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Journal of Physics A Mathematical and General

Hamilton's turn and geometric phase for two-level systems

R. Simon, Narasimhaiengar Mukunda

Published Nov 21, 1992• 13 citations
Publisher

Abstract

Hamilton, in the course of his studies on quaternions, introduced an elegant geometric representation for the composition of SU(2) elements, in terms of turns on the unit sphere S 2 . The authors use these turns to study two-level systems, with particular reference to geometric phase. The special roles played by piecewise geodesic circuits in the state space and evolution under constant Hamiltonian are recognized.

DOI
10.1088/0305-4470/25/22/034

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Hamilton, in the course of his studies on quaternions, introduced an elegant geometric representation for the composition of SU(2) elements, in terms of turns on the unit sphere S 2 . The authors use these turns to study two-level systems, with particular reference to geometric phase. The special roles played by piecewise geodesic circuits in the state space and evolution under constant Hamiltonian are recognized.
Geometric phases are important in quantum physics and are now central to fault-tolerant quantum computation. For spin 1∕21∕21∕2, the Bloch sphere S2{S}^{2}S2, together with a U(1) phase, provides a complete SU(2) description. We generalize to NNN-level systems and SU(N)\mathrm{SU}(N)SU(N) in terms of a $2(N\ensuremath{-}1)$-dimensional base space and reduction to a $(N\ensuremath{-}1)$-level problem, paralleling closely the two-dimensional case. This iteratively solves the time evolution of an NNN-level system and gives $(N\ensuremath{-}1)$ geometric phases explicitly. A complete analytical construction of an S4{S}^{4}S4 Bloch-like sphere for two qubits is given for the Spin(5) or SO(5) subgroup of SU(4).
Unitary evolutions of a qubit are traditionally represented geometrically as rotations of the Bloch sphere, but the composition of such evolutions is handled algebraically through matrix multiplication [of SU(2) or SO(3) matrices]. Hamilton's construct, called turns, provides for handling the latter pictorially through the addition of directed great circle arcs on the unit sphere S${}^{2}\ensuremath{\subset}{\mathbb{R}}^{3}$, resulting in a non-Abelian version of the parallelogram law of vector addition of the Euclidean translation group. This construct is developed into a visual tool kit for handling the design of single-qubit unitary gates. As an application, it is shown, in the concrete case wherein the qubit is realized as polarization states of light, that all unitary gates can be realized conveniently through a universal gadget consisting of just two quarter-wave plates (QWP) and one half-wave plate (HWP). The analysis and results easily transcribe to other realizations of the qubit: The case of NMR is obtained by simply substituting $\ensuremath{\pi}/2$ and $\ensuremath{\pi}$ pulses respectively for QWPs and HWPs, the phases of the pulses playing the role of the orientation of fast axes of these plates.
We describe a recently developed generalization of the Poincaré sphere method, to represent pure states of a three-level quantum system in a convenient geometrical manner. The construction depends on the properties of the group SU(3) and its generators in the defining representation, and uses geometrical objects and operations in an eight-dimensional real Euclidean space. This construction is then used to develop a generalization of the well known Pancharatnam geometric phase formula, for evolution of a three-level system along a geodesic triangle in state space.
The geometric phase in quantum mechanics was originally elucidated in the context of the adiabatic theorem. This and other limitations were later removed. We show how this phase can be presented as the simplest functional of smooth open curves of unit vectors in Hilbert space, invariant under the two groups of local phase changes and monotonic reparametrisations. This viewpoint clarifies the role of the early Pancharatnam criterion for two Hilbert space vectors to be `in phase', in the later geometric phase context. Noncyclic evolution is handled in a very direct manner. Connections to the Bargmann invariants of quantum mechanics are immediate and can be exploited. A class of geometric phases arises in connection with unitary representations of Lie groups. In this case a surprisingly complete analysis of the structure of the geometric phase is possible, motivated by the Wigner-Eckart theorem. The interplay between geometric phase ideas and Lie group unitary representations is extraordinarily rich; the differential geometric structures of Lie groups and coset spaces play very important roles and allow classification of all possible geometric phases associated with a given Lie group. Examples involving SU(2), SU(3) and the metaplectic group Mp(2) are presented.
A two-sphere (``Bloch'' or ``Poincare'') is familiar for describing the dynamics of a spin-1∕21∕21∕2 particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedure that constructs evolution operators for higher-dimensional SU(N)\mathrm{SU}(N)SU(N) in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation and entangled states in quantum-information sciences. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various subgroups so that fewer parameters suffice, we derive Bloch-like rotation of unit vectors analogous to the one familiar for a single spin in a magnetic field. The unitary evolution of a quantal spin pair is thereby expressed as rotations of real, many-dimensional vectors. Correspondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a four-sphere for the SO(5) subgroup and an eight-dimensional Grassmannian manifold for the general SU(4). The latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on a five-sphere with an additional phase constraint. This geometrical picture for two spins provides the extension and generalization of the Bloch sphere that has proved invaluable for the understanding of the dynamics of a single spin.