Generalized spin Sutherland systems revisited

We present generalizations of the spin Sutherland systems obtained earlier by Blom and Langmann and by Polychronakos in two different ways: from SU(n) Yang--Mills theory on the cylinder and by constraining geodesic motion on the N-fold direct product of SU(n) with itself, for any N>1. Our systems are in correspondence with the Dynkin diagram automorphisms of arbitrary connected and simply connected compact simple Lie groups. We give a finite-dimensional as well as an infinite-dimensional derivation and shed light on the mechanism whereby they lead to the same classical integrable systems. The infinite-dimensional approach, based on twisted current algebras (alias Yang--Mills with twisted boundary conditions), was inspired by the derivation of the spinless Sutherland model due to Gorsky and Nekrasov. The finite-dimensional method relies on Hamiltonian reduction under twisted conjugations of N-fold direct product groups, linking the quantum mechanics of the reduced systems to representation theory similarly as was explored previously in the N=1 case.Comment: 21 page

Derivations of the trigonometric BC(n) Sutherland model by quantum Hamiltonian reduction

The BC(n) Sutherland Hamiltonian with coupling constants parametrized by three arbitrary integers is derived by reductions of the Laplace operator of the group U(N). The reductions are obtained by applying the Laplace operator on spaces of certain vector valued functions equivariant under suitable symmetric subgroups of U(N)\times U(N). Three different reduction schemes are considered, the simplest one being the compact real form of the reduction of the Laplacian of GL(2n,C) to the complex BC(n) Sutherland Hamiltonian previously studied by Oblomkov.Comment: 30 pages, LateX; v2: final version with minor stylistic modification

Error Rate of the Kane Quantum Computer CNOT Gate in the Presence of Dephasing

We study the error rate of CNOT operations in the Kane solid state quantum computer architecture. A spin Hamiltonian is used to describe the system. Dephasing is included as exponential decay of the off diagonal elements of the system's density matrix. Using available spin echo decay data, the CNOT error rate is estimated at approsimately 10^{-3}.Comment: New version includes substantial additional data and merges two old figures into one. (12 pages, 6 figures

Generalized Killing equations and Taub-NUT spinning space

The generalized Killing equations for the configuration space of spinning particles (spinning space) are analysed. Simple solutions of the homogeneous part of these equations are expressed in terms of Killing-Yano tensors. The general results are applied to the case of the four-dimensional euclidean Taub-NUT manifold.Comment: 10 pages, late

Remarks on Finite W Algebras

The property of some finite W algebras to be the commutant of a particular subalgebra of a simple Lie algebra G is used to construct realizations of G. When G=so(4,2), unitary representations of the conformal and Poincare algebras are recognized in this approach, which can be compared to the usual induced representation technique. When G=sp(2,R) or sp(4,R), the anyonic parameter can be seen as the eigenvalue of a W generator in such W representations of G. The generalization of such properties to the affine case is also discussed in the conclusion, where an alternative of the Wakimoto construction for sl(2) level k is briefly presented. This mini review is based on invited talks presented by P. Sorba at the ``Vth International Colloquium on Quantum Groups and Integrable Systems'', Prague (Czech Republic), June 1996; ``Extended and Quantum Algebras and their Applications to Physics'', Tianjin (China), August 1996; ``Selected Topics of Theoretical and Modern Mathematical Physics'', Tbilisi (Georgia), September 1996; to be published in the Proceedings.Comment: LaTeX, 16 pages, references adde

Dynamical supersymmetry of spin particle-magnetic field interaction

We study the super and dynamical symmetries of a fermion in a monopole background. The Hamiltonian also involves an additional spin-orbit coupling term, which is parameterized by the gyromagnetic ratio. We construct the superinvariants associated with the system using a SUSY extension of a previously proposed algorithm, based on Grassmann-valued Killing tensors. Conserved quantities arise for certain definite values of the gyromagnetic factor: $\N=1$ SUSY requires $g=2$; a Kepler-type dynamical symmetry only arises, however, for the anomalous values $g=0$ and $g=4$. The two anomalous systems can be unified into an $\N=2$ SUSY system built by doubling the number of Grassmann variables. The planar system also exhibits an $\N=2$ supersymmetry without Grassmann variable doubling.Comment: 23 page

Controlling spin relaxation with a cavity

Spontaneous emission of radiation is one of the fundamental mechanisms by which an excited quantum system returns to equilibrium. For spins, however, spontaneous emission is generally negligible compared to other non-radiative relaxation processes because of the weak coupling between the magnetic dipole and the electromagnetic field. In 1946, Purcell realized that the spontaneous emission rate can be strongly enhanced by placing the quantum system in a resonant cavity -an effect which has since been used extensively to control the lifetime of atoms and semiconducting heterostructures coupled to microwave or optical cavities, underpinning single-photon sources. Here we report the first application of these ideas to spins in solids. By coupling donor spins in silicon to a superconducting microwave cavity of high quality factor and small mode volume, we reach for the first time the regime where spontaneous emission constitutes the dominant spin relaxation mechanism. The relaxation rate is increased by three orders of magnitude when the spins are tuned to the cavity resonance, showing that energy relaxation can be engineered and controlled on-demand. Our results provide a novel and general way to initialise spin systems into their ground state, with applications in magnetic resonance and quantum information processing. They also demonstrate that, contrary to popular belief, the coupling between the magnetic dipole of a spin and the electromagnetic field can be enhanced up to the point where quantum fluctuations have a dramatic effect on the spin dynamics; as such our work represents an important step towards the coherent magnetic coupling of individual spins to microwave photons.Comment: 8 pages, 6 figures, 1 tabl