Bloch Sphere Representation and Logic Gate Analysis for a Heavy Hole Confined in Semiconductor Two-Dimensional Quantum Well Systems

The topological features of a heavy-mass hole (HH) confined in a 2D Si quantum well system via Bloch sphere (BS) representation and logic gate analysis are studied. Strong anisotropy-and-nonparabolicity in the semiconductor valence band causes the re-degeneracy between HHs having opposite spins. The newly yielded degenerate states function as topological singularities. By employing the semiclassical equation of cyclotron motion, the time-evolved (TE) Schrödinger equation as a rate equation is rewritten, and then the numerically solved TE wave function on the BS by determining the zenith (Formula presented.) and azimuthal (Formula presented.) angles is represented. The two-state rotating-wave (TSRW) approximation introduced reveals that the slope of (Formula presented.) against quantum state (time) evolution includes the information of the non-Abelian Berry connection. The TSRW further demonstrates that the slope of (Formula presented.) contains the information of the Abelian connection. The TE operator is expressed by the three rotation logic gates (Formula presented.) around the x-axis by an angle (Formula presented.), which is sandwiched by the two rotations (Formula presented.) and (Formula presented.) around the BS pole axis (z). The TSRW approach enables us to decompose (Formula presented.) into two partial rotations around the z-axis, with Abelian and non-Abelian Berry connections.