The Grad-Shafranov Reconstruction of Toroidal Magnetic Flux Ropes: Method Development and Benchmark Studies

We develop an approach of Grad-Shafranov (GS) reconstruction for toroidal structures in space plasmas, based on in-situ spacecraft measurements. The underlying theory is the GS equation that describes two-dimensional magnetohydrostatic equilibrium as widely applied in fusion plasmas. The geometry is such that the arbitrary cross section of the torus has rotational symmetry about the rotation axis $Z$, with a major radius $r_0$. The magnetic field configuration is thus determined by a scalar flux function $\Psi$ and a functional $F$ that is a single-variable function of $\Psi$. The algorithm is implemented through a two-step approach: i) a trial-and-error process by minimizing the residue of the functional $F(\Psi)$ to determine an optimal $Z$ axis orientation, and ii) for the chosen $Z$, a $\chi^2$ minimization process resulting in the range of $r_0$. Benchmark studies of known analytic solutions to the toroidal GS equation with noise additions are presented to illustrate the two-step procedures and to demonstrate the performance of the numerical GS solver, separately. For the cases presented, the errors in $Z$ and $r_0$ are 9$^\circ$ and 22\%, respectively, and the relative percent error in the numerical GS solutions is less than 10\%. We also make public the computer codes for these implementations and benchmark studies.Comment: submitted to Sol. Phys. late Dec 2016; under review; code will be made public once review is ove

Low-decoherence flux qubit

A flux qubit can have a relatively long decoherence time at the degeneracy point, but away from this point the decoherence time is greatly reduced by dephasing. This limits the practical applications of flux qubits. Here we propose a new qubit design modified from the commonly used flux qubit by introducing an additional capacitor shunted in parallel to the smaller Josephson junction (JJ) in the loop. Our results show that the effects of noise can be considerably suppressed, particularly away from the degeneracy point, by both reducing the coupling energy of the JJ and increasing the shunt capacitance. This shunt capacitance provides a novel way to improve the qubit.Comment: 4 pages, 4 figure

Power Set of Some Quasinilpotent Weighted shifts on lpl^p

For a quasinilpotent operator $T$ on a Banach space $X$, Douglas and Yang defined $k_x=\limsup\limits_{z\rightarrow 0}\frac{\ln\|(z-T)^{-1}x\|}{\ln\|(z-T)^{-1}\|}$ for each nonzero vector $x\in X$, and call $\Lambda(T)=\{k_x: x\ne 0\}$ the power set of $T$. $\Lambda(T)$ have a close link with $T$'s lattice of hyperinvariant subspaces. This paper computes the power set of quasinilpotent weighted shifts on $l^p$ for $1\leq p< \infty$. We obtain the following results: (1) If $T$ is an injective quasinilpotent forward unilateral weighted shift on $l^p(\mathbb{N})$, then $\Lambda(T)=\{1\}$ when $k_{e_0}=1$, where $\{e_n\}_{n=0}^{\infty}$ be the canonical basis for $l^p(\mathbb{N})$; (2) There is a class of backward unilateral weighted shifts on $l^p(\mathbb{N})$ whose power set is $[0,1]$; (3) There exists a bilateral weighted shift on $l^p(\mathbb{Z})$ with power set $[\frac{1}{2},1]$ for $1<p<\infty$.Comment: 22 page

Geometric entanglement from matrix product state representations

An efficient scheme to compute the geometric entanglement per lattice site for quantum many-body systems on a periodic finite-size chain is proposed in the context of a tensor network algorithm based on the matrix product state representations. It is systematically tested for three prototypical critical quantum spin chains, which belong to the same Ising universality class. The simulation results lend strong support to the previous claim [Q.-Q. Shi, R. Or\'{u}s, J. O. Fj{\ae}restad, and H.-Q. Zhou, New J. Phys \textbf{12}, 025008 (2010); J.-M. St\'{e}phan, G. Misguich, and F. Alet, Phys. Rev. B \textbf{82}, 180406R (2010)] that the leading finite-size correction to the geometric entanglement per lattice site is universal, with its remarkable connection to the celebrated Affleck-Ludwig boundary entropy corresponding to a conformally invariant boundary condition.Comment: 4+ pages, 3 figure