Study Of Nonlinear And Inhomogeneous Bloch Equation And Geometric Quantum Phase Of Mixed State

Quantum pure state and mixed state are origin essentially from application of therepresentation for the Poincaré manifold, where the pure state corresponds to point onthe sphere and the mixed state to inner point in the sphere. Therefore，it may be areasonable expansion for the geometric phase of mixed state by the Poincaré manifold.Thus a possible approach to the geometric phase of mixed state is to find a nonunitstate vector describing the mixed state.In this thesis, we firstly seek for a generic solution of density matrix fromnonlinear and inhomogeneous master equation, then the one-to one relation betweenthe inner point of Poincaré sphere and nounit state vector of Hilbert space is founded.By using the Bloch radius and parameters, we rescale the microscopic physicalquantities describing the evolution of mixed state and nonunit state vector of mixedstate in a unified way, we investigate the correlation between the microscopic physicalquantities and geometric phase in order to find a way about adjusting and controllingquantum memory in terms of the geometric phase.The results are applied to the fluorescent oscillation and nuclear spin polarizationsystem. We find that by adjusting the initial conditions and external controllingphysical parameters, we can obtain the conditional geometric phase and further realizea controllable fault-tolerant quantum memory in terms of this conditional geometricphase in the fluorescent oscillation and nuclear spin polarization system.