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

Abstract

The Bloch sphere is a familiar and useful geometrical picture of the dynamics of a single spin or two-level system's quantum evolution. 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 describing the time evolution operator. The first, called the base manifold, is the counterpart of the S^2 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.

19 pages, 9 figures

References in corpus (5)

• Quantum phases of a qutrit
• Four-level and two-qubit systems, sub-algebras, and unitary integration
• Geometric phases and Bloch sphere constructions for SU(N), with a complete description of SU(4)
• Geometric phase for SU(N) through fiber bundles and unitary integration
• Steering quantum transitions between three crossing energy levels

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• Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical Systems, a Lie-Algebraic Approach
• Three-level spin system under decoherence-minimizing driving fields: Application to nitrogen-vacancy spin dynamics
• Calculation of quantum discord in arbitrary dimensions, especially for X- and other specialized states
• Affine Maps of the Polarization Vector for Quantum Systems of Arbitrary Dimension