Geometric phases for SU (3) representations and three level quantum systems (original) (raw)
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International Journal of Modern Physics A, 2001
We analyze the geometric aspects of unitary evolution of general states for a multilevel quantum system by exploiting the structure of coadjoint orbits in the unitary group Lie algebra. Using the same methods in the case of SU (3) we study the effect of degeneracies on geometric phases for three-level systems. This is shown to lead to a highly nontrivial generalization of the result for two-level systems in which degeneracy results in a "monopole" structure in parameter space. The rich structures that arise are related to the geometry of adjoint orbits in SU (3). The limiting case of a two-level degeneracy in a three-level system is shown to lead to the known monopole structure.
Bloch sphere-like construction of SU (3) Hamiltonians using unitary integration
Journal of Physics A: Mathematical and Theoretical, 2009
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 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.
Bloch Sphere Like Construction of SU(3) Hamiltonians
Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, 2008
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 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.
Physical Review A, 2008
A two-sphere ("Bloch" or "Poincare") is familiar for describing the dynamics of a spin-1/2 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 in terms of lowerdimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various sub-groups 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 vectors. Correspondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a foursphere for the SO(5) sub-group and an eight-dimensional Grassmannian manifold for the general SU(4). This latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on a five-sphere with an additional phase constraint.
Differential geometry on SU(3) with applications to three state systems
Journal of Mathematical Physics, 1998
The left and right invariant vector fields are calculated in an "Euler angle" type parameterization for the group manifold of SU (3), referred to here as Euler coordinates. The corresponding left and right invariant one-forms are then calculated. This enables the calculation of the invariant volume element or Haar measure. These are then used to describe the density matrix of a pure state and geometric phases for three state systems.
Geometric phases and Bloch sphere constructions for SU(N), with a complete description of SU(4)
Bulletin of the American Physical Society, 2008
A two-sphere ("Bloch" or "Poincare") is familiar for describing the dynamics of a spin-1/2 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 in terms of lowerdimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation. 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 vectors. Correspondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a foursphere for the SO(5) subgroup and an eight-dimensional Grassmannian manifold for the general SU(4). This latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on a five-sphere with an additional phase constraint.
1997
The group SU (3) is parameterized in terms of generalized "Euler angles". The differential operators of SU (3) corresponding to the Lie Algebra elements are obtained, the invariant forms are found, the group invariant volume element is found, and some relevant comments about the geometry of the group manifold are made.
Geometric phase in the G3+ quantum state evolution
When quantum mechanical qubits as elements of two dimensional complex Hilbert space are generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space, geometrically formal complex plane becomes explicitly defined as an arbitrary, variable plane in 3D. The result is that the quantum state definition and evolution receive more detailed description, including clear calculations of geometric phase, with important consequences for topological quantum computing.
Geometric phases for three state systems
1999
The adiabatic geometric phases for general three state systems are discussed. An explicit parameterization for space of states of these systems is given. The abelian and non-abelian connection one-forms or vector potentials that would appear in a three dimensional quantum system with adiabatic characteristics are given explicitly. This is done in terms of the Euler angle parameterization of SU (3) which enables a straightforward calculation of these quantities and its immediate generalization.
Journal of Physics A: Mathematical and General, 1998
The "D" matrices for all states of the two fundamental representations and octet are shown in the generalized Euler angle parameterization. The raising and lowering operators are given in terms of linear combinations of the left invariant vector fields of the group manifold in this parameterization. Using these differential operators the highest weight state of an arbitrary irreducible representation is found and a description of the calculation of Clebsch-Gordon coefficients is given.