Bloch sphere-like construction of SU(3) Hamiltonians using unitary integration

The Bloch sphere is a familiar and useful geometrical picture of the time evolution of a single spin or a quantal two-level system. The analogous geometrical picture for three-level systems is presented with several applications. The relevant SU(3) group and su(3) algebra are eight-dimensional objects and are realized in our picture as two four-dimensional manifolds that describe the time evolution operator. The first, called the base manifold, is the counterpart of the S2 Bloch sphere, whereas the second, called the fiber, generalizes the single U(1) phase of a single spin. Now four dimensional, it breaks down further into smaller objects depending on alternative representations that we discuss. Geometrical phases are also developed and presented for specific applications. Arbitrary time-dependent couplings between three levels or between two spins (qubits) with SU(3) Hamiltonians can be conveniently handled through these geometrical objects. © 2009 IOP Publishing Ltd