Analytic description for synchronization of two-level quantum systems

Abstract

Synchronization of quantum mechanical systems have in recent years become a well-discussed topic. The most explored systems are the quantum van der Pol oscillator and two-level system (TLS) synchronizing to an external field and synchronizing to another TLS. However, it was claimed that two-level systems do not have a valid limit cycle, and they can therefore not be synchronized. It was suggested that we could circumvent this problem by realizing a mixed state as an ensemble of pure states. Using quantum trajectory theory (QTT), numerical evidence was found for the synchronization of a TLS. We want to find an analytic expression for the synchronization observed numerically. In this thesis we develop the framework to do so: We give results from QTT which helps analyze the model used, and we show how the flow of the state of a TLS from a QTT model can be easily visualized. Furthermore, we give an explicit expression for the space of all Hamiltonians giving the same Lindblad equation for the QTT model used in, and give a master equation which can be solved to find the analytic expression for the synchronization observed numerically.