Integrable lattices and their sublattices II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

An integrable self-adjoint 7-point scheme on the triangular lattice and an integrable self-adjoint scheme on the honeycomb lattice are studied using the sublattice approach. The star-triangle relation between these systems is introduced, and the Darboux transformations for both linear problems from the Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A geometric interpretation of the Laplace transformations of the self-adjoint 7-point scheme is given and the corresponding novel integrable discrete 3D system is constructed.Comment: 15 pages, 6 figures; references added, some typos correcte

Prospects for the measurement of the electron electric dipole moment using YbF

We discuss an experiment underway at Imperial College London to measure the permanent electric dipole moment (EDM) of the electron using a molecular beam of YbF. We describe the measurement method, which uses a combination of laser and radiofrequency resonance techniques to detect the spin precession of the YbF molecule in a strong electric field. We pay particular attention to the analysis scheme and explore some of the possible systematic effects which might mimic the EDM signal. Finally, we describe technical improvements which should increase the sensitivity by more than an order of magnitude over the current experimental limit.Comment: 6 pages, 2 figure

Franck-Condon Factors and Radiative Lifetime of the A^{2}\Pi_{1/2} - X^{2}\Sigma^{+} Transition of Ytterbium Monoflouride, YbF

The fluorescence spectrum resulting from laser excitation of the A^{2}\Pi_{1/2} - X^{2}\Sigma^{+} (0,0) band of ytterbium monofluoride, YbF, has been recorded and analyzed to determine the Franck-Condon factors. The measured values are compared with those predicted from Rydberg-Klein-Rees (RKR) potential energy curves. From the fluorescence decay curve the radiative lifetime of the A^{2}\Pi_{1/2} state is measured to be 28\pm2 ns, and the corresponding transition dipole moment is 4.39\pm0.16 D. The implications for laser cooling YbF are discussed.Comment: 5 pages, 5 figure

Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations

In this paper we consider three-dimensional quantum q-oscillator field theory without spectral parameters. We construct an essentially big set of eigenstates of evolution with unity eigenvalue of discrete time evolution operator. All these eigenstates belong to a subspace of total Hilbert space where an action of evolution operator can be identified with quantized discrete BKP equations (synonym Miwa equations). The key ingredients of our construction are specific eigenstates of a single three-dimensional R-matrix. These eigenstates are boundary states for hidden three-dimensional structures of U_q(B_n^1) and U_q(D_n^1)$.Comment: 13 page

Enhancement of the electric dipole moment of the electron in the YbF molecule

We calculate an effective electric field on the unpaired electron in the YbF molecule. This field determines sensitivity of the molecular experiment to the electric dipole moment of the electron. We use experimental value of the spin-doubling constant to estimate the admixture of the configuration with the hole in the 4f-shell of Ytterbium to the ground state of the molecule. This admixture reduces the field by 7%. Our value for the effictive field is 5.1 a.u. = 2.5 10^{10} V/cm.Comment: 5 pages, LATEX, uses revtex.st

Towards a Mg lattice clock: Observation of the 1S0−^1S_{0}-3P0^3P_{0} transition and determination of the magic wavelength

We optically excite the electronic state $3s3p~^3P_{0}$ in $^{24}$Mg atoms, laser-cooled and trapped in a magic-wavelength lattice. An applied magnetic field enhances the coupling of the light to the otherwise strictly forbidden transition. We determine the magic wavelength, the quadratic magnetic Zeeman shift and the transition frequency to be 468.463(207)$\,$nm, -206.6(2.0)$\,$MHz/T$^2$ and 655 058 646 691(101)$\,$kHz, respectively. These are compared with theoretical predictions and results from complementary experiments. We also developed a high-precision relativistic structure model for magnesium, give an improved theoretical value for the blackbody radiation shift and discuss a clock based on bosonic magnesium.Comment: 5 pages, 3 figure

Pseudospherical surfaces on time scales: a geometric definition and the spectral approach

We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains (e.g, the Cantor set). In particular, we present a new expression for the discrete Gaussian curvature which turns out to be valid for asymptotic nets on any time scale. We show that asymptotic Chebyshev nets on an arbitrary time scale have constant negative Gaussian curvature. We present also the quaternion-valued spectral problem (the Lax pair) and the Darboux-Backlund transformation for pseudospherical surfaces (in asymptotic coordinates) on arbitrary time scales.Comment: 20 page

Darboux transformations for a 6-point scheme

We introduce (binary) Darboux transformation for general differential equation of the second order in two independent variables. We present a discrete version of the transformation for a 6-point difference scheme. The scheme is appropriate to solving a hyperbolic type initial-boundary value problem. We discuss several reductions and specifications of the transformations as well as construction of other Darboux covariant schemes by means of existing ones. In particular we introduce a 10-point scheme which can be regarded as the discretization of self-adjoint hyperbolic equation