Non-Abelian Berry Phases for Holes Confined in SiGe Two-Dimensional Quantum Well System

By coupling the Bloch equation with the time-dependent Schrödinger equation, we formulate the Berry connection tensor for the non-adiabatic process and then define the non-Abelian Berry phase matrix for the multi-subband system. We apply these results to holes confined in a two-dimensional SiGe quantum well system, where the Rashba field induces structure-inversion-asymmetry and the alternating arrangement of Si and Ge heteroatoms causes the bulk-inversion-asymmetry, resulting in the Rashba and Dresselhaus spin-orbit interaction, respectively. Due to the violence of Stokes' theorem, we calculate the non-Abelian Berry phase matrix for the holes by carrying out the contour integration of the non-Abelian Berry connection tensor faithfully. A comparison of the Abelian and non-Abelian Berry phases reveals that the strong intersubband hybridization around the quasi-degenerate points destroys the adiabatic process. Consequently, the "π"-quantization in the Berry phase against energy is broken at these points. We further explore this non-adiabatic process by analyzing the intersubband transition matrix and decomposing it into quantum logic gates.