Logic gate analysis for spin precession and non-Abelian Berry phase in multi-subband system

In multi-subband systems lacking space inversion symmetry, the spin-orbit interaction (SOI) and inter-subband interaction produce multiple quasidegenerate points (QDPs) having characteristic non-Abelian Berry parameters, as well as time-reversal invariant momenta. In this study, we focus on spin precession and discuss spin features modulated by non-Abelian Berry phases. To extract the effects of non-Abelian Berry phases from spin precessions, we project the corresponding state vectors onto Bloch spheres, eliminating the dynamical phases. We find that the Abelian and non-Abelian Berry phases change the azimuthal and zenith angles, respectively. Furthermore, we interpret them as quantum logic operations and discuss their correspondence to the QDPs. We then apply this formalism to holes confined in SiGe quantum wells with Rashba and Dresselhaus SOIs. In the lower energy region far from the QDPs, R_z operations arise from the Abelian Berry phases. Contrary, we elucidate that, at the energies of the QDPs, the non-Abelian Berry phases cause strong inter-subband hybridizations, realizing NOT-gate operations.