Abstract:
. We consider the asymptotic almost sure behavior of the solution of the equation \begin{eqnarray*} u(t,x) &=& u_{0}(x) + \kappa \int_{0}^{t} \Delta u(s,x)ds + \int_{0}^{t}u(s,x)\partial B_{x}(s)\\ &&u(0,x)=u_0(x) \end{eqnarray*} where $\{B_xx \in \mathbf Z^d\}$ is a field of independent Brownian motions.

Abstract:
We report an experimental determination of the diamagnetic correction to the $^9$Be$^+$ ground state hyperfine constant $A$. We measured $A$ = $-625\,008\,837.371(11)$ Hz at a magnetic field $B$ of 4.4609 T. Comparison with previous results, obtained at lower values of $B$ (0.68 T and 0.82 T), yields the diamagnetic shift coefficient $k$ = $2.63(18) \times 10^{-11}$ T$^{-2}$, where $A(B)=A_0\times (1+k B^2)$. The zero-field hyperfine constant $A_0$ is determined to be $-625\,008\,837.044(12)$ Hz. The $g$-factor ratio ${g_I}^\prime/g_J$ is determined to be $2.134\,779\,852\,7(10) \times 10^{-4}$, which is equal to the value measured at lower $B$ to within experimental error. Upper limits are placed on some other corrections to the Breit-Rabi formula. The measured value of $k$ agrees with theoretical estimates.

Abstract:
the soluble and insoluble cotyledon (spf-co and ipf-co) and tegument (spf-te and ipf-te) cell wall polymer fractions of common beans (phaseolus vulgaris) were isolated using a chemical-enzymatic method. the sugar composition showed that spf-co was constituted of 38.6% arabinose, 23.4% uronic acids, 12.7% galactose, 11.2% xylose, 6.4% mannose and 6.1% glucose, probably derived from slightly branched and weakly bound polymers. the ipf-co was fractionated with chelating agent (cdta) and with increasing concentrations of naoh. the bulk of the cell wall polymers (29.4%) were extracted with 4.0m naoh and this fraction contained mainly arabinose (55.0%), uronic acid (18.9%), glucose (10.7%), xylose (10.3%) and galactose (3.4%). about 8.7% and 10.6% of the polymers were solubilised with cdta and 0.01m naoh respectively and were constituted of arabinose (52.0 and 45.9%), uronic acids (25.8 and 29.8%), xylose (9.6 and 10.2%), galactose (6.1 and 3.9%) and glucose (6.5 and 3.8%). the cell wall polymers were also constituted of small amounts (5.6 and 7.2%) of cellulose (cel) and of non-extractable cell wall polymers (necw). about 16.8% and 17.2% of the polymers were solubilised with 0.5 and 1.0m naoh and contained, respectively, 92.1 and 90.7% of glucose derived from starch (ist). the neutral sugar and polymers solubilization profiles showed that weakly bound pectins are present mainly in spf-co (water-soluble), cdta and 0.01-0.1m naoh soluble fractions. less soluble, highly cross-linked pectins were solubilised with 4.0m naoh. this pectin is arabinose-rich, probably highly branched and has a higher molecular weight than the pectin present in spf-co, cdta and 0.01-0.1m naoh fractions.

Abstract:
The soluble and insoluble cotyledon (SPF-Co and IPF-Co) and tegument (SPF-Te and IPF-Te) cell wall polymer fractions of common beans (Phaseolus vulgaris) were isolated using a chemical-enzymatic method. The sugar composition showed that SPF-Co was constituted of 38.6% arabinose, 23.4% uronic acids, 12.7% galactose, 11.2% xylose, 6.4% mannose and 6.1% glucose, probably derived from slightly branched and weakly bound polymers. The IPF-Co was fractionated with chelating agent (CDTA) and with increasing concentrations of NaOH. The bulk of the cell wall polymers (29.4%) were extracted with 4.0M NaOH and this fraction contained mainly arabinose (55.0%), uronic acid (18.9%), glucose (10.7%), xylose (10.3%) and galactose (3.4%). About 8.7% and 10.6% of the polymers were solubilised with CDTA and 0.01M NaOH respectively and were constituted of arabinose (52.0 and 45.9%), uronic acids (25.8 and 29.8%), xylose (9.6 and 10.2%), galactose (6.1 and 3.9%) and glucose (6.5 and 3.8%). The cell wall polymers were also constituted of small amounts (5.6 and 7.2%) of cellulose (CEL) and of non-extractable cell wall polymers (NECW). About 16.8% and 17.2% of the polymers were solubilised with 0.5 and 1.0M NaOH and contained, respectively, 92.1 and 90.7% of glucose derived from starch (IST). The neutral sugar and polymers solubilization profiles showed that weakly bound pectins are present mainly in SPF-Co (water-soluble), CDTA and 0.01-0.1M NaOH soluble fractions. Less soluble, highly cross-linked pectins were solubilised with 4.0M NaOH. This pectin is arabinose-rich, probably highly branched and has a higher molecular weight than the pectin present in SPF-Co, CDTA and 0.01-0.1M NaOH fractions.

Abstract:
We present a detailed experimental study of the Uhrig Dynamical Decoupling (UDD) sequence in a variety of noise environments. Our qubit system consists of a crystalline array of $^{9}$Be$^{+}$ ions confined in a Penning trap. We use an electron-spin-flip transition as our qubit manifold and drive qubit rotations using a 124 GHz microwave system. We study the effect of the UDD sequence in mitigating phase errors and compare against the well known CPMG-style multipulse spin echo as a function of pulse number, rotation axis, noise spectrum, and noise strength. Our results agree well with theoretical predictions for qubit decoherence in the presence of classical phase noise, accounting for the effect of finite-duration $\pi$ pulses. Finally, we demonstrate that the Uhrig sequence is more robust against systematic over/underrotation and detuning errors than is multipulse spin echo, despite the precise prescription for pulse-timing in UDD.

Abstract:
We present experimental measurements on a model quantum system that demonstrate our ability to dramatically suppress qubit error rates by the application of optimized dynamical decoupling pulse sequences in a variety of experimentally relevant noise environments. We provide the first demonstration of an analytically derived pulse sequence developed by Uhrig, and find novel sequences through active, real-time experimental feedback. These new sequences are specially tailored to maximize error suppression without the need for a priori knowledge of the ambient noise environment. We compare these sequences against the Uhrig sequence, and the well established CPMG-style spin echo, demonstrating that our locally optimized pulse sequences outperform all others under test. Numerical simulations show that our locally optimized pulse sequences are capable of suppressing errors by orders of magnitude over other existing sequences. Our work includes the extension of a treatment to predict qubit decoherence under realistic conditions, including the use of finite-duration, square $\pi$ pulses, yielding strong agreement between experimental data and theory for arbitrary pulse sequences. These results demonstrate the robustness of qubit memory error suppression through dynamical decoupling techniques across a variety of qubit technologies.

Abstract:
We discuss the use of two-dimensional $^{9}$Be$^{+}$ ion crystals for experimental tests of quantum control techniques. Our primary qubit is the 124 GHz ground-state electron spin flip transition, which we drive using microwaves. An ion crystal represents a spatial ensemble of qubits, but the effects of inhomogeneities across a typical crystal are small, and as such we treat the ensemble as a single effective spin. We are able to initialize the qubits in a simple state and perform a projective measurement on the system. We demonstrate full control of the qubit Bloch vector, performing arbitrary high-fidelity rotations ($\tau_{\pi}\sim$200 $\mu$s). Randomized Benchmarking demonstrates an error per gate (a Pauli-randomized $\pi/2$ and $\pi$ pulse pair) of $8\pm1\times10^{-4}$. Ramsey interferometry and spin-locking measurements are used to elucidate the limits of qubit coherence in the system, yielding a typical free-induction decay coherence time of $T_{2}\sim$2 ms, and a limiting $T_{1\rho}\sim$688 ms. These experimental specifications make ion crystals in a Penning trap ideal candidates for novel experiments in quantum control. As such, we briefly describe recent efforts aimed at studying the error-suppressing capabilities of dynamical decoupling pulse sequences, demonstrating an ability to extend qubit coherence and suppress phase errors. We conclude with a discussion of future avenues for experimental exploration, including the use of additional nuclear-spin-flip transitions for effective multiqubit protocols, and the potential for Coulomb crystals to form a useful testbed for studies of large-scale entanglement.

Abstract:
We demonstrated transferring the stability of one highly stable clock laser operating at 729 nm to another less stable laser operating at 698 nm. The two different wavelengths were bridged using an optical frequency comb. The improved stability of the clock laser at 698 nm enabled us to evaluate the systematic frequency shifts of the Sr optical lattice clock with shorter averaging time. We determined the absolute frequency of the clock transition 1S0 - 3P0 in 87Sr to be 429 228 004 229 873.9 (1.4) Hz referenced to the SI second on the geoid via International Atomic Time (TAI).

Abstract:
We propose a new method to reduce the frequency noise of a Local Oscillator (LO) to the level of white phase noise by maintaining (not destroying by projective measurement) the coherence of the ensemble pseudo-spin of atoms over many measurement cycles. This scheme uses weak measurement to monitor the phase in Ramsey method and repeat the cycle without initialization of phase and we call, "atomic phase lock (APL)" in this paper. APL will achieve white phase noise as long as the noise accumulated during dead time and the decoherence are smaller than the measurement noise. A numerical simulation confirms that with APL, Allan deviation is averaged down at a maximum rate that is proportional to the inverse of total measurement time, tau^-1. In contrast, the current atomic clocks that use projection measurement suppress the noise only down to the level of white frequency, in which case Allan deviation scales as tau^-1/2. Faraday rotation is one of the possible ways to realize weak measurement for APL. We evaluate the strength of Faraday rotation with 171Yb+ ions trapped in a linear rf-trap and discuss the performance of APL. The main source of the decoherence is a spontaneous emission induced by the probe beam for Faraday rotation measurement. One can repeat the Faraday rotation measurement until the decoherence become comparable to the SNR of measurement. We estimate this number of cycles to be ~100 cycles for a realistic experimental parameter.

Abstract:
In this article we make an explicit approach to the higher degree case of the problem: " For a given $CM$ field $M$, construct its maximal abelian extension $C(M)$ (i.e. the Hilbert class field) by the adjunction of special values of certain modular functions" in a restricted case. We make our argument based on Shimura's main result on the complex multiplication theory of his article in 1967. His main result is constructed for a quaternion algebra $B$ over a totally real number field $F$. We determine the modular function which gives the canonical model for the case $B$ is coming from an arithmetic triangle group. That is our main theorem. And we make an explicit case-study for $B$ corresponding to the triangle group $\Delta (3,3,5)$. The corresponding canonical model appears as a restriction of the Appell's hypergeometric modular function on a 2-dimensional hyperball to a hyperplane section. That is a modular function for the family of the Koike pentagonal curves $w^5=z(z-1)(z-\lambda_1)(z-\lambda_2)$ with two parameters $\lambda_1,\lambda_2$. We use the result of K. Koike in 2003 to get an theta representation of the canonical model function. By using this expression, we show several examples of the Hilbert class fields of the $CM$ fields those are embedded in the above $B$.