transition probabilities within quantum systems. Our research reveals that the transition probabilities between adjacent energy levels can be effectively modulated near the vicinity of anti-crossings by tuning the external controlling parameter λ. This shows a possibility of precise control of the occupation probability distribution of the quantum systems through the external control parameter. Through the PY formalism, we provide an explanation for the phenomenon occurred in our simulations that components of noise generally reinforce each other and point out the possibility that this mutually reinforcing effect can be reduced by changing the composition of the noise. In addition, we found a relation between the power spectral density of the expectation value of energy of the quantum system and its likelihood of transitioning from its current energy level, i.e., the broadening of the power spectral density is always associated with an increased probability of a quantum system escaping from the current energy level, especially when initialised in an edge state (the ground or the most excited energy level). Systems initialised in edge states have a lower tendency to transit compared to those initialled in intermediate energy levels. This observation of the relation between the broadening of the power spectral density of expectation value of energy and an increased likelihood for the system to escape from its current energy level provides a new research direction about the influence of dynamical complexity of the system on its occupation probability distribution. Moreover, since spectral analysis is considered as one of possible indicators of quantum chaotic behaviors, if combined with other diagnostic methods, our study can provide some help in future studies on the role of quantum chaos on the dynamics of quantum systems. Additionally, our results show that in some special cases, the external controlling field λ can drive the quantum system from the mixed excited states to the ground state with a high probability (exceeding 90%). And this staying in the ground state can be maintained for an extended duration (over 150 periods of λ). This is a discovery with promising implications for the design of adiabatic quantum computers.Statement: Chatgpt has been used to improve the language of this thesis.