Boussinesq-type equations from nonlinear realizations of W3W_3

We construct new coset realizations of infinite-dimensional linear $W_3^{\infty}$ symmetry associated with Zamolodchikov's $W_3$ algebra which are different from the previously explored $sl_3$ Toda realization of $W_3^{\infty}$. We deduce the Boussinesq and modified Boussinesq equations as constraints on the geometry of the corresponding coset manifolds.The main characteristic features of these realizations are:i. Among the coset parameters there are the space and time coordinates $x$ and $t$ which enter the Boussinesq equations, all other coset parameters are regarded as fields depending on these coordinates;ii. The spin 2 and 3 currents of $W_3$ and two spin 1 $U(1)$ Kac- Moody currents as well as two spin 0 fields related to the $W_3$currents via Miura maps, come out as the only essential parameters-fields of these cosets. The remaining coset fields are covariantly expressed through them;iii.The Miura maps get a new geometric interpretation as $W_3^{\infty}$ covariant constraints which relate the above fields while passing from one coset manifold to another; iv. The Boussinesq equation and two kinds of the modified Boussinesq equations appear geometrically as the dynamical constraints accomplishing $W_3^{\infty}$ covariant reductions of original coset manifolds to their two-dimensional geodesic submanifolds;v. The zero-curvature representations for these equations arise automatically as a consequence of the covariant reduction. The approach proposed could provide a universal geometric description of the relationship between $W$-type algebras and integrable hierarchies.Comment: 23 pages, LaTe

N=2 Super - W3W_{3} Algebra and N=2 Super Boussinesq Equations

We study classical $N=2$ super-$W_3$ algebra and its interplay with $N=2$ supersymmetric extensions of the Boussinesq equation in the framework of the nonlinear realization method and the inverse Higgs - covariant reduction approach. These techniques have been previously applied by us in the bosonic $W_3$ case to give a new geometric interpretation of the Boussinesq hierarchy. Here we deduce the most general $N=2$ super Boussinesq equation and two kinds of the modified $N=2$ super Boussinesq equations, as well as the super Miura maps relating these systems to each other, by applying the covariant reduction to certain coset manifolds of linear $N=2$ super-$W_3^{\infty}$ symmetry associated with $N=2$ super-$W_3$. We discuss the integrability properties of the equations obtained and their correspondence with the formulation based on the notion of the second hamiltonian structure.Comment: LaTeX, 30

Scalable quantum search using trapped ions

We propose a scalable implementation of Grover's quantum search algorithm in a trapped-ion quantum information processor. The system is initialized in an entangled Dicke state by using simple adiabatic techniques. The inversion-about-average and the oracle operators take the form of single off-resonant laser pulses, addressing, respectively, all and half of the ions in the trap. This is made possible by utilizing the physical symmetrie of the trapped-ion linear crystal. The physical realization of the algorithm represents a dramatic simplification: each logical iteration (oracle and inversion about average) requires only two physical interaction steps, in contrast to the large number of concatenated gates required by previous approaches. This does not only facilitate the implementation, but also increases the overall fidelity of the algorithm.Comment: 6 pages, 2 figure

Two-photon double ionization of helium in the region of photon energies 42-50 eV

We report the total integrated cross-section (TICS) of two-photon double ionization of helium in the photon energy range from 42 to 50 eV. Our computational procedure relies on a numerical solution of the time-dependent Schr\"odinger equation on a square-integrable basis and subsequent projection of this solution on a set of final states describing two electrons in continuum. Close to the threshold, we reproduce results previously known from the literature. The region 47-50 eV seems to have been previously unexplored. Our results suggest that TICS, as a function of the photon energy, grows monotonously in the region 42-50 eV. We also present fully resolved triple differential cross sections for selected photon energies.Comment: 12 pages, 3 figure